LECTURE: “THE ARITHMETIC OF THE LANGLANDS1

PROGRAM”2

JOHANNES ANSCHUTZ3

Contents4

1. Remarks on the lecture 25

2. A general introduction to the Langlands program 36

3. From modular forms to automorphic representations, part I 107

4. From modular forms to automorphic representations, part II 198

5. From modular forms to automorphic representations, part III 289

6. Langlands reciprocity for newforms 3810

7. Modular curves as moduli of elliptic curves (by Ben Heuer) 4611

8. The Eichler–Shimura relation (by Ben Heuer) 5312

9. Galois representations associated to newforms 6113

10. Galois representations for weight 1 forms (by Ben Heuer) 6914

11. The Langlands program for general groups, part I 7715

12. The Langlands program for general groups, part II 8316

References 9117

1

1. Remarks on the lecture18

• Aim: General introduction to the Langlands program19

• Focus: More on “picture”, proofs usually by reference. More “local, geomet-20

ric” than “global, analytic”.21

• Prerequisites: Vary a lot (algebraic number theory, algebraic groups, alge-22

braic geometry, harmonic analysis, representation theory,...). Usually detailed23

knowledge will not be necessary to understand the statements/picture, at least24

in some example.25

• Original plan: Yield background material for, now cancelled, trimester progam26

“The Arithmetic of the Langlands program” of the HIM, which was originally27

scheduled in 2020 (⇒ explains title of lecture)28

• Topics to be covered: Let’s see...29

• Hint: More analytic stuff on automorphic forms/Jacquet-Langlands in Edgar30

Assing’s lecture “Topics in Automorphic forms”31

• Disclaimer: I am by far not a person with serious knowledge/understanding32

of the Langlands program, thus I may oversimplify/overcomplicate things33

or miss subtelties. For definite/correct statements, one has to consult the34

references.35

• Reference: Getz/Hahn: “An introduction to automorphic representations36

with a view towards the trace formula” [GH19], more references during the37

course.38

• Style of lecture: I will explain this file via zoom (the layout is meant to be39

optimized for the use of the file in zoom). Any suggestion on improvement is40

welcome!41

• Video/Audio: Please turn off your video/audio (except for questions/comments).42

Do not record me!43

• Questions: via chat (in zoom), or orally (make me aware of you, e.g., raise44

hand via zoom, or speak), or by e-mail.45

• Problems with technic: This file will be uploaded on the webpage.46

Questions?47

2. A general introduction to the Langlands program48

Today:49

• Rough introduction to the Langlands program, and a first example of its50

arithmetic significance.51

Setup:52

• G reductive group over Q, most important example: G = GLn for some n ≥ 1.53

– an affine, algebraic group G/Q is reductive if it has no closed normal54

subgroups isomorphic to Ga.55

• Af := (xp)p ∈∏pQp | xp ∈ Zp for all but finitely many primes p 56

“ring of finite adeles”57

• A := Af × R58

“ring of adeles”59

Then:60

• A is a locally compact topological ring61

– On Af : unique ring topology such that∏pZp is open.62

– On A: product topology.63

• Q ⊆ A discrete subring (via diagonal embedding).64

• G(A) ∼= G(Af )×G(R) is a locally compact topological group65

– Choose an embedding G ⊆ Spec(Q[X1, . . . , Xm]) for some m ≥ 1, and66

take the subspace topology of induced embedding G(A) ⊆ Am, cf. [GH19,67

Theorem 2.2.1.].68

• G(Q) ⊆ G(A) is a discrete subgroup.69

Facts:70

• On any locally compact topological group H exists a right-invariant Haar71

measure, cf. [GH19, Section 3.2.].72

– unique up to a scalar in R>073

• On G(A) the right Haar measure is also left-invariant, and it descends to a74

G(A)-invariant measure on G(Q)\G(A), cf. [GH19, Lemma 3.5.4.], [GH19,75

Lemma 3.5.3].76

Definition 2.1. We set77

L2(G(Q)\G(A))

as the space of measurable functions f : G(Q)\G(A)→ C such that78 ∫G(Q)\G(A)

|f |2 <∞

where the integration is w.r.t. the G(A)-invariant measure on G(Q)\G(A).79

As usual: Two functions in the L2-space have to be identified if they agree outside80

a set of measure zero.81

Facts:82

• L2(G(Q)\G(A)) is a Hilbert space, i.e., it has an inner product.83

• the G(A)-action coming from right translation on it is unitary, i.e., preserves84

the inner product.85

Aim of the Langlands program (crude form):86

• Decompose the Hilbert space L2(G(Q)\G(A)) as a representation of G(A),87

according to arithmetic data.88

Questions?89

What does “decomposing” mean, roughly?90

• G := isomorphism classes of irreducible, unitary G(A)-representations.91

• Roughly, decomposing means to write (as a unitary G(A)-representation)92

L2(G(Q)\G(A)) “ ∼= ”⊕

[π]∈G

π⊕mπ

with mπ ∈ N ∪ ∞ multiplicity of π.93

• Looking for such a decomposition is too naive.94

Example: G = Gm = GL195

• In this case,96

G(Q)\G(A) ∼= Q×\A×

is a locally compact abelian group97

⇒ can use abstract Fourier theory to analyze L2(Q×\A×).98

Abstract Fourier theory (cf. [DE14, Chapter 3]):99

• A any locally compact abelian group100

• A = isom. classes of irreducible, unitary representations of A101

• A ∼= Homcts(A,S1) is again a locally compact abelian group.102

– via pointwise multiplication and the compact-open topology103

• A ∼= ˆA as topological groups via a 7→ (χ 7→ χ(a)).104

• A is discrete if and only if A is compact.105

“⇒” Choose⊕I

Z A, then A embeds into∏I

S1. “⇐” Use A ∼= ˆA and106

definition of compact-open topology.)107

Theorem 2.2 (Plancherel theorem, cf. [DE14, Chapter 3.4]). The Fourier transform108

F : L1(A) ∩ L2(A)→ L2(A)

defined by109

f 7→ (χ 7→∫A

f(x)χ(x)dµ(x))

extends to a unitary isomorphism110

F : L2(A) ∼= L2(A)

of Hilbert spaces.111

Fact:112

• F is an A-equivariant isomorphism where A acts on L2(A) via113

a · g(χ) := χ(a)g(χ)

for a ∈ A, g ∈ L2(A), χ ∈ A.114

Examples:115

• If A = S1, then116

Z ∼= A, n 7→ χn with χn(z) = zn, z ∈ S1.

Thus117

L2(S1) ∼=⊕n∈Z

Cχn

is the Hilbert space direct sum of the S1-equivariant subspaces118

Cχn ⊆ L2(S1).

Concretely: each L2-function f : S1 → C can uniquely be written as119

f =∑n∈Z

anχn, where an ∈ C,∑n∈Z|an|2 <∞.

⇒ We found a nice decomposition of the unitary S1-representation L2(S1)120

into irreducible subspaces.121

• If A = R, then R ∼= R via122

R→ R, x 7→ χx, where χx : R→ S1, y 7→ e2πixy.

The isomorphism123

F : L2(R) ∼= L2(R)

implies therefore that each f ∈ L2(R) can be written as124

f(x) =

∫R

g(y)χy(x)dy

for a unique function g ∈ L2(R). Thus,125

L2(R) ∼=∫R

χydy

is a Hilbert integral of representation. We cannot do better: χx /∈ L2(R)126

for any x ∈ R, and L2(R) contains no irreducible subrepresentation of R, cf.127

[GH19, Section 3.7].128

– Also no irreducible quotient, the statement about quotients in [GH19,129

Lemma 3.7.1] is wrong, I think.130

Back to G = Gm:131

• Factoring each n ∈ Q× into n = ±p1 . . . pk with pi prime implies132

Q×\A× ∼=∏p

Z×p × R>0.

• Let us ignore R>0 and look at133

L2(∏p

Z×p ) ∼=⊕χ

Cχ.

with χ :∏pZ×p → S1 all continuous characters of

∏pZ×p .134

• The characters χ are “arithmetic” data, namely135

Gal(Q(µ∞)/Q) ∼=∏p

Z×p ,

where Q(µ∞) =⋃n∈N

Q(e2πin ) is the cyclotomic extension of Q.136

• Thus the χα are (continuous) 1-dimensional Galois representations of Gal(Q/Q).137

– Namely: Gal(Q/Q)α

∏p prime

Z×pχ−→ S1 ⊆ C×138

• By Kronecker-Weber each 1-dimensional Galois representation139

σ : Gal(Q/Q)→ C×

is of the form σ = χ α for some χ.140

Theorem 2.3 (Kronecker-Weber). The field Q(µ∞) is the maximal abelian extension141

of Q, i.e., each finite Galois extension F/Q with abelian Galois group Gal(F/Q) is142

contained in Q(µ∞).143

The final decomposition for G = Gm:144

•L2(R>0Q×\A×) ∼=

⊕σ : Gal(Q/Q)→C×

Cχσ

with χσ characterised by χσ α = σ.145

• This (easy) example is prototypical for what one aims for in the Langlands146

program.147

Questions?148

Back to general G: Geometry of G(Q)\G(A)149

• There exists a central subgroup AG ⊆ G(R), with AG ∼= Rr>0, such that150

[G] := AGG(Q)\G(A)

has finite volume, cf. [GH19, Theorem 2.6.2.].151

– AG is the connected component of the maximal split subtorus of the152

center Z(G) of G, e.g., AGLn ⊆ GLn(R) is the group of scalar matrix153

with entries in R>0 while ASLn is trivial.154

• Tamagawa measure = some canonical measure on G(A), and155

τ(G) := vol([G]) = Tamagawa number of G.

Knowledge of τ(G) is arithmetically interesting. Cf. [Col11, Appendix B],156

[Lur, Lecture 1].157

– τ(SLn) = 1 ⇒ vol(SLn(R)/SLn(Z)) = ζ(2)ζ(3) . . . ζ(n), with ζ(s) =158

Riemann ζ-function159

• G(Q)\G(A) and [G] are profinite coverings of some real manifold, e.g.,160

GLn(Z)\(∏p

GLn(Zp)×GLn(R)) with diagonal action of GLn(Z)

which covers the real manifold161

GLn(Z)\GLn(R).

Similarly, [G] covers GLn(Z)R>0\GLn(R). Cf. [GH19, Section 2.6], [Del73,162

(0.1.4.1)].163

• K∞ ⊆ G(R) maximal compact connected subgroup ⇒ [G] is a principal K∞-164

bundle (or K∞-torsor) over [G]/K∞.165

– G = GLn ⇒ K∞ ∼= SOn(R)166

• G = GL2, then by Mobius transformations167

AG\GL2(R)/K∞∼−→ H±, g =

(a b

c d

)7→ g(i) :=

a · i+ b

c · i+ d

with168

H± := z ∈ C | Im(z) 6= 0

the upper/lower halfplane169

⇒ [GL2]/K∞ ∼= GL2(Z)\(GL2(∏p

Zp)×H±)

• K∞ ⊆ G(Af ) compact-open (“a level subgroup”)170

⇒ G(Q)\G(Af )/K∞ =

m∐i=1

G(Q)giRK∞

is finite (cf. [GH19, Theorem 2.6.1]) and171

[G]/K∞K∞ ∼=

m∐i=1

Γi\X

where172

X := AG\G(R)/K∞

is a real manifold, and173

Γi := G(Q) ∩G(R)giK∞g−1

i ⊆ G(R)

is a congruence subgroup, cf. [GH19, Section 2.6].174

– One usually has to assume that K∞ is sufficiently small, so that Γi acts175

freely on X, cf. [GH19, Definition 15.2].176

• G = GL2, then177

Γi\X ∼= Γ\H± (e.g., SL2(Z)\H)

with Γ ⊆ GL2(Z) a discrete subgroup containing178

Γ(m) := ker(GL2(Z)→ GL2(Z/m))

for some m ≥ 0. (later ⇒ relation to modular forms).179

• Usually dim(Γi\X) > 0⇒ geometry more complicated than in case G = Gm.180

• The Γi\X are interesting real manifolds, e.g., sometimes actually complex181

manifolds and canonically defined over number fields.182

• [G] compact (⇔ each Γi\X compact) if and only if any Gm ⊆ G lies in center,183

cf. [GH19, Theorem 2.6.2].184

– e.g., G the units in a non-split quaternion algebra over Q, i.e., units in a185

Q-algebra with presentation 〈x, y|x2 = a, y2 = b, xy = −yx〉, for suitable186

a, b ∈ Q×. Note that G(R) ∼= GL2(R) if ab < 0. Cf. [GS17, Chapter 1].187

More examples: Weil restrictions188

• F/Q finite extension, H reductive over F189

⇒ reductive group G := ResF/Q(H) over Q.190

– G represents the functor R 7→ H(R⊗QF ) on the category of Q-algebras,191

cf. [CGP15, Appendix A] or [GH19, Section 1.4].192

• G(Q)\G(A) = H(F )\H(AF ) with AF ring of adeles for F (⇒ no need to193

discuss general F ).194

• H = Gm,F ⇒ desired decomposition of L2([G]) is equivalent to class field195

theory for F .196

• F imaginary quadratic, H = GL2,F ⇒ G(R) ∼= GL2(C), K∞ ∼= U(n) unitary197

group and198

X = AG\G(R)/K∞

is the 3-dimensional hyperbolic space H3. The quotients Γi\X are arithmetic199

hyperbolic 3-manifolds, cf. [Thu82, Section 4 & 5].200

– H3 can be modelled on the quaternions q = x + y · i + z · j | x, y ∈201

R, z > 0 and

(a b

c d

)∈ GL2(C) acts as q 7→ (aq+ b)(cq+ d)−1, where202

C = x+ y · i | x, y ∈ R203

Questions?204

Definition 2.4 ([GH19, Definition 3.3]). Let G be a reductive group over Q. An auto-205

morphic representation (in the L2-sense) is an irreducibe unitary G(A)-representation206

π which is isomorphic to a subquotient of L2([G]).207

Facts:208

• G(A) acts on [G] = AGG(Q)\G(A) from the right ⇒ G(A) acts unitarily on209

L2([G])210

– After choice of a right-invariant Haar measure on G(A)...211

• Necessarily π is trivial on AG. Usually harmless: Schur’s lemma ⇒ can find212

for irreducible unitary G(A)-repr. π a character χ : G(R) → C× with χ ⊗ π213

trivial on AG. Cf. [GH19, Section 6.5].214

– For G = GLn take χ of the form g 7→ |det(g)|s) for some s ∈ R>0.215

• If [G] is not compact, L2([G]) will decompose into a “discrete” and a “con-216

tinuous” part, cf. [GH19, Section 10.4].217

• Recall G = isom. classes of irreducible unitary G(A)-representations.218

• G has a natural topology, the Fell topology, cf. [GH19, Section 3.8].219

– It generalizes the compact-open topology on A for A locally compact220

abelian.221

• Let π ∈ G, x ∈ π, and ϕ : π → C continuous C-linear. Then

fx,ϕ : G→ C, g 7→ ϕ(gx)

is a matrix coefficient of π. Roughly, two π, π′ ∈ G are close if their matrix222

coefficients agree for the compact-open topology.223

Theorem 2.5 ([GH19, Theorem 3.9.4]). There exists a measurable multiplicity func-

tion m : G→ 1, 2, . . . ,∞ and a measure µ on G such that

L2([G]) ∼=∫G

(⊕m(π)

π)dµ(π).

The m and µ are unique up changes on sets of measure 0.224

The discrete and continuous part of L2([G]):225

• We don’t need the exact meaning of the integral in 2.5, its occurence is due to226

the fact that some representations of G(A) “contribute” to L2([G]) without227

being subrepresentations.228

– Compare with the discussion of the Fourier transform for R.229

• Points π ∈ G with µ(π) > 0 appear (with multiplicitym(π)) as subrepresentations230

in L2([G]).231

• By Theorem 2.5232

L2([G]) ∼= L2disc([G])⊕ L2

cont([G])

where L2disc([G]) is the Hilbert sum of all irreducible, unitary subrepresentations233

of G(A), and L2cont([G]) its orthogonal complement, cf. [GH19, Section 9.1].234

• By definition, a closed, connected subgroup P ⊆ G is parabolic if G/P is235

proper over Spec(Q). The quotient of P by its unipotent radical is its Levi236

quotient M . Cf. [GH19, Section 1.9].237

– In GLn: take decomposition n = n1 + n2 + . . . + nk, P is subgroup of238

block upper triangular matrices in GLn with blocks of size n1, . . . , nk.239

The Levi quotient is M = block diagonal matrices. Up to conjugation240

all parabolics in GLn are of this form.241

• Langlands proved that the continuous part L2cont([G]) can be described via242

“inducing” the discrete parts L2disc([M ]) for all Levi quotients of parabolics243

P ⊆ G, P 6= G, cf. [GH19, Section 10.4]).244

– This is his theory of “Eisenstein series”, and it is a starting point of the245

Langlands program.246

• In this course, L2disc([G]) will be more important than L2([G]).247

• [G] is compact if and only G has no proper parabolics (defined over Q), this248

confirms that L2([G]) = L2disc([G]) in this case.249

Questions?250

3. From modular forms to automorphic representations, part I1

Today (and next time):2

• Construct automorphic representations for GL2 associated to modular forms3

• Keep some piece of paper ready!4

Last time:5

• G reductive over Q, A = Af × R ring of adeles6

• [G] = AGG(Q)\G(A) adelic quotient, has finite invariant volume7

• Langlands program: Decompose8

L2([G])

according to arithmetic data.9

• G = Gm ⇒10

L2(R>0Q×\A×) ∼=⊕

σ : Gal(Q/Q)→C×C χσ.

• Geometry of [G]:11

– K∞ ⊆ G(R) some maximal compact, connected12

– K = K∞ ⊆ G(Af ) some compact-open subgroup13

– Then14

[G]

K-torsor (in particular: a profinite covering)

[G]/K

K∞-torsor

a real manifold, at least for sufficiently small K

[G]/KK∞ =m∐i=1

Γi\X disjoint union of arithmetic manifolds

with Γi ⊆ G(R) some (discrete) congruence subgroups determined by K,15

and16

X := AG\G(R)/K∞.

• An irreducible unitary representation π of G(A) is automorphic if it is iso-17

morphic to a subquotient of L2([G]).18

• Examples:19

– G = Gm, then the automorphic representations of G(A) are the charac-20

ters of A× which are trivial on Q× · R>0 (by definition of [G])21

– [G] has finite volume22

⇒ the trivial representation (=constant functions on [G]) is automorphic,23

and in L2disc([G]).24

• Have the general orthogonal decomposition25

L2([G]) ∼= L2disc([G])⊕ L2

cont([G])

with L2disc([G]) the closure of the sum of all irreducible subrepresentations of26

G(A).27

• Langlands⇒ L2cont([G]) described via L2

disc([M ]) for M running through Levi28

quotients of parabolic subgroups P ⊆ G,P 6= G.29

Questions ?30

For G = GL2:31

• AGL2∼= R>0 embedded as scalar matrices into GL2(R)32

• We have33

GL2(Q)\GL2(A) ∼= GL2(Z)\(GL2(Z)×GL2(R)),

where34

Z := lim←−m

Z/m ∼=∏p

Zp.

• Indeed:35

– GL2(Af ) is the restricted product of the GL2(Qp) w.r.t. to the GL2(Zp),36

i.e.,37

GL2(Af ) = (Ap)p ∈∏p

GL2(Qp) | Ap ∈ GL2(Zp) for all but finitely many p

(cf. [GH19, Proposition 2.3.1]).38

– Q ⊆ Af is dense (use Chinese remainder theorem).39

– A×f = Q× · Z× (use prime factorization).40

– GL2(Zp),GL2(Qp) generated by elementary and diagonal matrices41

⇒ GL2(Af ) = GL2(Q)GL2(Z)42

⇒ GL2(Q)\GL2(Af ) ∼= GL2(Q) ∩GL2(Z)\GL2(Z) ∼= GL2(Z)\GL2(Z).43

⇒ GL2(Q)\GL2(A) ∼= GL2(Z)\(GL2(Z)×GL2(R))44

– The proof works for all n ≥ 1 and shows:45

GLn(Q)\GLn(A) ∼= GLn(Z)\(GLn(Z)×GLn(R)).

• Km = ker(GL2(Z)→ GL2(Z/m)) for m ≥ 1.46

– these groups are cofinal within all compact-open subgroups in GL2(Af ).47

– we have48

GL2(Q)\GL2(Af )/Km∼= GL2(Z)\GL2(Z)/Km

and GL2(Z)/Km∼= GL2(Z/m).49

– SL2(Z/m) is generated by elementary matrices: use the euclidean algo-50

rithm and the magic identity51 (a 0

0 a−1

)=

(1 a

0 1

)(1 0

−a−1 1

)(1 a

0 1

)(0 −1

1 0

)for any invertible element a in some ring R. Check!Check!52

⇒ SL2(Z)→ SL2(Z/m) surjective53

– We obtain54

GL2(Z)\GL2(Z)/Km∼= ±1\(Z/m)×

via the determinant.55

• K∞ = SO2(R) ⊆ GL2(R)56

• X = AGL2\GL2(R)/K∞ ∼= H± via57

AGL2\GL2(R)/K∞

∼−→ H±, g =

(a b

c d

)7→ g(i) :=

a · i+ b

c · i+ d

• In the end:58

GL2(Q)\(GL2(Af )/Km ×X) ∼=∐

±1\(Z/m)×

Γ(m)\H±

with59

Γ(m) := Km ∩GL2(Z) = ker(GL2(Z)→ GL2(Z/m)).

Finiteness of the volume of [G] for G = GL2: A direct argument60

• K∞Km compact61

⇒ It suffices to show that [GL2]/K∞Km has finite volume.62

⇒ Suffices to see that Γ(m)\H± has finite volume.63

• Up to scalar in R>0, the GL2(R)-invariant measure on H± is given by the64

volume form65

1

y2dx ∧ dy.

• Indeed:66

– Have to show67

g∗(1

y2dx ∧ dy) =

1

y2dx ∧ dy

for g ∈ GL2(R).68

– Write dx ∧ dy = i2dz ∧ dz, z = x+ iy.69

– Statement true for70

g =

(1 a

0 1

), a ∈ R.

– Statement true for AGL2.71

– Statement true for72

g =

(a 0

0 1

), a ∈ R×.

– Statement true for73

g =

(0 1

1 0

),

i.e., for z 7→ 1z = z

|z|2 .74

– From the magic identity we get invariance for lower triangular matrices,75

and thus for all of GL2(R). Check!Check!76

• A known fundamental domain for SL2(Z) on H is given by77

D := z ∈ H | |z| > 1, |Re(z)| < 1/2

and finiteness of the volume of GL2(Z)\H± follows from78 ∫D

1

y2dx ∧ dy <∞.

• Finally, Γ(m) ⊆ GL2(Z) is of finite index79

⇒ Γ(m)\H± is of finite volume, as desired.80

Questions ?81

Modular forms classically (cf. [DS05]):82

• Γ ⊆ SL2(Z) congruence subgroup, later sufficiently small.83

• k ∈ Z84

• A function f : H± → C is a modular form of weight k for Γ if85

– f(γ · z) = (cz + d)kf(z) for γ =

(a b

c d

)∈ Γ86

– f is holomorphic87

– f is holomorphic at the cusps of Γ88

• “holomorphic at cusp”89

• Mk(Γ) = the C-vector space of modular forms of weight k for Γ.90

Let us call, for the moment, a function weakly modular if it satisfies the first two91

conditions.92

Holomorphic at cusps:93

• For each cusp σ a weakly modular form f for Γ has a Fourier expansion, i.e.,94

up to translating the cusp to ∞ there is an equality95

f(z) =∑n∈Z

anqn/h

for q = e2πiz, h the smallest periodicity of f , and an ∈ C, n ∈ Z.96

• f is holomorphic at the cusp, i.e., a modular form, if an = 0 for n < 0.97

• In this case, f is a cusp form if a0 = 0.98

• Let Sk(Γ) ⊆Mk(Γ) be the subspace of cusp forms.99

Modular forms as sections of line bundles on modular curves:100

• The embedding H± ⊆ P1C, z 7→ [z : 1] is GL2(R)-equivariant.101

• The GL2(C)-action on P1C lifts (naturally) to OP1

C(k):102

– Sufficient for k = −1103

– But OP1C(−1) is associated to the Gm-torsor104

A2C \ 0 → P1

C, (z1, z2) 7→ [z1 : z2],

which is naturally GL2(C)-equivariant.105

• We obtain a Γ-equivariant morphism106

OH±(k) := OP1C(k)×P1

CH±

H±

• Modding out Γ yields:107

ω⊗k := Γ\OH±(k)

Γ\H±,

with ω⊗k a holomorphic line bundle over Γ\H±108

• The space109

H0(Γ\H±, ω⊗k)

of holomorphic sections of ω⊗k identifies with weakly modular forms for Γ.110

• Indeed:111

– For k ∈ Z define the holomorphic GL2(R)-equivariant line bundle112

Lk := H± × C

over H± with GL2(R)-acting on the left by113

g · (z, λ) := (az + b

cz + d, (cz + d)−kλ).

for g =

(a b

c d

)∈ GL2(R).114

– The map115

L∗−1 → OH±(−1)∗ = H± ×P1CA2

C \ 0,(z, λ) 7→ (z, (λz, λ))

is a holomorphic isomorphism of GL2(R)-equivariant line bundles over Check!Check!116

H±. Here (−)∗ is the complement of the zero section in a line bundle.117

– By taking tensor powers:118

Lk ∼= OH±(k)

for all k ∈ Z, as GL2(R)-equivariant holomorphic vector bundles.119

– A holomorphic section120

H± → Lk, z 7→ (z, f(z))

is Γ-equivariant if and only if f(z) satisfies121

f(az + b

cz + d) = (cz + d)kf(z)

for

(a b

c d

)∈ Γ. Check!Check!122

– This shows that123

H0(Γ\H±, ω⊗k)

identifies with the space of weakly modular forms for Γ.124

• In particular, we have a natural embedding125

Mk(Γ) ⊆ H0(Γ\H±, ω⊗k)

with image defined by condition of being “holomorphic at the cusps”:126

– ω extends canonically to the canonical compactification Γ\H± of the127

algebraic curve Γ\H±128

– Mk(Γ) is the image of H0(Γ\H±, ω⊗k)129

⇒ In particular, Mk(Γ) is finite-dimensional over C. Moreover, ω is130

ample on Γ\H± which implies that Mk(Γ) = 0 for k < 0 and Mk(Γ) gets131

“big” for k 0.132

– Sk(Γ) ∼= H0(Γ\H±, ω⊗k(−D)) where D = Γ\H± \Γ\H± is the (reduced)133

divisor at infinity.134

Questions ?135

From sections of ω⊗k to functions on [G] = [GL2]:136

• Let K ⊆ GL2(Af ) compact-open and write137

GL2(Q)\(GL2(Af/K)×H±) ∼=m∐i=1

Γi\H±

• ω⊗k defines by pullback from H± a (complex) line bundle ω⊗k on138

GL2(Q)\(GL2(Af/K)×H±).

• The spaces Mk(Γi) of modular forms for Γi, i = 1, . . . ,m, embed into the139

space of holomorphic sections140

H0(GL2(Q)\(GL2(Af )/K ×H±), ω⊗k).

• Set141

Mk(K) :=

m⊕i=1

Mk(Γi).

• Then Mk(K) ⊆ H0(GL2(Q)\(GL2(Af )/K×H±), ω⊗k) is defined by condition142

of being holomorphic at all cusps.143

• ω⊗k defines by pullback a G(Af )-equivariant line bundle, again written ω⊗k,144

on145

GL2(Q)\(GL2(Af )×H±)

• Define146

H0(GL2(Q)\(GL2(Af )×H±), ω⊗k)

:= lim−→K

H0(GL2(Q)\(GL2(Af )/K ×H±), ω⊗k).

and147

Mk := lim−→K

Mk(K).

• The line bundle ω⊗k is GL2(R)-equivariantly trivial after pullback along148

AGL2\GL2(R)→ H±, g 7→ g(i).

Namely, for k = −1, there is the GL2(R)-equivariant section Check!Check!149

s : AGL2\GL2(R)→ AGL2\GL2(R)×H± OH±(−1) ∼= AGL2\GL2(R)×P1CA2

C \ 0,

g =

(a b

c d

)7→ (g, |det(g)|−1/2(ai+ b, ci+ d)).

• In particular, pullback of sections defines a morphism150

Φ: H0(GL2(Q)\(GL2(Af )×H±), ω⊗k)→ C∞([GL2])

which on the infinite component is induced by pullback along151

AGL2\GL2(R) 7→ H±, g 7→ g(i)

and the isomorphism152

AGL2\GL2(R)×H± Lk ∼= AGL2

\GL2(R)×H± OH±(k) ∼= AGL2\GL2(R)× C

where the last one is induced by the section s. One checks, by reducing to153

the case k = −1, that each section f : H± → Lk is send to the function154

g 7→ ϕf (g) := |det(g)|k/2(ci+ d)−kf(ai+ b

ci+ d)

for g =

(a b

c d

)∈ GL2(R). Indeed, for k = −1, the pullback of155

H± → L−1, z 7→ (z, f(z))

and the corresponding section156

AGL2\GL2(R)→ AGL2

\GL2(R)× C, g 7→ (g, ϕf (g))

are related, as sections of AGL2\GL2(R)×P1

CA2

C \ 0 by the formula Check!Check!157

f(g(i))(g(i), 1) = ϕf (g)|det(g)|1/2(ai+ b, ci+ d)

which forces158

ϕf (g) = |det(g)|−1/2(ci+ d)f(g(i))

as desired.159

• Note that the factor |det(g)|−1/2 is necessary only because by definition ele-160

ments of L2([GL2]) or C∞([GL2]) are constant along AGL2.161

We used the following general definition.162

Definition 3.1 (Smooth functions on [G]). Let G be a reductive group over Q. Then163

a function f : [G]→ C is smooth if it is stable under some K ⊆ G(Af ) compact-open164

and the resulting function165

f : [G]/K → C

is smooth, i.e., C∞. (Note that [G]/K is a real manifold.)166

Upshot:167

• We constructed a canonical morphism168

Mk ⊆ H0(GL2(Q)\(GL2(Af )×H±), ω⊗k)Φ−→ C∞([GL2]),

where Mk = lim−→K

Mk(K) and Mk(K) is a sum of spaces of modular forms169

m⊕i=1

Mk(Γi) for various congruence subgroups Γi ⊆ GL2(Z).170

• The image of Mk in H0(GL2(Q)\(GL2(Af ) × H±), ω⊗k) is defined by the171

condition of “holomorphicity at the cusps”.172

Questions ?173

When do modular forms give rise to functions in L2([GL2])?174

• More generally, pick a section f ∈ H0(GL2(Q)\(GL2(Af )×H±, ω⊗k).175

• Question: When is176

Φ(f) ∈ C∞([GL2])

in L2([GL2])?177

• There exists K ⊆ GL2(Af ) compact-open such that178

f ∈ H0(GL2(Q)\(GL2(Af )/K ×H±), ωωk),

and Φ(f) is L2 if and only if Φ(f) ∈ C∞([GL2]/K) ⊆ C∞([GL2]) (as K is179

compact).180

⇒ Reduce to question:181

For Γ ⊆ GL2(Z) some congruence subgroup, f ∈ H0(Γ\H±, ω⊗k). When is182

ΓAGL2\GL2(R)→ C, g 7→ ϕf (g) = |det(g)|k/2(ci+ d)−kf(g(i))

square-integrable, i.e., in L2?183

• Claim:184

|ϕf (g)|2 = |f(g(i))|2|Im(g(i))|k

for any g ∈ GL2(R).185

• Equivalently,186

|det(g)|−1/2|ci+ d| ?= |Im(g(i))|1/2

for g =

(a b

c d

)∈ GL2(R).187

• This is true as both sides are invariant under188

– K∞ = SO2(R)-invariant from the right,189

– the subgroup

(1 ∗0 1

)⊆ GL2(R) acting from the left,190

– the diagonal matrices in GL2(R) acting from the left. Check these!Check these!191

• Hence192 ∫ΓAGL2\GL2(R)

|ϕf (g)|2dg =

∫Γ\H±

|f(z)|2|y|k 1

y2dx ∧ dy

(Recall that we integrate over “the” invariant measure.)193

• Γ\H± has finitely many cusps ⇒ Reduce to local statement at cusps194

• Write f in Fourier expansion, wlog at ∞:195

f(z) =∑n∈Z

anqn/h.

• Then:196

|y|kf(z) = 2π|log(|q|)|k∑n∈Z

anqn/h

as y = log(|q|). Check!Check!197

• Now for 0 < r ≤ 1198 ∫0<|q|≤r

|log(|q|)|k−2∑n∈Z

anqn/h i

2dq ∧ dq <∞

if an = 0 for n ≤ 0, i.e., if f is cuspidal.199

Upshot:200

• The map Mk ⊆ H0(GL2(Q)\(GL2(Af ) × H±), ωk)Φ−→ C∞([GL2]) induces a201

canonical inclusion202

Φ: Sk → L2([GL2]),

where Sk := lim−→K

Sk(K).203

Observations:204

• GL2(Af ) acts on H0(GL2(Q)\(GL2(Af ) × H±), ωk), C∞([GL2]), L2([GL2]),205

Mk, Sk and all inclusions, in particular Φ, are equivariant.206

• E.g., for g ∈ GL2(Af ) and varying K ⊆ GL2(Af ) the isomorphism207

Mk(K)g−→Mk(g−1Kg) ⊆Mk

defines the action Mkg−→Mk as Mk = lim−→

K

Mk(K).208

• GL2(Af ) does not act on Mk(K), Sk(K), . . . for any fixed K ⊆ GL2(Af ).209

Next tasks:210

• Describe Sk as a GL2(Af )-representation.211

• Characterize the image of Sk in L2([GL2]).212

Questions ?213

4. From modular forms to automorphic representations, part II1

Today:2

• Continue construction of automorphic representations associated to cusp forms.3

Questions from last time:4

1) Can one classify GL2(R)-equivariant line bundles on H±?5

2) Similarly: Can one classify GL2(Z)-equivariant line bundles on H±?6

• For 1):7

– Assume G Lie group, H a closed subgroup. Then the category8

G-equivariant complex vector bundles on G/H

is equivalent to9

finite dimensional C-representations of H.

“⇒” Look at the representation of H at the coset 1 ·H ∈ G/H.10

“⇐” Pass to the quotient11

V := (G× V )/H

for the action h · (g, v) := (gh−1, hv).12

– Useful:13

σ : H → GL(V )

finite-dimensional C-representation of H. Then:14

Continuous resp. smooth sections15

G/H → V

identify with continuous resp. smooth functions16

f : G→ V

such that f(gh) = ρ(h)−1 · f(v). Check!Check!17

– In our case, G = GL2(R), H = AGL2K∞ with K∞ ∼= SO2(R). Then18

H ∼= C×

and the irreducible representations of H are given by19

z 7→ zpzq, p, q ∈ Z.

⇒ There exist more GL2(R)-equivariant line bundles on H±.20

– Only the ones for q = 0 define holomorphic line bundles.21

– The SL2(R)-equivariant line bundles on H are all isomorphic to ω⊗k for22

some k ∈ Z.23

– Potentially confusing: Exists modular forms of half-integral weight.24

– Explanation: These are not sections of an SL2(R)-equivariant line bundle25

on H.26

– But sections of an SL2(R)-equivariant line bundle.27

– Here: SL2(R)→ SL2(R), the “metaplectic group”, the degree 2-covering28

of SL2(R) (a non-algebraic Lie group).29

– Note: π1(SL2(R), 1) ∼= π1(S1, 1) ∼= Z.30

– More details: [Del73].31

• For 2):32

– M1,1 = stack of elliptic curves over Spec(C).33

– Analytic stack associated to M1,1 is the stacky quotient [GL2(Z)\H±].34

– Mumford:35

Pic(M1,1) ∼= Z/12,

generated by ω, with the relation ω⊗12 ∼= O induced by the cusp form36

∆ ∈ S12(SL2(Z)), cf. [Mum65].37

– Note: Does not imply Mk(Γ) ∼= Mk+12(Γ) for k ∈ Z, Γ ⊆ SL2(Z) con-38

gruence subgroup.39

– Reason: The canonical extensions of ω⊗k and ω⊗k+12 to the compactifi-40

cation of M1,1 differ.41

– I don’t know if all (holomorphic) line bundles on [GL2(Z)\H±] arise from42

an algebraic line bundle on M1,1, i.e., are algebraic.43

Last time:44

• We constructed a GL2(Af )-equivariant embedding45

Φ: Mk → C∞([GL2]),

inducing a GL2(Af )-equivariant embedding46

Φ: Sk → L2([GL2]).

• Here47

Mk = lim−→K⊆GL2(Af )

Mk(K)

with48

Mk(K) =

m⊕i=1

Mk(Γi)

a sum of spaces of modular forms for congruence subgroups Γi ⊆ GL2(Z).49

• Similarly:50

Sk = lim−→K

Sk(K)

with Sk(K) a sum of spaces of cusp forms for congruence subgroups.51

• Today: How can one characterize the images52

Φ(Mk) ⊆ C∞([GL2])

resp.53

Φ(Sk) ⊆ L2([GL2])?

.54

• Next time: How does Sk decompose as a GL2(Af )-representation?55

Recall the construction of Φ:56

• Consider GL2(R)-equivariant line bundle57

Lk → H±

associated to the representation z 7→ zk of C× = AGL2SO2(R).58

• By pullback: Obtain line bundle ω⊗k on GL2(Q)\(GL2(Af )×H±).59

• Smooth, holomorphic,... sections of ω⊗k identify with smooth, holomorphic,...60

functions61

f : GL2(Af )×H± → C, (g, z) 7→ f(g, z)

satisfying62

f(γg, γz) = (cz + d)kf(g, z)

for63

γ =

(a b

c d

)∈ GL2(Q).

• Define the function64

j : GL2(R)×H± → C×, (

(a b

c d

), z) 7→ cz + d.

Then65

j(gh, z) = j(g, hz)j(h, z)

for all g, h ∈ GL2(R), z ∈ H±. Check!Check!66

• For f : GL2(Af )×H± → C as above the function67

ϕf : GL2(Af )×GL2(R)→ C, (g, g∞) 7→ j(g∞, i)−kf(g, g∞i)

is GL2(Q)-equivariant, i.e., lies in C∞(GL2(Q)\GL2(A)).68

• This is not the function we denoted by Φ(f) last time: It is not invariant by69

AGL2!70

• This can be fixed by multiplying it with the function71

GL2(R)→ C×, g 7→ |det(g)|k/2.

• Thus72

Φ(f)(g, g∞) := |det(g∞)|k/2ϕf (g, g∞)

is the associated function in C∞([GL2]) from last time.73

• Recall74

C× ∼= AGL2SO2(R) =

(a −bb a

) ⊆ GL2(R).

• Then75

ϕf (g, g∞z) = z−kϕf (g, g∞)

for all z ∈ C× ⊆ GL2(R), (g, g∞) ∈ GL2(A). Check!Check!76

• We obtain that f 7→ ϕf identifies smooth sections77

GL2(Q)\GL2(Af )×H± → ω⊗k

with smooth functions78

ϕ : GL2(Q)\GL2(A)→ C

such that79

ϕ(g, g∞z) = z−kϕ(g, g∞)

for all (g, g∞) ∈ GL2(Af )×GL2(R), z ∈ C×.80

Questions?81

Upshot and next aims:82

• Describe the image of Mk in C∞(GL2(Q)\GL2(A)) under f 7→ ϕf by condi-83

tions soley on ϕ ∈ C∞(GL2(Q)\GL2(A)).84

• An additional condition will determine the image of Sk.85

⇒ Obtain a description of Sk ⊆ L2([GL2]).86

• We already know that an element ϕ in the image of Mk must satisfy87

ϕ(g, g∞z) = z−kϕ(g, g∞)

for z ∈ C× and (g, g∞) ∈ GL2(A) and that such ϕ descent to a smooth88

function89

fϕ : GL2(Af )×H± → C, (g, z) 7→ f(g, z)

satisfying modularity for GL2(Q).90

• Need to check:91

– When is fϕ holomorphic?92

– When is fϕ holomorphic at the cusps?93

When is fϕ holomorphic?94

• Reduces to: Given a smooth function95

f : H± → C.

Which condition on96

ϕf : GL2(R)→ C, g 7→ j(g, i)−kf(gi)

guarantees that f is holomorphic?97

The strategy:98

• The Lie algebra99

g := gl2(R) ∼= Mat2,2(R)

of GL2(R) acts on C∞(GL2(R)) by deriving the action of GL2(R) by right100

translations.101

• Make this action explicit on ϕf .102

• Then construct an element Y ∈ gC such that103

Y ∗ ϕf = 0 ⇔ f is holomorphic.

Infinitesimal actions (cf. [GH19, Section 4.2]):104

• H any Lie group with Lie algebra h := T1H (“tangent space at identity”)105

• V any representation of H, with V a Hausdorff topological C-vector space.106

• For X ∈ h, defined by some path γ : (−ε, ε) → H with γ(0) = 1, and v ∈ V107

consider108

(−ε, ε) \ 0 → V, t 7→ γ(t)v − vt

.

If the limit exists, we write109

X ∗ v := lim−→t→0

γ(t)v − vt

.

• An element v ∈ V is called smooth if110

X1 ∗ (. . . ∗ (Xn ∗ v)) . . .)

exists for all X1, . . . , Xn ∈ h.111

• Vsm ⊆ V the subspace of smooth vectors in V .112

• Vsm stable under H.113

• Vsm is a representation of h, equivalently a module under the enveloping114

algebra for h:115

U(h) :=⊕n≥0

h⊗n/〈X ⊗ Y − Y ⊗X − [X,Y ] | X,Y ∈ h〉.

• Here: [−,−] the Lie bracket of h.116

Questions?117

The explicit action of g = gl2(R) on ϕf :118

• Recall: f ∈ H± → C a smooth function.119

• Recall:120

ϕf : GL2(R)→ C, g 7→ j(g, i)−kf(gi)

with121

j(g, z) := cz + d ∈ C

for122

g =

(a b

c d

)∈ GL2(R), z ∈ H±.

• Recall:123

j(gh, z) = j(g, hz)j(h, z)

for g, h ∈ GL2(R), z ∈ H±.124

• We know125

ϕf (gz) = z−kϕf (g)

for z ∈ C×, g ∈ GL2(R)126

⇒ C ∼= Lie(C×) ⊆ g acts via the linear form127

C 7→ C, z 7→ −kz

on ϕf (g). Note that128

Lie C× = 〈

(1 0

0 1

),

(0 −1

1 0

)〉R

and the anti-diagonal matrix acts by multiplying with −ki. Check!Check!129

• Consider the subgroup130

U :=

(∗ ∗0 1

)⊆ GL2(R).

Then131

Lie(U) = 〈

(1 0

0 0

),

(0 1

0 0

)〉R.

• Define132

ϕf : GL2(R)×H± → C, (g, z) 7→ j(g, z)−kf(gz).

• For a fixed g ∈ GL2(R):133

∂ϕf∂z

(g, z) = det(g)j(g, z)−k−2 ∂f

∂z(gz).

Check! Needs that

z 7→ gz has ∂∂z -

derivative det(g)j(g,z)2 .

Check! Needs that

z 7→ gz has ∂∂z -

derivative det(g)j(g,z)2 .

134

• Thus, f is holomorphic if and only if135

∂ϕf∂z

(g, i) = 0

for all g ∈ GL2(R).136

• For any path γ : (−ε, ε)→ U with γ(0) = 1 we get137

ϕf (gγ(t)) = ϕf (gγ(t), i) = ϕf (g, γ(t)i)

because j(γ(t), i) = 1. Check!Check!138

• Set139

Xγ :=∂γ

∂t(0) ∈ Lie(U).

• Thus140

Xγ ∗ ϕf= ∂

∂t (ϕf (gγ(t))|t=0

=∂ϕf∂z (g, i) ∂∂t (γ(t) · i)|t=0 +

∂ϕf∂z (g, i) ∂∂t (γ(t) · i)|t=0

.

Check! (Use the

chain rule for

Wirtinger deriva-

tives.)

Check! (Use the

chain rule for

Wirtinger deriva-

tives.)

141

• Consider142

γ1 : (−ε, ε)→ U, t 7→

(1 + t 0

0 1

).

Then γ1(t) · i = (1 + t)i and thus143 (1 0

0 0

)∗ ϕf (g) = i

∂ϕf∂z

(g, i)− i∂ϕf∂z

(g, i).

• Consider144

γ2 : (−ε, ε)→ U, t 7→

(1 t

0 1

),

then γ1(t) · i = i+ t and thus145 (0 1

0 0

)∗ ϕf (g) =

∂ϕf∂z

(g, i) +∂ϕf∂z

(g, i).

• In particular,146

i

(1 0

0 0

)∗ ϕf (g) +

(0 1

0 0

)∗ ϕf (g) = 2

∂ϕf∂z

(g, i)

• Thus147 (i 1

0 0

)∈ gC.

would be a candidate for Y .148

• But we can do better:149

Y := 12

(−1 i

i 1

)

= 12

(1 0

0 1

)+ 1

2 i

(0 −1

1 0

)+ i

(i 1

0 0

)satisfies150

Y ∗ ϕf (g) = −k2ϕf (g) + i(−ki

2)ϕf (g)︸ ︷︷ ︸

=0

+2i∂ϕf∂z

(g, i) = 2i∂ϕf∂z

(g, i).

Check!Check!151

• Why better?152

– Set153

H :=

(0 −ii 0

), X :=

1

2

(−1 −i−i 1

).

– Then H,X, Y is an sl2-triple in gC.154

– Namely, in the basis155 (−i1

),

(i

1

),

it is given by156 (1 0

0 −1

),

(0 1

0 0

),

(0 0

1 0

).

Check!Check!157

Questions?158

When is fϕ holomorphic at the cusps?159

• Consider a function in Fourier expansion:160

f(z) =∑n∈Z

ane2πizn,

say on161

z ∈ C | |Re(z)| < 1, Im(z) > 1.

• Recall162

|e2πiz| = e−2πy

with y = Im(z).163

• Then an = 0 for n < 0 if and only if164

|f(x+ iy)| ≤ CyN

for some C,N ∈ R>0 and y →∞.165

• By definition this means f is of moderate growth, or slowly increasing.166

Moderate growth for general reductive groups:167

• The moderate growth condition generalizes to arbitrary G reductive over Q.168

• For each embedding ρ : G → SLn,Q (not into GLn,Q) we obtain a norm169

|g| := supv

maxi,j=1,...,n

|ρ(g)i,jv|v

on G(A), where v runs through all primes and ∞.170

• A function ϕ : G(A)→ C is of moderate growth or slowly increasing if171

|ϕ(g)| ≤ C|g|N

for suitable C,N ∈ R>0, cf. [GH19, Definition 6.4.].172

• This definition does not depend on the embeddding ρ.173

Exercise:174

• Consider a (holomorphic) section175

f ∈ H0(GL2(Q)\(GL2(Af )×H±), ω⊗k).

• Then: Φ(f) ∈ C∞([GL2]) is of moderate growth176

⇔ f is holomorphic at the cusps177

• In other words, if and only if f lies in Mk.178

Upshot:179

• We can describe the image of180

Mk → C∞(GL2(Q)\GL2(A)), f 7→ ϕf .

• Namely, ϕ ∈ C∞(GL2(Q)\GL2(A)) lies in the image if and only if181

– ϕ(g, g∞z) = z−kϕ(g, g∞z) for (g, g∞) ∈ GL2(A), z ∈ C×.182

– ϕ is of moderate growth.183

– For each g ∈ GL2(Af )184

Y ∗ ϕ(g,−) = 0,

where Y ∈ gC = gl2(R)C is the element constructed before.185

• For k ≥ 1 we will see that there exists a unique, up to isomorphism, irreducible186

(gC,O2(R))-moduleDk−1 containing a non-zero element v, such that Y ∗v = 0,187

and C× acts on v via the character z 7→ z−k, cf. [Del73, Section 2.1].188

• Using Dk−1 one can rewrite our result as saying that189

Mk∼= Hom(gC,O2(R))(Dk−1, C

∞mg(GL2(Q)\GL2(A))),

where C∞mg(GL2(Q)\GL2(A)) ⊆ C∞(GL2(Q)\GL2(A)) denotes the subset of190

functions with moderate growth.191

A condition for cuspidality:192

• Let ϕ = ϕf ∈ C∞(GL2(Q)\GL2(A)) for some f ∈Mk.193

• Then f ∈Mk if and only if194

ϕB(g) :=

∫N(Q)\N(A)

ϕ(ng)dn = 0

for all g ∈ GL2(A), where195

N =

(1 ∗0 1

)is the unipotent radical in the standard Borel196

B =

(∗ ∗0 ∗

)and dn an N(A)-invariant measure on N(Q)\N(A).197

• Namely:198

– We may assume that f , and thus ϕ = ϕf , is invariant under some prin-199

cipal congruence subgroup K(m) ⊆ GL2(Af ).200

– Consider g = 1. Then using N(Q)\N(A) ∼= (mZ× R)/mZ:201 ∫N(Q)\N(A)

ϕf (n)dn

=∫

N(Q)\N(A)

j(n∞, i)−kf(n, n∞i)d(n, n∞)

=∫

R/mZ

∫mZ

f(a, i+ x)dadx

= µ(mZ)∫

R/mZf(1, i+ x)dx

= µ(mZ)a0

with µ(mZ) the volume of mZ and202

f(1, z) =

∞∑n=0

anqn/m

the Fourier expansion of the modular form f(1,−) for Γ(m) at the cusp203

∞.204

– Indeed: The crucial point is that205 ∫R/mZ

e2πi nmxdx = 0

if n > 0.206

– The condition ϕB(g) = 0 for all g ∈ GL2(A) is then equivalent to the207

vanishing of the constant Fourier coefficients at all cusps.208

– We leave the details as an exercise, cf. [Del73, Rappel 1.3.2.].209

• Note: Because ϕ is GL2(Q)-invariant for left translations, we can replace B210

by any conjugate gBg−1 with g ∈ GL2(Q).211

Questions?212

5. From modular forms to automorphic representations, part III1

Last time:2

• Let ϕ ∈ C∞(GL2(Q)\GL2(A)).3

• Then ϕ = ϕf for some f ∈Mk if and only if4

– ϕ(g, g∞z) = z−kϕ(g, g∞) for (g, g∞) ∈ GL2(A), z ∈ C×.5

– ϕ is of moderate growth.6

– For each g ∈ GL2(Af )7

Y ∗ ϕ(g,−) = 0,

where Y ∈ gC = gl2(R)C is a suitably constructed, natural element.8

• f ∈Mk is a cusp form if and only if for all B ⊆ GL2,Q proper parabolic, i.e.,9

Borel, with unipotent radical N ⊆ GL2,Q:10 ∫N(Q)\N(A)

ϕf (ng)dn = 0

for all g ∈ GL2(A).11

Today:12

• Describe Sk as a GL2(Af )-representation.13

• This will yield our main source of examples for automorphic representations.14

Observation:15

• Mk and Sk are smooth and admissible GL2(Af )-representations in the follow-16

ing sense.17

Definition 5.1. Let G be a locally profinite group (like GL2(Af )). A representation18

G on a C-vector space V is called smooth if19

V =⋃K⊆G

V K ,

where K runs through the compact-open subgroups of G and20

V K := v ∈ V | kv = v for all k ∈ K.

Equivalently, V is smooth if the action morphism21

G× V → V

is continuous with V carrying the discrete topology. Denote by22

Rep∞C G

the (abelian) category of smooth representations of G on C-vector spaces.23

Definition 5.2. Let G be a locally profinite group. A smooth representation V of G24

is called admissible if25

dimC V K <∞

for all K ⊆ G compact-open.26

Examples:27

• Mk =⋃

K⊆GL2(Af )

MKk with MK

k = Mk(K) is smooth and admissible because28

the Mk(K) are finite dimensional.29

• Similarly: Sk is an admissible representation of GL2(Af ).30

• ForG reductive over Q⇒ C∞(G(Q)\G(A)) is a smoothG(Af )-representation,31

but not admissible.32

• L2([G]) is not a smooth G(Af )-representation.33

The Hecke algebra (of a locally profinite group):34

• G a locally profinite group.35

– e.g., GL2(Af ), GLn(Qp), Gal(Q/Q), . . .36

• For K ⊆ G compact-open, exists a unique (left resp. right invariant) Haar37

measure µ on G with µ(K) = 1.38

• Concretely:39

– By translation invariance sufficient to define µ for K ′ ⊆ G compact-open40

subgroup.41

– K ′′ := K ′ ∩K compact-open ⇒ of finite index in K resp. K ′.42

– Set43

µ(K ′) :=[K ′ : K ′′]

[K : K ′′].

– This works and is also the unique possibility.44

– Note that µ takes actually values in Q.45

• Fix a left invariant Haar measure µ on G.46

• Define the “Hecke algebra”47

H(G) := C∞c (G)

of G as the set of locally constant, compactly supported functions G→ C.48

• H(G) is an algebra via convolution:49

f ∗ g(x) :=

∫G

f(xy)g(y−1)dy

for f, g ∈ H(G). Check associativ-

ity!

Check associativ-

ity!

50

• The multiplication is a purely algebraic operation:51

– Each f ∈ H(G) is a finite sum52

f =

m∑i=1

aiχgiKi

for some gi ∈ G, Ki ⊆ G compact-open subgroups.53

– Wlog: K1 = . . . = Km (by shrinking).54

– Let K ⊆ G be compact-open and g1, g2 ∈ G. Then55

g1Kg2K =

m∐j=1

hjK

for some hj ∈ G and56

χg1K ∗ χg2K =

m∑j=1

µ(K)χhjK .

Check!Check!57

• H(G) has a lot of idempotents:58

eK :=1

µ(K)χK

with K ⊆ G compact-open subgroups, χK the characteristic function of K,59

but in general no identity. Check!Check!60

Modules for the Hecke algebra (cf. [BRa]):61

• H(G) is a replacement for the group algebra C[G], which is better suited to62

study smooth representations of G.63

• If V ∈ Rep∞C G, then64

f ∗ v :=

∫G

f(h)hvdh

for f ∈ H(G), v ∈ V , defines an action of H(G) on V .65

• Concretely: If K ⊆ G is compact-open and sufficiently small, such that v ∈ V66

is fixed by K and f =m∑i=1

aiχgiK with ai ∈ C, gi ∈ G, then67

f ∗ v =

m∑i=1

aigiv.

• Assume K ⊆ G compact-open. Define the Hecke algebra relative to K68

H(G,K) := eKH(G)eK .

Then H(G,K) is an algebra with unit eK .69

• Let V be any smooth representation of G. Then70

V K = eK ∗ V

for any K ⊆ G compact-open. Check!Check!71

• In particular, H(G,K) ∼= C∞c (K\G/K) (K-biinvariant, compactly supported72

functions on G)73

Important property:74

• If V ∈ Rep∞C G is irreducible and V K 6= 0, then V K is irreducible as an75

H(G,K)-module.76

– Indeed: If M ⊆ V K is a non-trivial H(G,K)-submodule, then77

V = H(G) ∗M

by irreducibility and thus78

V K = eK(H(G) ∗M)) = eKH(G)eKM = M.

• Conversely, if V K is irreducible or zero for all K ⊆ G compact-open, then V79

is irreducible if V 6= 0. Check!Check!80

From irreducible, admissible GLn(Af )-representations to Hecke eigensys-81

tems:82

• V irreducible, admissible GLn(Af )-representation83

• Recall84

K(m) = ker(GLn(Z)→ GLn(Z/m)), m ≥ 0,

form basis of compact-open neighborhoods of the identity.85

• We know:86

– V K(m) is an irreducible H(GLn(Af ),K(m))-module for m 0.87

– Admissibility + Schur’s lemma ⇒ EndH(GLn(Af ),K(m))(VK(m)) ∼= C.88

• Set89

S :=⋂

m≥0, V K(m) 6=0

p prime dividing m,

90

ZS :=∏p/∈S

Zp

and91

ASf := Q⊗ ZS

the adeles away from S.92

• Note: For p /∈ S there exists 0 6= v ∈ V fixed by GLn(Zp) ⊆ GLn(Af ).93

• Define94

TS := H(GLn(ASf ),GLn(ZS)),

the “Hecke-algebra away from S”95

• Very important:96

TS is commutative!97

• Namely, consider98

σ : GLn(ASf )→ GLn(ASf ), g 7→ gtr.

Then:99

– σ preserves GLn(ZS).100

– σ(f1 ∗ f2) = σ(f2) ∗ σ(f1) for f1, f2 ∈ TS .101

– The cosets GLn(ZS)gGLn(ZS) with g ∈ GLn(Af ) a diagonal matrix span102

TS (by the elementary divisor theorem).103

– Conclusion: σ(f) = f for all f ∈ TS and thus104

f1 ∗ f2 = σ(f1 ∗ f2) = σ(f2) ∗ σ(f1) = f2 ∗ f1

as desired.105

• This argument is called Gelfand’s trick.106

• In fact, TS is central in H(GLn(Af ),K(m)) if all prime divisors of m lie in S.107

– If S1, S2 are two, possibly infinite, but disjoint sets of primes and gi ∈108

GLn(ASif ), then109

K(m)g1K(m)g2K(m) = K(m)g2K(m)g1K(m),

which implies the claim.110

• Claim: V defines a natural morphism111

σV : TS → C

of C-algebras.112

• Indeed:113

– For p /∈ S choose 0 6= v ∈ V fixed by GLn(Zp). Then V is fixed by K(m)114

for some m ≥ 0 with p - m.115

– V K(m) is irreducible, and thus H(GLn(Qp),GLn(Zp)) acts via a homo-116

morphism117

σV,p : H(GLn(Qp),GLn(Zp))→ EndH(GLn(Af ),K(m)(VK(m)) ∼= C,

which is independent of v.118

– The σV,p combine to σV .119

• We have not been very careful about Haar measures, or how the σV,p combine120

to σV .121

Definition 5.3. A Hecke eigensystem, or system of Hecke eigenvalues, is a maximal122

ideal in TS where S is a finite set of primes and TS := H(GLn(ASf ),GLn(ZS)).123

Observations:124

• TS is of countable dimension and thus each Hecke eigensystem has residue125

field C.126

• If S ⊆ S′ are two finite sets of primes, each Hecke eigensystem for S induces127

one for S′. ⇒ We call these Hecke eigensystems equivalent.128

• Each irreducible, admissible GLn(Af )-representation yields by the above con-129

struction a system of Hecke eigenvalues: V 7→ mV := ker(σV ).130

• A Hecke eigensystem for S is equivalently a collection of C-algebra homomor-131

phisms132

σp : H(GLn(Qp),GLn(Zp))→ C

for p /∈ S.133

A full description of H(GL2(Qp),GL2(Zp)):134

• Set G = GL2(Qp), K = GL2(Zp).135

• Normalize Haar measure such that µ(K) = 1.136

• Elementary divisor theorem: H(G,K) is free on the elements137

K

(pi 0

0 pj

)K

with (i, j) ∈ Z2,+ := (i, j) ∈ Z2 | i ≥ j.138

• Claim:139

f : C[X1, X±12 ]

'→ H(G,K)

via140

X1 7→ K

(p 0

0 1

)K,

X2 7→ K

(p 0

0 p

)K.

• Here we identify a double coset with its characteristic function.141

• Indeed:142

– The element143

K

(p 0

0 p

)K

is central in H(G,K) as144

KzK ·KgK = KgK ·KzK

for all g ∈ G, and z ∈ G in the center.145

– Moreover, let us check that each146

K

(pi 0

0 1

)K

with i ≥ 1 lies in the image of f .147

– Using induction we get that the n-fold product148

K

(p 0

0 1

)K · . . . ·K

(p 0

0 1

)K

agrees with the set of matrices in Mat2,2(Zp) of determinant of valuation149

n.150

– Thus, the product151

K

(p 0

0 1

)K · . . . ·K

(p 0

0 1

)K

is the characteristic function of this set.152

– As this characterstic function has the characteristic function of153

K

(pn 0

0 1

)K

as a summand with coefficient 1 we are done.154

– In particular, we see that H(G,K) must be commutative without using155

Gelfand’s trick.156

– Now it is not difficult to conclude that f must be an isomorphism. In-157

deed,158

Spec(H(G,K)) ⊆ Spec(C[X1, X±12 ])

defines a subscheme whose projection to Spec(C[X±12 ]) is free of infinite159

rank, but this is only possible for Spec(C[X1, X±12 ]) itself.160

Upshot:161

• Each irreducible, admissible GL2(Af )-representation V yields canonically a162

system of Hecke eigenvalues163

ap ∈ C, bp ∈ C×

for all primes p outside a finite set of primes S, which is naturally determined164

by V .165

• Concretely: If v ∈ GL2(Af ) is fixed by K(m) and p - m, then166

Tp(v) := GL2(Zp)

(p 0

0 1

)GL2(Zp) ∗ v = apv,

“the Hecke operator at p”, and167

Sp(v) := GL2(Zp)

(p 0

0 p

)GL2(Zp) ∗ v = bpv.

• Let Z ⊆ GL2 be the center, i.e., the scalar matrices.168

• Let ψ : A×f ∼= Z(Af ) → C× be the central character of V (exists by Schur’s169

lemma). Then170

bp = ψ((1, . . . , 1, p, 1, . . . , 1)

for p prime such that Z×p ⊆ A×f acts trivially on V .171

General results on the GL2(Af )-representation Sk:172

• The GL2(Af )-representation Sk decomposes into a direct sum173

Sk ∼=⊕i∈I

Vi

of irreducible, admissible representations Vi.174

• Indeed:175

– SkΦ⊆ L2([GL2]) and thus the C-vector space Sk is equipped with the176

non-degenerate GL2(Af )-equivariant hermitian pairing177

(·, ·) : Sk × Sk → C, (f, g) 7→∫

[GL2]

Φ(f)Φ(g)

– Each irreducible GL2(Af )-subquotient of Sk is already a direct summand178

(use orthogonal complements).179

– Using Zorn’s lemma each smooth non-zero GL2(Af )-representation V180

has an irreducible subquotient (Wlog: V generated by some v ∈ V .181

Then apply Zorn’s lemma to the set of submodules not containing v.)182

• Shalika/Piatetski-Shapiro: For each finite set S of primes and each systems183

of Hecke eigenvalues ap, bpp/∈S there exists at most one irreducible subrep-184

resentation V ⊆ Sk with this system of Hecke eigenvalues.185

– This is a special case of the ”strong multiplicty one theorem” for cus-186

pidal automorphic representations for GLn, cf. [GH19, Theorem 11.7.2]187

(maybe we’ll prove this later for n = 2).188

• Consequence: Assume that 0 6= f ∈ Sk is fixed by K(m), and an eigenvector189

for the Tp, Sp−operators for p - m. Then V := 〈f〉GL2(Af ) is irreducible.190

– Reason: Each v ∈ V yields a system of Hecke eigenvalues, which is191

equivalent to the one for f . Now apply strong multiplicity one.192

The decomposition of Sk indexed by newforms:193

• Atkin-Lehner/Casselman: For each irreducible GL2(Af )-subrepresentation194

V ⊆ Sk, there exists some m ∈ N and some 0 6= v ∈ V such that v is195

fixed by the subgroup196

K1(m) :=

(a b

c d

)| c ≡ 0, d ≡ 1 mod m ⊆ GL2(Z)

• Set197

Γ1(m) := GL2(Z) ∩K1(m).

• Remarks:198

– Our notation is different than in many sources, where Γ1(m) ⊆ SL2(Z).199

– The minimal m for which there exists such a non-zero v ∈ V is called200

the conductor of V . In this case, v is unique up to a scalar.201

– Maybe more details later, cf. [Del73].202

• The morphism203

Γ1(m)\H± '−→ GL2(Q)\(GL2(Af )/K1(m)×H±)

[z] 7→ [(1, z)]

is an isomorphism, compatible with ω⊗k.204

• In particular,205

Mk(Γ1(m)) ∼= Mk(K1(m)).

• Let f ∈Mk(K1(m)).206

• Let f ∈Mk(Γ1(m)) be the section corresponding to f .207

• Then208

f(g, z) = f(γ, z) = j(γ−1, z)−kf(γ−1z).

where g = γh with γ ∈ GL2(Q), h ∈ K1(m). Check!Check!209

• Fix p prime, p - m.210

• Tp, Sp act on Mk(K1(m)). Thus, by transport of structure on Mk(Γ1(m)).211

• Let us make Sp on Mk(Γ1(m)) explicit.212

• We have to find an expression213 (p 0

0 p

)= γh, with γ ∈ GL2(Q), h ∈ K1(m).

• Because p,m are prime there exists a, b ∈ Z, such that214

A :=

(a b

m p

)∈ GL2(Z).

• Then215

Adiag ·

(p−1 0

0 p−1

)diag

·

(p 0

0 p

)p

∈ K1(m),

where (−)diag denotes the diagonal embedding for GL2(Q) and (−)p the em-216

bedding at p.217

• In other words, we can take218

γ :=

(p 0

0 p

)A−1

• Thus Sp acts on Mk(Γ1(m)) via219

f 7→ (z 7→ j(γ−1, z)−kf(γ−1z)).

• The group220

Γ0(m) :=

(a b

c d

)| c ≡ 1 mod m

normalizes Γ1(m) and thus there is an action of Γ0(m)/Γ1(m) ∼= (Z/m)× via221

(B, f) 7→ j(B, z)−kf(Bz).

for B ∈ Γ0(m), f ∈Mk(Γ1(m)).222

• Thus, we obtain a decomposition223

Mk(Γ1(m)) =⊕

χ : (Z/m)×→C×Mk(Γ0(m), χ)

with Mk(Γ0(m), χ) the space of modular forms for Γ0(m) with nebentypus χ,224

i.e., modular forms for Γ1(m) satisfying225

f(az + b

cz + d)) = χ(d)(cz + d)kf(z) for

(a b

c d

)∈ Γ0(m).

• For each χ : (Z/m)× → C× there exists a unique character226

ψχ : Q×\A×/(1 +mZ) ∩ Z× → C×

such that ψχ(r) = r−k for r ∈ R>0.227

• We get that228

Mk(Γ0(m), χ) ∼= Mk(K1(m), ψχ)

where the RHS denotes the ψχ-eigenspace for the action of A× onMk(K1(m)).229

• Let us finish the description of Sp on Mk(Γ1(M)).230

• The action of Sp is given on Mk(Γ0(m), χ) ⊆Mk(Γ1(m)) by231

γ−1 =

(p−1 0

0 p−1

)A

where

(p−1 0

0 p−1

)acts by pk and A ∈ Γ0(m) by χ(p).232

• Let us describe now the action (by transport of structure) of Tp onMk(Γ0(m), χ).233

• We have234

GL2(Zp)

(p 0

0 1

)GL2(Zp) =

p−1∐j=0

(p j

0 1

)GL2(Zp)

∐(1 0

0 p

)GL2(Zp).

• We again have to find factorizations into235

γ · h, γ ∈ GL2(Q), h ∈ K1(m).

• We can take the factorization236 (p j

0 1

)p

=

(p j

0 1

)diag

· h

for j = 0, . . . , p− 1, with h ∈ K1(m), i.e.,237

γ−1 =

(p−1 −p−1j

0 1

).

• For238 (1 0

0 p

)we find, similar to the argument above,239

γ−1 =

(a b

m p

)·

(1 0

0 p−1

)=

(1 0

0 p−1

)·

(a p−1b

pm p

).

• Putting these together we obtain240

Tp(f) = pkχ(p)f(pz) +

p−1∑j=0

f(z − jp

).

• If f ∈Mk(Γ0(m), χ) is written in Fourier expansion241

f(z) =

∞∑n=0

anqn, q = e2πiz,

then242

Tp(f)(z) =

∞∑n=0

panpqn +

∞∑n=0

pkχ(p)anqpn.

• This follows from243

p−1∑j=0

e2πi−jnp =

p, p|n

0, otherwise.

Check!Check!244

• In particular, if f ∈Mk(Γ0(m), χ) is an eigenvector for Tp and a1 = 1, then245

Tp(f) = apf,

where246

Tp =1

pTp.

In other words, the Hecke eigenvalues are the Fourier coefficients of the nor-247

malized eigenform.248

Theorem 5.4. The GL2(Af )-representation Sk decomposes into irreducibles as249

Sk ∼=⊕f

〈f〉GL2(Af )

with f running through the set of (normalized) newforms for Γ0(N) with nebentypus250

χ for varying N and Dirichlet characters χ : (Z/N)× → C×.251

Newforms (roughly):252

• A newform is an eigenform realising the minimal N among all eigenforms253

with an equivalent system of Hecke eigenvalues.254

Proof. After the above preparation we can now quote [DS05, Proposition 5.8.4],255

[DS05, Theorem 5.8.2] which imply that for each system of Hecke eigenvalues ap-256

pearing in Sk there exists a unique normalized (i.e., in Fourier expansion a1 = 1)257

newform with equivalent system of Hecke eigenvalues. 258

Questions?259

6. Langlands reciprocity for newforms1

Last time:2

• We proved the decomposition3

Sk ∼=⊕f

〈f〉GL2(Af )

into irreducibles.4

• Here f is running through the set of (normalized) newforms of weight k for5

Γ0(N) with nebentypus χ for varying N and varying Dirichlet characters6

χ : (Z/N)× → C×.

• To each irreducible, admissible GL2(Af )-representation π we associated a7

system of Hecke eigenvalues8

ap(π), bp(π)p/∈S

for S := p prime with πGLn(Zp) = 0. In fact, ap(π) resp. bp(π) is the eigen-9

value of the double coset10

GL2(Zp)

(p 0

0 1

)GL2(Zp)

resp.11

GL2(Zp)

(p 0

0 p

)GL2(Zp)

when acting on a non-zero vector fixed by GL2(Zp).12

• If π = 〈f〉GL2(Af ) for a normalized newform f ∈ Sk(Γ0(N), χ), then S =13

prime divisors of N and14

ap(π) = pap, bp(π) = pkχ(p)

if p /∈ S, where f(q) =∞∑n=1

anqn is the Fourier expansion of f .15

Today:16

• Traces of Frobenii for `-adic Galois representations.17

• Statement of Langlands program for Sk.18

Traces of Frobenii for `-adic representations:19

• ` some prime.20

• Q` = lim−→E/Q`

E is a topological field via colimit topology.21

• Let W be a finite dimensional Q`-vector space.22

• Let σ : GQ := Gal(Q/Q)→ GL(W ) be a continuous representation (an “`-adic23

Galois representation”)24

• Recall: For each prime p there exists an embedding25

GQp := Gal(Qp/Qp)→ GQ,

well-defined up to conjugacy in GQ.26

• GQp sits in the exact sequence:27

1→ Ip → GQp → Gal(Fp/Fp) = (Frobgeomp )Z → 1

with Ip the inertia subgroup, and28

Frobgeomp : Fp → Fp, x 7→ x1/p

the geometric Frobenius at p (= inverse of usual Frobenius).29

• Let S be a finite set of primes. We say σ is unramified outside S if σ(Ip) = 130

for all p /∈ S.31

• If σ unramified outside S and n := dimQ`W , we can associate to σ a system32

c1,p, . . . , cn,pp/∈S

of elements c1,p, . . . , cn,p ∈ Q`. Namely,33

ci,p := Tr(σ(Frobgeomp )|ΛiW )

• Note: The ci,p are well-defined because σ(Ip) = 1.34

• Note: The ci,p = Tr(σ(Frobgeomp )|ΛiW ) depend only on p as the trace is35

invariant under conjugation.36

• If W is semisimple, then σ is determined by the37

c1,p, . . . , cn,pp/∈S

for each finite set of primes S such that W is unramified outside S, cf. [Ser97,38

I-10].39

• In fact, the collection c1,pp/∈S is sufficient (+ W semisimple and unramified40

outside S).41

• Indeed:42

– Consider Λ := Q`[GQ,S ], where GQ,S is the quotient of GQ by the closure43

of the subgroup generated by the Ip, p /∈ S.44

– Assume W,W ′ are two semisimple, continuous GQ,S-representations with45

Tr(Frobgeomp |W ) = Tr(Frobgeom

p |W ′)

for all p /∈ S.46

– By Chebotarev density and continuity this implies47

Tr(λ|W ) = Tr(λ|W ′),

for all λ ∈ Λ.48

– For irreducible, pairwise non-isomorphic Λ-modules W1, . . . ,Wm and49

each i = 1, . . . ,m, there exists µi ∈ Λ, such that50

µi = 1

on Wi, but51

µi = 0

on Wj , j 6= i (this is a version of the Chinese remainder theorem).52

– Write53

W =

m⊕i

W⊕nii , W ′ =

m⊕i=1

W⊕n′ii

in isotypic components withW1, . . . ,Wm irreducible, pairwise non-isomorphic,54

and ni, n′i ∈ N, possibly zero.55

– Then56

ni dimQ`Wi = Tr(µi|W ) = Tr(µi|W ′) = n′i dimQ`Wi

for all i = 1, . . . ,m, i.e., ni = n′i, i = 1, . . . ,m and W ∼= W ′ as desired.57

• This statement can be seen as an analog of “strong multiplicity one” for58

cuspidal automorphic representations for GLn mentioned last time.59

Arithmetic significance of the traces of Frobenii:60

• Let σ : GQ → GL(W ) be an `-adic representation, unramified outside S.61

• The traces62

c1,p := Tr(σ(Frobgeomp )|W )

encode significant arithmetic information.63

• For example, assume F/Q is Galois with Gal(F/Q) ∼= S3 and64

σ : GQ Gal(F/Q)→ GL2(Q`)

is inflated from the unique irreducible, 2-dimensional representation of S3.65

Set66

S := p ramified in F

and for p /∈ S67

FrobgeomF,p

as the image of Frobgeomp ∈ GQ,S in Gal(F/Q) ∼= S3.68

• Then:69

∗ Tr(Frobgeomp |W ) = 2⇔ Frobgeom

F,p ∈ 1 ⇔ p splits completely in OF ,70

∗ Tr(Frobgeomp |W ) = 0⇔ Frobgeom

F,p ∈ (1, 2), (1, 3), (2, 3) ⇔ p splits into three distinct primes in OF ,71

∗ Tr(Frobgeomp |W ) = −1⇔ Frobgeom

F,p ∈ (1, 2, 3), (1, 3, 2) ⇔ p splits into two distinct primes in OF .72

• Thus: Knowledge of traces of Frobenii for all `-adic Galois representations73

(with finite image) implies knowledge of the decomposition of unramified74

primes in all finite extensions of Q.75

• Other examples of `-adic representations:76

Hiet(XQ,Q`)

for proper, smooth schemes X over Q, and i ≥ 0.77

• TheGQ-action is induced by functoriality ofHiet(−,Q`) from the (right) action78

of GQ on79

XQ = X ×Spec(Q) Spec(Q).

• Proper, smooth base change + Grothendieck-Lefschetz trace formula ⇒ If X80

has good reduction at p and p 6= `, then81

2dim(X)∑i=0

(−1)iTr(Frobgeomp |Hi

et(XQ,Q`)) = ]X (Fp),

where X → Spec(Z(p)) is a proper, smooth model of X at p.82

• The last example motivates why we took the geometric Frobenius and not83

the arithmetic Frobenius Frobarithp = (Frobgeom

p )−1.84

Questions?85

The whole point of Langlands reciprocity:86

• Traces of Frobenii are supposed to match Hecke eigenvalues!87

• Thus, arithmetic information is expected to be encoded in automorphic in-88

formation, which might be more concrete.89

• Example (Hecke):90

– Let F be the splitting field of x3 − x− 1.91

– Then F is the Hilbert class field of Q(√−23), and Galois over Q with92

Galois group S3.93

– Consider as before the associated 2-dimensional, irreducible Galois rep-94

resentation σ : GQ → Gal(F/Q) → GL2(Q`).95

– Hecke proved that there exists a normalized newform f(q) =∞∑i=1

anqn

96

such that97

ap = Tr(σ(Frobgeomp ))

for all primes p with p - 23.98

– One can make f more explicit: Considering ramification, f must lie in99

S1(Γ0(23), χ) with χ : (Z/23)× → C× the quadratic character determin-100

ing101

Q(√−23) ⊆ Q(ζ23),

and people knowing modular forms will tell us that we must have102

f(q) = q

∞∏i=1

(1− qi)(1− q23i).

– We can now calculate the developement of f (or look it up in some103

database like LMFDB):104

f(q) = q − q2 − q3 + . . .+ q58 + 2q59 + . . .

and deduce how primes decompose in F , e.g., 59 splits completely!105

– This example was posted by Matthew Emerton on mathoverflow. More106

insightful posts by him can be found via his webpage: http://www.107

math.uchicago.edu/~emerton/ . Find and read

them!

Find and read

them!

108

• Fermat’s theorem was proved by Wiles/Taylor using the following strategy109

(initiated by Frey, and complemented by Serre, Ribet):110

– Assume up + vp + wp = 0 with u, v, w ∈ Q, uvw 6= 0 and p ≥ 3.111

– Consider the elliptic curve E over Q with (affine) Weierstraß equation112

y2 = x(x+ up)(x− vp)

(after possibly manipulating u, v, w a bit).113

– Consider the Galois representation114

σ : GQ → GL(H1et(EQ,Q`)) ∼= GL2(Q`).

– Show that σ is modular, i.e., there exists a normalized newform f1(q) =115∞∑n=1

anqn of weight 2 such that116

Tr(σ(Frobgeom)) = ap

for almost all primes p.117

– Using a theorem of Ribet, one concludes that f1 must be congruent to a118

normalized newform f2 in119

S2(Γ0(2)),

i.e., the Fourier coefficients of f1, f2 are algebraic integers and congruent120

modulo some prime.121

– But S2(Γ0(2)) = 0, and thus f2 cannot exist.122

– This yields the desired contradiction.123

– For more details, cf. [Wil95], [Rib90].124

Langlands reciprocity for newforms:125

• Fix a prime ` and an isomorphism126

ι : Q` ∼= C.

• For newforms the Langlands program combined with the Fontaine-Mazur127

conjecture predicts a bijection128

LL: Amod'−→ Gmod

from the set129

Amod := irreducible GL2(Af )-subrepresentations π ⊆⊕k≥1

Sk,

to a certain set130

Gmod

consisting of irreducible, 2-dimensional Galois representations σ : GQ → GL2(Q`)131

• LL should satisfy that for all π ∈ Amod with σ := LL(π) we have132

ι(Tr(σ(Frobarithp ))) = ap(π), ι(det(σ(Frobarith

p ))) = bp(π)

for p outside some specified finite set S of primes. In a previous ver-

sion I considered

the geometric

Frobenius, which

was wrong.

In a previous ver-

sion I considered

the geometric

Frobenius, which

was wrong.

133

• Here, the eigenvalues ap(π) resp. bp(π) are used and not the ap(π), bp(π).134

• Note: Such a bijection LL is uniquely determined (by the respective multi-135

plicity one theorems), if it exists.136

• How is S specified? Let us call a prime p unramified for σ resp. π if137

σ(Ip) = 1

resp.138

πGL2(Zp) 6= 0.

• Then a prime p 6= ` is conjecturally unramified for π if and only if it is139

unramified for σ := LL(π).140

• The matching of Hecke eigenvalues states then more precisely that σ(Frobarithp )141

should have characteristic polynomial142

X2 − ι−1(ap(π))X + ι−1(bp(π))

for each unramified prime p 6= ` for π.143

• It is the next aim of the course to indicate how the map LL can be constructed,144

i.e., how to associate Galois representations to normalized newforms. For this145

we follow results of Deligne, cf. [Del71b], and Deligne/Serre, cf. [DS74].146

The set Gmod:147

• The set Gmod was defined by Fontaine/Mazur in relation with their remarkable148

conjecture on geometric Galois representations, cf. [FM95].149

• Let σ : GQ → GL2(Q`) be an irreducible, 2-dimensional `-adic representation.150

• Then by definition σ ∈ Gmod if and only if151

– σ unramified at almost all p, i.e., σ(Ip) = 1 for p outside some finite set152

of primes S.153

– σ is odd, i.e., for each complex conjugation c ∈ Gal(C/R) ⊆ GQ we have154

det(σ(c)) = −1.

– The restriction155

σ` := σ|GQ`: GQ` → GL2(Q`)

is de Rham with Hodge-Tate weights 0, ω for some ω ∈ N (note that156

ω = 0 is allowed).157

• If σ = LL(π) with π ⊆ Sk, then conjecturally ω = k − 1.158

• Using the work of many people (Wiles, Kisin, Emerton, Skinner-Wiles, Pan,. . . )159

any σ ∈ Gmod with distinct Hodge-Tate weights is modular if ` ≥ 5, cf. [Pan19,160

Theorem 1.0.4.].161

• If σ ∈ Gmod has finite image, then σ is modular, cf. [PS16], [GH19, Section162

13.4.] (and the references therein).163

• Conjecturally: HT-weights 0 ⇔ image of σ finite.164

de Rham representations:165

• p prime (the previous `).166

• K/Qp a discretely valued, non-archimedean extension (e.g., K/Qp finite).167

• σ : GK := Gal(K/K)→ GLm(Qp) a continuous representation.168

• Then σ has coefficients in some finite extension E/Qp (by compactness of GK169

as Qp carries the colimit topology here).170

• We obtain a Qp-linear representation, a “local p-adic Galois representation”,171

ρ : GK → GLm(E) ⊆ GLn(Qp),

where n := m · dimQpE.172

• Upshot: Let V be a finite dimensional Qp-vector space and ρ : GK → GL(V )173

a continuous representation.174

• Then V is called de Rham if175

dimK(BdR ⊗Qp V )GK = dimQp V

for a certain field extension BdR of K with GK-action, cf. [BC09, Section 6].176

• Namely: BdR is Fontaine’s field of p-adic periods, cf. [Fon94], [BC09, Defini-177

tion 4.4.7]. Abstractly,178

BdR∼= CK((t)),

where CK is the completion of K for its p-adic valuation (this topology is179

coarser than the colimit topology on K).180

• BdR is a discretely valued field, with residue field CK , the deduced decreasing181

filtration Fil•BdR is GK-stable and as CK-semilinear GK-representations182

gr•BdR∼=⊕j∈Z

CK(j),

where CK(j) := CK ⊗Zp Zp(j), with183

Zp(1) := Tpµp∞(CK)

the p-adic Galois representation associated to the cyclotomic character184

χcyc : GK → Z×p ∼= Aut(µp∞(CK)).

• Important Theorem of Tate, cf. [BC09, Theorem 2.2.7]: Let χ : GK → Q×p be185

a character. Then186

H0(GK ,CK(χ)) =

K, if χ|IK has finite image

0, otherwise ,

here IK ⊆ GK is the inertia subgroup.187

• In particular:188

H0(GK ,CK(j)) = 0, j 6= 0

as χjcyc|IKhas infinite image if j 6= 0.189

• Conclusion: If dimC V = 1, then V is de Rham if and only if V is isomorphic190

to χ ·χjcyc for some character χ : GK → Q×p with IK of finite order, and j ∈ Z191

⇒ There exists many representations, which are not de Rham: (χp−1cyc )a with192

a ∈ Zp \ Z.193

• Facts:194

– ρ : GK → GL(V ) is de Rham if ρ|GK′ is de Rham for some finite extension195

K ′ of K, cf. [BC09, Proposition 6.3.8].196

– de Rham representations are stable under subquotients, duals and tensor197

products, cf. [BC09, Section 6.1].198

• Let V be a de Rham representation of dimension n. Then199

V ⊗Qp CK ∼= CK(j1)⊕ . . .⊕ CK(jn)

as CK-semilinear GK-representations, where j1, . . . , jn ∈ Z. The unordered200

collection (j1, . . . , jn) is called the collection of Hodge-Tate weights for V .201

Strategy for constructing LL (roughly):202

• Let π ∈ Amod, with associated system of Hecke eigenvalues203

ap, bpp/∈S .

• Assume for simplicity that π ⊆ S2, i.e., associated to a weight 2 newform.204

• Then we will construct a GL2(Af )-equivariant embedding205

S2 → H1(GL2(Q)\(GL2(Af )×H±),C) := lim−→K

H1(GL2(Q)\(GL2(Af )/K ×H±),C)

of S2 into cohomology.206

• The given isomorphism ι : Q` ∼= C yields an isomorphism207

H1(GL2(Q)\(GL2(Af )×H±),C) ∼= H1(GL2(Q)\(GL2(Af )×H±),Q`)

• For K ⊆ GL2(Af ) compact-open, the complex manifold208

GL2(Q)\(GL2(Af )/K ×H±)

is actually algebraic, i.e., given by XK(C) for some quasi-projective scheme209

XK → Spec(C).210

• The etale comparison theorem implies211

H1(GL2(Q)\(GL2(Af )×H±),Q`) ∼= H1et(XK ,Q`).

• Now the miracle happens: For K ⊆ GL2(Af )212

the quasi-projective scheme XK → Spec(C) is canonically defined over Spec(Q)!213

In other words, XK∼= XK ×Spec(Q) Spec(C) for (compatible) quasi-projective214

schemes XK → Spec(Q).215

• In particular,216

H1et(XK ,Q`) ∼= H1

et(XK,Q,Q`)

by invariance of etale cohomology under change of algebraically closed base217

fields. But the RHS carries an action of GQ!218

• Look now at the GQ ×GL2(Af )-module219

H1et(XQ,Q`) := lim−→

K

H1et(XK,Q,Q`).

• Then we will check that the GQ-module220

LL(π) := HomGL2(Af )(π,H1et(XQ,Q`))

is 2-dimensional and that we can express the traces of Frobenii by Hecke221

eigenvalues.222

• Replacing Q` by some local system a similar strategy works for weight k ≥ 2.223

• However, this does not help for k = 1. Here one has to use that a weight 1224

modular form can be congruent to some modular form of weight k ≥ 2, and225

use the previous case.226

Questions?227

7. Modular curves as moduli of elliptic curves (by Ben Heuer)1

Last time:2

• First rough sketch of construction of3

LL: Amod'−→ Gmod

Amod :=irreducible GL2(Af )-subrepresentations π ⊆⊕k≥1

Sk,

Gmod :=σ : GQ → GL2(Q`), unramified outside finite set S, odd, de Rham

• Crucial point for construction: The complex manifold4

GL2(Q)\(GL2(Af )/K ×H±)

(1) is algebraic over Spec(C),5

(2) admits canonical model XK over Spec(Q).6

This gives rise to the Galois action on l-adic etale cohomology H1et(XK,Q,Q`).7

Today:8

• Explain why the algebraic model XK exists.9

• Reinterpret notions defined before in algebro-geometric terms: modular forms,10

compactification, q-expansions, adelic Hecke action, ...11

Reminder on elliptic curves Reference: [Silverman: The Arithmetic...]12

• Let K be any field. We have the following equivalent definitions:13

Definition 7.1. An elliptic curve over K is a14

– connected smooth projective curve E|K of genus 1 with point O ∈ E(K).15

– connected smooth projective algebraic group E|K of dimension 1.16

– non-singular plane cubic curve, defined by a Weierstraß equation17

E : y2 = x3 + ax+ b, a, b ∈ K

(if charK 6∈ 2, 3, otherwise more terms are required).18

• Fact: The group scheme is automatically commutative (!).19

• Fact: The non-singularity can be expressed as a condition on the discriminant:20

E non-singular ⇔ ∆(a, b) = −16(4a3 + 27b2) 6= 0.

• Fact: Given E, the Weierstraß equation is not unique. But the j-invariant21

j(E) := j(a, b) = 17284a3

4a3 + 27b2

is independent of choice of Weierstraß equation.22

• Fact: If K is algebraically closed, we have23

E ∼= E′ ⇔ j(E) = j(E′).

• More generally, we can replace Spec(K) by any base scheme S.24

Definition 7.2. An elliptic curve over S is a proper smooth curve E → S25

with geometrically connected fibres of genus 1, together with a point 0 : S → E.26

• This can be thought of as family of elliptic curves parametrised by S.27

• Again, this automatically has a group structure:28

Theorem (Abel). There is a unique way to endow an elliptic curve E → S29

with the structure of a commutative S-group scheme s.t. 0 = identity section.30

• Zariski-locally on S, one can describe E in terms of a Weierstraß equation.31

Definition 7.3. A morphism of elliptic curves E → E′ over S is a homomor-32

phism of S-group schemes. An isogeny is an fppf-surjective homomorphism33

with finite flat kernel.34

Example 7.4. For any N ∈ Z, the group scheme structure gives the mor-35

phism “multiplication by N”, which is denoted by [N ] : E → E.36

• Fact: [N ] is finite flat of rank N2. It is finite etale if N is invertible on S.37

• In particular, [N ] is an isogeny.38

Definition 7.5. E[N ] := ker[N ] ⊆ E, the N -torsion subgroup. This is a39

finite flat closed subgroup scheme. It is finite etale if N is invertible on S.40

• More generally, for any finite flat subgroup scheme D ⊆ E[N ] of exponent N ,41

∃ unique elliptic curve E/D with isogeny ϕ : E → E/D with kernel D.42

• [N ] induces a dual isogeny ϕ∨ : E/D → E with kernel E[N ]/D.43

Elliptic curves over K = C Reference: [Diamond–Shurman: A first course...]44

• Let E|C be an elliptic curve. Then E(C) has the structure of a compact45

complex Lie group. As such, it can be canonically uniformised:46

• Let LieE be the tangent space at 0 ∈ E(C). Then LieE ∼= C and there is an47

isomorphism of complex Lie groups48

E(C) = LieE/H1(E,Z) ∼= C/Λ, where Λ ∼= Z2.

• For any N ∈ Z, we have49

E(C)[N ] = ( 1NΛ)/Λ ⊆ C/Λ.

• Note: E(C)[N ] = ( 1NZ/Z)2 = (Z/N)2 is of rank N2, as claimed.50

• Upshot: E(C) is a 1-dimensional complex torus: A quotient of C by a51

lattice Z2 ∼= Λ ⊆ C such that Λ⊗Z R = C.52

• morphisms of complex tori := morphisms of complex Lie groups53

Theorem. There is an equivalence of categories54

elliptic curves over C → 1-dim complex tori

• Essential surjectivity: Let Λ ⊆ C as above. Define the Weierstraß ℘-function55

℘(z,Λ) =1

z2+∑w∈Λw 6=0

(1

z2− 1

(z − w)2

)

• Then ℘(z) = ℘(z,Λ) is holomorphic on C\Λ with complex derivative56

℘′(z) = −2∑w∈Λ

1

(z − w)3.

• This is clearly Λ-periodic ℘ and ℘′ are meromorphic functions on C/Λ.57

Theorem. The functions ℘ and ℘′ are related by a Weierstraß equation

EΛ : ℘′2 =4(℘3 + g2(Λ)℘+ g3(Λ))

for explicit g2(Λ), g3(Λ) ∈ C. We thus get an isomorphism of Lie groups58

(℘, ℘′) : C/Λ ∼−→ EΛ(C).

• Sending C/Λ 7→ EΛ defines a quasi-inverse59

1-dim complex tori → elliptic curves over C.

Complex moduli spaces of elliptic curves60

• We can now parametrise complex elliptic curves up to isomorphism61

• For this, we just have to determine when two lattices Λ1,Λ2 satisfy62

C/Λ1∼= C/Λ2.

• Fact: Any homomorphism of complex tori is multiplication by a ∈ C.63

• ⇒ this happens iff Λ1 = aΛ2 for some a ∈ C×.64

• Can now uniformise as follows: Write Λ = w1Z⊕ w2Z.65

• The condition Λ⊗Z R = C implies w1/w2 ∈ C \ R = H±.66

• ⇒ For every complex torus E, there is τ ∈ H± such that67

E ∼= Eτ := C/Λτ , Λτ = Z⊕ τZ

• Finally, for τ, τ ′ ∈ H±, we have68

Z + τZ = Z + τ ′Z⇔ ∃γ =(a bc d

)∈ GL2(Z) : γτ =

aτ + b

cτ + d= τ ′

• This shows: τ 7→ Eτ defines69

GL2(Z)\H± ∼= complex tori/ ∼∼= elliptic curves over C/ ∼

This gives our space from earlier,70

SL2(Z)\H = GL2(Z)\H± = GL2(Q)\(GL2(Af )/K ×H±) for K = GL2(Z)

an interpretation as a “moduli space” (so far, in a weak sense):71

SL2(Z)\H = elliptic curves over C/ ∼

• C algebraically closed⇒ j-invariant defines bijection (in fact biholomorphism)72

j : SL2(Z)\H ∼−→ C.

• Via the covering map H± → GL2(Z)\H±, we get from the above:

H± =complex tori with ordered basis α : Z2 ∼−→ Λ/ ∼

=elliptic curves E over C with α : Z2 ∼−→ H1(E,Z)/ ∼

• What about moduli interpretations of other levels? Recall for N ∈ N, we had73

the level K = K(N),74

XK := GL2(Q)\(GL2(Af )/K ×H±) =∐

Γ(N)\H±

where75

Γ(N) = γ ∈ GL2(Z)|γ ≡ ( 1 00 1 ) mod N.

• These are precisely the automorphisms of Λ that preserve the subgroup76

EΛ[N ] = 1NΛ/Λ ⊆ C/Λ.

Consequently, we have the more general statement

Γ(N)\H± =complex tori with ordered basis α : (Z/N)2 ∼−→ 1NΛ/Λ/ ∼

=elliptic curves E over C with α : (Z/N)2 ∼−→ E(C)[N ]/ ∼ .

• In between GL2(Z) and Γ(N), we had the groups77

Γ1(N) = γ ∈ GL2(Z)|γ ≡ ( ∗ ∗0 1 ) mod N,78

Γ0(N) = γ ∈ GL2(Z)|γ ≡ ( ∗ ∗0 ∗ ) mod N.

• The modular curves associated to these have moduli interpretations

Γ1(N)\H± =elliptic curves E|C with point Q ∈ E(C)[N ] of exact order N/ ∼

Γ0(N)\H± =elliptic curves E|C with cyclic subgroup C ⊆ E(C)[N ] of rank N/ ∼ .

Motivating Example: The Legendre family79

• Consider the complex manifold U = P1\0, 1,∞ in the variable λ.80

• Over U , we have a complex family of elliptic curves:81

• The Legendre family is cut out of P2U → U by the Weierstraß equation82

Eλ : Y 2 = X(X − 1)(X − λ).

Proposition 7.6. (1) We have Eλ[2] = O, (0, 0), (1, 0), (λ, 0). In partic-83

ular, there is a natural isomorphism Eλ[2] = (Z/2Z)2.84

(2) For any elliptic curve E over C together with a trivialisation85

α : (Z/2Z)2 ∼−→ E[2],

there is a unique point x ∈ U such that E = (Eλ)x.86

• Upshot: Eλ → U is universal elliptic curve with level Γ(2), in some sense.87

• This is all algebraic! Can make sense of Eλ → U as morphism of schemes C.88

• Even better: These schemes are already defined over Q.89

• Can elaborate this argument to show that90

Γ(2)\H± =: XΓ(2)(C) = P1\0, 1,∞.

• NB: In particular, the compactification is X∗Γ(2) = P1.91

• This shows that S2(Γ0(2)) = 0, which we used for Fermat’s Last Theorem.92

Idea:93

Get algebraic model of Γ(N)\H± over Q by passing from C to moduli of elliptic

curves over general Q-schemes S.

• Question: Is there a scheme representing the functor94

PSL2(Z) : S 7→ elliptic curves over S/ ∼

on schemes over Q? This would be a scheme X → Q with C-points SL2(Z)\H.95

• Answer: No. An easy way to see this is the phenomenon of twists:96

• Can have two non-isomorphic E 6∼= E′ over Q that become isomorphic over a97

quadratic extension K. In particular, j(E) = j′(E).98

• Clearly, two Q-points of X agree iff they do over K. So X cannot exist!99

• Same problem for the Legendre family above: This does not represent100

S 7→ elliptic curves E over S with α : (Z/2Z)2 ∼−→ E[2]/ ∼

because this fails to account for quadratic twists.101

• Fact: Twists of elliptic curves E over a field K correspond to 1-cocycles102

GK|K → Aut(E).

Quadratic twists come from the involution ([−1] : E → E) ∈ Aut(E).103

• Slogan: Nontrivial automorphisms are bad for representability104

• Upshot: In order to get a representable functor, need additional data.105

Moduli problems of elliptic curves [Katz–Mazur: Arithmetic moduli...]106

• Consider the moduli functors on schemes over Z[ 1N ]:

PΓ0(N) : S 7→ (E|S,C ⊆ E[N ] cyclic subgroup scheme of rank N)/ ∼,

PΓ1(N) : S 7→ (E|S,Q ∈ E[N ](S) point of exact order N)/ ∼,

PΓ(N) : S 7→ (E|S, α : (Z/NZ)2 ∼−→ E[N ] isom of group schemes)/ ∼ .

• We can also form “mixed moduli problems” like107

PΓ1(N)∩Γ0(p) := PΓ1(N) ×PSL2(Z)PΓ0(p) : S 7→ (E,Q,C)/ ∼

• Crucial point: Let (E,C) ∈ PΓ0(N)(S). The automorphism [−1] : E → E108

sends C 7→ C. In particular, [−1] ∈ Aut(E,C).109

• In contrast, let (E,Q) ∈ PΓ1(N)(S), then [−1] : E → E sends Q ∈ E[N ] to110

−Q ∈ E[N ], which is different if N ≥ 3. In particular, [−1] 6∈ Aut(E,Q).111

• Similarly for PΓ(N)(S). This explains the problem with the Legendre family:112

N = 2 is too small to deal with the automorphism [−1]! We need N ≥ 3.113

• Upshot: The moduli problems PΓ(N) and PΓ1(N) are “rigid” (:= have no114

non-trivial automorphisms) for N ≥ 3, in contrast to PΓ0(N) and PΓ(2).115

Theorem. Let N ≥ 3 and p any prime.

(1) The moduli problems PΓ(N) and PΓ1(N) are each representable by smooth

affine curves XΓ(N) and XΓ1(N) over Z[ 1N ].

(2) For any n ∈ N, the moduli problem PΓ(N)∩Γ0(pn) is representable by a flat

affine curve XΓ1(N)∩Γ0(pn) over Z[ 1N ] that is smooth over Z[ 1

pN ].

• Remark: In order to represent PSL2(Z), one can pass from schemes to the116

bigger category of stacks. Get “moduli stack of elliptic curves” M1,1.117

• Between varying level structures Γ ⊆ Γ′ ⊆ Γ(N), have forgetful morphisms118

XΓ′ → XΓ.

These are all finite etale over Z[ 1N ].119

• GL2(Z/NZ) acts from the right on PΓ(N) by precomposition120

α 7→ α γ.

Alternatively, this is a left action via precomposition with γ∨ = γ−1 det γ.121

• This induces a GL2(Z/NZ)-action on XΓ(N).122

• Similarly, there is a natural (Z/NZ)×-action on XΓ1(N).123

• We can then define a modular curve124

XΓ0(N) := XΓ1(N)/(Z/NZ)×.

• This does not represent P(Γ0(N)), but it is “as close as possible”.125

• In particular, XΓ0(N)(K) = P(Γ0(N))(K) for algebraically closed K.126

Compactification127

• The j-invariant associated to Weierstraß equations defines a finite flat function128

j : X → A1.

• By normalisation in A1 → P1, can define129

j : X∗ → P1,

still a finite flat morphism. Think of X∗ as X plus a finite divisor of points.130

• Fact: X∗ is smooth and proper.131

• Reason: This can be seen using the Tate curve over Z[ 1N ][[q]].132

• X∗(C) is the smooth compactification X∗C of Γ\H± mentioned earlier.133

• Remark: X∗ is a moduli space of “generalised elliptic curves”.134

Geometric Modular forms References: [Katz], [Loeffler: lectures notes]135

• Fix N ≥ 3. Let X = XΓ1(N).136

• Goal: Geometric reinterpretation of modular forms as sections of sheaves on137

X:138

Definition 7.7. Let ω := e∗Ω1E|X where e : X → E is the identity section.139

• Since E → X is smooth of dimension 1, this is a line bundle.140

• Fact: This extends uniquely to a line bundle ω on X∗. Reason: The universal141

elliptic curve extends to a group scheme E∗ → X∗, take e∗Ω1E∗|X∗ .142

Proposition 7.8. The analytification of ω⊗k is naturally isomorphic to the143

sheaf ωk of modular forms of weight k on X∗(C).144

Sketch of proof. Recall: H± is moduli space of E|C with α : Z2 ∼−→ H1(E,Z).145

Integration∫α(1,0)

defines an isomorphism e∗Ω1E|C

∼−→ C.146

Get canonical trivialisation of ω over H±. Check GL2(Z)-action coincides. 147

• In particular, the sheaf ω already has an algebraic model over Z[ 1N ].148

Definition 7.9. For any Z[ 1N ]-algebra R, can define149

Mk(Γ1(N), R) = Γ(X∗Γ1(N) ×Spec(Z[1/N ]) Spec(R), ω⊗k),

the R-module of modular forms of weight k with coefficients in R.150

• Fact: This is a finite free Z[ 1N ]-module.151

• Fact: Using Tate curves, get q-expansions in R[[q]]. Can define cusp forms.152

• Fact: If R,S are flat Z[ 1N ]-algebras (e.g. any Q-algebras), then153

Mk(Γ1(N), R)⊗R S = Mk(Γ1(N), S).

• Upshot: Modular forms are manifestly objects of algebraic geometry!154

8. The Eichler–Shimura relation (by Ben Heuer)1

Last time:2

• There is a GL2(Z)-equivariant bijection.3

H± → elliptic curves E over C with α : Z2∼=−→ H1(E,Z)/ ∼

τ 7→ (Eτ := C/(Z + Zτ), α : Z2 → Z + Zτ, (1, 0) 7→ 1, (0, 1) 7→ τ)

• Corollary:4

GL2(Z)\(GL2(Z)×H±) ∼= elliptic curves E over C with α : Z2 ∼−→ TE/ ∼,

where TE := lim←−N∈N

E[N ] is the full adelic Tate module of E.5

• This can be defined over Q!6

• Set X → Spec(Q) as the moduli scheme representing the functor7

Q−Sch→ Sets, S 7→ elliptic curves E over S with α : Z2 ∼−→ TE/ ∼ .

Here, TE = lim←−N

E[N ] is an inverse limit of finite, etale group schemes over S8

• −1 ∈ GL2(Z) acts trivially on X.9

• For K ⊆ GL2(Z) compact-open with K ⊆ K1(3) (note in particular, −1 /∈ K)10

have smooth quasi-projective curve11

XK = X/K.

• E.g.: If K = K(N), N ≥ 3, then12

XK(N)

parametrizes elliptic curves E together with trivialization (Z/N)2 ∼−→ E[N ].13

• Other examples, K = K1(N) Z/N → E[N ] , K = K0(N) subgroups.14

Next goal:15

• Use modular curves to reinterpret Hecke operators geometrically.16

• From geometry of modular curves, deduce fundamental relation between Ga-17

lois action and Hecke action: The Eichler–Shimura relation.18

The adelic action:19

• Clearly, GL2(Z) acts on X, by precomposition on α : Z2 ∼= TE.20

• But: Want GL2(Af )-action to get Hecke operators21

⇒ work with elliptic curves up to quasi-isogeny.22

• Recall: Af = Z⊗Z Q and23

GL2(Z)\(GL2(Z)×H±) ∼= GL2(Q)\(GL2(Af )×H±).

• On elliptic curves can interpret this as follows.24

• Recall that the LHS geometrizes to X, which represents the functors

P : Q−Sch→Sets, S 7→(E|S, α : Z2 ∼−→ TE)/isomorphism

• Let V E = TE ⊗Z Q be the rational adelic Tate module, and define

P ′ : Q−Sch→Sets, S 7→(E|S, α : A2f∼−→ V E)/quasi-isogeny

Definition 8.1. Recall: An isogeny is a surjective homomorphism E → E′25

with finite flat kernel. A quasi-isogeny is an element of26

E → E′ isogeny ⊗Z Q.

A p-quasi-isogeny is an element of27

E → E′ isogeny of degree a power of p ⊗Z Z[ 1p ].

• Then P ∼= P ′. Indeed:28

– Clearly have natural transformation P → P ′. Sketch for inverse:29

– Take (E,α : A2f∼−→ V E) ∈ P ′(S). Multiply by isogeny N : E → E until30

α−1 restricts to31

TE → Z2.

Reduce both sides mod M ∈ N. Let D be the kernel of the induced map32

E[M ]→ (Z/M)2,

this stabilises for M 0. The dual isogeny E/D → E induces33

α : Z2 ∼−→ T (E/D).

– This defines unique representative in P (S). Thus P (S) P ′(S).34

• This shows that the natural GL2(Z)-action extends to a GL2(Af )-action.35

• BThis does not fix the projection to any XK because it changes E.36

• Example:(p 00 1

)acts by sending E 7→ E/D where D ⊆ E[p] is the subgroup37

generated by the image of α(1, 0) under TpE E[p].38

• Fact: All of this still works after compactification.39

Geometric interpretation of Hecke operators40

• Let V = C∞(GL2(Af ) × H±) be the smooth GL2(Af )-representation of41

smooth functions42

f : GL2(Af )×H± → C, (g, z) 7→ f(g, z),

i.e., f is fixed under some compact-open subgroup in GL2(Af ) and z 7→ f(g, z)43

is smooth for any g ∈ GL2(Af ).44

• Fix a prime p.45

• Let ϕ ∈ H(GL2(Qp)) be any element of the Hecke algebra at p, i.e., a locally46

constant function ϕ : GL2(Qp)→ C with compact support.47

• Recall that for f ∈ V we have48

ϕ ∗ f(g, z) =

∫GL2(Qp)

ϕ(h)f(gh, z)dh,

where we chose the Haar measure on GL2(Qp) with volume 1 on Kp.49

• The integral above can be rewritten as follows:50

• Consider the following diagram:51

GL2(Af )×H± ×GL2(Qp)

GL2(Af )×H± GL2(Af )×H±,

q2 q1

q1(g, z, h) =(g, z),

q2(g, z, h) =(gh, z).

• Such a diagram is called a correspondence.52

• We also have the projection to the third factor:53

π : GL2(Af )×H± ×GL2(Qp)→ GL2(Qp)

Lemma 8.2. For f : GL2(Af )×H± → C in V we have54

ϕ ∗ f = q1,!(q∗2(f) · π∗(ϕ)),

where (−)∗ means the pullback of a function and (−)! integrating along the55

fiber (which makes sense because ϕ has compact support).56

• Proof. The fiber of q1 over (g, z) ∈ GL2(Af )×H± is57

q−11 (g, z) = (g, z, h)| h ∈ GL2(Qp) ∼= GL2(Qp)

and q∗2(f) · π∗(ϕ) is the function58

(g, z, h) 7→ f(gh, z)ϕ(h).

Applying q1,!, the integral over this fibre, gives59 ∫GL2(Qp)

ϕ(h)f(gh, z)dh = ϕ ∗ f(g, z).

Note that this is well-defined because ϕ has compact support. 60

Passage to finite level61

• Let N be coprime to p and let K := K1(N) ⊆ GL2(Af ). Then Kp = GL2(Zp).62

• Write Kp for the prime-to-p-part of K, i.e. K = KpKp and Kp ∩Kp = 1.63

• Assume that f is fixed under K, i.e., arises by pullback from a function64

f : GL2(Af )/K ×H± → C.

• Assume ϕ ∈ H(GL2(Qp),Kp), i.e., that ϕ is Kp-biinvariant.65

• ϕ is the pullback of a function with finite support66

ϕ : Kp\GL2(Qp)/Kp → C.

• In this case, ϕ ∗ f is again K-invariant, i.e., the pullback of a function67

ϕ ∗ f : GL2(Af )/K ×H± → C.

• We can change the above diagram (while retaining notation) to:68

(GL2(Af )/Kp ×H±)×Kp GL2(Qp)/K

GL2(Af )/K ×H± GL2(Af )/K ×H±.

q2 q1

Here − ×Kp − is the contracted product (i.e., the quotient for the action69

k · (g, z, h) := (gk−1, z, kh)) and70

q2([g, z, h]) = [gh, z], q1([g, z, h]) = [g, z],

where square brackets indicate equivalence classes.71

• We again have a projection

π : (GL2(Af )/Kp ×H±)×Kp GL2(Qp)/Kp →Kp\GL2(Qp)/Kp,

[g, z, h] 7→[h].

• Next, mod out by GL2(Q) on the left: Recall this gives a complex manifold72

XΓ1(N)(C) = GL2(Q)\(GL2(Af )/K ×H±)

• We then need to replace functions f by sections in ωk, k ∈ Z on this.73

• More precisely, we obtain74

(1)

GL2(Q)\(GL2(Af )/Kp ×H±)×Kp GL2(Qp)/Kp

XΓ1(N)(C) XΓ1(N)(C).

q2 q1

and for f ∈ H0(XΓ1(N)(C), ωk), i.e., a weakly modular form of level K1(N),75

ϕ ∗ f = q1,!(q∗2(f) · π∗(ϕ)).

• To make this precise, use: We have a canonical isomorphism76

q∗2(ωk) ∼= q∗1(ωk)

since ωk is defined via pullback of a GL2(Q)-equivariant line bundle on H±.77

• Thus can regard q∗2(f) · π∗(ϕ) ∈ q∗2(ωk) as section of q∗1(ωk).78

• Sum up along the fibers to obtain the section79

q1,!(q∗2(f) · π∗(ϕ)) ∈ Γ(XΓ1(N)(C), ωk).

Interpretation in terms of moduli80

• On elliptic curves, (1) corresponds to a Hecke correspondence:81

(ι : E1 99K E2, α)

(E2, α′) (E1, α)

q2 q1 where

– (E1, α) is an elliptic curve over C with a Γ1(N)-level structure82

α : Z/NZ → E1[N ]

– Equivalently, this is the datum of a point Q := α(1) ∈ E1[N ](C).83

– (ι : E1 99K E2, α) is the data of elliptic curves E1, E2 over C with α a84

Γ1(N)-level structure on E1, and ι a p-quasi-isogeny:85

– q1 : (ι : E1 99K E2, α) 7→ (E1, α).86

– q2 : (ι : E1 99K E2, α) 7→ (E2, α′) where α′ is the composition87

Z/NZ E1[N ] E2[N ]α∼ι

(use: ι is an isomorphism on N -torsion as (p,N) = 1). Equivalently, this88

sends Q to the image Q′ in E2[N ].89

– The map π sends (ι : E1 99K E2, α) to the class of the isomorphism90

Tp(ι) : TpE1 ⊗Zp Qp∼−→ TpE2 ⊗Zp Qp.

More concretely, choose any bases of TpE1 resp. TpE2. Then Tp(ι) is rep-91

resented by a matrix A ∈ GL2(Qp). The class of A ∈ Kp\GL2(Qp)/Kp92

is independent of the chosen basis.93

• Problem: Functor of p-quasi-isogenies not representable (only by ind-scheme).94

• However: Mainly interested in ϕ given by the characteristic functions for95

Kp

(p 0

0 p

)Kp ↔ Sp,

96

Kp

(p 0

0 1

)Kp ↔ Tp.

• Then term ϕ(h) in defining integral is supported on this double coset.97

• Hecke correspondences are supported on subspaces represented by schemes!98

• For Sp, consider tuples (ι : E1 99K E2, α) of the form99

([p] : E → E,α)

i.e. E1 = E2 and ι : E1 → E2 is multiplication [p].100

• For Tp, instead need to consider101

(ι : E1 → E2, α)

where ι : E1 → E2 is an isogeny of degree p.102

• Equivalently, this is the datum of a cyclic subgroup ker ι ⊆ E1 of rank p.103

• Last time: These are now representable by schemes, even over Spec(Q)!104

Hecke action on modular curves – perspective of algebraic geometry105

• We now pass to the finite level K = K1(N) for some p - N .106

• Corresponding modular curve XΓ1(N) over Z[ 1N ] represents pairs (E,Q) over107

Z[ 1N ]-schemes S of an elliptic curve E over S with a point108

Q ∈ E[N ](S).

• We start with 〈p〉 := Sp. By the above, just need to multiply Q by p.109

• For this, use action of (Z/NZ)× on XΓ1(N).110

• By the above, the correspondence for Sp now becomes111

XΓ1(N)

XΓ1(N) XΓ1(N).

p·∼

id

• Fact: For any a ∈ (Z/NZ)×, have canonical isomorphism a∗ω = ω.112

• Since the right map is an isomorphism, the integration over fibers is trivial.113

• Everything extends to compactifications.114

• Putting everything together, we have proved:115

Proposition 8.3. Consider the operator defined as the composition116

Mk(Γ1(N),Z[ 1N ]) = H0(X∗Γ1(N), ω

k)p∗−→ H0(X∗Γ1(N), p

∗ωk) = Mk(Γ1(N),Z[ 1N ]).

Then its base-change to C is the Hecke operator Sp.117

• For Tp, instead of isogenies (ι : E → E′, α), parametrise triples (E,D,Q),

where D = ker ι finite cyclic subgroup of rank p. Get maps of moduli functors

π1 :(E,D,Q)7→(E,Q),

π2 :(E,D,Q)7→(E/D,Q′), Q′ := Q+D = image of Q under E[N ]→ (E/D)[N ]

• These induce a Hecke correspondence118

XΓ0(p)∩Γ1(N)

XΓ1(N) XΓ1(N).

π2 π1

• Fact: π1 and π2 extend to compactifications X∗Γ0(p)∩Γ1(N) → X∗Γ1(N).119

• Fact: π1 and π2 are finite flat of degree p+ 1, even on compactifications.120

• Fact: There are canonical isomorphisms π∗1ω = π∗2ω121

• Since π1 has finite fibers, the integral occurring in the definition of the Hecke122

operator becomes a discrete sum, which we can interpret as a trace map123

Trπ : π1,!π∗1ω → ω

• Combining all this, we have proved:124

Proposition 8.4. The base-change to C of the operator125

H0(X∗Γ1(N), ω⊗k)

π∗2−→ H0(X∗Γ1(N)∩Γ0(p), ω⊗k)

Trπ1−−−→ H0(X∗Γ1(N), ω⊗k).

is the Hecke operator Tp.126

• Note: This is all defined already over Q (even over Z[ 1N ])! Consequence:127

Corollary 8.5. The eigenvalues of Sp, Tp on Mk(Γ1(N),C) are algebraic.128

• To give a more explicit description of Tp, reinterpret in terms of divisors:129

• Let Div(XΓ1(N)) = set of divisors on X (≈ formal sums of points).130

• We can then reinterpret 〈p〉 := Sp and Tp as operators on Div(XΓ1(N)) :

Sp : Div(XΓ1(N))→ Div(XΓ1(N)), [E,Q] 7→ [E, p ·Q],

Tp : Div(XΓ1(N))→ Div(XΓ1(N)), [E,Q] 7→∑

D⊆E[p]

[E/D,Q+D],

where E is an elliptic curve, and Q ∈ E[N ] a point of order N .131

Modular curves in characteristic p132

• Let E be an elliptic curve over a scheme S of characteristic p.133

• Then E[N ] is etale for all p - N .134

• The morphism [p] : E → E factors into135

EF−→ E(p) V−→ E,

the Frobenius and Verschiebung isogeny. Here E(p) = E ×S,F S.136

• ⇒ always have kerF ⊆ E[p]. This is a connected, i.e. kerF (S) = 0.137

Definition 8.6. Over S = Fp, there are two cases:138

(1) E(Fp) = Z/pZ. Then E is called ordinary, and E[p] = µp × Z/pZ.139

(2) E(Fp) = 0. Then E is called supersingular, and E[p] is connected.140

• B “supersingular” is not about smoothness. It just means “really special”:141

• Fact: There are only finitely many isomorphism classes of supersingular el-142

liptic curves over Fp. All others are ordinary.143

• Recall: XΓ1(N) is defined over Z[ 1N ].144

• Can therefore reduce to Fp to get modular curve XΓ1(N),Fp → Spec(Fp).145

• Similarly for XΓ1(N)∩Γ0(p) representing (E,D,Q) with D ⊆ E[p] of rank p.146

• Deligne–Rapoport: The fibre XΓ1(N)∩Γ0(p),Fp → Spec(Fp) is of the form147

148

id F

D = µp

D = Z/pZXΓ1(N)∩Γ0(p),Fp

XΓ1(N),Fp149

• Two copies of XΓ1(N),Fp , corresponding to D = µp or D = Z/pZ as subgroup150

of E[p], with transversal intersections at supersingular points (black).151

• Projection is identity (deg=1) on one copy, and Frobenius (deg=p) on other.152

Eichler–Shimura relation [Diamond–Shurman], [Conrad: Appendix to Serre’s...]153

• Eichler–Shimura relation expresses reduction Tp mod p in terms of Frobenius:154

• Base change to algebraic closure Fp.155

• Consider morphism Div0(X∗Γ1(N),Fp

)→ Pic0(X∗Γ1(N),Fp

)156

• Fact: Pic0(X∗Γ1(N),Fp

) is points of a group scheme over Fp. In particular,157

multiplication by p factors through Frobenius F : there is V = F∨ s.t.158

p = V F = F V

Theorem. (Eichler–Shimura relation) In Pic0(X∗Γ1(N),Fp

), we have159

Tp = F + 〈p〉V.

• Note: There is a version for X∗Γ0(N),Fp

, which famously reads160

Tp = F + V

We get a second important variant by multiplying by F and using FF∨ = p.161

F 2 − TpF + 〈p〉p = 0.

Proof. (Sketch) Let E be an ordinary elliptic curve over Zp = W (Fp). Let C ⊆ E[p]162

be the “canonical subgroup”:= generated by kernel of163

E[p](Zp)→ E[p](Fp).

• Key fact: For D ⊆ E[p] cyclic subgroup scheme of rank p,164

(1) the isogeny E → E/D reduces to F : E → E(p) ⇔ D = C.165

(2) the isogeny E → E/D reduces to V : E → E(p−1) ⇔ D 6= C.166

• Here E(p−1)

:= E ×Fp,F−1 Fp = base-change along inverse of Frobenius.167

• F sends Q ∈ E[N ] to base-change Q(p) ∈ E(p)[N ] = E[N ](p).168

• Since V F = p on E[N ], this implies: V sends Q to pQ(p−1) ∈ E(p−1)[N ].169

• Recall: There are p+ 1 different subgroups D of E[p] over W (Fp).170

• Thus Tp([E,Q]) =∑D⊆E[p][E/D,Q+D] in Div(XΓ1(N)) reduces to171

[E(p), Q(p)] + p[E(p−1), pQ(p−1)] in Div(XΓ1(N),Fp).

• The first summand is F [E,Q].172

• By definition, 〈p〉[E(p−1), Q(p−1)] = [E(p−1), pQ(p−1)].173

• Now pass to Pic0(X∗Γ1(N),Fp

). Here: p = V F . We therefore have174

p[E(p−1), Q(p−1)] = pF−1[E,Q] = V [E,Q].

(technically, we need to work with [E,Q]− [E′, Q′] to be in Div0). 175

9. Galois representations associated to newforms1

Exams:2

• Write me to arrange an oral exam on 31.7. (second week after end of lectures).3

Next aim:4

• Realize newforms in cohomology.5

• Finish construction of Galois representations associated to newforms of weight6

k = 2.7

• Give hints on how to proceed in the case that k ≥ 2.8

De Rham cohomology:9

• X a real manifold10

• C∞(X) = space of smooth functions f : X → C.11

• Ai(X) = space of smooth i-forms on X, i ≥ 0.12

• de Rham complex of X:13

A•(X) : A0(X)=C∞(X)

d−→ A1(X)d−→ A2(X)→ . . .

with d the exterior derivative.14

• de Rham cohomology of X:15

H∗dR(X) := H∗(A•(X))

• de Rham comparison isomorphism:16

H∗dR(X) ∼= H∗(X,C)

with RHS the sheaf cohomology of the constant sheaf C on X.17

• Moreover,18

H∗(X,C) ∼= H∗sing(X,C).

with RHS = singular cohomology of X with values in C.19

• If Γ is a discrete group acting properly discontinuously and freely on X, then20

A•(Γ\X) ∼= A•(X)Γ = (A0(X)Γ

=C∞(X)Γ

d−→ A1(X)Γ d−→ A2(X)Γ → . . .).

• Finally, we set21

H∗dR(GL2(Q)\(GL2(Af )×H±)) := lim−→K

H∗dR(GL2(Q)\(GL2(Af )/K ×H±))

and22

H∗(GL2(Q)\(GL2(Af )×H±),C) := lim−→K

H∗(GL2(Q)\(GL2(Af )/K ×H±),C),

where the colimit is over all compact-open subgroups K ⊆ GL2(Af ), and the23

transition maps are induced by pullback.24

• Then, for i ≥ 0, the (GL2(Af )-equivariant) de Rham comparison25

HidR(GL2(Q)\(GL2(Af )×H±)) ∼= Hi(GL2(Q)\(GL2(Af )×H±),C)

holds by passing to the colimit.26

Upshot:27

• Can construct classes in H∗(X,C) using (closed) differential forms.28

Cohomology classes associated to modular forms of weight 2:29

• Note that30

GL2(Q)\(GL2(Af )×H±) ∼= GL2(Q)+\(GL2(Af )×H),

where GL2(Q)+ ⊆ GL2(Q) is the subgroup of elements of positive determi-31

nant.32

• Pick f ∈ H0(GL2(Q)+\(GL2(Af )×H), ω⊗k), k ∈ Z.33

• We view f as a function34

f : GL2(Af )×H→ C, (g, z) 7→ f(g, z)

satisfying35

f(γg, γz) = (cz + d)kf(g, z)

for γ ∈

(a b

c d

)∈ GL2(Q)+.36

• Assume that f is of weight k = 2.37

• Then the differential form f(g, z)dz on GL2(Af )×H is closed and satisfies38

γ∗(f(g, z)dz) = f(γg, γz)γ∗dz = det(γ)f(g, z)dz

for γ ∈ GL2(Q)+.39

• Indeed:40

– Closedness follows from holomorphicity as41

d(f(g, z)dz) =∂

∂zf(g, z)dz ∧ dz − ∂

∂zf(z)dz ∧ dz = 0.

– If γ =

(a b

c d

)∈ GL2(Q), then42

γ∗(dz) =det(γ)

(cz + d)2dz.

• Let43

| − | : Q×\A× → R>0

be the adelic norm, i.e.,44

|(x2, x3, . . . , x∞)| :=∏p

|xp|p · |x∞|∞

with | − |p : Qp → R≥0, x 7→ p−vp(x) the p-adic norm, and | − |∞ : R → R≥045

the real norm.46

• | − | defines the character47

χno := | − | det

of GL2(A), which is trivial on GL2(Q).48

• Thus, for49

f(g, z) := χno((g, 1))f(g, z)

(with (g, 1) ∈ GL2(A) ∼= GL2(Af )×GL2(R)), the form50

ηf := f(g, z)dz

on GL2(Af )×H is GL2(Q)+-equivariant.51

• Get a map, which is not GL2(Af )-equivariant,52

α : M2 → H1dR(GL2(Q)+\(GL2(Af )×H)) ∼= H1(GL2(Q)+\(GL2(Af )×H),C)

by sending f to [ηf ].53

• For a smooth GL2(Af )-representation V and a smooth character54

χ : GL2(Af )→ C×

define V (χ) := V ⊗C χ.55

• Then56

α : M2(χno)→ H1(GL2(Q)+\(GL2(Af )×H),C)

is GL2(Af )-equivariant.57

• Let K ⊆ GL2(Af ) be compact-open, and set58

XK := GL2(Q)+\(GL2(Af )/K ×H).

with canonical compactification59

X∗K := GL2(Q)+\(GL2(Af )×H∗)

where H∗ := H ∪ P1(Q) (equipped with Satake topology).60

• If f ∈ S2(K), then ηf extends to a holomorphic differential form on X∗K .61

– Indeed: If q = e2πiz, then62

dq = 2πiqdz,

i.e.,63

dz =1

2πiqdq.

Now express f(g, z) at each cusp in Fourier expansion, i.e., as a function64

of q.65

• The diagram66

H1dR(X∗K)

'

// H1dR(XK)

'

H1(X∗K ,C) // H1(XK ,C)

commutes.67

• Moreover, the morphism68

H1c (XK ,C)→ H1(X∗K ,C) = H1

c (X∗K ,C)

is surjective as its cokernel embeds into H1(cusps,C) = 0.69

• Thus, α(S2(K)) lies in the interior cohomology H1(XK ,C) ∼= H1(X∗K ,C) of70

XK , which by definition is the image of H1c (XK ,C)→ H1(XK ,C).71

• Note that the above discussion applies similarly to antiholomorphic modular72

forms, i.e., complex conjugates of modular forms.73

• This yields the map74

α : S2(K)→ H1(XK ,C), f 7→ [χnofdz],

where S2(K) denotes the C-linear space of antiholomorphic modular forms.75

• We can pass to infinite level and obtain the morphism76

α⊕ α : S2 ⊕ S2 → H1(GL2(Q)\(GL2(Af )×H±),C)

with (hopefully) self-explaining notation.77

Theorem 9.1 (Eichler–Shimura). The map78

α⊕ α : S2(χno)⊕ S2(χno)→ H1(GL2(Q)\(GL2(Af )×H±),C)

is a GL2(Af )-equivariant isomorphism.79

• The statement is equivalent to the analogous statement for all (sufficiently80

small) compact-open subgroup K ⊆ GL2(Af ).81

• Thus, fix some K ⊆ GL2(Af ) compact-open.82

• The proof will exploit the ∪-product pairing83

H1c (XK ,C)×H1(XK ,C)→ H2

c (XK ,C)integrate−−−−−→ C,

which under the de Rham comparison is induced from the pairing84

(η1, η2) 7→∫XK

η1 ∧ η2

of differential forms, where η1 has compact support.85

• We will use the following observation:86

– H∗c (XK ,C) ∼= H∗(A•c(XK)), where A•c denotes differential forms with87

compact supports.88

– The canonical map89

(2) A•c(XK)→ A•rd(XK)

is a quasi-isomorphism, where the RHS denotes differentials forms of90

rapid decay, cf. [Bor80, Theorem 2].91

• For f ∈ S2 the form ηf is rapidly decreasing.92

– Use that |e2πinz| = e−2πnIm(z) decreases rapidly if Im(z)→∞ and n ≥ 1.93

• Thus α lifts naturally to a map94

α : S2(K)→ H1c (XK ,C).

• Consider f ∈ S2(K), g ∈ S2(K). Then:95

∗ α(f) ∪ α(f) = 0 as dz ∧ dz = 0.96

∗ Similarly for g.97

∗ Using Stokes’ theorem, one proves98

tr(α(f) ∪ α(g)) =

∫XK

ηf ∧ ηg.

∗ This implies that the Poincare pairing induces (up to a scalar) the Pet-99

terson scalar product aka L2-pairing 〈−,−〉 on S2.100

• We can now prove that β := α⊕ α is injective.101

• Indeed:102

– If β(f + g) = 0, then103

β(f + g) ∪ α(f) = 〈f, f〉 = 0,

i.e., f = 0. Similarly, g = 0.104

• Now we check H1(XK ,C) = 2dim(S2(K)).105

⇒ Proof of the Eichler-Shimura isomorphism is finished.106

• Namely (we use lower case latters to denote dimensions):107

∗ h1(XK ,C)) = h1(X∗K ,C) = 2 · h0(X∗K ,C)− χtop(X∗K)108

∗ Here: χtop is the topological Euler characteristic.109

∗ On the other hand:

dim(S2(K))

= h0(X∗K ,Ω1X∗K

)RR= deg(Ω1

X∗K) + χhol(X

∗K) + h1(X∗K ,Ω

1X∗K

)

= −χhol(X∗K) + h0(X∗K ,C)

∗ Here: χhol is the holomorphic Euler characteristic.110

∗ χtop(X∗K) = 2χhol(X∗K)111

Galois representations associated to newforms:112

• Recall that in the last two lectures we introduced a scheme113

X → Spec(Q)

with GL2(Af )-action such that naturally114

X(C) ∼= GL2(Q)\(GL2(Af )×H±).

• ForK ⊆ GL2(Af ) compact-open (plus sufficiently small), get a quasi-projective,115

smooth curve116

XK → Spec(Q)

with117

XK(C) ∼= GL2(Q)\(GL2(Af )/K ×H±).

(note clash in notation with previous section, where XK was equal to RHS).118

• Fix a prime `.119

• For K ⊆ GL2(Af ) can consider the interior etale cohomology120

H1et(XK,Q,Q`) := Im(H1

c,et(XK,Q,Q`)→ H1et(XK,Q,Q`)) ∼= H1

et(X∗K,Q,Q`).

• In the limit, we get121

H1et(XQ,Q`) := lim−→

K

H1et(XK,Q,Q`)

• By naturality of H1c,et, H

1et this space has a natural action of122

Gal(Q/Q)×GL2(Af ),

i.e., an action of Gal(Q/Q) and an action of GL2(Af ), and these two actions123

commute.124

• Fix an isomorphism ι : Q` ∼= C.125

• Using etale comparison theorems (and ι) we get the isomorphisms126

H1et(XQ,Q`) ∼= H1

et(XC,Q`) ∼= H1(X(C),Q`)ι∼= H1(X(C),C).

• These isomorphisms are GL2(Af )-equivariant.127

• Let π ⊆ S2 be an irreducible GL2(Af )-representation.128

• Using ι, we will view each smooth GL2(Af )-representation over C as a smooth129

representation over Q`.130

• The space131

ρπ := HomGL2(Af )(π(χno), H1et(XQ,Q`))

is two-dimensional by the Eichler-Shimura isomorphism132

H1(X(C),C) ∼= S2(χno)⊕ S2(χno)

and because S2 is multiplicity free as a GL2(Af )-representation.133

• In summary, as a Gal(Q/Q)×GL2(Af )-representation134

(3) H1et(XQ,Q`) ∼=

⊕π

ρπ ⊗Q` π(χno),

where the sum is running over all irreducible GL2(Af )-representations π ⊆ S2135

(note that there is a natural morphism from the RHS to the LHS).136

The Eichler–Shimura relation in etale cohomology:137

B We need to show that Hecke eigenvalues match with traces of Frobenii. B138

• Recall: To π ⊆ Sk with system of Hecke eigenvalues139

ap(π), bp(π) = χ(p)pk−1p/∈S ,

we want to attach a 2-dimensional `-adic representation140

ρπ : Gal(Q/Q)→ GL2(Q`),

such that for p /∈ S each arithmetic Frobenius Frobarithp at p has characteristic141

polynomial Confession: I

stated the Lang-

lands reciprocity

for newforms

wrongly!

Confession: I

stated the Lang-

lands reciprocity

for newforms

wrongly!

142

X2 − ap(π)X + χ(p)pk−1

(we omit ι, ρπ in the following), i.e.,143

(Frobarithp )2 − ap(π)Frobarith

p + χ(p)pk−1 = 0.

• By passing to K1(N)-invariants in (3), we have (for N ≥ 3)144

H1et(X

∗Γ1(N),Q,Q`) ∼=

⊕π

ρπ ⊗Q` π(χno)K1(N)

(note XK1(N)∼= XΓ1(N)).145

• Let p be a prime, p - `N .146

• Fix a place of Q over p. This determines an algebraic closure Fp of Fp.147

• Recall: For an abelian variety A over a field L, and ` a prime, we write148

T`A = lim←−A[`n](L), V`A = T`A⊗Z` Q`

for the `-adic Tate module resp. the rationalized `-adic Tate module.149

Lemma 9.2. Let p be prime, p - `N . Then we have150

H1et(X

∗Γ1(N),Q,Q`) = V`Pic0(X∗

Γ1(N),Fp)∨,

where (−)∨ denotes the Q`-dual.151

Proof. The Kummer sequence 1→ µ`n → Gm`n−→ Gm → 1 implies152

H1et(X

∗Γ1(N),Q,Q`(1)) ∼= V`Pic0(X∗

Γ1(N),Q).

The Weil pairing yields a canonical isomorphism153

V`Pic0(X∗Γ1(N),Q)∨ ∼= V`Pic0(X∗

Γ1(N),Q)(−1).

Finally, because ` 6= p154

V`Pic0(X∗Γ1(N),Q) ∼= V`Pic0(X∗

Γ1(N),Qp) ∼= V`Pic0(X∗

Γ1(N),Fp)

by lifting torsion points. 155

• In particular, the Galois representation156

H1et(X

∗Γ1(N),Q,Q`)

is unramified at p.157

• On V`Pic0(X∗Γ1(N),Fp

), we have the Eichler–Shimura relation, which we recall158

reads159

F 2 − TpF + pSp = 0

with F the arithmetic Frobenius.160

• Last time it was written Tp, but the action of the double coset161

GL2(Zp)

(p 0

0 1

)GL2(Zp)

was considered, which yields Tp.162

• Moreover, instead of Sp, which corresponds to163

GL2(Zp)

(p 0

0 p

)GL2(Zp),

it was written 〈p〉.164

Proposition 9.3. Let p be a prime with p - `N . Then on H1et(X

∗Γ1(N),Q,Q`), we have165

(Frobgeomp )2 − TpFrobgeom

p + pSp = 0.

• For a normalized newform f ∈ S2(Γ0(N), χ) we define finally166

ρf := (ρπ)∨

with π the irreducible GL2(Af )-representation generated by f , and167

ρπ := HomGL2(Af )(π(χno), H1et(XQ,Q`)).

Some analysis of ρf , cf. [Rib77]:168

• Write the (normalized) newform f ∈ S2(Γ0(N), χ) in Fourier expansion169

f(q) =∑n≥1

anqn.

Then for p - `N , Frobarithp has characteristic polynomial170

X2 − apX + χ(p)p

on ρf .171

• Indeed:172

– By the (dual of the) Eichler–Shimura relation173

(Frobarithp )2 − TpFrobarith

p + pSp

on ρf .174

– Recall that Tp has eigenvalue pap on πK1(N), while Sp has eigenvalue175

χ(p)p2.176

– We had to twist the GL2(Af )-action on S2 by χno = |det(−)|. Thus Tp177

has eigenvalue ap on178

ρπ ⊗Q` π(χno)K1(N) ⊆ H1et(X

∗Γ1(N),Q,Q`).

while Sp has eigenvalue 1p2χ(p)p2 = χ(p).179

• The determinant of ρf is180

ψ · χcyc

where181

χcyc : GQ → Q`×

denotes the cyclotomic character, and182

ψ : Q×R>0\A× → Q`× ι∼= C×

the (finite order) adelic character determined by χ.183

• Indeed:184

– Use Chebotarev and check that both sides agree on arithmetic Frobenius185

elements for p - `N .186

• In particular, ρf is odd, i.e., the determinant of each complex conjugation is187

−1.188

– This uses that for all k ≥ 1 non-triviality of Sk(Γ0(N), χ) implies χ(−1) =189

(−1)k.190

• ρf is irreducible, cf. [Rib77, Theorem (2.3)].191

• Because, ρf is realized in the etale cohomology of a proper, smooth scheme192

over Q, the de Rham comparison theorem in `-adic Hodge theory implies that193

ρf |GQ`is de Rham.194

Outline of the construction in weights ≥ 2:195

• Instead of196

H1et(XQ,Q`)

use197

H1et(XQ,L

k),

with the local system198

Lk := Symk−1R1f∗(Q`)

for f : E → X the universal elliptic curve.199

• Problem: Lk does not extend to a local system on the compactification. Thus200

one needs an additional argument to justify base change to Fp.201

• Use similar strategy to obtain analogs of the Eichler–Shimura relation/isomorphism.202

• Also replace the de Rham comparison/etale comparsion theorems by their203

versions with coefficients.204

10. Galois representations for weight 1 forms (by Ben Heuer)1

Last time: Constructed Galois representations attached to newforms in weight ≥ 2:2

Theorem 10.1 (Deligne ’71). Let N ≥ 3 and k ≥ 2. Let ε : (Z/NZ)× → C× be a3

Dirichlet character. Let f ∈ Mk(Γ0(N), ε) be a newform of weight k. Let K be the4

number field generated by the ap(f) and ε. Then for any place λ of K over a prime5

` - N , there is a continuous, irreducible Galois representation6

ρ : GQ → GL2(Kλ),

unramified outside N , such that the characteristic polynomial of Frobp is7

X2 − apX + ε(p)pk−1 for all p - N`.

Galois reps associated to weight 1 modular forms8

• Goal of today: prove the following Theorem9

Theorem 10.2 (Deligne–Serre ’73). Let N ≥ 3. Let ε : (Z/NZ)× → C× be a

Dirichlet character. Let f ∈M1(Γ0(N), ε) be a newform of weight 1. Then there

is a continuous, odd, irreducible Galois representation

ρ : GQ → GL2(C),

unramified outside N , such that the characteristic polynomial of Frobp is

X2 − apX + ε(p) for all p - N.

• This finishes the construction of newforms Galois reps.10

• Note that ρ has image in GL2(C), i.e. is an Artin representation, in contrast11

to the ones for weight ≥ 2, which were `-adic. Why is that?12

– Easy fact: Any Artin representation has finite image.13

– In fact, ρ is defined already over a number field, so could embed into p-14

adic field Kλ as before. But using C is a way of signaling “finite image”.15

– On the Galois side, see that ρ can only be finite for k = 1 since the16

determinant is required to be det ρ = εχk−1cycl , and χcycl has infinite image.17

• If ε is even, then M1(Γ0(N), ε) = 0. Hence ε is odd, which implies ρ odd.18

First sketch of construction19

Compared to last time, the proof will be less geometry, more Galois representations:20

• Reduce f mod p. This gives a “mod p modular form”21

• Use congruences of modular forms to show there is a mod p Hecke eigenform22

f ′ with same eigenvalues but higher weight ≥ 2.23

• Lift f ′ back to characteristic 0 using “Deligne–Serre lifting”24

• Attach Galois representation ρf ′ using weight ≥ 2 construction25

• Reduce Galois representation mod `26

• Lift Galois representation from F` to C.27

Introduction to mod p modular forms28

• Let N ≥ 3. Let S be any Z[ 1N ]-scheme S.29

• Recall: We had defined modular forms over S of weight k by30

Mk(Γ1(N);S) := Γ(X∗Γ1(N),S , ωk).

• ω is the ample line bundle of relative differentials on universal elliptic curve31

• Can define q-expansions: Let S = Spec(A), then have the Tate curve T (q)32

over Z((q))⊗Z A. Over Spec(Z((q))⊗Z A), have canonical isomorphism33

ωGm = ωT (q)

• bundle ωGm is trivial: the canonical differential dxx on Gm is invertible section34

• Via this trivialisation, get for every cusp ofX∗Γ1(N),A an associated q-expansion35

Mk(Γ1(N);S)→ A[[q]].

Theorem 10.3 (Katz’ q-expansion principle). This is injective for all k at each cusp.36

• Now apply this all to S = Spec(F) where F is a finite field of characteristic p:37

The resulting modular forms are called mod p modular forms.38

• The following fact perhaps helps you get a feeling for mod p modular forms:39

Lemma 10.4. For weight k ≥ 2, any mod p modular form is the reduction40

of a modular form over a number field modulo p. In particular, in this case41

Mk(Γ1(N);F) = Mk(Γ1(N);Z[ 1N ])⊗Z F.

Proof. Using Riemann–Roch, one can show that H1(X∗Γ1(N), ω⊗k) = 0. 42

• However, it is often better to have a more intrinsic definition. For example:43

Example of mod p modular form: The Hasse invariant44

• Recall: For any elliptic curve in characteristic p, have Verschiebung isogeny45

V : E(p) → E.

• In particular, have this for the universal elliptic curve.46

• On relative differentials, this defines a pullback map in the other direction47

V ∗ : ω → ω⊗p.

• Tensoring with ω−1 yields48

V ∗ : O → ω⊗(p−1) V ∗ ∈ Γ(ω⊗(p−1)) = Mp−1(Γ1(N);F).

• This is the Hasse invariant, denoted by Ha.49

• It is a mod p modular form of weight p− 1. In fact: an eigenform of level 1.50

Proposition 10.5. The Hasse invariant has constant q-expansion = 1 ∈ Fp[[q]].51

Proof. Sketch: Use ωT (q) = ωGm : On Gm have F = [p] and thus V = 1. 52

• Note: 1 ∈M0(Γ1(N);Fp) is also a mod p modular form with q-expansion 1.53

• This does not contradict the q-expansion principle as the weights are different.54

Upshot: In characteristic p, we have a new phenomenon:55

• Modular forms of different weight can have same q-expansion!56

• E.g. for any modular form f of weight k, any n ∈ N, have modular form57

Han · f of weight k + n(p− 1) with same q-expansion.58

• In particular, this preserves eigenforms a system of eigenvalues for Hecke59

operators can correspond to eigenforms of different weights.60

• BThe multiplication map61

Mk(Γ1(N);S)·Ha−−→Mk+p−1(Γ1(N);S)

is injective (q-expansion principle) but not surjective!62

• Reason: Ha has zeros, precisely on supersingular points of X∗Γ1(N)(Fp)63

• This is one motivation for removing those points p-adic modular forms.64

Eisenstein series65

• There is a more classical perspective on this that we want to at least mention:66

Definition 10.6.67

For any even k ≥ 4, let68

Gk :=∑

(n,m)∈Z2

(n,m) 6=(0,0)

1

(m+ nτ)k.

This is a modular eigenform of weight k and level 1 with q-expansion69

2ζ(k)(1−∞∑n=0

4

Bkσk−1(n)qn)

where σk−1(n) =∑d|n d

k−1 and Bk is the k-th Bernoulli number.70

Define the normalised Eisenstein series as71

Ek :=1

2ζ(k)Gk = 1− 4

Bk

∞∑n=0

σk−1(n)qn ∈ Z(p)[[q]]

Lemma 10.7. Assume p ≥ 5, then in terms of q-expansions, we have72

Ep−1 ≡ 1 mod p.

Proof. This follows from Kummer’s congruences for Bernoulli numbers. 73

Upshot: For p ≥ 5, the reduction mod p of the Eisenstein series Ep−1 is = Ha:74

Ep−1 ≡ Ha mod p “Deligne’s congruence”

Strategy (Deligne–Serre):75

• Given eigenform of weight 1, reduce mod p.76

• Multiply with Ha so it becomes eigenform of weight ≥ 2.77

• Use construction from last lecture in this case.78

Galois reps attached to mod p modular forms79

Theorem 10.8. Let k ≥ 1 and let f ∈ Mk(Γ1(N), ε;Fp) be a mod p cuspidal eigen-80

form. Let Ff be the (finite) field generated over Fp by the ap(f) and ε. Then there is81

a semi-simple Galois representation82

ρf : GQ → GL2(Ff ),

unramified outside N` such that for all ` - Np,83

charpoly(Frobp|ρf ) = X2 − apX + ε(p)pk−1.

Step 1: Reduce to case of k ≥ 2: If k = 1, multiply with Ha. This does not change84

the q-expansion, hence is still an eigenform. But its weight is ≥ 2.85

Step 2: Lift the Hecke eigensystem to characteristic 0. This is always possible in86

weight ≥ 2, as we shall discuss next.87

Deligne–Serre lifting lemma88

• Let R be a Z[ 1N ]-algebra without zero-divisors. Let89

TR ⊆ End(Mk(Γ1(N);R))

be the Hecke algebra: R-subalgebra generated by the Hecke operators Tl, Sl.90

• A Hecke eigensystem is a ring homomorphism91

ψ : TR → R.

The archetypal example of a Hecke eigensystem is:92

• Let g be an eigenform in Mk(Γ1(N);R), this gives rise to a Hecke eigensystem93

ψg : TR → R, T 7→ aT s.t. T (g) = aT g.

Lemma 10.9. Assume that R = K is a field. Then any Hecke eigensystem94

ψ : TK → K

comes from a unique Hecke eigenform.95

Proof. • TK is an Artinian K-algebra.96

• After passing to extension, it is the product of Artinian local rings Ti.97

• Since TK acts faithfully on M := Mk(Γ1(N);K), the submodule Mi := Ti ·M98

is non-zero (commutative algebra fact).99

• Take any eigenvector in Mi. Then this has eigensystem ψ.100

• Uniqueness: The Hecke eigenvalues determine the q-expansion. 101

• Let K be a number field, p a place over p - N . Let Fp be the residue field.102

• Write OK,(p) for the valuation subring of K of p-integral elements.103

Lemma 10.10 (Deligne–Serre Lifting Lemma). Let k ≥ 2 and let g ∈Mk(Γ1(N);Fp)104

be an eigenform. Then there is a finite extension K ′|K, an extension p′|p and an105

eigenform g ∈Mk(Γ1(N);OK′,(p′)) such that g reduces to g mod p′.106

Proof. • Consider the ring homomorphism107

TOK → TFp

ψg−−→ Fp.

• This corresponds to a maximal ideal m. Since108

Spec(TOK )→ Spec(OK)

is locally free, can find prime ideal q ⊆ m such that q ∩ O = 0.109

• This defines a non-zero prime ideal of TK , corresponding to Hecke eigensystem110

TK → K ′.

• By Lemma 10.9, this corresponds to an eigenform.111

• Use uniqueness in Lemma 10.9 to see that this reduces to g. 112

Proof of Theorem 10.8113

• Let f be mod p eigenform of weight ≥ 2, defined over extension Ff of Fp.114

• Lift to eigenform f over some number field K. Let p be a place over p.115

• Associated to this, have Galois representation116

ρf : GQ → GL2(Kp).

• Since image is compact, can always find isomorphic representation of the form117

ρf : GQ → GL2(OK,p).

• Reduce this mod p to get representation118

ρf : GQ → GL2(Fp).

• This might not yet be semi-simple Define ρf as semi-simplification119

ρf := ρssf

(:=direct sum of Jordan–Holder factors). Same trace and determinant as ρf .120

• Since ker ρf ⊆ ker ρf , this is unramified outside of pN .121

• Remains to show: ρf already defined over Ff : For this, STP:122

ρf = σ ρf for all σ ∈ Gal(Fp|Ff ).

• Certainly, all characteristic polynomials of Frob` for ` - pN ,123

X2 − a`X + ε(`),

are preserved by σ, since a`, ε(`) ∈ Ff by definition. Now use:124

Theorem 10.11 (Brauer–Nesbitt). Let F be a perfect field. Let G be a finite125

group. Let V,W be two semi-simple finite dimensional representations of G126

over F . Then V ∼= W if and only if127

charpoly(g|V ) = charpoly(g|W ) for all g ∈ G.

• This finishes the construction of f ρf . 128

Serre’s conjecture129

• A quick digression before we continue with weight 1 forms:130

• In 73’, Serre gave a conjecture that described exactly which mod p Galois131

representations arise in this way. It became known as132

Conjecture 10.12 (Serre’s Modularity Conjecture, or Serre’s Conjecture). Let ρ :133

GQ → GL2(Fp) be a continuous Galois representation that is odd and irreducible.134

Then there is a mod p eigenform g such that135

ρ = ρg.

In 1987, Serre published an article refining this conjecture:136

• He predicted the optimal level of g (in terms of the Artin conductor of ρ)137

• He predicted the optimal weight (recipe in terms of ramification of ρ at p)138

• He showed that his conjecture would imply Fermat’s Last Theorem.139

• Serre’s Conjecture is now a theorem by Khare–Wintenberger (2008).140

Lifting ρf to C141

• Back to our earlier setup: f ∈M1(Γ0(N), ε) eigenform142

• K number field generated by eigenvalues and ε.143

• Let L := set of primes of Q which decompose completely in K. This is an144

infinite set by Chebotarev.145

• Let ` ∈ L and λ a place over ` in K. Let OK,(λ) ⊆ K be λ-integral elements.146

• BNow apply previous section with role of p played by varying ` ∈ L.147

• For any such λ, consider reduction f of f mod λ. This is a mod ` eigenform.148

• Theorem 10.8 associates a semi-simple Galois representation149

ρf,` : GQ → GL2(F`)

(over F` as λ completely split) with charpoly of Frobp for any p - `N given by150

charpoly(Frobp|ρf,`) = X2 − apX + ε(p).

• Zeros of this are n-th units roots, which we may assume are all in OK (by the151

next lemma) and thus in Fl. Thus152

= (X − a)(X − b) for some a, b ∈ F×` .

Definition 10.13. Let G` := im(ρf,`) ⊆ GL2(F`). This is a finite group.153

Lemma 10.14. If f is of weight 1, there is A ∈ N such that |G`| ≤ A for all ` ∈ L.154

Proof. After using an estimate on the coefficients of weight 1 modular forms this155

becomes purely group theoretical: use classification of subgroups of GL2(F`). We156

shall not discuss it, but if you want to see it: [DS74, Lemma 8.4]. 157

• Enlarge K so that it contains all n-th unit roots for n ≤ A. This shrinks L.158

• Also exclude the primes ≤ A from L. Then L is still infinite.159

Lemma 10.15. Let G be a finite group of order prime to `. Then any representation160

ρ : G→ GL2(F`) lifts to a representation161

ρ : G→ GL2(OK,(λ)).

Proof. Step 1: Lift to GL2(Z`):162

• STP: any G→ GL2(Z/`nZ) lifts to G→ GL2(Z/`n+1Z).163

• For this, consider the short exact sequence164

1→ 1 + `nM2(F`)→ GL2(Z/`n+1Z)→ GL2(Z/`nZ)→ 1

• The first group is isomorphic to M2(F`) via 1 + `nx 7→ x.165

• Now use non-abelian group cohomology: Consider the left-action166

G acts on Map(G,GL2(Z/`nZ) via g · ϕ := ϕ(g)ϕ(g)−1.

• Then167

Hom(G,GL2(Z/`nZ)) = Map(G,GL2(Z/`nZ))G.

• Apply Map to the above short exact sequence, then G gives a long exact168

sequence of non-abelian group cohomology169

· · · → Hom(G,GL2(Z/`n+1))→ Hom(G,GL2(Z/`n))→ H1(G,Map(G,M2(F`)))

• Fact in group cohomology: H1(G,A) = 0 if |G|, |A| finite and coprime.170

Step 2: Already defined over number field171

• Since G is finite, ρ is semi-simple.172

• Traces must be sums of roots of unity of order ≤ |G| ≤ A contained in K.173

• Brauer–Nesbitt ensures that ρ is already defined over K.174

• Finally, K ∩ Z` = OK,(λ) 175

Back to prove of Theorem 10.2 (Galois representation for weight 1 forms)176

• Apply this to ρg : G` → GL2(F`), get G` → GL2(OK,(λ))177

• Compose with GQ → G` to get Galois representation178

ρg,` : GQ → GL2(OK,(λ)).

Proposition 10.16. The lift ρg,` : GQ → GL2(OK,(λ)) is such that for all p - `N ,179

charpoly(Frobp|ρg,`) = X2 − apX + ε(p).

The key is to vary `: Recall that we get ρg,` for all ` ∈ L.180

Definition 10.17. Let Y be the set of polynomials of the form (X − α)(X − β) for181

α, β ∈ OK,(λ) roots of unity of order ≤ |A|. This is a finite set.182

• Note: Given n ≤ A, ` ∈ L, and n-th root of unity x ∈ F×` , there is exactly183

one lift to root of unity in K (existence: n ≤ A, uniqueness: ` ≥ A).184

• Consequence: If F,G ∈ Y , then F ≡ G mod λ implies F = G.185

• Idea: Apply this to F = X2 − apX + ε(p) and G = charpoly(Frobp|ρg,`)186

Lemma 10.18. X2 − apX + ε(p) ∈ Y .187

Proof. For all λ over a prime in L, there is F ∈ Y for which188

charpoly(Frobp|ρg,`) ≡ F mod λ

since ρg,` has finite image. Thus also189

X2 − apX + ε(p) ≡ F mod λ.

Since Y is finite, there is F for which this holds for infinitely many λ. Thus190

X2 − apX + ε(p) = F ∈ Y

Proof of Proposition. • As G` finite of order ≤ A, have191

charpoly(Frobp|ρg,`) ∈ Y

• charpoly(Frobp|ρg,`) ≡ charpoly(Frobp|ρg,`) ≡ X2 − apX + ε(p) mod λ192

• Thus both sides are in Y , which implies193

charpoly(Frobp|ρg,`) = X2 − apX + ε(p).

Remains to prove:194

• ρ is odd: This is simply because ε is odd and c has order 2.195

• ρ is irreducible: Use complex analytic estimate due to Rankin 196

This finishes the construction of eigenforms Galois representations!197

Summary of the weight 1 constructioneigenforms

of weight 1

eigenforms of

weight ≥ 2

`-adic Galois

representations

complex Galois

representations

mod ` eigenforms

of weight 1

mod ` eigenforms

of weight ≥ 2

mod ` Galois

representations

reduce

congruence Deligne’s

construction

reduce

·Ha

DS-lifting

mod ` construction

Theorem 10.8

lift

11. The Langlands program for general groups, part I1

Next aims:2

• More or less precise statements (for parts of) Langlands program for GLn, or3

even general G reductive over Q.4

Langlands(-Clozel-Fontaine-Mazur) reciprocity for GLn,F :5

• F arbitrary number field6

• ` some prime. Fix an isomorphism C ∼= Q`.7

• Then for any n ≥ 1, there exists a (unique) bijection between8

i) the set of L-algebraic cuspidal automorphic representations of GLn(AF ),9

ii) the set of (isomorphism classes) of irreducible continuous representations10

Gal(F/F )→ GLn(Q`) which are almost everywhere unramified, and de11

Rham at places dividing `,12

such that the bijection matches Satake parameters with eigenvalues of Frobe-13

nius elements.14

Comments:15

• This is only a part of the Langlands program for G = ResF/QGLn,F .16

• This does not describe all automorphic representations for GLn,F , nor all17

Galois representations.18

• Not known for F = Q, n ≥ 2, e.g., there exists L-algebraic cuspidal automor-19

phic representations of GL2(A) (associated to Maassforms), which should cor-20

respond to even irreducible, 2-dimensional `-adic representations of Gal(Q/Q)21

(with finite image).22

• Langlands-Tunnell: If the image is solvable, automorphicity of the Galois rep-23

resentation is known, cf. [GH19, Theorem 13.4.5.]. This does not cover almost24

everywhere unramified 2-dimensional, irreducible Galois representations with25

image SL2(F5).26

• Unicity follows from the (strong) multiplicity one theorems on both sides.27

• Today: Explain “L-algebraic”.28

Automorphic forms for general groups:29

• G an arbitrary reductive group over Q.30

• Will introduce a convenient (=more algebraic) replacement for L2([G]).31

• Recall for G = GL2 = GL2,Q, k ∈ Z, the embedding32

Φ: Mk → C∞(GL2(A)), f 7→ ϕf .

• Recall the description of the image of Φ(Mk) resp. Φ(Sk), namely:33

– For k ∈ Z an element ϕ ∈ C∞(GL2(A)) lies in the image of34

Φ: Mk → C∞(GL2(A))

if and only if35

∗ ϕ(g, g∞z) = z−kϕ(g, g∞) for (g, g∞) ∈ GL2(A), z ∈ C× ⊆ GL2(R).36

∗ ϕ is of moderate growth (implies the vanishing of negative Fourier37

coefficients).38

∗ For each g ∈ GL2(Af )39

Y ∗ ϕ(g,−) = 0,

where Y ∈ gC = gl2(R)C is a suitably constructed, natural element40

(implies holomorphicity).41

∗ ϕ(γg) = ϕ(g) for all γ ∈ GL2(Q).42

– f is moreover cuspidal if and only if43 ∫N(Q)\N(A)

ϕf (ng)dn = 0

for all unipotent radicals N in proper parabolics P ⊆ G (defined over44

Q).45

• Fix a maximal compact (usually not connected) subgroup K∞ of G(R), e.g.,46

O2(R) = SO2(R) ∪ sSO2(R) ⊆ GL2(R) with s =

(−1 0

0 1

).47

Definition 11.1 (cf. [GH19, Definition 6.5]). Let G be a reductive group over Q. An48

(adelic) automorphic form for G is a smooth function ϕ : G(A)→ C such that49

1) ϕ is of moderate growth (was defined in Section 4, not important for this50

lecture).51

2) ϕ is right K∞-finite, i.e., the functions g 7→ ϕ(gk) for k ∈ K∞ ⊆ G(A) span52

a finite dimensional vector space.53

3) ϕ is killed by an ideal in Z(gC) of finite codimension, where gC denotes the54

(complexified) Lie algebra of G(R), and Z(gC) the center of the enveloping55

algebra U(gC) of gC.56

4) ϕ is left G(Q)-invariant, i.e., ϕ(γg) = ϕ(g) for all γ ∈ G(Q).57

The Casimir element for GL2:58

• Let us check that 3) is indeed verified for ϕf : C∞(GL2(A))→ C with f ∈Mk.59

• Namely, the element Y was not in Z(gl2,C).60

• Set61

H :=

(0 −ii 0

), X :=

1

2

(−1 −i−i 1

), Y :=

1

2

(−1 i

i 1

), Z :=

(1 0

0 1

).

• Then62

∗H ∗ ϕf = −kϕf

as z · ϕf = z−kϕf for z ∈ C×.63

∗ Y ∗ ϕf = 0.64

• [X,Y ] = XY − Y X = H (because H,X, Y form an sl2-triple)65

• Define the Casimir operator66

∆ :=1

4(H2 + 2XY + 2Y X).

Then ∆ ∈ Z(gl2,C).67

• In fact, Z(gl2,C) is generated, as a C-algebra, by ∆ and Z ∈ gl2,C (this follows68

from the Harish-Chandra isomorphism, cf. [GH19, Theorem 4.6.1]).69

• One calculates (using Y ∗ ϕf = 0)70

∆ ∗ ϕf =1

4(H2 − 2XY + 2Y X) ∗ ϕf =

1

4(H2 − 2H) ∗ ϕf =

1

4(k2 − 2k)ϕf ,

thus ϕf is indeed killed by the ideal 〈∆ − 14 (k2 − 2k), Z + k〉 ⊆ Z(gl2,C) of71

finite codimension.72

More on automorphic forms:73

• G arbitrary reductive over Q.74

• Set A(G) as the space of automorphic forms, and A([G]) as the subspace of75

automorphic forms, which are invariant under AG (acting via left or right76

translations on G(A)).77

• Unfortunately, the spaces A([G]) and A(G) are not stable under G(R), be-78

cause K∞-finiteness need not be preserved.79

More on automorphic forms (continued):80

• Clearly, G(Af ) acts on A(G) via ϕ 7→ (g 7→ ϕ(gh)) for h ∈ G(Af ).81

• Similarly, K∞, even AGK∞, acts on A(G)82

• Less clear, but true (cf. [GH19, Proposition 4.4.2.]): gC acts on A(G).83

⇒ U(gC), in particular Z(gC) ⊆ U(gC), acts on A(G) by differential operators.84

• The gC and K∞-action make A(G) into a (gC,K∞)-module.85

Definition 11.2 (cf. [GH19, Definition 4.5.]). A (gC,K∞)-module is a C-vector space86

V , which is simultaneously a gC- and K∞-module, such that87

1) V is a union of finite dimensional K∞-stable subspaces.88

2) For X ∈ k := Lie(K∞)C and ϕ ∈ V we have89

X ∗ ϕ =∂

∂t(exp(tX)ϕ)t=0.

3) For h ∈ K∞, X ∈ gC and ϕ ∈ V we have90

h(X ∗ (h−1ϕ) = (Ad(h)(X)) ∗ ϕ.

Comments:91

• We required no topology on V , thus (gC,K∞)-modules are much more alge-92

braic than G(R)-representations, and thus easier to handle.93

• In 2) the limit is taken for the natural topology of a finite dimensional K∞-94

stable subspace of V containing ϕ.95

• For each representation of G(R) on some Hilbert space C-vector space, the96

space of smooth and K∞-finite vectors is naturally a (gC,K∞)-module, cf.97

[GH19, Proposition 4.4.2.], and on unitary representations one does not loose98

any informations, cf. [GH19, Theorem 4.4.4.], namely: isomorphism classes of99

irreducible unitary representations of G(R) inject into isomorphism classes of100

irreducible (gC,K∞)-modules.101

• Moreover, the (gC,K∞)-modules obtained in this way are admissible, i.e., for102

each (finite dimensional) irreducible representation σ of K∞, the σ-isotypic103

component is finite dimensional, cf. [GH19, Theorem 4.4.1.].104

• Version of Schur’s lemma⇒ On each irreducible, admissible (gC,K∞)-module105

π the center Z(gC) of U(gC) acts via some morphism106

ωπ : Z(gC)→ C

of C-algebras, called the “infinitesimal character of π”.107

• The Harish-Chandra isomorphism, cf. [GH19, Theorem 4.6.1.], furnishes108

Z(gC) ∼= U(tC)W

for each Cartan subalgebra t ⊆ g and W the corresponding Weyl group. Note109

U(tC) ∼= Sym•tC because tC is an abelian Lie algebra.110

• As W is finite, C-algebra morphisms111

Z(gC) ∼= U(tC)W → C

correpond bijectively to W -orbits of C-algebra homomorphisms112

U(tC) ∼= Sym•tC → C.

Moreover,113

HomC−alg(U(tC),C) ∼= HomC−lin(tC,C) =: t∨C .

U(tC)W → U(tC) is finite.114

• Thus, the infinitesimal character defines a canonical W -orbit of elements in115

the “weight space” t∨C .116

• Now assume that t = Lie(T ) for some maximal torus T ⊆ GR. Then W -117

equivariantly118

X∗(TC) ⊆ t∨C .

• An irreducible, admissible (gC,K∞)-module π is called L-algebraic if the W -119

orbit in t∨C corresponding to the infinitesimal character ωπ lies in the finite120

free Z-module X∗(TC) ⊆ t∨C , cf. [BG10, Definition 2.3.1.], [Tho, Definition 93].121

(gl2,C,O2(R))-modules generated by modular forms:122

• Let f ∈ Mk. We will analyze the (gl2,C,O2(R))-module V ⊆ A(GL2) gener-123

ated by ϕ := ϕf .124

• Recall125

Y ∗ ϕ = 0, z · ϕ = z−kϕ for z ∈ C×.

• It is not difficult to see that there exists a unique, up to isomorphism, (gl2,C,O2(R))-126

module D′k−1 (the dual of Dk−1 in [Del73, Section 2.1]) generated by such an127

element ϕ, and that D′k−1∼= V is irreducible.128

• In particular, it only depends on k, not f .129

• We obtain, as was mentioned some time ago,130

Mk∼= Hom(gl2,C,O2(R))(D

′k−1,A(GL2)),

(note: the space H(GA) in [Del73, Scholie 2.1.3.] agrees with our A(GL2)131

only up to inversion on GL2(A)).132

• We calulated133

∆ ∗ ϕ =1

4(k2 − 2k)ϕ, Z ∗ ϕ = −kϕ,

and this determines the infinitesimal character ωD′k−1.134

• The W -orbit of C-algebra homomorphisms U(tC) ∼= C[Z,H]→ C correspond-135

ing to ωD′k−1under the Harish-Chandra isomorphism is given by the homo-136

morphisms137

Z 7→ −k, H 7→ ±(k − 1).

• The lattice X∗(T ) ⊆ 〈Z,H〉∨C is given by C-linear maps138

Z 7→ a, H 7→ b

with a, b ∈ Z and a ≡ b mod 2.139

• Thus, D′k−1 is not L-algebraic (it is C-algebraic in the sense of [BG10]).140

• However,141

D′k−1 ⊗C |det|1/2+a

is L-algebraic for any a ∈ Z because (the (gl2,C,O2(R))-module associated142

with) |det|1/2 has infinitesimal character corresponding to143

Z 7→ 1, H 7→ 0.

Upshot:144

• For any G there exists the space A(G) of (adelic) automorphic forms, which145

for GL2 naturally contains the ϕf with f ∈Mk.146

• The space A(G) carries a (gC,K∞) × G(Af )-action, and is a more algebraic147

replacement for L2([G]) with its G(A)-action.148

A new notion of automorphic representations:149

• We redefine the notion of an automorphic representation using A(G) instead150

of L2([G]).151

• Namely, an automorphic representation for G is an irreducible G(Af ) ×152

(gC,K∞)-subquotient of A(G), cf. [GH19, Definition 6.8.].153

• The former automorphic representations will from now on be called “L2-154

automorphic representations”.155

• For informations how both notions relate, cf. [GH19, Section 6.5.].156

Flath’s theorem (cf. [GH19, Theorem 5.7.1.]):157

• For almost all primes p the reductive group GQp := G ×Spec(Q) Spec(Qp) is158

“unramified”, i.e., extends to a reductive group scheme159

Gp → Spec(Zp).

• Equivalently, GQp is quasi-split (=contains a Borel subgroup defined over Qp)160

and split(=contains a maximal and split torus) over an unramified extension.161

• Then162

G(A) =

′∏p

(G(Qp),Gp(Zp))×G(R),

where Gp(Zp) ⊆ G(Qp) is a compact-open subgroup (cf. [GH19, Proposition163

2.3.1.]), called “hyperspecial”.164

• Flath’s theorem (cf. [GH19, Section 5.7.] and [Fla79]) states that irreducibe,165

admissible G(Af ) × (gC,K∞)-modules π decompose (uniquely) into a “re-166

stricted tensor product”167

π ∼=′⊗

p prime

πp ⊗ π∞

of irreducible, smooth (even admissible) GQp -representations πp and an irre-168

ducible, admissible (gC,K∞)-module π∞.169

• Moreover, for almost all primes p the representation πp is unramified, i.e., the170

group GQp is unramified and πGp(Zp)p 6= 0, where Gp is a reductive model of171

GQp .172

• Important, cf. [GH19, Theorem 7.5.1.]: H(G(Qp),Gp(Zp)) is again commuta-173

tive! (⇒ If πGp(Zp)p 6= 0, then its dimension is one.)174

Upshot:175

• Thus irreducible, admissible G(Af ) × (gC,K∞)-modules are a collection of176

“local data”.177

• Being automorphic puts strong relations among these local data.178

• For almost all primes p get homomorphism179

χp : H(G(Qp),Gp(Zp))→ C,

similarly to case for GL2 ⇒ this will yield analog of a “system of Hecke180

eigenvalues”.181

• For π set182

πf :=

′⊗p

πp,

thus183

π ∼= πf ⊗ π∞.

Definition 11.3 (L-algebraic automorphic representation). An automorphic repre-184

sentation185

π ∼= πf ⊗ π∞is L-algebraic if the irreducible, admissible (gC,K∞)-module π∞ is L-algebraic, cf.186

[BG10, Definition 3.1.1.].187

Examples:188

• An automorphic representation π ⊆ A(GL2) which is generated by a modular189

form is not L-algebraic, but the twist190

π ⊗C |det|1/2+aadelic

is for any a ∈ Z.191

• A continuous character ψ : Q×\A× → C is L-algebraic if and only if192

ψ = χ| − |kadelic

for some k ∈ Z, | − |adelic the adelic norm and χ : Q×\A× → C a character of193

finite order.194

12. The Langlands program for general groups, part II1

Last time:2

• F arbitrary number field3

• The Langlands(-Clozel-Fontaine-Mazur) reciprocity conjecture for GLn,F says:4

– ` some prime. Fix an isomorphism C ∼= Q`.5

– Then for any n ≥ 1, there exists a (unique) bijection between6

i) the set of L-algebraic cuspidal automorphic representations of GLn(AF ),7

ii) the set of (isomorphism classes) of irreducible continuous repre-8

sentations Gal(F/F ) → GLn(Q`) which are almost everywhere9

unramified, and de Rham at places dividing `,10

such that the bijection matches Satake parameters with eigenvalues of11

Frobenius elements.12

• Introduced space of (adelic) automorphic forms A(G) for any G.13

• Flath’s theorem⇒ automorphic representations π factorize: π ∼=⊗′

p πp⊗π∞.14

• Explained “L-algebraic” (=infinitesimal character of π∞ is “integral”).15

Today:16

• Introduce “cuspidal” automorphic representations.17

• Explain Satake parameters.18

• Discuss some expectations of the Langlands program for general G.19

Definition 12.1 ([GH19, Definition 9.2]). Let G/Q be reductive. We call ϕ ∈ L2([G])20

resp. ϕ ∈ A(G) cuspidal if21 ∫N(Q)\N(A)

ϕ(ng)dn = 0

for all unipotent radicals N of proper parabolic subgroups P ⊆ G (defined over Q),22

and almost all g ∈ G(A).23

Cuspidal automorphic representations:24

• Set L2cusp([G]) resp. Acusp(G) resp. Acusp([G]) as the subspaces of cuspidal25

elements.26

• There is an embedding with dense image27

Acusp([G]) ⊆ L2cusp([G]),

i.e., cuspidal automorphic forms satisfy a growth condition strong enough to28

make them L2 (they are “rapidly decreasing”), cf. [GH19, Section 6.5.].29

• An L2-automorphic representation of G(A) is cuspidal if it is isomorphic to a30

subquotient (actually subrepresentation) of L2cusp([G]).31

• Similarly, an automorphic representation is cuspidal if it occurs as a subquo-32

tient of Acusp(G).33

• Gelfand, Piatetski-Shapiro: As a unitary G(A)-representation34

L2cusp([G]) ∼=

⊕π∈G(A)

mππ

with each mπ finite, i.e., mπ ∈ N ∪ 0, cf. [GH19, Corollary 9.1.2].35

• In particular, L2cusp([G]) ⊂ L2

disc([G]) (note: the trivial representation occurs36

in L2disc([G]) \ L2

cusp([G])).37

• From38

L2cusp([G]) ∼=

⊕π∈G(A)

mππ

on deduces the decomposition (into irreducible, admissible G(Af )×(gC,K∞)-39

modules)40

Acusp([G]) ∼=⊕

π∈G(A)

mππK∞−finite,

where πK∞−finite ⊆ π denotes the (dense) subspace of K∞-finite vectors, cf.41

[GH19, Theorem 4.4.4.].42

• Thus, the decomposition of L2cusp([G]) is “the same as” the decomposition of43

Acusp([G]).44

• Piatetski-Shapiro, Shalika, cf. [GH19, Theorem 11.3.4]: F number field, G =45

ResF/QGLn,F . Then the G(A)-representation on L2cusp([G]) is multiplicity46

free (“multiplicity one”).47

• “Strong multiplicity one”: Assume π, π′ are cuspidal automorphic represen-48

tations for GLn,F . If πp ∼= π′p for almost all primes p, then π ∼= π′, cf. [GH19,49

11.7.2].50

• Not true without cuspidality, cf. [BG10, page 38].51

• Moeglin, Waldspurger, cf. [GH19, Theorem 10.7.1.], [MW89]: G = ResF/QGLn,F .52

Then L2disc([G]) is known, up to parametrizing cuspidal automorphic repre-53

sentations of GLd,F for d|n, and turns out to be multiplicity free.54

Satake parameters:55

• Describe πp if π is unramified at p, i.e.,56

πGp(Zp)p 6= 0

with Gp → Spec(Zp) a reductive model of GQp (which exists for almost all57

primes p).58

• Thus, describe C-algebra homomorphisms59

H(G(Qp),K)→ C

for K := Gp(Zp).60

• Langlands/Satake, cf. [GH19, Theorem 7.5.1., Corollary 7.5.2.], [BG10, Sec-61

tion 2.1.]: There exists62

– a “natural” algebraic group G over Q,63

– an automorphism Frp : G(Q) ∼= G(Q),64

– and a bijection65

HomC−alg(H(G(Qp),K),C)1:1↔

Frobenius semisimple G(C)− conjugacy classes in G(C) o FrZp.

• The group GC has the dual root datum as G. Thus for example, cf. [Bor79]:66

– if G = GLn, then G = GLn,C,67

– if G = SLn, then G = PGLn,C,68

– if G = SO2n, then G = SO2n,C,69

– if G = SO2n+1, then G = Sp2n,C,70

– if G = GSp2n, then G = GSpin2n+1,C.71

• If GQp is split, then the automorphism Frp is trivial, and Frobenius semisimple72

conjugacy classes are just conjugacy classes of semisimple elements in G(C).73

• If G = GLn, then semisimple conjugacy classes are uniquely determined by74

their characteristic polynomials.75

• Let π =⊗′

p πp⊗π∞ be an automorphic representation of G. Then we obtain76

the following analog of a system of Hecke eigenvalues:77

– a finite set S of primes, such that πp is unramified for p /∈ S,78

– for each p /∈ S a Frobenius semisimple conjugacy class cp(π) in G(C) o79

FrZp .80

• For GLn the eigenvalues of (each element in) cp(π) are the Satake parameters81

of π.82

• Concretely, if the coset83

GLn(Zp)Diag(p, . . . , p︸ ︷︷ ︸i−times

, 1, . . . , 1︸ ︷︷ ︸(n−i)−times

)GLn(Zp),

has eigenvalue ap,i on πGLn(Zp)p , then the elements in cp(π) have characteristic84

polynomial85

Xn − p(1−n)

2 ap,1Xn−1 + . . .+ (−1)ip

i(i−n)2 ap,iX

i + . . .+ (−1)nap,n,

cf. [GH19, Section 7.2.].86

• Even more concrete, for π generated by a newform f =∞∑i=1

anqn ∈ Sk(Γ0(N), χ)87

the cp(π), p - N, have characteristic polynomial88

X2 − p−1/2papX + χ(p)pk.

Recall that π ⊗C |det|1/2adelic is L-algebraic. For π ⊗C |det|1/2adelic we obtain the89

polynomial90

X2 − apX + χ(p)pk−1,

which (after choosing an isomorphism Q` ∼= C) was the characteristic poly-91

nomial of an arithmetic Frobenius at p.92

The L-group:93

• G reductive over Q94

• ` a prime.95

• With a little work (cf. [GH19, Section 7.3.]) the group Gal(Q/Q) acts on the96

reductive group G over Q (the action is trivial if G is split).97

• Set98

LG := G(Q`) o Gal(Q/Q).

• An L-parameter is by definition a G(Q`)-conjugacy class of continuous homo-99

morphisms100

Gal(Q/Q)→ LG,

whose projection to Gal(Q/Q) is the identity.101

• If F/Q is finite and G = ResF/QGLn,F , then an L-parameter identifies with102

an isomorphism class of an n-dimensional `-adic Galois representation of103

Gal(F/F ).104

The Buzzard–Gee conjecture for L-algebraic automorphic representations:105

106

• Let ` be a prime.107

• Fix an isomorphism ι : C ∼= Q`.108

• Let π be an L-algebraic automorphic representation of G.109

• Then Buzzard–Gee (cf. [BG10, Conjecture 3.2.2.]) conjecture that there exists110

an L-parameter111

ρπ : Gal(Q/Q)→ LG

such that (in particular)112

– if p 6= ` is unramified for π, then each arithmetic Frobenius Frobp ∈113

Gal(Qp/Qp) maps to the conjugacy class of114

ι(cp(π)) ∈ G(Q`) o FrobZp

under115

Gal(Qp/Qp)→ G(Q`) o Gal(Qp/Qp) G(Q`) o FrobZp ,

– for each continuous representation LG → GLN (Q`), which is algebraic116

on G(Q`), the composition117

Gal(Q`/Q`)ρ−→ LG→ GLN (Q`)

is de Rham.118

• This generalizes the association of Galois representations to (a twist of) the119

automorphic representation associated to some newform.120

• Contrary to the case of GLn the L-parameter ρπ is not conjectured to be121

unique, cf. [BG10, Remark 3.2.4.].122

• Moreover, we may chose the same L-parameter ρ for different L-algebraic123

automorphic representations π.124

• For GLn the Buzzard–Gee conjecture predicts in addition to the previous125

Langlands-Clozel-Fontaine-Mazur conjecture the existence of Galois repre-126

sentations attached to possibly non-cuspidal (L-algebraic) automorphic rep-127

resentations. These Galois representations are then no longer conjectured to128

be irreducible.129

The Langlands group L:130

• Conjecturally, there should exist a (very big) locally compact group L, the131

(global) Langlands group, with (at least) the following properties, cf. [GH19,132

Section 12.6.], [Art02], [LR87]:133

– There exists a canonical surjection134

L Gal(Q/Q)

(in fact, LWQ with WQ the global Weil group).135

– For each n ≥ 0 the isomorphism classes Irrn(L) of continuous irreducible136

n-dimensional C-representations of L are naturally in bijection with the137

set Cspn of cuspidal automorphic representations of GLn(A).138

– Fix an isomorphism Q` ∼= C. Then there is an injection of the set Ggeom139

of isomorphism classes of irreducible, almost everywhere unramified `-140

adic Galois representations Gal(Q/Q) which are de Rham at ` to the set141

Irr of isomorphism classes of irreducible representations of L, with image142

corresponding to L-algebraic cuspidal automorphic representations.143

– For each place v of Q there is an injection144

Lv → L

of the local Langlands groups, which is well-defined up to conjugation in145

L. Here:146

∗ Lv ∼= WR, the non-split extension of C× by Gal(C/R), if v =∞.147

∗ Lv ∼= WQp × SU2(R) if v = p prime and148

WQp∼= IQp o Z ⊆ Gal(Qp/Qp) ∼= IQp o Z

is the subgroup of elements mapping to integral powers of Frobe-149

nius.150

– If π ∼=⊗′

v πv ∈ Cspn corresponds to a continuous representation ρπ : L →151

GLn(C), then πv should correspond to ρπ |Lv under the (conjectural) local152

Langlands correspondence, cf. [GH19, Section 12.5.]. Next semester Pe-

ter is giving a lec-

ture on this ⇒Follow it!

Next semester Pe-

ter is giving a lec-

ture on this ⇒Follow it!

153

• If this conditions hold, then necessarily Lab ∼= Q×\A×. Note that by global154

class field theory there exists a surjection Q×\A× → Gal(Q/Q).155

• The existence of L is currently completely out of reach, even more conjec-156

turally L should surject onto the “motivic Galois group” Gmot over C, cf.157

[Art02], [LR87].158

The global Langlands correspondence for general G (very, very rough):159

• No precision is claimed, and all statements are close to being empty!160

• For precise statements one should consult [GH19, Chapter 12] or [Art94].161

• In order to parametrize non L-algebraic automorphic representations (at least162

those relevant for the decomposition of L2disc([G])) one should replace the163

previous Galois version of L-parameters by (certain) L-parameters164

L → LG

(actually in the Weil form WQ n G(C) of the L-group).165

• Then one hopes to construct a surjective map (cf. [GH19, Conjecture 12.6.2.])166

(certain) automorphic representations LL−−→ (certain) L− parameters.

• The fibers of LL are called L-packets .167

• A precise construction of a local Langlands correspondence (cf. [GH19, Con-168

jecture 12.5.1.]) should yield a parametrization of the L-packets, cf. [GH19,169

(12.23)].170

• For each π ⊆ L2disc([G]) its multiplicity should be computable via the L-171

parameter LL(π) and the precise parametrizations of the L-packets, cf. [GH19,172

Conjecture 12.6.3.].173

• However, for non-tempered representations the above should not be reason-174

able and one should consider Arthur parameters175

L × SL2(C)→ LG

instead of L-parameters, cf. [Art94].176

Important stuff, which was not mentioned in the lecture:177

• L-functions, [Bor79], [GH19, Chapter 11, Section 12.7.],178

• Functoriality, [GH19, Section 12.6.],179

• The trace formula, [Art05], [GH19, Section 18],180

• The Langlands program for function fields,181

• ...182

A glimpse on Shimura varieties:183

• Let K∞ ⊆ G(R) be a maximal (connected) compact subgroup.184

• Let K ⊆ G(Af ) be a (sufficiently small) compact-open subgroup.185

• Recall that186

–

[G]→ [G]/K = G(Q)AG\G(A)/K

is a profinite covering of a real manifold187

–

[G]/K → XK := [G]/KK∞ = G(Q)AG\G(A)/KK∞

is a K∞-bundle over XK , which is a disjoint union of arithmetic mani-188

folds (=quotients of a symmetric spaces by arithmetic subgroups)189

• Note that implicitly XK depends on K∞.190

• Let us set (just to simplify some notations later)191

X := lim←−K

XK .

Stuff we encountered for G = GL2:192

(1) The upper/lower halfplane H± ∼= G(R)/K∞, which is naturally a complex193

manifold194

(2) The holomorphic embedding195

H± → P1C

(3) The G(R)-equivariant vector bundles196

ωk

on H±, or by pullback on XK for K ⊆ G(Af ) compact-open.197

(4) For k ∈ Z the space of modular forms198

Mk ⊆ H0(X, ωk),

where the RHS denotes holomorphic sections.199

(5) The canonical compactification X∗K of XK , with the extension of ωk on it.200

(6) A scheme X → Spec(Q), whose C-valued points are naturally isomorphic to201

X.202

(7) The Gal(Q/Q)×G(Af )-representation203

H1et(XQ,L)

associated to certain G(Af )-equivariant Q`-local systems L on X , e.g., L ∼=204

Q`.205

(8) The Eichler–Shimura isomorphism relating Sk for k ≥ 2, to certain206

H1et(XQ,L).

(9) The Eichler–Shimura relation expressing a relation of the Gal(Q/Q)-action207

and the G(Af )-action on208

H1et(XQ,L),

which was a consequence of the mod p geometry of some modular curve.209

(10) A map210

LL: Amod → Gmod

with (conjecturally) describable image.211

For general G there are unfortunately no analogs - but wait: there exists212

G giving rise to Shimura varieties!213

• Already (1) fails for general G: The space214

G(R)/K∞

need not be complex manifold! (E.g., if G = GL3 , then the real dimension215

dim(GL3(R)/AGL3(R)SO3(R)) = 3(3+1)2 − 1 = 5 is odd)216

• In particular, everything related to holomorphicity has no analog for such G,217

e.g., 2, 3, 4, 6, 7,...218

• That is bad news!219

• Good news: For groups G giving rise to Shimura data (like ResF/QGL2,F for220

F/Q totally real, or GSp2n,... , cf. [Lan17], [Mil05], [Del71a], [Del79]), the221

real manifold222

G(R)/K∞AG

is a complex manifold.223

• A Shimura datum is a pair (G,X) with G a reductive group G over Q, and X224

a G(R)-conjugacy classes of morphisms h : S := ResC/RGm → GR such that225

1) For each h ∈ X, the characters of S(R) = C× acting on Lie(GR)C (via226

Ad h) are either z 7→ zz , z 7→ 1, or z 7→ z

z .227

2) For each h ∈ X, the group g ∈ G(C) | h(i)gh(i)−1 = g is compact228

modulo its center.229

3) The adjoint group Gad has no factor, defined over Q, for which the230

projection of h ∈ X is trivial.231

• E.g., take G = GL2, and X as the conjugacy class of the usual embedding232

C× → GL2(R). Note that X ∼= H±.233

• In general, for h ∈ X the stabilizer Kh of h in G(R) is compact modulo234

the center of G(R) (this follows from condition 2)) and X ∼= G(R)/Kh(∼=235

G(R)/AGK∞).236

• For K ⊆ G(Af ) recall237

XK := G(Q)\(G(Af )/K ×X).

Other notation: ShK(G,X) the Shimura variety of levelK attached to (G,X).238

• We get:239

(1) X is a disjoint union of hermitian symmetric domains, cf. [Mil05], [Del79].240

(2) To each h ∈ X is naturally attached a parabolic subgroup Ph ⊆ G(C)241

(cf. [CS17, Section 2.1]), and sending h→ Ph defines an open embedding242

π : X ∼= G(R)/Kh → F l ∼= G(C)/Ph

of X into a flag variety (this generalizes the embedding H± → P1(C)). In243

particular, X carries a natural complex structure. More is true (Baily-244

Borel): XK is naturally a quasi-projective variety, cf. [Lan17, Theorem245

2.4.1.].246

(3) The pullback of G(C)-equivariant vector bundles on F l gives rise to247

G(Q)-equivariant vector bundles on G(Af )×X. By descent one one ob-248

tains automorphic vector bundles E on XK for any K ⊆ G(Af ) compact-249

open, i.e., analogs of the ωk, k ∈ Z, [Har88]. Note that G(C)-equivariant250

vector bundles on F l are equivalent to (algebraic) representations of Ph.251

(4), (5) If dimC(XK) > 1, there do exist compactifications XK , but these are252

no longer canonical. However, it is possible to define subspaces in the253

cohomology of H∗(XK , E)), which generalize Mk for k ∈ Z, cf. [Har88].254

(6) The space X (=inverse limit of complex manifolds) arises again as the255

C-points of a scheme X → Spec(E), the canonical model, defined over256

some number field (the “reflex field of the Shimura data”), cf. [Lan17,257

Theorem 2.4.3.].258

(8),(9),(10) Having the canonical model, one can attempt (sometimes sucessfully)259

to obtain analogs of points (18),(19),(20) and finally relate (certain)260

automorphic representations to (variants of) Galois representations, cf.261

[Har88], [BRb]. Needless to say that everything (compactifications, Eichler–262

Shimura isomorphism/relation, integral models, mod p geometry, interior263

vs intersection cohomology,...) is much more complicated!264

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