Higher rank Kolyvagin systems - krubin/lectures/lausanne.pdf · Higher rank Kolyvagin systems Karl...

Higher rank Kolyvagin systems

Karl Rubin, UC Irvine

joint with Barry Mazur

Lausanne, November 2012

Karl Rubin Higher rank Kolyvagin systems Lausanne, 11/2012 1 / 39

Goals

Study Selmer groups . . .

Selmer groups of elliptic curves and abelian varietiesideal class groupsmore general Selmer groups

. . . in families . . .

families of “twists” (controlled change of local conditions)statistics

. . . and use the full families to study the original Selmer groups

ideally, relate the order of a Selmer group to an L-valuethis is where Euler systems and Kolyvagin systems come in


General Selmer groups

Fix a number field K and a power pk of a rational prime p. We beginwith the following data:

T , a free Z/pkZ-module of finite rank, with a continuous actionof GK (think of T = T∞/p

kT∞, with a free Zp-module T∞)

a Selmer structure F on T , i.e., a choice of subgroup

H1F(Kv, T ) ⊂ H1(Kv, T )

for every place v, such that for all but finitely many v we haveH1F(Kv, T ) = H1

ur(Kv, T ).

Define the Selmer group Sel(T ) by

Sel(T ) = Sel(T,F) := {c ∈ H1(K,T ) : cv ∈ H1F(Kv, T ) for all v}.


General Selmer groups

A Selmer structure F on T induces a Selmer structure F∗ on theCartier dual T ∗ as follows. Define

T ∗ := Hom(T,µpk)

and define F∗ by

H1F∗(Kv, T

∗) := H1F(Kv, T )⊥

orthogonal complement under the local Tate pairing.

Our goal is to study the Selmer group Sel(T ∗,F∗).


Examples of Selmer groups

Example (Elliptic curves and abelian varieties)

Suppose E is an elliptic curve over K, and let T := E[pk] with thenatural Selmer structure:

H1F(Kv, E[pk]) := image [E(Kv)→ H1(Kv, E[pk])].

Then T ∗ = T , F∗ = F , and Sel(E[pk]) is the classical Selmer groupSelpk(E).

Similarly, if A is an abelian variety, and T = A[pk] with the analogousSelmer structure, then Sel(A[pk]) = Selpk(A).



Example (Global units and ideal class groups)

Let T := µpk , so T ∗ = Z/pkZ. The natural Selmer structure here is

H1F(Kv,µpk) :=image [O×v → H1(Kv,µpk)],

H1F∗(Kv,Z/pkZ) =Homur(Gv,Z/pkZ) ⊂ H1(Kv,Z/pkZ)

where Ov = integers of Kv, Gv ⊂ GK a decomposition group at v.Then

0→ O×K/(O×K)p

k → Sel(T )→ Cl(K)[pk]→ 0

Sel(T ∗) = Hom(Cl(K),Z/pkZ) = 0.



Example (Global units and ideal class groups)

More generally, let T := µpk ⊗ χ, where χ is a character of GK offinite order. Let F be the cyclic extension of K cut out by χ. Then

H1(K,T ) ∼= (F×/(F×)pk

)χ,

H1(K,T ∗) ∼= Hom(GF ,Z/pkZ)χ,

0→ (O×F /(O×F )p

k

)χ → Sel(T )→ Cl(F )[pk]χ → 0

Sel(T ∗) ∼= Hom(Cl(F )χ−1

,Z/pkZ).


Twisted Selmer groups

To simplify notation, suppose now that K = Q.

Let P denote the set of rational primes ` satisfying

T is unramified at `,

H1F(Q`, T ) = H1

ur(Q`, T ),

` ≡ 1 (mod pk),

T/(Frob` − 1)T is free of rank one over Z/pkZ.

The first two conditions rule out only finitely many primes. The thirdis satisfied by a set of primes of positive density, and so is the last ifthere is a γ ∈ GQ such that T/(γ − 1)T is free of rank one overZ/pkZ.



If ` ∈ P , define

H1tr(Q`, T ) := ker [H1(Q`, T )→ H1(Q`(µ`), T )].

LemmaFor every ` ∈ P ,

H1(Qp, T ) = H1ur(Q`, T )⊕H1

tr(Q`, T ),

H1ur(Q`, T ), H1

tr(Q`, T ) are both free of rank one over Z/pkZ,

There is an isomorphism

φ` : H1ur(Q`, T )

∼−→ H1tr(Q`, T )

depending only on a choice of generator of (Z/`Z)×.



DefineN := {squarefree products of primes ` ∈ P}.

For n ∈ N , define a new Selmer structure F(n) by

H1F(n)(Q`, T ) :=

{H1F(Q`, T ) if ` - n,

H1tr(Q`, T ) if ` | n.

We will abbreviate

Sel(T, n) := Sel(T,F(n)), Sel(T ∗, n) := Sel(T ∗,F(n)∗)

(Fact: F(n)∗ = F∗(n).)



Example (2-Selmer groups of elliptic curves)

Take an elliptic curve E over Q, pk = 2, and T = E[2].

If n ∈ N , n ≡ 1 (mod 8), and n is a square modulo ` for everyprime ` where E has bad reduction, then

Sel(E[2], n) = Sel2(En),

the classical 2-Selmer group of the quadratic twist of E by n.

Thus the family of Selmer groups Sel(E[2], n) is closely related tothe family of 2-Selmer groups of quadratic twists of E.



Example (p -Selmer groups of elliptic curves)

We can also consider T = E[p], for odd primes p.

Suppose n ∈ N , and let χ be a primitive Dirichlet character of orderp modulo n. The twist Eχ of E by χ is an abelian variety ofdimension p− 1, with Z[ζp] ⊂ End(Eχ).

If n is a p-th power in Qp and in Q` for for every prime ` where Ehas bad reduction, then

Sel(E[p], n) = Sel(1−ζp)(Eχ).

Thus the family of Selmer groups Sel(E[p], n) is closely related tothe family of (1− ζp)-Selmer groups of twists of E by characters oforder p.


Statistics of families of Selmer groups

Suppose pk = p, i.e., T is an Fp-vector space. If n` ∈ N , then∣∣dimFp Sel(T, n)− dimFp Sel(T, n`)∣∣ ≤ 1.

LemmaLet Fn be the smallest extension of Q(T ) in which all cocycles inSel(T, n) become trivial.

Then the value of dimFp Sel(T, n)− dimFp Sel(T, n`) is determinedby the Frobenius of ` in Gal(Fn/Q).

Using this lemma, for each n we can compute the density of

{` : dimFp Sel(T, n`) = dimFp Sel(T, n) + i}for i = −1, 0, 1. In this way we can study the densities of the sets

{n ∈ N : dimFp Sel(T, n) = r}.Karl Rubin Higher rank Kolyvagin systems Lausanne, 11/2012 13 / 39

Kolyvagin systems

DefinitionA (rank one) Kolyvagin system is a collection

{κn ∈ Sel(T, n) : n ∈ N}

such that if n` ∈ N , then

(κn`)` = φ`((κn)`) ∈ H1tr(Q`, T ).

Here ( · )` denotes localization at `, and recall that we have

φ` : H1ur(Q`, T )

∼−→ H1tr(Q`, T ).


Assumptions

Assume from now on:

the image of GQ in Aut(T ) is big enough

T/pT is irreducible and nontrivialH1(Q(T )/Q, T ) = 0there is a γ ∈ GQ(µ

pk) such that T/(γ − 1)T is free of rank one

similarly for T ∗

. . .

(this will ensure that N is large)

F is “nice”

for example, T = T∞/pkT∞ and F is induced by a Selmer

structure on T∞ such that H1(Q`, T∞)/H1F (Q`, T∞) is

torsion-free

These hold when T = E[pk] and GQ → Aut(E[pk]) is surjective, orwhen T = µpk ⊗ χ for “most” χ.


Core rank

LemmaThere is an r ∈ Z such that for every n ∈ N , there is a noncanonicalisomorphism

Sel(T, n) ∼= (Z/pkZ)r ⊕ Sel(T ∗, n).

(If r < 0 we move it to the other side.)

DefinitionThe core rank of T (and F) is the integer r of the Lemma.

The core rank can be calculated from local data.


Core rank

ExampleSuppose K is a totally real number field, and we work over K insteadof Q. Take

T = µpk ⊗ χ, with a nontrivial, totally even character χ of GK ,

F is the natural Selmer structure.

Recall

0→ (O×K/(O×K)p

k

)χ → Sel(T )→ Cl(K)[pk]χ → 0

Sel(T ∗) ∼= Hom(Cl(F )χ−1

,Z/pkZ)

so the core rank of T is rank ((O×K/(O×K)p

k) = [K : Q].


Core rank

ExampleSuppose

T = A[pk], where A is an abelian variety over Q,

F is the natural Selmer structure, except that the localcondition at p is relaxed: H1

F(Qp, A[pk]) = H1(Qp, A[pk]).

Then the core rank of (A[pk],F) is dim(A).

(Without relaxing the local condition at p, the core rank would bezero.)


Results, when the core rank is one

LetKS1(T ) = {Kolyvagin systems of rank one for T}.

Theorem 1Suppose the core rank of T is one, and κ ∈ KS1(T ). Then for everyn ∈ N ,

κn ∈ |Sel(T ∗, n)| · Sel(T, n).

Corollary

If κ ∈ KS1(T ), then |Sel(T ∗)| ≤ pd, where

d = max{t : κ1 ∈ ptH1(Q, T )}.



Further, if κ1 6= 0 then (with more work) κ determines the full groupstructure of Sel(T ∗).

Let ω(n) denote the number of prime factors of n ∈ N .

Corollary

Suppose κ ∈ KS1(T ) and κ1 6= 0. For every i ≥ 0, let

di := max{t : κn ∈ ptSel(T, n) for every n ∈ N with ω(n) = i}.

Then d0 ≥ d1 ≥ d2 · · · ,

Sel(T ∗) ∼=⊕i≥0

Z/pdi−di+1 .



RemarkRecall that if the core rank of T is one, then

Sel(T, n) ∼= Z/pkZ⊕ Sel(T ∗, n)

so |Sel(T ∗, n)| · Sel(T, n) is a canonical cyclic subgroup of Sel(T, n).

Idea of proof: each κn gives a relation on Sel(T ∗). If c ∈ Sel(T ∗),then

0 =∑`

〈c, κn〉` =∑`|n

〈c, κn〉`



Theorem 2Suppose the core rank of T is one. Then KS1(T ) is free of rank oneover Z/pkZ.

Further, if κ generates KS1(T ), then κn generates|Sel(T ∗, n)| · Sel(T, n) for every n.

So Kolyvagin systems exist. How do we find them, and relate themto L-values?


Euler systems of rank one

DefinitionAn Euler system (of rank one) for T is a collection

{ξn ∈ H1(Q(µn), T ) : n ∈ N · pZ≥0}

such that if n` ∈ N · pZ≥0 , then

NQ(µn`)/Q(µn)ξn` =

{(det(1− Frob−1

` x|T ∗)|x=Frob−1`

)ξn if ` 6= p,

ξn if ` = p.

Let ES1(T ) = {Euler systems of rank one for T}.



Theorem 3There is a canonical map

ES1(T )→ KS1(T )

{ξn} 7→ {κn}.

The map ES1(T )→ KS1(T ) is Kolyvagin’s derivative construction.One can describe explicitly how κn is related to ξn; in particular

κ1 = ξ1 ∈ H1(Q, T ).



In applications, one proceeds as follows.

Produce, somehow, an Euler system ξ related to L-values.

By Theorem 3 there is a Kolyvagin system κ with κ1 = ξ1.

Then κ1 is related to an L-value, and Theorem 1 gives an upperbound on |Sel(T ∗)| in terms of the p-divisibility of κ1.

Applications

K = Q, T = µpk ⊗ χ, χ even: Euler system of cyclotomic units,

E an elliptic curve over Q, T = E[pk]: Kato’s Euler system


Higher core rank

What if the core rank is greater than one?

T = µpk ⊗ χ over a totally real field K ) Q (to study ideal classgroups of abelian extensions of K)

T = A[pk] with an abelian variety A over Q of dimensiongreater than one (to study the p-power Selmer groups of A)

Clue: conjectures about L-functions don’t see individual elements(unless r = 1), they see regulators. So don’t look for individualelements of H1, look for “determinants”, i.e., elements of ∧rH1.


Higher core rank

Conjecture (Perrin-Riou)

Let r be the core rank of T . There is a collection

{ξn ∈ ∧rH1(Q(µn), T )⊗Q : n ∈ N · pZ≥0}

related to p-adic L-functions attached to T .

DefinitionAn Euler system of rank r for T is a collection

{ξn ∈ ∧rH1(Q(µn), T ) : n ∈ N · pZ≥0} such that

NQ(µn`)/Q(µn)ξn` =

{(det(1− Frob−1

` x|T ∗)|x=Frob−1`

)ξn if ` 6= p,

ξn if ` = p.


Higher core rank

This suggests that one should define a Kolyvagin system of rank r tobe a collection

{κn ∈ ∧rSel(T, n) : n ∈ N}.

But:

What is the correct relation between κn and κn`? We don’t wantto compare them in ∧rH1

ur(Q`, T ) or ∧rH1tr(Q`, T ), because if

` ∈ P then ∧rH1ur(Q`, T ) = ∧rH1

tr(Q`, T ) = 0 for r ≥ 1.

How can we use such a collection to bound Sel(T ∗)? Anelement of ∧rSel(T, n) does not give a relation in Sel(T ∗).(Note that we’re not particularly interested in ∧rSel(T ∗).)


Higher core rank

DefinitionA Kolyvagin system of rank r for T is a collection

{κn ∈ ∧rSel(T, n) : n ∈ N}

such that if n` ∈ N , then the images of κn and κn` coincide in

κn ∈ ∧rSel(T, n)

loc`��

H1ur(Q`, T )⊗ ∧r−1Sel(T, n)

φ`⊗1��

κn` ∈ ∧rSel(T, n`)loc` // H1

tr(Q`, T )⊗ ∧r−1Sel(T, n)

κn ∈ ∧rSel(T, n)

loc`��

H1ur(Q`, T )⊗ ∧r−1Sel(T, n)triv-at-`

φ`⊗1��

κn` ∈ ∧rSel(T, n`)loc` // H1

tr(Q`, T )⊗ ∧r−1Sel(T, n)triv-at-`

When r = 1, this is the same as before.


Higher core rank

The map λ : ∧rSel(T, n)→ H1ur(Q`, T )⊗ ∧r−1Sel(T, n) is given by

λ(c1 ∧ · · · ∧ cr) =r∑i=1

(−1)i(ci)` ⊗ (c1 ∧ · · · ∧ ci−1 ∧ ci+1 ∧ · · · ∧ cr).

Since H1ur(Q`, T ) is cyclic, the exact sequence

0→ Sel(T, n)triv-at-` → Sel(T, n)→ H1ur(Q`, T )

allows us to decompose Sel(T, n) = M ⊕N , where M is cyclic andN ⊂ Sel(T, n)triv-at-`. Using this we can show that λ factors(uniquely) through

∧rSel(T, n) λ //

((

H1ur(Q`, T )⊗ ∧r−1Sel(T, n)

H1ur(Q`, T )⊗ ∧r−1Sel(T, n)triv-at-`

44


Results, for general core rank

Let KSr(T ) = {Kolyvagin systems of rank r for T}.

Theorem 1′

Suppose the core rank of T is r ≥ 1, and κ ∈ KSr(T ). Then forevery n ∈ N ,

κn ∈ |Sel(T ∗, n)| · ∧rSel(T, n).

Corollary

If κ ∈ KSr(T ), then |Sel(T ∗)| ≤ pd, where

d = max{t : κ1 ∈ pt ∧r H1(Q, T )}.

As when r = 1, if κ1 6= 0 we can recover the full group structure ofSel(T ∗) from κ.



RemarkRecall that if the core rank of T is r, then

Sel(T, n) ∼= (Z/pkZ)r ⊕ Sel(T ∗, n)

so |Sel(T ∗, n)| · ∧rSel(T, n) is a canonical cyclic subgroup of∧rSel(T, n).

Theorem 2′

Suppose the core rank of T is r. Then KSr(T ) is free of rank oneover Z/pkZ.

If κ generates KSr(T ), then κn generates |Sel(T ∗, n)| · ∧rSel(T, n)for every n.



Idea of proof: we want show that

κn ∈ |Sel(T ∗, n)| · ∧rSel(T, n). (*)

Let λ(n) be the p-rank of Sel(T ∗, n). We prove (*) by induction onλ(n).

If λ(n) = 0, then (*) is vacuous.

If λ(n) = t > 0, then we use Cebotarev to choose ` ∈ P suchthat λ(n`) = t− 1, and use our induction hypothesis and globalduality to prove (*).



We don’t know how to prove the analogue of Theorem 3 (Kolyvagin’sderivative construction) when r > 1.

Conjecture 3′

There is a canonical map

ESr(T ) → KSr(T )

{ξn} 7→ {κn}

with κn related to ξn in an explicitly specified way. In particular

κ1 = ξ1 ∈ ∧rH1(Q, T ).



We know that Kolyvagin systems exist in cases with core rankgreater than 1, but we don’t know any nontrivial examplesexplicitly.

We expect that Euler systems exist in cases with core rankgreater than 1, but we don’t know any nontrivial examples.

Perrin-Riou’s conjecture predicts “explicit” Euler systems in greatgenerality. But even if we assume that conjecture, with no analogueof Theorem 3 we have no explicit Kolyvagin systems when the corerank is greater than 1.


Example: Stark units

Take T = µpk ⊗ χ, and let r be the core rank.

Stark units in abelian extensions of K give an Euler systemξStark−Euler of rank r for T .

Stark S-units of K, as the set S of primes varies, can be used toconstruct a Kolyvagin system κStark−Kolyvagin of rank r for T .

We expect that the missing map ESr → KSr of the Conjecturesends ξStark−Euler 7→ κStark−Kolyvagin. This is equivalent to aspecific concrete relation between the Stark units “upstairs” andthe Stark units “downstairs”. When K = Q and χ is an evenquadratic character (so in particular r = 1), this relation wasconjectured by Darmon as a “refined class number formula”.

When r = 1 Conjecture 3′ is proved (Theorem 3), so in this casewe can prove this conjectured relation (including Darmon’soriginal conjecture).



F/K: abelian extension of totally realnumber fields,

r := [K : Q],

f := conductor of F/K,

K(n) := ray class field of K mod n,

F (n) := FK(n),

Γ := Gal(F/K),

G := Gal(F (n)/K),

H := Gal(F (n)/F ).

F (n)

F

H

K

Γ

G

Q

r



For every n ∈ N , an integral version of Stark’s conjecture predictsthe existence of

εF (n)/K ∈ ∧rO×F (n),fn, εF/K,n ∈ ∧r+ω(n)O×F,fn

where ω(n) is the number of prime divisors of n. These are”L-function evaluators”, in the sense that there are natural regulatormaps Rχ such that

Rχ(εF (n)/K) = lims→0

Lfn(F (n)/K, ψ, s)

srfor all χ ∈ Hom(G,C×),

Rχ(εF/K,n) = lims→0

Lfn(F/K, χ, s)

sr+ω(n)for all χ ∈ Hom(Γ,C×).



Let IH denote the augmentation ideal of Z[H].

Conjecture

1

∑h∈H

εhF (n)/K ⊗ h ∈ IrH(∧rF (n)× ⊗Z Z[H]),

2

∑h∈H

εhF (n)/K ⊗ h ≡ R(εK,n) (mod Ir+1H (∧rF (n)× ⊗Z Z[H])).

Here R is an algebraic regulator map ∧r+ω(n)O×F,fn → ∧rO×F,fn ⊗ IrH .

This conjecture implies (and is more or less equivalent to) theassertion that ξStark−Euler maps to κStark−Kolyvagin under thecorrespondence of Conjecture 3′.


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