The Birch and Swinnerton-Dyer Conjecture · The Conjecture BSD Conjecture (part 1) Conjecture...

MotivationElliptic curves

L-functions of Elliptic CurvesProgress and Application

The Birch and Swinnerton-Dyer Conjecture

Sunil Chetty

Department of Mathematics

October 9, 2014

Sunil Chetty BSD



CurvesCommon Strategies

Definition

There are two objects which could be used to define a curve:

1. A geometric set of (e.g. rational) points

C(Q) =

(a, b) ∈ Q2 : f (a, b) = 0.

2. An algebraic equation in two variables, i.e. C : f (x, y) = 0,for some polynomial f (x, y) with coefficients in Q.

Sunil Chetty BSD




General Results

For a general curve C, sometimes C(Q) is easy to describequalitatively.

I (1890) Hilbert and Hurwitz show that for linear andquadratic curves C, the set C(Q) is non-empty if and only ifit is “everywhere locally non-empty.”

I (1983) Faltings proves that for curves of sufficiently highdegree, C(Q) must be finite.

Note: The invariant that technically matters is ‘genus.’

Sunil Chetty BSD




Quadratic curves: Generating solutions

Consider the quadratic curve C : x2 + y2 = 1.

Geometry helps to see C(Q) is infinite.

Sunil Chetty BSD




Quadratic curves: general form

For Q : ax2 + bxy + cy2 + dx + ey + f = 0 there is a generalprocedure:

I Decide if some point (x, y) ∈ Q(Q) exists.I If not, Q(Q) = ∅.I If (x0, y0) ∈ Q(Q) then Q(Q) is infinite, in the same way as

with C above.

QuestionHow does one decide if any (x, y) ∈ Q(Q) exist?

Sunil Chetty BSD




Reduction

Consider C : x2 + y2 = 3. What can one say about C(Q)?

I If C(Q) 6= ∅ then D(Z) 6= ∅ for D : x2 + y2 = 3z2.

I If (a, b, c) ∈ D(Z) then by scaling we may assumegcd(a, b, c) = 1, giving a, b 6≡ 0 (mod 3).

I Now, a2 ≡ b2 ≡ 1 (mod 3) and so a2 + b2 ≡ 2 (mod 3).

QuestionCan reduction (mod p) also help show many solutions exist?

Sunil Chetty BSD



DefinitionsStructure on PointsTorsionReduction modulo p

Cubic curves: singular

E : y2 = x3 + ax + b, discriminant ∆ = 4a3 + 27b2.

Singular cases: ∆ = 0, x3 + ax + b has repeated roots.E(Q) is determined as quadratics were.

Sunil Chetty BSD




Cubic curves: non-singular

E : y2 = x3 + ax + b, discriminant ∆ = 4a3 + 27b2.

Non-singular cases: ∆ 6= 0, x3 + ax + b has no repeated roots.E(Q) is more interesting.

Sunil Chetty BSD




Elliptic curves: more generally

Alternatively, elliptic curves can be defined asI (Non-singular) solution set to the equation

y2 + a1xy + a3y = x3 + a2x2 + a4x + a6.

I Compact genus 1 Riemann surface over C.

I Smooth (projective) genus 1 curve with a defined point O.

I One-dimensional abelian variety.

Sunil Chetty BSD




Structure: geometrically

The set of points E(Q) admits a commutative addition operation

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Structure: algebraically

Addition formula: Let P = (x1, y1), Q = (x2, y2). Then

P + Q = (α2 − x1 − x2, αx3 + β)

where y = αx + β is the line connecting P and Q.

Theorem (Mordell 1922)The group E(Q) is finitely generated. Thus, E(Q) decomposes

E(Q) ∼= Etors(Q)× Zr(E,Q).

The quantity r(E,Q) is known as the (algebraic) rank of E(Q).

Sunil Chetty BSD




Mordell’s proof

The main steps:1. Show that E(Q)/2E(Q) is finite.

I Let Q1, . . . ,Qn represent E(Q)/2E(Q).I Every P := P0 ∈ E(Q) is related to some 2P1 ∈ E(Q), ...

Pi − Qi+1 = 2Pi+1.

2. R ∈ E(Q) : ht(R) ≤ c is finite for any c > 0.I height ht(P) measures complexity: ht

( ab

)=max(|a|, |b|).

I ht(2P) ≥ 4ht(P)− (some uniform constant)I Sequence P = P0,P1, . . . ,Pn is decreasing in height,

eventually lands in a finite set.

Sunil Chetty BSD




Torsion: Classification

From Mordell’s Theorem, one piece of E(Q) is the set of torsionpoints

Etors(Q) = P ∈ E(Q) : nP = O for some n > 0 .

Theorem (Mazur 1977)Etors(Q) must be (isomorphic to) one of the following:

Z/NZ 1 ≤ N ≤ 10 or N = 12Z/NZ× Z/(2N)Z 1 ≤ N ≤ 4

Moreover, all of these possibilities do occur.

Sunil Chetty BSD




Torsion: Computing points

Theorem (Nagell 1935, Lutz 1937)Recall E : y2 = x3 + ax + b, a, b ∈ Z.If P ∈ E(Q)tors then

1. P ∈ E(Z)

2. y(P) = 0 and P ∈ E(Q)[2], or y(P) | (4a3 + 27b2).

Theorem (Siegel 1928)The set E(Z) is finite.

Sunil Chetty BSD




Reduction modulo p

For a prime number p, let Fp denote Z/pZ (a finite field).

If E : y2 = x3 + ax + b has a, b ∈ Z then one can reducea, b (mod p), and define a new curve E.

The set E(Fp) still admits an addition law:I The geometric picture is no longer meaningful.I The algebraic formulas for addition still hold.I E(Fp) is automatically finite.

QuestionHow many points can E(Fp) have?

Sunil Chetty BSD




Number of points: Hasse bound

There are p choices for x(P) in Fp:I x(P) = 0 yields (0, 0) ∈ E(Fp)

I x(P) 6= 0 and not a square in E(Fp) yields no points.I x(P) 6= 0 and a square in E(Fp) yields two points.

One expects half of the values in Fp to be squares, hence p + 1points in E(Fp).

Theorem (Hasse 1933)|#E(Fp)− (p + 1)| < 2

√p.

Sunil Chetty BSD




EDSAC

Sunil Chetty BSD




Birch and Swinnerton-Dyer

Let Np denote #E(Fp).

I By Hasse’s Theorem, Np ∼ p as p→∞.I Birch and Swinnerton-Dyer computed

∏p≤M

Npp .

Sunil Chetty BSD




Computational Evidence

Idea: If #E(Q) is infinite then Np > p for many p.

Sunil Chetty BSD



BasicsStructureThe Conjecture

Local factors for E

Given E, for every prime p, define ap = p + 1− Np.

DefinitionThe local factor at p is defined to be

Lp(T) =

1− apT + pT2 if E has good red at p1− T if E has split mult red at p1 + T if E has non-split mult red at p1 if E has additive red at p

I Lp(1/p) = Np/p.I The conditions are determined by the structure of E(Fp).

Sunil Chetty BSD




Hasse-Weil L-function

DefinitionThe L-function of the elliptic curve E is

L(E, s) =∏p≥2

1Lp(p−s)

,

where s is a complex variable.

I L(E, 1) =∏

p (Lp(1/p))−1 =∏

pp

Np.

I This should be seen as the elliptic curve analog of theRiemann ζ-function.

I Hasse’s Theorem implies that L(E, s) converges (and isanalytic) for Re(s) > 3/2.

Sunil Chetty BSD




Hasse-Weil L-function

Two nice properties:I Continuation: L(E, s) has an analytic continuation to the

entire complex plane.

I Functional Equation: Λ(E, s) = w · Λ(E, 2− s) with w = ±1and

Λ(E, s) = (NE)s/2 (2π)−sΓ(s)L(E, s).

Note: The functional equation again indicates that s = 1 is asignificant value for L(E, s).

Sunil Chetty BSD




Series representation

Suppose again we are given an elliptic curve E.Define a sequence (ai) : a1 = 1. For each p ≥ 2, let

ap =

p + 1− Np if E has good red at p1 if E has split mult red at p−1 if E has non-split mult red at p0 if E has additive red at p

Define apr recursively by

apr+1 = ap · apr − p · apr−1 ,

for good reduction and apr = (ap)r for bad reduction. Lastly, ifgcd(m, n) = 1 then define amn = am · an.

Sunil Chetty BSD




Series representation

PropositionGiven an elliptic curve and (ai) defined as above

L(E, s) =∑n≥1

an

ns .

RemarkThe recurrence relation satisfied by an is very similar torecurrence relations satisfied by Hecke operators andeigenvalues of eigenforms. These similarities are the first clueto a connection with the theory of modular forms.

Sunil Chetty BSD




BSD Conjecture (part 1)

Conjecture (Birch and Swinnerton-Dyer)Suppose E is an elliptic curve defined over Q and L(E, s) is theL-function associated to E. Then L(E, s) has a zero at s = 1 oforder exactly equal to the rank r(E,Q) of E(Q).

In other words, the Taylor expansion of L(E, s) about s = 1 is

L(E, s) = C0(s− 1)r(E,Q) + C1(s− 1)r(E,Q)+1 + · · · ,

with C0 6= 0.

Sunil Chetty BSD




BSD Conjecture (part 2)

Conjecture (Birch and Swinnerton-Dyer)With the notation as above and r = r(E,Q),

C0 = lims→1

L(E, s)(s− 1)r =

#X · ΩE · Reg(E/Q) ·∏

p cp

(#Etors(Q))2 .

NoteI The symbol X refers to the Tate-Shafarevic group

associated to E.I The finiteness of X is another prominent conjecture in the

theory of elliptic curves.

Sunil Chetty BSD



ProgressBSD and Triangles

L-functions and Modular forms

Modularity Theorem (Wiles, et al 2001)All elliptic curves E/Q come from some modular form.

I The L-function of L(E, s) of E can be viewed instead as anL-function L(f , s) of a modular form f .

I L-functions of modular forms were already known to haveanalytic continuation to the entire complex plane andsatisfy a functional equation.

Sunil Chetty BSD




Progress on BSD

TheoremIf L(E, s) has a zero of order ≤ 1 then part 1 of the BSDConjecture is true.

(1977) Coates and Wiles prove that for CM elliptic curves,L(E, 1) 6= 0 implies E(Q) finite.

(1986) Gross and Zagier prove that if E/Q is modular and L(E, s)has a simple zero at s = 1 then E(Q) is infinite.

(1987) Rubin proves that for CM elliptic curves |X(E/Q)| <∞and if r ≥ 2 then the order of vanishing of L(E/Q, 1) is ≥ 2.

(1988) Kolyvagin proves that the theorems of Coates and Wilesand of Rubin hold if E is modular (weaker than being CM).

Sunil Chetty BSD




More recent progress

Theorem (Bhargava-Shankar-Skinner-Zhang 2014)Part 1 of the BSD Conjecture is true for a positive proportion (amajority even) of elliptic curves.

(2014) Bhargava, Skinner, and Zhang show that

lim infX→∞

# E/Q : BSD is true, height < X# E/Q : height < X

> 0.6648

(2013) Bhargava and Shankar show that the average rank of allelliptic curves is at most 0.885.

(2010) Dokchitser and Dokchitser show that the Parity Conjectureis true.

Sunil Chetty BSD




Congruent Numbers

Consider the set T of right triangles with all three sides ofrational length.

If n = 12 XY then n is called congruent.

Question

1. Which natural numbers n are congruent?2. Given n, how can one check if n is congruent?

Sunil Chetty BSD




Connections with elliptic curves

PropositionA number n is congruent iff En : y2 = x3 − n2x has a point(x, y) ∈ E(Q) with y 6= 0.

I From X2 + Y2 = Z2, 12 XY = n one has (X ± Y)2 = Z2 ± 4n.

I These multiplied together give

((X2 − Y2)/4)2 = (Z/2)4 − n2.

I By substitution, this is v2 = u4 − n2 and multiplying bothsides by u2, one has (uv)2 = u6 − n2u2.

I En(Q)tors consists of only 2-torsion.

Sunil Chetty BSD




Tunnell’s Theorem

Theorem (Tunnell 1983)If n is an odd, square-free number and is congruent then

2·#

x, y, z | n = 2x2 + y2 + 32z2 = #

x, y, z | n = 2x2 + y2 + 8z2 .Also, if part 1 of the BSD Conjecture is true then the conversealso holds.

I This gives an unconditional test to show n is not congruent.I In light of the above proposition, Tunnell’s brilliant

contribution is connecting the above equality with theL-function of En.

Sunil Chetty BSD




Good resources:I Lozano-Robeldo. Elliptic Curves, Modular Forms, and Their

L-functions. AMS, 2011.I Silverman J., Tate J. Rational Points on Elliptic Curves. Springer,

1992.Sunil Chetty BSD

The Birch and Swinnerton-Dyer Conjecture · The Conjecture BSD Conjecture (part 1) Conjecture...

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