Heegner Points, Stark-Heegner points, and values of L -series - - PDF document

Heegner Points, Stark-Heegner points, and values of L-series

Number Theory Section Talk International Congress of Mathematicians August 2006, Madrid.

Elliptic Curves

E= elliptic curve over a number field F L(E/F, s) = its Hasse-Weil L-function. Birch and Swinnerton-Dyer Conjecture.

• rds=1 L(E/F, s) = rank(E(F)).

Theorem (Gross-Zagier, Kolyvagin) Suppose ords=1 L(E/Q, s) ≤ 1. Then the Birch and Swinnerton-Dyer conjecture is true. Key special case: if L(E/Q, 1) = 0 and L′(E/Q, 1) = 0, then E(Q) is infnite. Essential ingredient: Heegner points

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Modularity

Write L(E/Q, s) =

n≥1 ann−s.

Consider f(τ) = sumnane2πinτ, quadτ ∈ cH. Theorem The function f is a modular form of weight two on Γ0(N), where N is the conduc- tor of E. Modular parametrisation attached to E: Φ : H/Γ0(N) − → E(C). Φ∗(ω) = 2πif(τ)dτ logE(Φ(τ)) =

τ

i∞ 2πif(z)dz = ∞

• n=1

an n e2πinτ.

2

CM points

K = Q(√−D)subsetC a quadratic imaginary field.

• Theorem. If τ belongs to H ∩ K, then Φ(τ)

belongs to E(Kab). This theorem produces a systematic and well- behaved collection of algebraic points on E de- fined over class fields of K.

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Heegner points

Let D be a negative discriminant. Heegner hypothesis: D ≡ s2 (mod N). F(N)

D

= {Ax2+Bxy+Cy2 such thatB2−4AC = D, N|A, B ≡ s (mod N)} Gaussian Composition: Γ0(N)\F(N)

D

= SL2(Z)\FD = GD is an abelian group under composition, and is identified with the class group of the order of discrimiannt D. Given F ∈ F(N)

D

, the point PF := Φ(tau), where F(τ, 1) = 0, is called the Heegner point (of discriminant D) attached to F.

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Heegner points

Class field theory: rec : GD − → Gal(HD/K), where HD is the ring class field attached to D. Write Γ0(N)F(N)

D

= {F1, . . . , Fh}. Theorem The Heegner points PFj belong to E(HD) and PσF = rec(σ−1)PF. In particular, letting D = disc(K), PK := PF1 + · · · + PFh

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belongs to E(K). Theorem (Gross-Zagier) L′(E/K, OK, 1) = ˆ h(PK) · (period)

Kolyvagin’s theorem

Theorem (Kolyvagin) If PK is of infinite order, then E(K) has rank

• ne and L

L I(E/K) is finite. (Hence, BSD holds for E/K.) Main ingredient: PK does not come alone, but is part of a norm-compatible collection of points in E(Kab).

• Corollary. If ords=1 L(E, s) ≤ 1, then the Birch

and Swinnerton-Dyer conjecture holds for E. Sketch of Proof. Choose a quadratic field K satisfying the Heegner hypothesis, for which

• rds=1 L(E/K, s) = 1.

By Gross-Zagier, PK is of infinite order. By Kolyvagin, the BSD conjecture holds for E/K. BSD for E/Q follows.

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Totally real fields

Question: Does the above scheme generalise to other number fields? Suppose E is defined over a totally real field F. Definition: E is arithmetically uniformisable if [F : Q] is odd or if N is not a square. If E is modular, and arithmetically uniformis- able, there is a Shimura curve parametrisation Φ : Jac(X) − → E defined over F. Also, X is equipped with a collection of CM points attached to orders in CM extensions of F. Theorem (Zhang, Kolyvagin). Suppose that E is modular and arithmetically unifomisable. If ords=1 L(E/F, s) ≤ 1, then BSD holds for E/F.

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Non arithmetically uniformisable curves

Theorem (Longo, Tian). Suppose that E is modular. If ords=1 L(E/F, s) = 0, then BSD holds for E/F. Sketch of proof: Let f be the modular form on

GL2(F) attached to E. One can produce mod-

ular forms that are congruent to f, and cor- respond to quotients of Shimura curves. For each n ≥ 1, there is a Shimura curve Xn for which Jn[pn] has E[pn] as a constitutent. Key formula: Relate Heegner points attached to K, on Xn, to L(EK, 1) modulo pn.

• Question. If E is not arithmetically uniformis-

able, and ords=1 L(E/F, s) = 1, show that rank(E(F)) = 1? E.g. If E has everywhere good reduction over a real quadratic field.

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Stark-Heegner points

Wish: There should be generalisations of Heeg- ner points making it possible to a) prove BSD for elliptic curves in analytic rank ≤ 1, for more general E/F; b) Construct class fields of K; Paradox: Sometimes we can write down pre- cise formulae for points whose existence is not proved. General setting: E defined over a number field F; K = auxiliary quadratic extension of F; I will present three contexts.

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• 1. F = Q, K = real quadratic field;
• 2. F = totally real field, K = ATR extension

(“Almost Totally Real”). (Logan)

• 3. F = imaginary quadratic field. (Trifkovic)

Real quadratic fields

Set-up: E has conductor N = pM, with p |M. Hp := Cp − Qp (A p-adic analogue of H) K = real quadratic field, embedded both in R and Cp. Naive motivation for Hp: H∩K = ∅, but Hp∩K need not be empty! Goal: Define a p-adic “modular parametrisa- tion” Φ : HD

p /Γ0(M) ?

− → E(HD), for positive discriminants D.

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Modular symbols

Set ωf := Re(2πif(z)dz). Fact: There exists a real period Ω such that If{r → s} := 1 Ω

s

r ωfmboxbeongstoZ,

for all r, s ∈ P1(Q). Mazur-Swinnerton-Dyer measure: There is a measure on Zp defined by µf(a + pnZp) = If{a/pn → ∞}.

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Systems of measures

Let Γ = {

• a

b c d

• ∈ SL2(Z) such that M|c}.

Proposition There exists a unique collection

• f measures µ{r → s} on P1(Qp) satisfying
• 1. µ{r → s}|Zp = µf.

2. gamma∗µ{γr → γs} = µ{r → s}, for all γ ∈ Γ.

• 3. µ{r → s} + µ{s → t} = µ{r → t}.

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Rigid analytic functions

f{r → s}(z) :=

• ¶1(Qp)dµ{r→s

(

t) z − t.

Properties :

• 1. f{γr → γs}(γz) = (cz + d)2f{r → s}(z), for

all γ ∈ Γ.

• 2. f{r → s} + f{s → t} = f{r → t}.

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Stark’s conjecture

K= number field. v1, v2, . . . , vn = Archimedean place of K. Assume: v2, . . . , vn real. s(x) = sign(v2(x)) · · · sign(vn(x)). ζ(K, A, s) = N(A)s

• x∈A/(O+

K)×

s(x)N(x)−s. H = Narrow Hilbert class field of K. ˜ v1 : H − → C extending v1 : K − → C. Conjecture (Stark) There exists u(A) ∈ O×

H

such that ζ′(K, A, 0) . = log |˜ v1(u(A))|. u(A) is called a Stark unit attached to H/K.

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Is there a stronger form?

Stark Question: Is there an explicit analytic formula for ˜ v1(u(A)), and not just its absolute value? Some evidence that the answer is “Yes”: Sczech-

• Ren. (Also, ongoing work of Charollois-D.)

If ˜ v1 is real, ˜ v1(u(A)) ? = ±exp(ζ′(K, A, 0)). If ˜ v1 is complex, it is harder to recover ˜ v1(u(A)) from its absolute value. log(˜ v1(u(A))) = log |˜ v1(u(A))|+iθ(A) ∈ C/2πiZ. Applications to Hilbert’s Twelfth problem ⇒ Explicit class field theory for K. The Stark Question has an analogue for el- liptic curves.

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Elliptic Curves

E= elliptic curve over K L(E/K, s) = its Hasse-Weil L-function. Birch and Swinnerton-Dyer Conjecture. If L(E/K, 1) = 0, then there exists P ∈ E(K) such that L′(E/K, 1) = ˆ h(P) · ( explicit period). Stark-Heegner Question: Fix v : K − → C. Ω = Period lattice attached to v(E). Is there an explicit analytic formula for P, or rather, for logE(v(P)) ∈ C/Ω? A point P for which such an explicit analytic recipe exists is called a Stark-Heegner point.

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The prototype: Heegner Points

Modular parametrisation attached to E: Φ : H/Γ0(N) − → E(C). K = Q(√−D) ⊂ C a quadratic imaginary field. logE(Φ(τ)) =

τ

i∞ 2πif(z)dz = ∞

• n=1

an n e2πinτ.

• Theorem. If τ belongs to H ∩ K, then Φ(τ)

belongs to E(Kab). This theorem produces a systematic and well- behaved collection of algebraic points on E de- fined over class fields of K.

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Heegner points

Given τ ∈ H ∩ K, let Fτ(x, y) = Ax2 + Bxy + Cy2 be the primitive binary quadratic form with Fτ(τ, 1) = 0, N|A. Define Disc(τ) := Disc(Fτ). HD := {τ s.t. Disc(τ) = D.}. HD = ring class field of K attached to D. Theorem 1. If τ belongs to HD, then PD := Φ(τ) belongs to E(HD).

• 2. (Gross-Zagier)

L′(E/K, OK, 1) = ˆ h(PD) · (period)

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The Stark-Heegner conjecture

General setting: E defined over F; K = auxiliary quadratic extension of F; The Stark-Heegner points belong (conjecturally) to ring class fields of K. So far, three contexts have been explored:

• 1. F = totally real field, K = ATR extension

(“Almost Totally Real”).

• 2. F = Q, K = real quadratic field
• 3. F = imaginary quadratic field.

(Trifkovic, Balasubramaniam, in progress).

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ATR extensions

E of conductor 1 over a totally real field F, ωE = associated Hilbert modular form on (H1 × · · · × Hn)/SL2(OF). K = quadratic ATR extension of F; (“Almost Totally Real”): v1 complex, v2, . . . , vn real. D-Logan: A “modular parametrisation” Φ : H/SL2(OF) − → E(C) is constructed, and Φ(H ∩ K)

?

⊂ E(Kab). Φ defined analytically from periods of ωE.

• Experimental evidence (Logan);
• Replacing ωE with a weight two Eisenstein

series yields a conjectural affirmative answer to the Stark Question for K (work in progress with Charollois).

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Real quadratic fields

Set-up: E has conductor N = pM, with p |M. Hp := Cp − Qp (A p-adic analogue of H) K = real quadratic field, embedded both in R and Cp. Motivation for Hp: H∩K = ∅, but Hp ∩K need not be empty! Goal: Define a p-adic “modular parametrisa- tion” Φ : HD

p /Γ0(M) ?

− → E(HD), for positive discriminants D. In defining Φ, I follow an approach suggested by Dasgupta’s thesis.

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Hida Theory

U = p-adic disc in Qp with 2 ∈ U; A(U) = ring of p-adic analytic functions on U.

• Hida. There exists a unique q-expansion

f∞ =

∞

• n=1

anqn, an ∈ A(U), such that ∀k ≥ 2, k ∈ Z, k ≡ 2 (mod p − 1), fk :=

∞

• n=1

an(k)qn is an eigenform of weight k on Γ0(N), and f2 = fE. For k > 2, fk arises from a newform of level M, which we denote by f†

k.

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Heegner points for real quadratic fields

• Definition. If τ ∈ Hp/Γ0(M), let γτ ∈ Γ0(M)

be a generator for StabΓ0(M)(τ). Choose r ∈ P1(Q), and consider the “Shimura period” attached to τ and f†

k:

J†

τ(k) := Ω−1 E

γτr

r

(z − τ)k−2f†

k(z)dz.

This does not depend on r. Proposition. There exist λk ∈ C× such that λ2 = 1 and Jτ(k) := λ−1

k (ap(k)2 − 1)J† τ(k)

takes values in ¯

Q ⊂ Cp and extends to a p-adic

anaytic function of k ∈ U.

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The definition of Φ

Note: Jτ(2) = 0. We define: logE Φ(τ) := d dkJτ(k)|k=2. There are more precise formulae giving Φ(τ) itself, and not just its formal group logarithm. Conjecture 1. If τ belongs to HD

p , then

PD := Φ(τ) belongs to E(HD).

• 2. (“Gross-Zagier”)

L′(E/K, OK, 1) = ˆ h(PD) · (period)

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Computational Issues

The definition of Φ is well-suited to numerical

• calculations. (Green (2000), Pollack (2004)).

Magma package shp: software for calculat- ing Stark-Heegner points on elliptic curves of prime conductor.

http://www.math.mcgill.ca/darmon/programs/shp/shp.html

• H. Darmon and R. Pollack. The efficient cal-

culation of Stark-Heegner points via overcon- vergent modular symbols. Israel Math Journal, submitted. The key new idea in this efficient algorithm is the theory of overconvergent modular symbols developped by Stevens and Pollack.

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Numerical examples

E = X0(11) : y2 + y = x3 − x2 − 10x − 20.

> HP,P,hD := stark heegner points(E,8,Qp); The discriminant D = 8 has class number 1 Computing point attached to quadratic form (1,2,-1) Stark-Heegner point (over Cp) = (−2088624084707821, 1566468063530870w + 2088624084707825) + O(1115) This point is close to [9/2, 1/8(7s − 4), 1] (9/2 : 1/8(7s − 4) : 1) is a global point on E(K).

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A second example E = 37A : y2 + y = x3 − x, D = 1297.

> ,,hD := stark heegner points(E,1297,Qp); The discriminant D = 1297 has class number 11 1 Computing point for quadratic form (1,35,-18) 2 Computing point for quadratic form (-4,33,13) 3 Computing point for quadratic form (16,9,-19) 4 Computing point for quadratic form (-6,25,28) 5 Computing point for quadratic form (-8,23,24) 6 Computing point for quadratic form (2,35,-9) 7 Computing point for quadratic form (9,35,-2) 8 Computing point for quadratic form (12,31,-7) 9 Computing point for quadratic form (-3,31,28) 10 Computing point for quadratic form (12,25,-14) 11 Computing point for quadratic form (14,17,-18) Sum of the Stark-Heegner points (over Cp) = (0 : −1 : 1)) + (37100) This p-adic point is close to [0, −1, 1] (0 : −1 : 1) is indeed a global point on E(K).

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Polynomial hD satisfied by the x-ccordinates: 961x11 − 4035x10 − 3868x9 + 19376x8 + 13229x7 − 27966x6 − 21675x5 + 11403x4 + 11859x3 + 1391x2 − 369x − 37 > G := GaloisGroup(hD); Permutation group G acting on a set of cardinality 11 (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) (1, 10)(2, 9)(3, 8)(4, 7)(5, 6) > #G; 22

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A theoretical result

χ : GD := Gal(HD/K) − → ±1 ζ(K, χ, s) = L(s, χ1)L(s, χ2). P(χ) :=

• σ∈GD

χ(σ)Φ(τσ), τ ∈ HD

p .

H(χ) := extension of K cut out by χ. Theorem (Bertolini, D). If ap(E)χ1(p) = −sign(L(E, χ1, s)), then

• 1. logE P(χ) = logE ˜

P(χ), with ˜ P(χ) ∈ E(H(χ)).

• 2. The point ˜

P(χ) is of infinite order, if and

• nly if L′(E/K, χ, 1) = 0.

The proof rests on an idea of Kronecker (“Kro- necker’s solution of Pell’s equation in terms of the Dedekind eta-function”).

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Kronecker’s Solution of Pell’s Equation

D = negative discriminant. Replace HD

p /Γ0(N) by HD/SL2(Z).

Replace Φ by η∗(τ) := |D|−1/4

• Im(τ)|η(τ)|2.

χ = genus character of Q( √ D), associated to D = D1D2, D1 > 0, D2 < 0. Theorem (Kronecker, 1865).

• σ∈GD

η∗(τσ)χ(σ) = ǫ2h1h2/w2, where hj = class number of Q(

• Dj).

ǫ = Fundamental unit of O×

D1.

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Kronecker’s Proof

Three key ingredients:

• 1. Kronecker limit formula:

ζ′(K, χ, 0) =

• σ∈GD

χ(σ) log η∗(τσ).

• 2. Factorisation Formula:

ζ(K, χ, s) = L(s, χD1)L(s, χD2). In particular ζ′(K, χ, 0) = L′(0, χD1)L(0, χD2).

• 3. Dirichlet’s Formula.

L′(0, χD1) = h1 log(ǫ), L(0, χD2) = 2h2/w2. Note: Complex multiplication is not used!

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The Stark-Heegner setting

Assume χ = trivial character. PK = “trace” to K of PD.

• 1. A “Kronecker limit formula”

d2 dk2Lp(fk/K, k/2) = 1 4 logp(PK + ap(E) ¯ PK)2. If ap(E) = −sign(L(E/Q, s), then d2 dk2Lp(fk/K, k/2) = logp(PK)2.

• 2. Factorisation formula:

Lp(fk/K, k/2) = Lp(fk, k/2)Lp(fk, χD, k/2). Lp(fk, k/2) = specialisation to the critical line s = k/2 of Lp(fk, k, s) (Mazur’s two-variable p-adic L-function.)

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An analogue of Dirichlet’s Formula

Suppose ap = −sign(L(E/Q, s)) = 1. Theorem over Q (Bertolini, D) The function Lp(fk, k/2) vanishes to order ≥ 2 at k = 2, and there exists PQ ∈ E(Q) ⊗ Q such that 1.

d2 dk2Lp(fk, k/2) = − log2(PQ).

• 2. PQ is of infinite order iff L′(E/Q, 1) = 0.

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Proof of theorem over Q Introduce a suitable auxiliary imaginary quadratic field K. A “Kronecker limit formula” d2 dk2Lp(fk/K, k/2) = logp(PK)2, where PK is a Heegner point arising from a Shimura curve parametrisation. Key Ingredients: Cerednik-Drinfeld Theorem.

• M. Bertolini and H. Darmon, Heegner points,

p-adic L-functions and the Cerednik-Drinfeld uniformisation, Invent. Math. 131 (1998).

• M. Bertolini and H. Darmon, Hida families and

rational points on elliptic curves, in prepara- tion.

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End of Proof

We now use the factorisation formula L′′

p(fk/K, k/2) = L′′ p(fk, k/2)Lp(fk, χD, 1)

to conclude. The structure of the argument Heegner points + Cerednik-Drinfeld ⇒ Theorem for K imaginary quadratic ⇒ Theorem for Q ⇒ Theorem for K real quadratic. This argument seems to shed no light on the rationality of the Stark-Heegner point PD (un- less the class group has exponent two).

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