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. ord_{s=1} L(E/F, s) = rank (E(F)). Theorem (Gross-Zagier, Kolyvagin) Suppose ord_{s=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

Modularity

Write L(E/Q, s) = ∑_{n≥1} a_{n}n^{−s}. Consider

f(τ) = sum_{n}a_{n}e^{2πinτ} , quadτ ∈ cH. Theorem The function f is a modular form of weight two on Γ_{0}(N), where N is the conductor of E. Modular parametrisation attached to E:

Φ : H/Γ_{0}(N) ⟶ E(C).

Φ^{∗}(ω) = 2πif(τ)dτ

log_{E}(Φ(τ)) = ∫_{i∞}^{τ} 2πif(z)dz = ∞ ; ∑ ; n=1 \frac{a_{n}}{n}e^{2πinτ} .

CM points

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

Theorem. If τ belongs to H ∩ K, then Φ(τ) belongs to E(K^{ab}).

This theorem produces a systematic and wellbehaved collection of algebraic points on E defined over class fields of K.

Heegner points

Let D be a negative discriminant. Heegner hypothesis: D ≡ s^{2} (mod N). F_{D}^{(N)} = {Ax^{2}+Bxy+Cy^{2} such that B^{2}−4AC = D, N|A, B ≡ s (mod N)} Gaussian Composition: Γ_{0}(N)\F_{D}^{(N)} = SL_{2}(Z)\F_{D} = G_{D} is an abelian group under composition, and is identified with the class group of the order of discrimiannt D. Given F ∈ F_{D}^{(N)} , the point

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

Heegner points

Class field theory: rec : G_{D} ⟶ Gal(H_{D}/K), where H_{D} is the ring class field attached to D. Write Γ_{0}(N)F_{D}^{(N)} = {F_{1}, …, F_{h}}. Theorem The Heegner points P_{F_{j}} belong to E(H_{D}) and

P_{σF} = rec (σ^{−1})P_{F} .

In particular, letting D = disc(K), P_{K} := P_{F_{1}} + ⋯ + P_{F_{h}} belongs to E(K). Theorem (Gross-Zagier)

L′(E/K, O_{K}, 1) = ĥ(P_{K}) · (period)

Kolyvagin’s theorem

Theorem (Kolyvagin) If P_{K} is of infinite order, then E(K) has rank one and LLI(E/K) is finite. (Hence, BSD holds for E/K.) Main ingredient: P_{K} does not come alone, but is part of a norm-compatible collection of points in E(K^{ab}). Corollary. If ord_{s=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 ord_{s=1} L(E/K, s) = 1. By Gross-Zagier, P_{K} is of infinite order. By Kolyvagin, the BSD conjecture holds for E/K. BSD for E/Q follows.

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 uniformisable, 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 ord_{s=1} L(E/F, s) ≤ 1, then BSD holds for E/F.

Non arithmetically uniformisable curves

Theorem (Longo, Tian). Suppose that E is modular. If ord_{s=1} L(E/F, s) = 0, then BSD holds for E/F.

Sketch of proof: Let f be the modular form on GL_{2}(F) attached to E. One can produce modular forms that are congruent to f, and correspond to quotients of Shimura curves. For each n ≥ 1, there is a Shimura curve X_{n} for which J_{n}[p^{n}] has E[p^{n}] as a constitutent.

Key formula: Relate Heegner points attached to K, on X_{n}, to L(EK, 1) modulo p^{n}.

Question. If E is not arithmetically uniformisable, and ord_{s=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.

Stark-Heegner points

Wish: There should be generalisations of Heegner 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 precise 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.

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. H_{p} := C_{p} − Q_{p} (A p-adic analogue of H) K = real quadratic field, embedded both in R and C_{p}. Naive motivation for H_{p}: H∩K = ∅, but H_{p}∩K need not be empty! Goal: Define a p-adic “modular parametrisation”

Φ : H_{p}^{D}/Γ_{0}(M) ⟶^{?} E(H_{D}),

for positive discriminants D.

Modular symbols

Set ω_{f} := Re(2πif(z)dz). Fact: There exists a real period Ω such that

I_{f}\{r → s\} := \frac{1}{Ω} ∫_{r}^{s} ω_{f}mboxbeongstoZ,

for all r, s ∈ P_{1}(Q). Mazur-Swinnerton-Dyer measure: There is a measure on Z_{p} defined by

μ_{f}(a + p^{n}Z_{p}) = I_{f}{a/p^{n} → ∞}.

Systems of measures

Let Γ = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ SL_{2}(Z) such that M|c\}.

Proposition There exists a unique collection of measures μ{r → s} on P_{1}(Q_{p}) satisfying

1. μ{r → s}|_{Z_{p}} = μ_{f}.

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

3. μ{r → s} + μ{s → t} = μ{r → t}.

Rigid analytic functions

f\{r → s\}(z) := ∫_{¶_{1}(Q_{p})\frac{dμ\{r→s}{(}t)} z − t.

Properties :

1. f{γr → γs}(γz) = (cz + d)^{2}f{r → s}(z), for all γ ∈ Γ.

2. f{r → s} + f{s → t} = f{r → t}. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxa

Stark’s conjecture

K= number field. v_{1}, v_{2}, …, v_{n} = Archimedean place of K. Assume: v_{2}, …, v_{n} real.

s(x) = sign (v_{2}(x)) ⋯ sign (v_{n}(x)). ζ(K, A, s) = N(A)^{s} _{x∈A/(O_{K}^{+})^{×}}^{∑} s(x)N(x)^{−s}.

H = Narrow Hilbert class field of K. ṽ_{1} : H ⟶ C extending v_{1} : K ⟶ C. Conjecture (Stark) There exists u(A) ∈ O_{H}^{×} such that

ζ′(K, A, 0) =^{.} log |ṽ_{1}(u(A))|.

u(A) is called a Stark unit attached to H/K.

Is there a stronger form?

Stark Question: Is there an explicit analytic formula for ṽ_{1}(u(A)), and not just its absolute value?

Some evidence that the answer is “Yes”: Sczech-Ren. (Also, ongoing work of Charollois-D.)

If ṽ_{1} is real,

ṽ_{1}(u(A)) =^{?} ±exp (ζ′(K, A, 0)).

If ṽ_{1} is complex, it is harder to recover ṽ_{1}(u(A)) from its absolute value.

log (ṽ_{1}(u(A))) = log |ṽ_{1}(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 elliptic curves.

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) = ĥ(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

log_{E}(v(P)) ∈ C/Ω?

A point P for which such an explicit analytic recipe exists is called a Stark-Heegner point.

The prototype: Heegner Points

Modular parametrisation attached to E:

Φ : H/Γ_{0}(N) ⟶ E(C).

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

log_{E}(Φ(τ)) = ∫_{i∞}^{τ} 2πif(z)dz = ∞ ; ∑ ; n=1 \frac{a_{n}}{n}e^{2πinτ} .

Theorem. If τ belongs to H ∩ K, then Φ(τ) belongs to E(K^{ab}).

This theorem produces a systematic and wellbehaved collection of algebraic points on E defined over class fields of K.

Heegner points

Given τ ∈ H ∩ K, let

F_{τ}(x, y) = Ax^{2} + Bxy + Cy^{2}

be the primitive binary quadratic form with

F_{τ}(τ, 1) = 0, N|A.

Define Disc (τ) := Disc (F_{τ}).

H^{D} := {τ s.t. Disc (τ) = D.}.

H_{D} = ring class field of K attached to D. Theorem 1. If τ belongs to H^{D}, then P_{D} := Φ(τ) belongs to E(H_{D}).

2. (Gross-Zagier)

L′(E/K, O_{K}, 1) = ĥ(P_{D}) · (period)

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).

ATR extensions

E of conductor 1 over a totally real field F, ω_{E} = associated Hilbert modular form on (H_{1} × ⋯ × H_{n})/SL_{2}(O_{F}). K = quadratic ATR extension of F; (“Almost Totally Real”): v_{1} complex, v_{2}, …, v_{n} real. D-Logan: A “modular parametrisation”

Φ : H/SL_{2}(O_{F}) ⟶ E(C)

is constructed, and Φ(H ∩ K) ⊂^{?} E(K^{ab}). Φ defined analytically from periods of ω_{E}.

• Experimental evidence (Logan);

• Replacing ω_{E} with a weight two Eisenstein series yields a conjectural affirmative answer to the Stark Question for K (work in progress with Charollois).

Real quadratic fields

Set-up: E has conductor N = pM, with p |̸M. H_{p} := C_{p} − Q_{p} (A p-adic analogue of H) K = real quadratic field, embedded both in R and C_{p}. Motivation for H_{p}: H∩K = ∅, but H_{p} ∩K need not be empty! Goal: Define a p-adic “modular parametrisation”

Φ : H_{p}^{D}/Γ_{0}(M) ⟶^{?} E(H_{D}),

for positive discriminants D. In defining Φ, I follow an approach suggested by Dasgupta’s thesis.

Hida Theory

U = p-adic disc in Q_{p} with 2 ∈ U; A(U) = ring of p-adic analytic functions on U. Hida. There exists a unique q-expansion

f_{∞} = ∞ ; ∑ ; n=1 a_{n}q^{n}, a_{n} ∈ A(U),

such that ∀k ≥ 2, k ∈ Z, k ≡ 2 (mod p − 1),

f_{k} := ∞ ; ∑ ; n=1 a_{n}(k)q^{n}

is an eigenform of weight k on Γ_{0}(N), and

f_{2} = f_{E}.

For k > 2, f_{k} arises from a newform of level M, which we denote by f_{k}^{†}.

Heegner points for real quadratic fields

Definition. If τ ∈ H_{p}/Γ_{0}(M), let γ_{τ} ∈ Γ_{0}(M) be a generator for Stab _{Γ_{0}(M)} (τ).

Choose r ∈ P_{1}(Q), and consider the “Shimura period” attached to τ and f_{k}^{†}:

J_{τ}^{†}(k) := Ω_{E}^{−1} ∫_{r}^{γ_{τ}r}(z − τ)^{k−2}f_{k}^{†}(z)dz.

This does not depend on r.

Proposition. There exist λ_{k} ∈ C^{×} such that λ_{2} = 1 and

J_{τ}(k) := λ_{k}^{−1}(a_{p}(k)^{2} − 1)J_{τ}^{†}(k)

takes values in Q̄ ⊂ C_{p} and extends to a p-adic anaytic function of k ∈ U.

The definition of Φ

Note: J_{τ}(2) = 0. We define:

log_{E} Φ(τ) := \frac{d}{dk}J_{τ}(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 H_{p}^{D}, then P_{D} := Φ(τ) belongs to E(H_{D}).

2. (“Gross-Zagier”)

L′(E/K, O_{K}, 1) = ĥ(P_{D}) · (period)

Computational Issues

The definition of Φ is well-suited to numerical calculations. (Green (2000), Pollack (2004)).

Magma package shp: software for calculating 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 calculation of Stark-Heegner points via overconvergent 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.

Numerical examples

E = X_{0}(11) : y^{2} + y = x^{3} − x^{2} − 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(11^{15} )

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).

A second example

E = 37A : y^{2} + y = x^{3} − 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)) + (37^{100} ) This p-adic point is close to [0, −1, 1] (0 : −1 : 1) is indeed a global point on E(K).

Polynomial hD satisfied by the x-ccordinates:

961x^{11} − 4035x^{10} − 3868x^{9} + 19376x^{8} + 13229x^{7}
− 27966x^{6} − 21675x^{5} + 11403x^{4} + 11859x^{3}
+ 1391x^{2} − 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

A theoretical result

χ : G_{D} := Gal(H_{D}/K) ⟶ ±1 ζ(K, χ, s) = L(s, χ_{1})L(s, χ_{2}).

P (χ) := _{σ∈G_{D}}^{∑} χ(σ)Φ(τ^{σ}), τ ∈ H_{p}^{D}.

H(χ) := extension of K cut out by χ. Theorem (Bertolini, D). If a_{p}(E)χ_{1}(p) = −sign (L(E, χ_{1}, s)), then

1. log_{E} P (χ) = log_{E} P̃ (χ), with P̃ (χ) ∈ E(H(χ)).

2. The point P̃ (χ) is of infinite order, if and only if L′(E/K, χ, 1) ≠ 0. The proof rests on an idea of Kronecker (“Kronecker’s solution of Pell’s equation in terms of the Dedekind eta-function”).

Kronecker’s Solution of Pell’s

Equation

D = negative discriminant. Replace H_{p}^{D}/Γ_{0}(N) by H^{D}/SL_{2}(Z). Replace Φ by

η^{∗}(τ) := |D|^{−1/4} √Im(τ)|η(τ)|^{2}.

χ = genus character of Q(√D), associated to

D = D_{1}D_{2}, D_{1} > 0, D_{2} < 0. Theorem (Kronecker, 1865).

_{σ∈G_{D}}^{∏} η^{∗}(τ^{σ})^{χ(σ)} = ϵ^{2h_{1}h_{2}/w_{2}} ,

where h_{j} = class number of Q(√D_{j}). ϵ = Fundamental unit of O_{D_{1}}^{×}.

Kronecker’s Proof

Three key ingredients:

1. Kronecker limit formula:

ζ′(K, χ, 0) = _{σ∈G_{D}}^{∑} χ(σ) log η^{∗}(τ^{σ}).

2. Factorisation Formula:

ζ(K, χ, s) = L(s, χ_{D_{1}})L(s, χ_{D_{2}}).

In particular

ζ′(K, χ, 0) = L′(0, χ_{D_{1}})L(0, χ_{D_{2}}).

3. Dirichlet’s Formula. L′(0, χ_{D_{1}}) = h_{1} log (ϵ), L(0, χ_{D_{2}}) = 2h_{2}/w_{2}. Note: Complex multiplication is not used!

The Stark-Heegner setting

Assume χ = trivial character. P_{K} = “trace” to K of P_{D}.

1. A “Kronecker limit formula”

\frac{d^{2}}{dk^{2}}L_{p}(f_{k}/K, k/2) = \frac{1}{4} log_{p}(P_{K} + a_{p}(E)P̄_{K})^{2}.

If a_{p}(E) = −sign (L(E/Q, s), then

\frac{d^{2}}{dk^{2}}L_{p}(f_{k}/K, k/2) = log_{p}(P_{K})^{2}.

2. Factorisation formula:

L_{p}(f_{k}/K, k/2) = L_{p}(f_{k}, k/2)L_{p}(f_{k}, χ_{D}, k/2). L_{p}(f_{k}, k/2) = specialisation to the critical line s = k/2 of L_{p}(f_{k}, k, s) (Mazur’s two-variable p-adic L-function.)

An analogue of Dirichlet’s Formula

Suppose a_{p} = −sign (L(E/Q, s)) = 1. Theorem over Q (Bertolini, D) The function L_{p}(f_{k}, k/2) vanishes to order ≥ 2 at k = 2, and there exists P_{Q} ∈ E(Q) ⊗ Q such that

1. \frac{d^{2}}{dk^{2}}L_{p}(f_{k}, k/2) = − log^{2}(P_{Q}).

2. P_{Q} is of infinite order iff L′(E/Q, 1) ≠ 0.

Proof of theorem over Q

Introduce a suitable auxiliary imaginary quadratic field K.

A “Kronecker limit formula”

\frac{d^{2}}{dk^{2}}L_{p}(f_{k}/K, k/2) = log_{p}(P_{K})^{2},

where P_{K} 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 preparation.

End of Proof

We now use the factorisation formula

L′′_{p}(f_{k}/K, k/2) = L′′_{p}(f_{k}, k/2)L_{p}(f_{k}, χ_{D}, 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 P_{D} (unless the class group has exponent two).