Stark-Heegner points: a status report Invited talk Midwest Number - PDF document

Stark-Heegner points: a status report Invited talk Midwest Number Theory Conference University of Chicago, Chicago October 23-24 2004

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 )) . s ( x )N( x ) − s . ζ ( K, A , s ) = N( A ) s � x ∈A / ( O + K ) × H = Narrow Hilbert class field of K . ˜ v 1 : H − → C extending v 1 : K − → C . Conjecture (Stark) There exists u ( A ) ∈ O × H such that ζ ′ ( K, A , 0) . = log | ˜ v 1 ( u ( A )) | . u ( A ) is called a Stark unit attached to H/K . 1

Is there a stronger form? Stark Question: Is there an explicit analytic formula for ˜ v 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 ˜ v 1 is real, v 1 ( u ( A )) ? = ± exp( ζ ′ ( K, A , 0)) . ˜ If ˜ v 1 is complex, it is harder to recover ˜ v 1 ( u ( A )) from its absolute value. log(˜ v 1 ( u ( A ))) = log | ˜ v 1 ( u ( A )) | + iθ ( A ) ∈ C / 2 πi Z . Applications to Hilbert’s Twelfth problem ⇒ Explicit class field theory for K . The Stark Question has an analogue for el- liptic curves. 2

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 log E ( v ( P )) ∈ C / Ω? A point P for which such an explicit analytic recipe exists is called a Stark-Heegner point . 3

The prototype: Heegner Points Modular parametrisation attached to E : Φ : H / Γ 0 ( N ) − → E ( C ) . K = Q ( √− D ) ⊂ C a quadratic imaginary field . � τ ∞ a n n e 2 πinτ . � log E (Φ( τ )) = i ∞ 2 πif ( z ) dz = n =1 Theorem . If τ belongs to H ∩ K , then Φ( τ ) belongs to E ( K ab ). This theorem produces a systematic and well- behaved collection of algebraic points on E de- fined over class fields of K . 4

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) = ˆ h ( P D ) · (period) 5

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). 6

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 ) ? ⊂ E ( K ab ). is constructed, and Φ( H ∩ K ) 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). 7

Real quadratic fields E defined over Q , of conductor pM . K = real quadratic field in which p is non-split. ⇒ p -adic construction of points on E over ring class fields of K . Advantages of a p -adic context : 1. The setting is more basic. 2. More tools at our disposal: • Iwasawa Theory • p -adic uniformisation • Hida Theory • Overconvergent modular forms • Deformations of Galois representations... 8

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 parametrisa- tion” ? Φ : H D p / Γ 0 ( M ) − → E ( H D ) , for positive discriminants D . In defining Φ, I follow an approach suggested by Dasgupta’s thesis . 9

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 ∞ a n q n , � f ∞ = a n ∈ A ( U ) , n =1 such that ∀ k ≥ 2, k ∈ Z , k ≡ 2 (mod p − 1), ∞ a n ( k ) q n � f k := n =1 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 . 10

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 : � γ τ r ( z − τ ) k − 2 f † J † τ ( k ) := Ω − 1 k ( z ) dz. E r This does not depend on r . There exist λ k ∈ C × such that Proposition . λ 2 = 1 and k ( a p ( k ) 2 − 1) J † J τ ( k ) := λ − 1 τ ( k ) takes values in ¯ Q ⊂ C p and extends to a p -adic anaytic function of k ∈ U . 11

The definition of Φ Note: J τ (2) = 0. We define: log E Φ( τ ) := d dkJ 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 D p , then P D := Φ( τ ) belongs to E ( H D ). 2. (“Gross-Zagier”) L ′ ( E/K, O K , 1) = ˆ h ( P D ) · (period) 12

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. 13

Numerical examples E = X 0 (11) : y 2 + y = x 3 − x 2 − 10 x − 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 , 1566468063530870 w + 2088624084707825) + O (11 15 ) This point is close to [9 / 2 , 1 / 8(7 s − 4) , 1] (9 / 2 : 1 / 8(7 s − 4) : 1) is a global point on E(K). 14

A second example E = 37 A : 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). 15

Polynomial hD satisfied by the x-ccordinates: 4035 x 10 − 3868 x 9 + 19376 x 8 + 13229 x 7 961 x 11 − 27966 x 6 − 21675 x 5 + 11403 x 4 + 11859 x 3 − 1391 x 2 − 369 x − 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 16

A theoretical result χ : G D := Gal( H D /K ) − → ± 1 ζ ( K, χ, s ) = L ( s, χ 1 ) L ( s, χ 2 ) . χ ( σ )Φ( τ σ ) , τ ∈ H D � P ( χ ) := p . σ ∈ G 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 ˜ P ( χ ) , with ˜ P ( χ ) ∈ E ( H ( χ )). 2. The point ˜ P ( χ ) is of infinite order, if and only 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”). 17

Kronecker’s Solution of Pell’s Equation D = negative discriminant. Replace H D p / 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). η ∗ ( τ σ ) χ ( σ ) = ǫ 2 h 1 h 2 /w 2 , � σ ∈ G D where � h j = class number of Q ( D j ). ǫ = Fundamental unit of O × D 1 . 18

Kronecker’s Proof Three key ingredients: 1. Kronecker limit formula: ζ ′ ( K, χ, 0) = χ ( σ ) log η ∗ ( τ σ ) . � σ ∈ G D 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( ǫ ) , L (0 , χ D 2 ) = 2 h 2 /w 2 . Note : Complex multiplication is not used! 19