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 . 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 1

Modularity n ≥ 1 a n n − s . Write L ( E/ Q , s ) = � Consider f ( τ ) = sum n a n e 2 πinτ , quadτ ∈ cH. Theorem The function f is a modular form of weight two on Γ 0 ( N ), where N is the conduc- tor of E . Modular parametrisation attached to E : Φ : H / Γ 0 ( N ) − → E ( C ) . Φ ∗ ( ω ) = 2 πif ( τ ) dτ � τ ∞ a n n e 2 πinτ . � log E (Φ( τ )) = i ∞ 2 πif ( z ) dz = n =1 2

CM points K = Q ( √− D ) subset C 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 well- behaved collection of algebraic points on E de- fined over class fields of K . 3

Heegner points Let D be a negative discriminant. Heegner hypothesis : D ≡ s 2 (mod N ) . F ( N ) = { Ax 2 + Bxy + Cy 2 such that B 2 − 4 AC = D D, N | A, B ≡ s (mod N ) } Gaussian Composition: Γ 0 ( N ) \F ( N ) = SL 2 ( Z ) \F D = G D D is an abelian group under composition, and is identified with the class group of the order of discrimiannt D . Given F ∈ F ( N ) , the point D P F := Φ( tau ) , where F ( τ, 1) = 0 , is called the Heegner point (of discriminant D ) attached to F . 4

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 ( N ) = { F 1 , . . . , F h } . D 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 5

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

Kolyvagin’s theorem Theorem (Kolyvagin) If P K is of infinite order, then E ( K ) has rank one and L I ( E/K ) is finite. (Hence, BSD holds L 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. 6

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 ord s =1 L ( E/F, s ) ≤ 1, then BSD holds for E/F . 7

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 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 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 uniformis- able, 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. 8

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. 9

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 parametrisa- tion” ? Φ : H D p / Γ 0 ( M ) − → E ( H D ) , for positive discriminants D . 10

Modular symbols Set ω f := Re (2 πif ( z ) dz ). Fact : There exists a real period Ω such that � s I f { r → s } := 1 r ω f mboxbeongsto Z , Ω 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 → ∞} . 11

Systems of measures Let � � a b Γ = { ∈ SL 2 ( Z ) such that M | c } . c d 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 . gamma ∗ µ { γr → γs } = µ { r → s } , for all 2. γ ∈ Γ. 3. µ { r → s } + µ { s → t } = µ { r → t } . 12

Rigid analytic functions � f { r → s } ( z ) := t ) z − t. ¶ 1 ( Q p ) dµ { r → s ( 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 } . 13

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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 . 14

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. 15

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 . 16

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 . 17

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) 18

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). 19

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). 20