Numerical Computation of StarkHegner points in higher level Xevi - PowerPoint PPT Presentation

Numerical Computation of Stark–Hegner points in higher level Xevi Guitart 1 Marc Masdeu 2 1 Max Planck Institute/U. Politècnica de Catalunya 2 Columbia University Rational points on curves: A p -adic and computational perspective Mathematical College, Oxford 2012 X. Guitart, M. Masdeu (UPC/MPIM, CU) Stark–Heegner points in higher level 2012 1 / 21

Stark–Heegner points in higher level E / Q elliptic curve of conductor N = pM , with p ∤ M . K / Q real quadratic field in which ◮ p is inert ◮ all primes dividing M are split Darmon’s construction of Stark–Heegner points P 1 ( K p ) \ P 1 ( Q p ) = H p − → E ( K p ) τ �− → P τ P τ is defined in terms of certain p -adic periods of f = f E ∈ S 2 ( N ) Conjecture (Darmon, 2001) P τ is a global point: P τ ∈ E ( H τ ) where H τ is a Ring Class Field of K Explicit computations and numerical evidence: ◮ Darmon–Green (2002): algorithm for computing P τ ◮ Darmon–Pollack (2006): more efficient calculations with OMS The algorithm needs to assume M = 1 ( P τ ’s only computed on curves of conductor p ) In this talk: remove the requirement M = 1 in this algorithm, so that P τ in curves of composite conductor can be computed. X. Guitart, M. Masdeu (UPC/MPIM, CU) Stark–Heegner points in higher level 2012 3 / 21

Integration on H p × H � τ 2 � y τ 1 , τ 2 ∈ H p , x , y ∈ P 1 ( Q ) Double integrals ω f ∈ K p , x τ 1 Definition � � ◮ Γ 0 ( M ) = γ ∈ SL 2 ( Z [ 1 ⊂ SL 2 ( Z [ 1 p ]): γ ≡ ( ⋆ ⋆ 0 ⋆ ) ( mod M ) p ]) � y ◮ � τ 2 � � t − τ 2 � x ω f := P 1 ( Q p ) log d µ f { x → y } ( t ) ∈ K p t − τ 1 τ 1 ◮ µ f { x → y } measure in P 1 ( Q p ) � γ − 1 y 1 µ f { x → y } ( γ Z p ) = γ − 1 x Re ( 2 π if ( z ) dz ) ∈ Z for γ ∈ Γ 0 ( M ) ⋆ Ω + Double multiplicative integral: � τ 2 � y � � t − τ 2 � d µ f { x → y } ( t ) ∈ K × × ω f := × p t − τ 1 x P 1 ( Q p ) τ 1 � y � y ◮ � τ 2 � � τ 2 � x ω f = log x ω f × τ 1 τ 1 Effective computation: ◮ They can be very efficiently computed (up to a prescribed p -adic precision) using overconvergent modular symbols X. Guitart, M. Masdeu (UPC/MPIM, CU) Stark–Heegner points in higher level 2012 5 / 21

Semi-indefinite integrals � τ � y ω f ∈ K × p / q Z , τ ∈ H p , x , y ∈ P 1 ( Q ) , x = γ y some γ ∈ Γ 0 ( M ) × x Definition ◮ Cohomology of measured valued modular symbols Properties � τ � y � τ � z � τ � z 1. × x ω f × × y ω f = × x ω f (multiplicative in the limits) � τ � y � γτ � γ y 2. × γ x ω f = × x ω f for all γ ∈ Γ 0 ( M ) (invariance under Γ 0 ( M ) ) � τ 2 � y � τ 1 � y � y � τ 2 3. × x ω f ÷ × x ω f = × x ω f (Relation with double integrals) τ 1 Definition of Stark–Heegner points � τ � γ τ ∞ � � P τ = Φ Tate × ω f , Stab Γ 0 ( M ) ( τ ) = � γ τ � ∞ Computing P τ boils down to compute semi-indefinite integrals ◮ Direct computation (using the very definition) seems to be difficult ◮ Darmon-Green-Pollack: use 1, 2 and 3 to transform semi-indefinite integrals into definite double integrals. ◮ This is the only stage where the assumption M = 1 is needed. ◮ We give a different method, that works with M > 1 X. Guitart, M. Masdeu (UPC/MPIM, CU) Stark–Heegner points in higher level 2012 6 / 21

Reduction to Γ 1 ( M ) � a b � 1 ⋆ � � Γ 1 ( M ) = { γ = ∈ SL 2 ( Z [ 1 / p ]): γ ≡ ( mod M ) } ⊂ Γ 0 ( M ) c d 0 1 γ τ ∈ Γ 0 ( M ) , but we can reduce to the case where γ τ ∈ Γ 1 ( M ) � τ � γ m � � τ ∞ ◮ If m = [Γ 0 ( M ): Γ 1 ( M )] , computing mP τ = Φ Tate × ∞ p − n 0 ◮ if a ≡ p n ( mod M ) we let α = � � and p n 0 � τ � γ τ ∞ � ατ � αγ τ ∞ P τ = × = × ∞ ∞ with αγ τ ∈ Γ 1 ( M ) X. Guitart, M. Masdeu (UPC/MPIM, CU) Stark–Heegner points in higher level 2012 8 / 21

� τ � γ ∞ We are reduced to compute × with γ ∈ Γ 1 ( M ) ∞ SL 2 ( Z [ 1 p ]) has the congruence subgroup property � � 1 x r � � � � � 1 0 1 x 2 1 0 � γ = · · · , γ ∈ Γ 1 ( M ) Mx r − 1 1 Mx 1 1 0 1 0 1 � τ � 0 � τ � 0 � E − 1 · τ � E − 1 � τ � γ ∞ � τ � γ ∞ γ ·∞ 1 1 ω f = × ω f = × × ω f × × ω f × × ω f 0 0 ∞ ∞ ∞ � τ � 0 � E − 1 � E − 1 · τ � E − 1 � E − 1 � E − 1 · τ � E − 1 · τ � ∞ � ∞ γ ·∞ · τ γ ·∞ 1 1 1 1 1 1 = × ω f = × ω f × × ω f × × ω f × × ω f 0 0 ∞ ∞ τ ∞ Small problem: 0 and ∞ are not Γ 0 ( M ) -equivalent if M > 1 � 0 − 1 � But W d · 0 = ∞ , W d = d 0 Assumption There exists d | M such that W d ( f ) = f ◮ For instance, if M has at least two factors this is always true Then semi-indefinite integrals are also defined on W d -equivalent cusps. X. Guitart, M. Masdeu (UPC/MPIM, CU) Stark–Heegner points in higher level 2012 9 / 21

The problem is reduced to finding an algorithm for computing � � 1 x r � � � � � 1 0 1 x 2 1 0 � γ = · · · , γ ∈ Γ 1 ( M ) . Mx r − 1 1 Mx 1 1 0 1 0 1 Remark: if M = 1 then the x i ’s are the quotients of the contineued � a b � fraction of a / c , if γ = . c d For M > 1 we need another algorithm. X. Guitart, M. Masdeu (UPC/MPIM, CU) Stark–Heegner points in higher level 2012 10 / 21

A more general setting F : number field with at least a real place S a set of places of F containing the archimedean ones O S ring of S -integers, M ⊂ O S an ideal � 1 ⋆ � Γ 1 ( M ) = { γ ∈ SL 2 ( O S ): γ ≡ ( mod M ) } 0 1 ( p -adic Stark–Heegner points F = Q , S = { p , ∞} , M = M · Z [ 1 p ] ) Theorem (Serre, Vaserstein): If O × S is infinite (i.e. if # S > 1) then Γ 1 ( M ) is generated by the matrices � 1 x � 1 0 � � with x ∈ O S , with x ∈ M , 0 1 x 1 � a b � Problem: given γ = ∈ Γ 1 ( M ) , write it as a product of c d elementary matrices Simple case: if c = u + ta with u ∈ O × S and t ∈ O S then � � 1 x � � � � � 1 0 1 − u − 1 1 0 � γ = . (1) c + t ( 1 − a ) 1 u ( 1 − a ) 1 0 1 0 1 � a b � 1 λ � a + λ c b + λ d � � � We can replace γ = by γ = c d 0 1 c d X. Guitart, M. Masdeu (UPC/MPIM, CU) Stark–Heegner points in higher level 2012 12 / 21

Effective Congruence Subgroup Problem Theorem (Cooke–Weinberger) Assume GRH. Then the set of prime ideals in O S of the form a + λ c → ( O S / ( a + λ c ) O S ) × is onto has positive density. such that O × S − Algorithm for elmentary matrix decomposition � a b � Given γ = ∈ Γ 1 ( M ) c d → ( O S / ( a + λ c ) O S ) × is onto Find λ ∈ O S such that O × S − 1 � 1 λ � a + λ c b + λ d Set γ ′ = � � γ = 2 0 1 c d Find u ∈ O × S representing the class of c modulo a + λ c 3 Compute the explicit decomposition (1) to γ ′ . 4 Corollary Assuming GRH, every matrix in Γ 1 ( M ) can be expressed as a product of 5 elementary matrices. X. Guitart, M. Masdeu (UPC/MPIM, CU) Stark–Heegner points in higher level 2012 13 / 21

Computing the double integrals � τ 2 � ∞ 0 ω f We need to compute integrals of the form × τ 1 � � t − τ 1 � d µ f ( t ) The hard part is P 1 ( Q p ) log t − τ 2 Darmon–Pollack: P 1 ( Q p ) = � L i = 1 g i Z p , g i ∈ GL 2 ( Q ) � � t − τ 1 � � � ( g − 1 t ) n d µ f ( t ) log d µ f ( t ) = · · · = α n i t − τ 2 g i Z p g i Z p n ≥ 1 g i Z p ( g − 1 t ) n d µ f ( t ) : the moments can be efficiently computed via ◮ � i overconvergent modular symbols ◮ Number of g i ’s depends on the affinoid H n p containing τ 1 , τ 2 H 0 p = { τ ∈ P 1 ( K p ): red ( τ ) / ∈ P 1 ( Z / p Z ) } ∈ P 1 ( Z / p n + 1 Z ) } \ H n − 1 H n p = { τ ∈ P 1 ( K p ): red ( τ ) / p ◮ We can take a covering of size ( p + 1 ) + n ( p − 1 ) ◮ This increases the running time with respect to the M = 1 case (when M = 1, then τ 1 , τ 2 ∈ H 0 p so p + 1 evaluations is enough), but it is not critical in the range of values we tested. X. Guitart, M. Masdeu (UPC/MPIM, CU) Stark–Heegner points in higher level 2012 15 / 21

Implementation We have written a SAGE implementation of the method ◮ we use Robert Pollack’s implementation in SAGE for computing the moments with overconvergent modular symbols ◮ we adapt part of the code written by Darmon and Pollack in Magma for the M = 1 case ◮ we added the routines for the elementary matrix decomposition, for transforming semi-indefinite into definite integrals, and for integrating over the appropriate open covers. ( x , y ) = Φ Tate ( J τ ) and we can actually recognize x , y ∈ K p as elements of the expected number field H τ (usually we choose τ so that H τ is the Hilbert class field of K ) X. Guitart, M. Masdeu (UPC/MPIM, CU) Stark–Heegner points in higher level 2012 16 / 21