Numerical Computation of Stark Hegner points in higher level

Transcription

1 Numerical Computation of Stark Hegner points in higher level Xevi Guitart Marc Masdeu 2 Max Planck Institute/U. Politècnica de Catalunya 2 Columbia University Rational points on curves: A p-adic and computational perspective Mathematical College, Oxford 202 X. Guitart, M. Masdeu UPC/MPIM, CU Stark Heegner points in higher level 202 / 2

2 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 K p \ P Q p = H p EK p τ P τ P τ is defined in terms of certain p-adic periods of f = f E S 2 N Conjecture Darmon, 200 P τ is a global point: P τ EH τ where H τ is a Ring Class Field of K Explicit computations and numerical evidence: Darmon Green 2002: algorithm for computing Pτ Darmon Pollack 2006: more efficient calculations with OMS The algorithm needs to assume M = P τ s only computed on curves of conductor p In this talk: remove the requirement M = 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 / 2

3 Integration on H p H Double integrals τ2 y τ ω f K p, τ, τ 2 H p, x, y P Q x Definition { } Γ0 M = γ SL 2 Z[ p ]: γ 0 mod M SL 2 Z[ p ] τ2 y τ x ω f := P Q log t τ2 p t τ dµ f {x y}t K p µ f {x y} measure in P Q p µ f {x y}γz p = γ y Re2πif zdz Z forγ Γ Ω + γ x 0M Double multiplicative integral: τ2 y t τ2 ω f := dµ τ x P Q p t τ f {x y}t K p τ2 y x ω τ2 y f = log x ω f τ τ 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 / 2

4 Semi-indefinite integrals τ y ω f K p /q Z, τ H p, x, y P Q, x = γy some γ Γ 0 M x Definition Cohomology of measured valued modular symbols Properties τ y. x ω τ z f y ω τ z f = x ω f multiplicative in the limits γτ γy 2. γx ω τ y f = x ω f for all γ Γ 0 M invariance under Γ 0 M τ2 y 3. x ω τ y f x ω τ2 y f = τ x ω f Relation with double integrals Definition of Stark Heegner points τ γτ P τ = Φ Tate ω f, Stab Γ0 Mτ = 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, 2 and 3 to transform semi-indefinite integrals into definite double integrals. This is the only stage where the assumption M = is needed. We give a different method, that works with M > X. Guitart, M. Masdeu UPC/MPIM, CU Stark Heegner points in higher level / 2

5 Reduction to Γ M Γ M = {γ = a b c d SL2 Z[/p]: γ 0 mod M} Γ 0 M γ τ Γ 0 M, but we can reduce to the case where γ τ Γ M τ γ m τ If m = [Γ 0 M: Γ M], computing mp τ = Φ Tate if a p n mod M we let α = p n 0 and 0 p n with αγ τ Γ M τ γτ ατ αγτ P τ = = X. Guitart, M. Masdeu UPC/MPIM, CU Stark Heegner points in higher level / 2

6 τ γ We are reduced to compute with γ Γ M SL 2 Z[ p ] has the congruence subgroup property γ = 0 x2 0 xr, γ Γ M Mx 0 Mx r 0 τ γ τ 0 τ γ τ 0 ω f = ω f ω f = ω f 0 τ 0 E τ E τ E γ = ω f ω f ω f = 0 τ E τ E γ ω f 0 E τ E τ E γ ω f ω f 0 Small problem: 0 and are not Γ 0 M-equivalent if M > But W d 0 =, W d = 0 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 / 2

7 The problem is reduced to finding an algorithm for computing γ = 0 x2 Mx 0 xr 0 Mx r 0, γ Γ M. Remark: if M = then the x i s are the quotients of the contineued fraction of a/c, if γ = a b c d. For M > we need another algorithm. X. Guitart, M. Masdeu UPC/MPIM, CU Stark Heegner points in higher level / 2

8 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 Γ M = {γ SL 2 O S : γ 0 mod M} p-adic Stark Heegner points F = Q, S = {p, }, M = M Z[ p ] Theorem Serre, Vaserstein: If O S is infinite i.e. if #S > then Γ M is generated by the matrices x 0 0 with x OS, x with x M, Problem: given γ = a b c d Γ M, write it as a product of elementary matrices Simple case: if c = u + ta with u O S and t O S then 0 γ = u 0 x c+t a 0 u a 0. We can replace γ = a b c d by λ 0 γ = a+λc b+λd c d X. Guitart, M. Masdeu UPC/MPIM, CU Stark Heegner points in higher level / 2

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

10 Computing the double integrals τ2 We need to compute integrals of the form τ 0 ω f The hard part is P Q p log t τ t τ 2 dµ f t Darmon Pollack: P Q p = L i= g iz p, g i GL 2 Q t τ log dµ g i Z p t τ f t = = α n g i t n dµ f t 2 n g i Z p g i Z p g i t n dµ f t: the moments can be efficiently computed via overconvergent modular symbols Number of gi s depends on the affinoid Hp n containing τ, τ 2 Hp 0 = {τ P K p : redτ / P Z/pZ} Hp n = {τ P K p : redτ / P Z/p n+ Z} \ Hp n We can take a covering of size p + + np This increases the running time with respect to the M = case when M =, then τ, τ 2 Hp 0 so p + 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 / 2

11 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 = 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 / 2

12 Curve 2A p=7, M=3, prec=7 80, K = Q D D h P τ , , , , , , , , D h h D x 65 2 x x X. Guitart, M. Masdeu UPC/MPIM, CU Stark Heegner points in higher level / 2

13 Curve 33A p =, M = 3, prec=3 80, K = Q D D h P , , , , , , , , D h h D x 40 2 x x x x x x x X. Guitart, M. Masdeu UPC/MPIM, CU Stark Heegner points in higher level / 2

14 Curve 5A p=3, M=7, prec=3 80, K = Q D D h P + 8 2, , , , , , , , , x x x x x x X. Guitart, M. Masdeu UPC/MPIM, CU Stark Heegner points in higher level / 2

15 Curve 05A p = 3, M = 5 7, prec=3 80, K = Q D D h P , , , X. Guitart, M. Masdeu UPC/MPIM, CU Stark Heegner points in higher level / 2

16 Numerical Computation of Stark Hegner points in higher level Xevi Guitart Marc Masdeu 2 Max Planck Institute/U. Politècnica de Catalunya 2 Columbia University Rational points on curves: A p-adic and computational perspective Mathematical College, Oxford 202 X. Guitart, M. Masdeu UPC/MPIM, CU Stark Heegner points in higher level / 2