Computing modular equations for Shimura curves

2006

Proposition 1.0.2. i.e., φ 1 is the q-expansion of E 1 (τ, 1 , B). 2 pending on B. 1 Alternatively, φ1(τ ) can be obtained by calculating the integral over M(C) of a theta function valued in (1, 1) forms; this amounts to a very special case of the results of . The analogous computation in the case of modular curves was done by Funke .

2006

2003

We present explicit models for Shimura curves X_D and Atkin-Lehner quotients X_D/w_m of them of genus 2. We show that several equations conjectured by Kurihara are correct and compute for them the kernel of Ribet's isogeny J_0(D)^{new} --> J_D between the new part of the Jacobian of the modular curve X_0(D) and the Jacobian of X_D.

Mathematische Annalen, 2013

We use mock modular forms to compute generating functions for the critical values of modular L-functions, and we answer a generalized form of a question of Kohnen and Zagier by deriving the "extra relation" that is satisfied by even periods of weakly holomorphic cusp forms. To obtain these results we derive an Eichler-Shimura theory for weakly holomorphic modular forms and mock modular forms. This includes two "Eichler-Shimura isomorphisms", a "multiplicity two" Hecke theory, a correspondence between mock modular periods and classical periods, and a "Haberland-type" formula which expresses Petersson's inner product and a related antisymmetric inner product on M ! k in terms of periods. 2000 Mathematics Subject Classification. 11F67, 11F03.

1997

In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the threedimensional adjoint representation ad(f) of a twodimensional modular Galois representation f. We start with the p-adic Galois representation f0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(f0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(f0)) from the proof of Wiles of the Shimura–Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the onevariable case, we let f denote the p-ordinary Galois representation with values ...

arXiv (Cornell University), 2017

We prove two formulas in the style of the Gross-Zagier theorem, relating derivatives of L-functions to arithmetic intersection pairings on a unitary Shimura variety. We also prove a special case of Colmez's conjecture on the Faltings heights of abelian varieties with complex multiplication. These results are derived from the authors' earlier results on the modularity of generating series of divisors on unitary Shimura varieties.

Lecture Notes in Computer Science, 2002

We give a method for finding rational equations of genus 2 curves whose jacobians are abelian varieties A f attached by Shimura to normalized newforms f ∈ S2(Γ0(N)). We present all the curves corresponding to principally polarized surfaces A f for N ≤ 500.

Astérisque, 2020

We form generating series, valued in the Chow group and the arithmetic Chow group, of special divisors on the compactified integral model of a Shimura variety associated to a unitary group of signature pn ´1, 1q, and prove their modularity. The main ingredient in the proof is the calculation of vertical components appearing in the divisor of a Borcherds product on the integral model.

Journal of Mathematical Analysis and Applications, 2013

A Hecke action on the space of periods of cusp forms, which is compatible with that on the space of cusp forms, was first computed using continued fraction[19] and an explicit algebraic formula of Hecke operators acting on the space of period functions of modular forms was derived by studying the rational period functions[8]. As an application an elementary proof of the Eichler-Selberg trace formula was derived[26]. Similar modification has been applied to period space of Maass cusp forms with spectral parameter s[21, 22, 20]. In this paper we study the space of period functions of Jacobi forms by means of Jacobi integral and give an explicit description of Hecke operator acting on this space. a Jacobi Eisenstein series E 2,1 (τ, z) of weight 2 and index 1 is discussed as an example. Periods of Jacobi integrals are already appeared as a disguised form in the work of Zwegers to study Mordell integral coming from Lerch sums[27] and mock Jacobi forms are typical example of Jacobi integral[9].