Computing modular equations for Shimura curves

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].

The Open Book Series, 2019

The standard approach to evaluate Hecke eigenvalues of a Siegel modular eigenform F is to determine a large number of Fourier coefficients of F and then compute the Hecke action on those coefficients. We present a new method based on the numerical evaluation of F at explicit points in the upper half-space and of its image under the Hecke operators. The approach is more efficient than the standard method and has the potential for further optimization by identifying good candidates for the points of evaluation, or finding ways of lowering the truncation bound. A limitation of the algorithm is that it returns floating point numbers for the eigenvalues; however, the working precision can be adjusted at will to yield as close an approximation as needed. (2) computing Siegel modular forms does not immediately give you the associated L-functions since the Hecke eigenvalues of Siegel modular forms, unlike in the classical case, are not equal to the Fourier coefficients and because the Euler factors of the L-function involve knowing both the p-th and the p 2-th eigenvalues. To give an idea of the difficulty of computing the L-function of a Siegel modular form, we consider an example. Let ϒ 20 be the unique normalized Siegel modular form of degree 2 and weight 20 that is a Hecke eigenform and not a Saito-Kurokawa lift. Skoruppa [15] gave an explicit formula for ϒ 20 in terms MSC2010: 11F46, 11F60.

Proceedings of The London Mathematical Society, 2013

We study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner product on modular forms via a formula of Haberland, and with an action of Hecke operators, defined algebraically by Zagier. We generalize Haberland's formula to (not necessarily cuspidal) modular forms for finite index subgroups, and we show that it conceals two stronger formulas. We extend the action of Hecke operators to period polynomials of modular forms, we show that the pairing on period polynomials appearing in Haberland's formula is nondegenerate, and we determine the adjoints of Hecke operators with respect to it. We give a few applications for Γ1(N ): an extension of the Eichler-Shimura isomorphism to the entire space of modular forms; the determination of the relations satisfied by the even and odd parts of period polynomials associated with cusp forms, which are independent of the period relations; and an explicit formula for Fourier coefficients of Hecke eigenforms in terms of their period polynomials, generalizing the Coefficients Theorem of Manin.

Rocky Mountain Journal of Mathematics, 2007

We present explicit equations of semi-stable elliptic surfaces (i.e., having only type In singular fibers) which are associated to the torsion-free genus zero congruence subgroups of the modular group as classified by A. Sebbar.

2008

The classical theory of elliptic modular equations is refor mulated and extended, and many new rationally parametrized modular equations are dis covered. Each arises in the context of a family of elliptic curves attached to a genus-zero congrue nce subgroupΓ0(N), as an algebraic transformation of elliptic curve periods, which are parame triz d by a Hauptmodul (function field generator). Since the periods satisfy a Picard–Fuchs equat ion, which is of hypergeometric, Heun, or more general type, the new equations can be viewed as algeb r ic transformation formulas for special functions. The ones for N = 4, 3, 2 yield parametrized modular transformations of Ramanujan’s elliptic integrals of signatures 2, 3, 4. The case of signature 6 will require an extension of the present theory, to one of modular equations for general elliptic surfaces.

Analytic Number Theory, 2007