Differential operators on Hilbert modular forms

We investigate differential operators and their compatibility with subgroups of SL₂n(R). In particular, we construct Rankin-Cohen brackets for Hilbert modular forms, and more generally, multilinear differential operators on the space of Hilbert modular forms. As an application, we explicitly determine the Rankin-Cohen bracket of a Hilbert-Eisenstein series and an arbitrary Hilbert modular form. We use this result to compute the Petersson inner product of such a bracket and a Hilbert modular cusp form.     

On bounding Hecke-Siegel eigenvalues

We use the action of the Hecke operators (1≤j≤n) on the Fourier coefficients of Siegel modular forms to bound the eigenvalues of these Hecke operators. This extends work of Duke-Howe-Li and of Kohnen, who provided bounds on the eigenvalues of the operator T(p).     

Lifting maps from a vector space of Jacobi cusp forms to a subspace of elliptic modular forms

In this paper we construct a lifting map from a vector space of generalized Jacobi cusp forms to a certain subspace of elliptic cusp forms and vice versa such that both mappings are adjoint with respect to the Petersson scalar products.     

On the Birch and Swinnerton-Dyer conjecture for abelian varieties attached to Hilbert modular forms

In this paper we prove that if the Birch and Swinnerton-Dyer conjecture holds for abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character, then the Birch and Swinnerton-Dyer conjecture holds for abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character regarded over arbitrary totally real number fields.     

On the congruences of Jacobi forms

In this paper, we study congruence properties of coefficient of Jacobi forms. The result for elliptic modular form case was studied by Sturm (Lecture Notes in Mathematics, Springer, Berlin Heidelberg New York 1987).     

Cohomological relation between Jacobi forms and skew‐holomorphic Jacobi forms

Eichler and Zagier developed a theory of Jacobi forms to understand and extend Maass' work on the Saito‐Kurokawa conjecture. Later Skoruppa introduced skew‐holomorphic Jacobi forms, which play an important role in understanding liftings of modular forms and Jacobi forms. In this paper, we explain a relation between Jacobi forms and skew‐holomorphic Jacobi forms in terms of a group cohomology. More precisely, we introduce an isomorphism from the direct sum of the space of Jacobi cusp forms on and the space of skew‐holomorphic Jacobi cusp forms on with the same half‐integral weight to the Eichler cohomology group of with a coefficient module coming from polynomials.     

Weight-dependent congruence properties of modular forms

In this paper, we study congruence properties of modular forms in various ways. By proving a weight-dependent congruence property of modular forms, we give some sufficient conditions, in terms of the weights of modular forms, for a modular form to be non-p-ordinary. As applications of our main theorem we derive a linear relation among coefficients of new forms. Furthermore, congruence relations among special values of Dedekind zeta functions of real quadratic fields are derived.     

Superspecial abelian varieties and the Eichler Basis Problem for Hilbert modular forms

Let p be an unramified prime in a totally real field L such that h⁺(L)=1. Our main result shows that Hilbert modular newforms of parallel weight two for Γ₀(p) can be constructed naturally, via classical theta series, from modules of isogenies of superspecial abelian varieties with real multiplication on a Hilbert moduli space. This may be viewed as a geometric reinterpretation of the Eichler Basis Problem for Hilbert modular forms.     

On the p-adic L-function of Hilbert modular forms at supersingular primes

In this paper, I discuss the construction of the p-adic L-function attached to a Hilbert modular form f, supersingular or ordinary, which turns out to be the non-archimedean Mellin transform of an h-admissible measure. And h is explicitly given. As a special case, when the Fourier coefficient of f at p|p is zero, plus/minus p-adic L-functions are furthermore defined as bounded functions, and they interpolate special values of L(f,χ,s) for cyclotomic characters χ. This can be used to formulate Iwasawa main conjecture for supersingular elliptic curve defined over a totally real field.     

Jacobi Theta Functions over Number Fields

We use Jacobi theta functions to construct examples of Jacobi forms over number fields. We determine the behavior under modular transformations by regarding certain coefficients of the Jacobi theta functions as specializations of symplectic theta functions. In addition, we show how sums of those Jacobi theta functions appear as a single coefficient of a symplectic theta function.     

Differential operators for Hermitian Jacobi forms and Hermitian modular forms

Kim (Arch Math (Basel) 79(3):208–215, 2002) constructs multilinear differential operators for Hermitian Jacobi forms and Hermitian modular forms. However, her work relies on incorrect actions of differential operators on spaces of Hermitian Jacobi forms and Hermitian modular forms. In particular, her results are incorrect if the underlying field is the Gaussian number field. We consider more general spaces of Hermitian Jacobi forms and Hermitian modular forms over \(\mathbb {Q}(i)\), which allow us to correct the corresponding results in Kim (2002).     

Congruences for modular forms of weights two and four

We prove a conjecture of Calegari and Stein regarding mod p congruences between modular forms of weight four and the derivatives of modular forms of weight two.     

Vanishing periods of cusp forms over modular symbols

partially supported by NSF grant #DMS-8919696     

L-functions for Jacobi forms and the basis problem

This paper deals with Jacobi forms Φ on ?×ℂ. The Rankin–Selberg doubling method is employed to study properties of the standard L-function of Hecke–Jacobi eigenforms. It is shown that every analytic Klingen–Jacobi Eisenstein series attached to Φ has a meromorphic continuation on the whole complex plane. Hecke–Jacobi cusp eigenforms of weight k > 4 and k≡ 0 mod 4 can written explicitly as a linear combination of theta series. Finally the basis problem of Jacobi forms of square-free index is solved. Received: 12 March 2000 / Revised version: 17 September 2001     

Variants of recognition problems for modular forms

We consider the problem of distinguishing two modular forms, or two elliptic curves, by looking at the coefficients of their L-functions for small primes (compared to their conductor). Using analytic methods based on large-sieve type inequalities we give various upper bounds on the number of forms having the first few coefficients equal to those of a fixed one. In addition, we consider similar questions of recognizing symmetric squares and CM forms from the behavior of small primes.Received: 30 August 2004     

Linear dependence among Siegel modular forms

Theorems are given which describe when high enough vanishing at the cusps implies that a Siegel modular cusp form is zero. Formerly impractical computations become practical and examples are given in degree four. Vanishing order is described by kernels, a type of polyhedral convex hull. Received: November 19, 1998 / Revised: July 5, 1999 / Published online: September 5, 2000