Shimura lifts of half-integral weight modular forms arising from theta functions

In 1973, Shimura (Ann. Math. (2) 97:440–481, 1973) introduced a family of correspondences between modular forms of half-integral weight and modular forms of even integral weight. Earlier, in unpublished work, Selberg explicitly computed a simple case of this correspondence pertaining to those half-integral weight forms which are products of Jacobi’s theta function and level one Hecke eigenforms. Cipra (J. Number Theory 32(1):58–64, 1989) generalized Selberg’s work to cover the Shimura lifts where the Jacobi theta function may be replaced by theta functions attached to Dirichlet characters of prime power modulus, and where the level one Hecke eigenforms are replaced by more generic newforms. Here we generalize Cipra’s results further to cover theta functions of arbitrary Dirichlet characters multiplied by Hecke eigenforms.      

Differential eigenforms

The aim of this paper is to show how differential characters of Abelian varieties (in the sense of [A. Buium, Differential characters of Abelian varieties over p-adic fields, Invent. Math. 122 (1995) 309-340]) can be used to construct differential modular forms of weight 0 and order 2 (in the sense of [A. Buium, Differential modular forms, Crelle J. 520 (2000) 95-167]) which are eigenvectors of Hecke operators. These differential modular forms will have “essentially the same” eigenvalues as certain classical complex eigenforms of weight 2 (and order 0).     

Algebraic Hecke characters and strictly compatible systems of abelian mod p Galois representations over global fields

In [10] (C R Acad Sci Paris Ser I Math 323(2) 117–120, 1996), [11] (Math Res Lett 10(1):71–83 2003), [12] (Can J Math 57(6):1215–1223 2005), Khare showed that any strictly compatible systems of semisimple abelian mod p Galois representations of a number field arises from a unique finite set of algebraic Hecke characters. In this article, we consider a similar problem for arbitrary global fields. We give a definition of Hecke character which in the function field setting is more general than previous definitions by Goss and Gross and define a corresponding notion of compatible system of mod p Galois representations. In this context we present a unified proof of the analog of Khare’s result for arbitrary global fields. In a sequel we shall apply this result to strictly compatible systems arising from Drinfeld modular forms, and thereby attach Hecke characters to cuspidal Drinfeld Hecke eigenforms.     

Products of nearly holomorphic eigenforms

We prove that the product of two nearly holomorphic Hecke eigenforms is again a Hecke eigenform for only finitely many choices of factors.     

Hecke actions on certain strongly modular genera of lattices

We calculate the action of some Hecke operators on spaces of modular forms spanned by the Siegel theta-series of certain genera of strongly modular lattices closely related to the Leech lattice. Their eigenforms provide explicit examples of Siegel cusp forms.Received: 10 September 2004     

Simultaneous sign change and equidistribution of signs of Fourier coefficients of two cusp forms

We study the simultaneous sign change of Fourier coefficients of a pair of distinct normalized newforms of integral weight supported on primes power indices, we also prove some equidistribution results. Finally, we consider an analogous question for Fourier coefficients of a pair of half-integral weight Hecke eigenforms.     

A remark on the normalization of the standard zeta values for Siegel modular forms

In (Katsurada in Math. Z. 259:97–111, 2008), we gave a certain type of normalization of the standard zeta values for Siegel modular forms, and considered the relationship between such values and congruence of cuspidal Hecke eigenforms. In this paper we give more reasonable normalization for such values and improve our previous result.     

Poincaré Series and Hilbert Modular Forms

In this paper, we bound the square moment of the linear form in the Fourier coefficients of Hilbert modular forms by means of Poincaré series, and obtain sharp estimate on the critical line for the fourth moment of L-functions associated with Hilbert cusp forms which are primitive Hecke eigenforms.     

Spectral approximation by L 2 discretization of the Laplace operator on manifolds of bounded geometry

The author considers a discretization of the p-form Laplacian on open complete Riemannian manifolds of bounded geometry. Following Dodziuk and Patodi [8], the eigenvalues below the essential spectrum together with their eigenforms are approximated by eigenvalues and eigencochains of a semicombinatorical Laplacian acting on L ₂-cochains. We obtain a similar result for a ray which is contained in the essential spectrum. An example of a manifold of bounded geometry which admits eigenvalues below the essential spectrum is constructed.     

Products of Hecke eigenforms

Let f(z) and g(z) be Hecke eigenforms for Γ₀(p), where p is a prime. If both f(z) and g(z) are non-cuspidal forms and p?7, then the product is a Hecke eigenform only if it comes trivially from a level 1 solution. If g(z) is a cuspform and p?5, then in addition to the level 1 solutions, there are 8 new cases where the product of Hecke eigenforms is a Hecke eigenform.     

Shintani lifts and fractional derivatives for harmonic weak Maass forms

In this paper, we construct Shintani lifts from integral weight weakly holomorphic modular forms to half-integral weight weakly holomorphic modular forms. Although defined by different methods, these coincide with the classical Shintani lifts when restricted to the space of cusp forms. As a side effect, this gives the coefficients of the classical Shintani lifts as new cycle integrals. This yields new formulas for the L-values of Hecke eigenforms. When restricted to the space of weakly holomorphic modular forms orthogonal to cusp forms, the Shintani lifts introduce a definition of weakly holomorphic Hecke eigenforms. Along the way, auxiliary lifts are constructed from the space of harmonic weak Maass forms which yield a “fractional derivative” from the space of half-integral weight harmonic weak Maass forms to half-integral weight weakly holomorphic modular forms. This fractional derivative complements the usual ξ-operator introduced by Bruinier and Funke.     

Koecher-maaß-series for modular forms of quaternions

In this paper we demonstrate the meromorphic continuation and functional equation for Koecher-Maaß-series attached to modular forms of quaternions. The Koecher-Maaß-series associated with modular forms in the Maaß space, which are simultaneous eigenforms under all Hecke operators, possess Euler product expansions.     

Classicality for small slope overconvergent automorphic forms on some compact PEL Shimura varieties of type C

We study the rigid cohomology of the ordinary locus in some compact PEL Shimura varieties of type C with values in automorphic local systems and use it to prove a small slope criterion for classicality of overconvergent Hecke eigenforms. This generalises work of Coleman, and is a first step in an ongoing project to extend the cohomological approach to classicality to higher-dimensional Shimura varieties.     

On Wronskians of weight one Eisenstein series

We describe the span of Hecke eigenforms of weight four with nonzero central value of L-function in terms of Wronskians of certain weight one Eisenstein series.     

Arithmetic behaviour of Hecke eigenvalues of Siegel cusp forms of degree two

The Ramanujan Journal - Let F and G be Siegel cusp forms for $${\mathrm{Sp}}_4({{\mathbb {Z}}})$$ and weights $$k_1, k_2$$ , respectively. Also let F and G be Hecke eigenforms lying in distinct...     

Some notes on distribution of Hecke eigenvalues for Maass cusp forms

In this paper, we get some results on the distribution of Hecke eigenvalues for Maass cusp forms. We consider the diagonal version of Sato–Tate conjecture, a central limit theorem, and a quantitative result on the Ramanujan conjecture.     

A note on Hecke eigenvalues of Hermitian Siegel Eisenstein series

In this note we compute Hecke eigenvalues of Siegel Eisenstein series on the unitary group \(\operatorname{U}(2,2)\) induced from a character. We do this for Hecke operators at primes not dividing the level of the Eisenstein series.     

Comparing Hecke eigenvalues of newforms

Given two distinct newforms with real Fourier coefficients, we show that the set of primes where the Hecke eigenvalues of one of them dominate the Hecke eigenvalues of the other has density \(\ge \)1/16. Furthermore, if the two newforms do not have complex multiplication, and neither is a quadratic twist of the other, we also prove a similar result for the squares of their Hecke eigenvalues.     

A Spectral Correspondence for Maaß Waveforms

Let be a (cocompact) Fuchsian group, given as the group of units of norm one in a maximal order in an indefinite quaternion division algebra over . Using the (classical) Selberg trace formula, we show that the eigenvalues of the automorphic Laplacian for and their multiplicities coincide with the eigenvalues and multiplicities of the Laplacian defined on the Maa? newforms for the Hecke congruence group , when d is the discriminant of the maximal order . We also show the equality of the traces of certain Hecke operators defined on the Laplace eigenspaces for and the newforms of level d, respectively. Submitted: July 1998, final version: January 1999.