Hecke eigenvalues and relations for Siegel Eisenstein series of arbitrary degree, level, and character

We evaluate the action of Hecke operators on Siegel Eisenstein series of arbitrary degree, level and character. For square-free level, we simultaneously diagonalise the space with respect to all the Hecke operators, computing the eigenvalues explicitly, and obtain a multiplicity-one result. For arbitrary level, we simultaneously diagonalise the space with respect to the Hecke operators attached to primes not dividing the level, again computing the eigenvalues explicitly.Comment: to appear, Internat. J. Number Theor

Action of Hecke operators on Siegel theta series II

Given a Siegel theta series and a prime p not dividing the level of the theta series, we apply to the theta series the n+1 Hecke operators that generate the local Hecke algebra at p. We show that the average theta series is an eigenform and we compute the eigenvalues

Some relations on Fourier coefficients of degree 2 Siegel forms of arbitrary level

We extend some recent work of D. McCarthy, proving relations among some Fourier coefficients of a degree 2 Siegel modular form $F$ with arbitrary level and character, provided there are some primes $q$ so that $F$ is an eigenform for the Hecke operators $T(q)$ and $T_1(q^2)$

Hecke operators on Hilbert-Siegel modular forms

We define Hilbert-Siegel modular forms and Hecke "operators" acting on them. As with Hilbert modular forms, these linear transformations are not linear operators until we consider a direct product of spaces of modular forms (with varying groups), modulo natural identifications we can make between certain spaces. With Hilbert-Siegel forms we identify several families of natural identifications between certain spaces of modular forms. We associate the Fourier coefficients of a form in our product space to even integral lattices, independent of a basis and choice of coefficient rings. We then determine the action of the Hecke operators on these Fourier coefficients, paralleling the result of Hafner and Walling for Siegel modular forms (where the number field is the field of rationals)

Explicitly realizing average Siegel theta series as linear combinations of Eisenstein series

We find nice representatives for the 0-dimensional cusps of the degree $n$ Siegel upper half-space under the action of \Gamma_0(\stufe). To each of these we attach a Siegel Eisenstein series, and then we make explicit a result of Siegel, realizing any integral weight average Siegel theta series of arbitrary level \stufe and Dirichlet character $\chi_L$ modulo \stufe as a linear combination of Siegel Eisenstein series

Constructing Simultaneous Hecke Eigenforms

It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie-Kohnen who considered diagonalization of “bad” Hecke operators on spaces with square free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character