Lynne H. Walling

Twists of Hilbert modular forms

The theory of newforms for Hilbert modular forms is summarized including a statement of a strong multiplicity-one theorem and a characterization of newforms as eigenfunctions for a certain involution whose Dirichlet series has a prescribed Euler product. The general question of twisting Hilbert modular newforms by arbitrary Fecke characters is considered and the exact level of a character twist of a Hilbert modular form is determined. Conditions under which the twist of a newform is a newform are given. Applications include a strengthening in the elliptic modular case of a theorem of Atkin and Lis regarding the characterization of imprimitive newforms as well as its generalization to the Hilbert modular case, and a decomposition theorem for certain spaces of newforms as the direct sum of twists of spaces of newforms of lower level

ACTION OF HECKE OPERATORS ON SIEGEL THETA SERIES, II

We apply the Hecke operators T(p)2 and (1 ≤ j ≤ n ≤ 2k) to a degree n theta series attached to a rank 2k ℤ-lattice L equipped with a positive definite quadratic form in the case that L/pL is regular. We explicitly realize the image of the theta series under these Hecke operators as a sum of theta series attached to certain sublattices of , thereby generalizing the Eichler Commutation Relation. We then show that the average theta series (averaging over isometry classes in a given genus) is an eigenform for these operators. We explicitly compute the eigenvalues on the average theta series, extending previous work where we had the restrictions that χ(p) = 1 and n ≤ k. We also show that for j > k when χ(p) = 1, and for j ≥ k when χ(p) = -1, and that θ(gen L) is an eigenform for T(p)2.

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 diagonalize the space with respect to all the Hecke operators, computing the eigenvalues explicitly, and obtain a multiplicity-one result. For arbitrary level, we simultaneously diagonalize the space with respect to the Hecke operators attached to primes not dividing the level, again computing the eigenvalues explicitly.

Indefinite quadratic forms and Eisenstein series

Abstract We use geometric algebra and the theory of automorphic forms to realize theta series attached to an indefinite quadratic form as the sum of a specific Eisenstein series and an L 2-function. From this we obtain explicit formulas for the measure of the representation of an integer by an indefinite quadratic form.

Hecke operators on half-integral weight Siegel Eisenstein series

We construct a basis for the space of half-integral weight Siegel Eisenstein series of level 4N where N is odd and square-free. Then we restrict our attention to those Eisenstein series generated from elements of

The Ubiquitous Heat Kernel

\Gamma_0(4)

Explicit Action of Hecke Operators on Siegel Modular Forms

, commenting on why this restriction is necessary for our methods. We directly apply to these forms all Hecke operators attached to odd primes, and we realize the images explicitly as linear combinations of Siegel Eisenstein series. Using this information, we diagonalize the subspace of Eisenstein series generated from elements of

Hecke operators on theta series attached to lattices of arbitrary rank

\Gamma_0(4)

Sums of squares over function fields

, obtaining a multiplicity-one result.