Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables

Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary form. In particular, we can show that for growing $d$ a positive proportion of the binary quadratic forms of discriminant $-d$ is represented by the given quaternary quadratic form.Comment: v5: Some typos correcte

Equivariant holomorphic differential operators and finite averages of values of L-functions

AbstractUsing pullback formulas for both Siegel–Eisenstein series and Jacobi–Eisenstein series the second author obtained relations between critical values of certain L-functions. To extend these relations to other critical values we use holomorphic differential operators for both types of pullbacks. The differential operators in question are well known in the Siegel case whereas for the Jacobi case they have to be developed from scratch. To compare the two pullbacks, we have furthermore to establish a relation of unexpected nature between the two types of differential operators