67 research outputs found
Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables
Let be a a negative discriminant and let vary over a set of
representatives of the integral equivalence classes of integral binary
quadratic forms of discriminant . We prove an asymptotic formula for for the average over of the number of representations of 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 which are represented by a given quaternary form. In
particular, we can show that for growing a positive proportion of the
binary quadratic forms of discriminant 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 SiegelâEisenstein series and JacobiâEisenstein 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
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