EULER PRODUCTS FOR CONGRUENCE SUBGROUPS OF THE SIEGEL GROUP OF GENUS 2

Abstract

In this paper the construction is begun of a theory of Dirichlet series with Euler expansion which correspond to analytic automorphic forms for congruence subgroups of the integral symplectic group of genus 2. Namely, for an arbitrary positive integer a connection is revealed between the eigenvalues of an eigenfunction of all the Hecke operators , where is the principal congruence subgroup of degree of the group , and its Fourier coefficients. This connection can be written in the language of Dirichlet series in the form of identities; here an infinite sequence of identities arises, indexed by classes of positive definite integral primitive binary quadratic forms equivalent modulo the principal congruence subgroup of degree of . Bibliography: 15 titles.