Explicit Shimura's conjecture for Sp4

Abstract

Shimura's conjectire (1963) concerns the rationality of the generating series for Hecke operators for the symplectic group of genus g. This conjecture wes proved by Andrianov for arbitrary genus g. For genus g=4, we explicify the rational fraction in this conjecture. Using formulas for images of double cosets, we first compute the sum of the generating series under the Satake spherical map, which is a rational fraction with polynomial coefficients. Then we recover the coefficients of this fraction as elements of the Hecke algebra using polynomial representation of basic Hecke operators under spherical map. Numerical examples of these fractions for special choice of Satake parameters are given.