Explicit Hecke series for symplectic group of genus 4
Résumé
Shimura conjectured the rationality of the generating series for Hecke operators for the symplectic group of genus $n$. This conjecture was proved by Andrianov for arbitrary genus $n$, but the explicit expression was out of reach for genus higher than 3. For genus $n=4$, we explicitly compute the rational fraction in this conjecture. Using formulas for images of double cosets under the Satake spherical map, we first compute the sum of the generating series, 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 basis Hecke operators under the spherical map. Numerical examples of these fractions for special choice of Satake parameters are given.