Hecke Theory and L-Functions

Abstract

In this chapter, we introduce the symplectic Hecke algebra and discuss its action on the Siegel modular forms. This allows us to consider a basis of $M_k(\Gamma _n)$ consisting of simultaneous eigenforms of the Hecke algebra. We explicate the relation between the Hecke eigenvalues and the Fourier coefficients of the modular forms. For genus greater than 1, this relation is very complicated. Finally, we introduce the two L-functions associated with Hecke eigenforms—the spin L-function and the standard L-function.