The Euler product expression of the Dirichlet series of Fourier coefficients of an elliptic modular eigenform follows from a formal identity in the Hecke algebra for GL(2) with full level. In the case of Siegel modular forms of degree two with paramodular level, the situation is more delicate. In this talk, I will present two rationality results. The first concerns the Dirichlet series of radial Fourier coefficients for an eigenform of paramodular level divisible by the square of a prime. This result is an application of the theory of stable Klingen vectors (joint work with Brooks Roberts and Ralf Schmidt). While we are able to calculate the action of certain Hecke operators on eigenforms, the structure of the Hecke algebra of deep level is not known in general. However, in the case of prime level, there is a robust description of the local Hecke algebra which yields a rationality result for a formal power series of Hecke operators (joint work with Joshua Parker and Brooks Roberts). In both cases, we obtain the expected local L-factor as the denominator of the rational function.

[pre-talk at 3:00pm]