| Popis: |
In this work, we make a detailed study of the Fourier coefficients of cuspidal Siegel modular forms of degree 2. We derive a very general relation between the Fourier coefficients that extends previous work in this direction by Andrianov, Kowalski-Saha-Tsimerman and others. The basis for our relation is the dependence between values of global Bessel periods and averages of Fourier coefficients. Consequently our relation applies also to Bessel periods of more general automorphic forms on GSp4(A). We use our relation to prove that cuspidal Siegel modular forms associated to P-CAP representations (Saito-Kurokawa lifts with level) satisfy the so-called Maass relations. This is the first result of this kind for Siegel modular forms with respect to general congruence subgroups. Another important corollary of our work is the existence of non-zero Fourier coefficients of the simplest form possible (often fundamental or primitive) for a wide family of cuspidal Siegel modular forms of degree 2. Finally, using classical methods, we are able to prove that paramodular newforms of square-free level have infinitely many non-zero fundamental Fourier coefficients. This result extends previous work by Saha in the full-level case, and is especially interesting because of the paramodular conjecture connecting paramodular newforms of weight 2 and rational abelian surfaces. |
