Zeta Integrals

As we saw in the first chapter, if (π, V ) is a generic, irreducible, admissible representation of GSp(4, F) with trivial central character, then a theory of zeta integrals for π exists. This theory is used to define the L- and ε-factors for π. In this chapter we consider zeta integrals of paramodular vectors and prove central results required to fully exploit zeta integrals as a tool for investigating paramodular vectors. A major obstruction is the existence of degenerate vectors, i.e., paramodular vectors with vanishing zeta integrals; this phenomenon does not occur in the GL(2) theory. We prove the important η Principle, which fully accounts for degenerate vectors via the level raising operator η. The η Principle is proved using P3-theory. To apply P3-theory, we will prove a result that relates poles of the L-functions of generic representations to certain irreducible subquotients in the associated P3-filtration; this is a general result that has nothing to do with paramodular vectors. In the last section of this chapter we also use P3-theory to prove the existence of non-zero paramodular vectors in any generic representation. These results are proved after some basic observations about the zeta integrals of paramodular vectors.