Local Newforms for GSp(4) pp 123-149 | Cite as

# 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 *P*3-theory. To apply *P*3-theory, we will prove a result that relates poles of the *L*-functions of generic representations to certain irreducible subquotients in the associated *P*3-filtration; this is a general result that has nothing to do with paramodular vectors. In the last section of this chapter we also use *P*3-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.

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