On Modular Forms for the Paramodular Group

Contents 1 Definitions 3 2 Linear independence at different levels 6 3 The level raising operators 8 4 Oldforms and newforms 13 5 Saito–Kurokawa liftings 16 6 Two theorems 23 Appendix 26 References 28 Introduction Let F be a p–adic field, and let G be the algebraic F –group GSp(4). In our paper [RS] we presented a conjectural theory of local newforms for irreducible, admissible, generic representations of G(F) with trivial central character. The main feature of this theory is that it considers fixed vectors under the paramodular groups K(p n), a certain family of compact-open subgroups. The group K(p 0) is equal to the standard maximal compact subgroup G(o), where o is the ring of integers of F. In fact, K(p 0) and K(p 1) represent the two conjugacy classes of maximal compact subgroups of G(F). In general K(p n) can be conjugated into K(p 0) if n is even, and into K(p 1) if n is odd. Our theory is analogous to Casselman's well-known theory for representations of GL(2, F); see [Cas]. The main conjecture made in [RS] states that for each irreducible, admissible, generic representation (π, V) of PGSp(4, F) there exists an n such that the space V (n) of K(p n) invariant vectors is non-zero; if n 0 is the minimal such n then dim C (V (n 0)) = 1; and the Novodvorski zeta integral of a suitably normalized vector in V (n 0) computes the L–factor L(s, π) (for this last statement we assume that V is the Whittaker model of π). We recently proved all parts of this conjecture; it is now a theorem 1. Parts of the main theorem 1 It was still a conjecture at the time of the Arakawa conference.