On Modular Forms for the Paramodular Group
On Modular Forms for the Paramodular Group Brooks Roberts and Ralf Schmidt 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(pn), a certain family of compact-open subgroups. The group K(p0) is equal to the standard maximal compact subgroup G(o), where o is the ring of integers of F . In fact, K(p0) and K(p1) represent the two conjugacy classes of maximal compact subgroups of G(F ). In general K(pn) can be conjugated into K(p0) if n is even, and into K(p1) 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(pn) invariant vectors is non-zero; if n0 is the minimal such n then dimC(V (n0)) = 1; and the Novodvorski zeta integral of a suitably normalized vector in V (n0) computes the L–factor L(s, π) (for this last statement we assume that V is the Whittaker model of π).
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