Iwahori–spherical representations of GSp(4) and Siegel modular forms of degree 2 with square-free level

A theory of local old– and newforms for representations of GSp(4) over a p–adic field with Iwahori– invariant vectors is developed. The results are applied to Siegel modular forms of degree 2 with square-free level with respect to various congruence subgroups. Introduction For representations of GL(2) over a p–adic field F there is a well-known theory of local newforms due to Casselman, see [Cas]. This local theory together with the global strong multiplicity one theorem for cuspidal automorphic representations of GL(2) is reflected in the classical Atkin– Lehner theory for elliptic modular forms. On the other hand, there is currently no satisfactory theory of local newforms for the group GSp(4, F ). As a consequence, there is no analogue of Atkin–Lehner theory for Siegel modular forms of degree 2. It is the goal of this paper to provide such theories for the “square-free” case. In the local context this means that the representations in question are assumed to have non-trivial Iwahori–invariant vectors. In the global context it means that we are considering various congruence subgroups of square-free level. This paper is organized into three parts. In the first part we shall take from [ST] the complete list of irreducible, admissible representations of GSp(4, F ) supported in the minimal parabolic subgroup and list their basic properties (Table 1). We shall describe the local Langlands correspondence for these representations and give all the local parameters and local factors (Table 2). Assuming the inducing characters are unramified, we shall compute the dimensions of the spaces of fixed vectors under any parahoric subgroup for each of these representations (Table 3). In the second part of this paper we shall define local new– and oldforms with respect to a parahoric subgroup. Our main local result is Theorem 2.3.1, saying that, with respect to a fixed parahoric subgroup, a representation has either oldforms or newforms, but never both. In Table 3 the spaces of newforms have been indicated by writing their dimensions in bold face. We see that in almost all cases the space of newforms (with respect to a fixed parahoric subgroup) is one-dimensional, but there are two exceptions. In the third part we will apply the previously obtained local results to prove several theorems on classical Siegel modular forms “of square-free level”. We will need the spin (degree 4) L– function of GSp(4) as a global tool. Even though we only need the usual analytic properties of this L–function for global representations whose local components at finite places are all Iwahori–spherical, none of the current results on this L–function seems to satisfy all our needs. We shall therefore assume that an L–function theory with the desired properties exists. Under MSC: 11F46, 11F70, 11F85