Brooks Roberts
Department of Mathematics
PO Box 441103
University of Idaho
Moscow ID 83844-1103
USA
The nonarchimedean local analogues of modular forms of half-integral weight with level and character are certain vectors in irreducible, admissible, genuine representations of the metaplectic group over a nonarchimedean local field of characteristic zero. Two natural level raising operators act on such vectors, leading to the concepts of oldforms and newforms. We prove that the number of newforms for a given representation and character is finite and equal to the number of square classes with respect to which the representation admits a Whittaker model.
Let F be a totally real number field. We define global L-packets for GSp(4) over F which should correspond to the elliptic tempered admissible homomorphisms from the conjectural Langlands group of F to the L-group of GSp(4) which are reducible, or irreducible and induced from a totally real quadratic extension of F. We prove that the elements of these global L-packets occur in the space of cusp forms on GSp(4) over F as predicted by Arthur's conjecture. This can be regarded as the GSp(4) analogue of the dihedral case of the Langlands-Tunnell theorem. To obtain these results we prove a nonvanishing theorem for global theta lifts from the similitude group of a general four dimensional quadratic space over F to GSp(4) over F.
Let X be an even dimensional symmetric bilinear space defined over a totally real number field F with adeles A, and let be an irreducible tempered cuspidal automorphic representation of O(X,A). We give a sufficient condition for the nonvanishing of the theta lift of to the symplectic group Sp(n,A) (2n by 2n matrices) for 2n dim X for a large class of X. As a corollary, we show that if 2n = dim X and all the local theta lifts are nonzero, then is nonzero if the standard L-function is nonzero at 1, and is nonzero if has a pole at 1. The proof uses only essential structural features of the theta correspondence, along with a new result in the theory of doubling zeta integrals.
In this paper we consider the theta correspondence between the sets Irr(GSp(4,k)) and Irr(GO(X)) when k is a nonarchimedean local field and dim X = 4. Our main theorem determines all the elements of Irr(GO(X)) that occur in the correspondence. The answer involves distinguished representations. As a corollary, we characterize all the elements of Irr(O(X)) that occur in the theta correspondence between Irr(Sp(4,k)) and Irr(O(X)). We also apply our main result to prove a case of a new conjecture of S.S. Kudla concerning the first occurrence of a representation in the theta correspondence.
Let V be an even dimensional nondegenerate symmetric bilinear space defined over a nonarchimedean field F of characteristic zero, and let n be a nonnegative integer. Suppose that in Irr(O(V)) and in Irr(Sp(n,F)) correspond under the theta correspondence. Assuming that is tempered, we investigate the problem of determining the Langlands quotient data for .
Let V be a nondegenerate even dimensional symmetric bilinear space over a nonarchimedean local field F of characteristic zero. Let in Irr(O(V)) be pre-unitary, and assume that corresponds to a tempered element of Irr(Sp(n_{0}, F)) with respect to the theta correspondence for some n_{0} with 2n_{0} ≥ dim V. We show that if 2n > 2n_{0}, and in Irr(Sp(n, F)) corresponds to σ, then the doubling L-function of twisted by the quadratic character of F^{×} associated to V has L(s,|•|^{-(n - }dimV/2)) as a factor, and so has a pole at n-dimV/2. The existence of this pole has an application to the important nonvanishing problem for global theta lifts.
In this paper we investigate the theta correspondence for similitudes over a nonarchimedean field. We show that the two main approaches to a theta correspondence for similitudes from the literature are essentially the same, and we prove that a version of strong Howe duality holds for both constructions.