Brooks Roberts

Research Associate Professor
University of Idaho
PhD University of Chicago 1992

Contact

Research



Contact

brooksr@uidaho.edu

Brooks Roberts
Department of Mathematics
PO Box 441103
University of Idaho
Moscow ID 83844-1103
USA


Research

On the number of local newforms in a metaplectic representation

A paper with Ralf Schmidt

To appear in the volume in honor of S.S. Kudla.

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.


Local Newforms for GSp(4)

A research monograph with Ralf Schmidt

Springer Lecture Notes in Mathematics Vol 1918 (2007), 312 pages.

A decomposition of the spaces Sk0(N)) in degree 2 and the construction of hypercuspidal modular forms

A paper with Ralf Schmidt

in Proceedings of the 9th Autumn Workshop on Number Theory, Hakuba, Japan, (2006) (46 pages).

An alternative proof of a theorem about local newforms for GSp(4)

A paper with Ralf Schmidt

in Proceedings of the 9th Autumn Workshop on Number Theory, Hakuba, Japan, (2006) (29 pages).

On modular forms for the paramodular groups

A paper with Ralf Schmidt

in Automorphic forms and zeta functions (Arakawa Memorial Conference), World Sci. Publ., Hackensack, NJ, (2006), 334-364 (31 pages).


New vectors for GSp(4): a conjecture and some evidence

A paper with Ralf Schmidt

Surikaisekikenkyusho Kokyuroku (Research Institute for Mathematical Sciences, Kyoto University), 1338, (2003), 107-121 (15 pages).


Global L-packets for GSp(2) and theta lifts

This paper uses the classical notation, so that GSp(2) means 4 by 4 matrices, i.e., what is usually called GSp(4).

Documenta Mathematica, 6, (2001), 247-314 (68 pages).

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.


Nonvanishing of global theta lifts from orthogonal groups

This paper uses the classical notation, so that Sp(n) means n by n matrices, i.e., what is usually called Sp(2n).

Journal of the Ramanujan Mathematical Society, 14, (1999), 153-216 (63 pages).

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.


The nonarchimedean theta correspondence for GSp(2) and GO(4)

This paper uses the classical notation, so that GSp(2) means 4 by 4 matrices, i.e., what is usually called GSp(4).

Transactions of the AMS, 351, (1999), 781-811 (31 pages).

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.


Tempered representations and the theta correspondence

This paper uses the classical notation, so that Sp(n) means n by n matrices, i.e., what is usually called Sp(2n).

Canadian Journal of Mathematics, 50, (1998), 1105-1118 (13 pages).

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 .


Poles of local L-functions and the theta correspondence

This paper uses the classical notation, so that Sp(n) means n by n matrices, i.e., what is usually called Sp(2n).

Preprint, (1997), (28 pages).

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(n0, F)) with respect to the theta correspondence for some n0 with 2n0dim V. We show that if 2n > 2n0, 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.


Nonvanishing of GL(2) automorphic L functions at 1/2

Mathematische Annalen, 312, (1998), 575-598 (23 pages).

The theta correspondence for similitudes

Israel Journal of Mathematics, 94, (1996), 285-317 (32 pages).

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.


Lifting of Automorphic Forms on the Units of a Quaterion Algebra to Automorphic Forms on the Symplectic Groups

Ph.D. Thesis, University of Chicago, 1992.