Brooks Roberts

Department of Mathematics, University of Idaho

Email addresses and telephone numbers:
brooksr@uidaho.edu



509-332-4946 (home: I usually work here)
208-885-6742 (office)
208-885-5843 (fax)

Mailing addresses:
University:
Brooks Roberts
Department of Mathematics
PO Box 441103
University of Idaho
Moscow ID 83844-1103
USA
 Home:
Brooks Roberts
1065 NE Alfred Lane
Pullman WA 99163
USA

Curriculum Vitae
Research publications
Expository publications
Some expository talks
Notes from some conferences

Research publications:

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 S_k(\Gamma_0(N)) in degree 2 and the construction of hypercuspidal modular forms (a joint 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 joint 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 joint paper with Ralf Schmidt)

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

New vectors for GSp(4): a conjecture and some evidence (a joint paper with Ralf Schmidt)

Surikaisekikenkyusho Kokyuroku (Research Institute for Mathematical Sciences, Kyoto University), 1338, (2003), 107--121. pdf file (500 K).

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

Documenta Mathematica, 6, (2001), 247-314 (68 pages) (Journal of the Deutsche Mathematiker-Vereinigung).

Let $F$ be a totally real number field. We define global $L$-packets for $\GSp(2)$ over $F$ which should correspond to the elliptic tempered admissible homomorphisms from the conjectural Langlands group of $F$ to the $L$-group of $\GSp(2)$ 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 $\GSp(2)$ over $F$ as predicted by Arthur's conjecture. This can be regarded as the $\GSp(2)$ 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 $\GSp(2)$ over $F$.

Nonvanishing of global theta lifts from orthogonal groups

Journal of the Ramanujan Mathematical Society, 14, (1999), 153-216 (63 pages). pdf file (720 K).

Let $X$ be an even dimensional symmetric bilinear space defined over a totally real number field $F$ with adeles $\A$, and let $\sigma=\otimes_v \sigma_v$ be an irreducible tempered cuspidal automorphic representation of $\OO(X,\A)$. We give a sufficient condition for the nonvanishing of the theta lift $\Theta_n(\sigma)$ of $\sigma$ to the symplectic group $\SSp(n,\A)$ ($2n$ by $2n$ matrices) for $2n \geq \dim X$ for a large class of $X$. As a corollary, we show that if $2n=\dim X$ and all the local theta lifts $\Theta_n(\sigma_v)$ are nonzero, then $\Theta_n(\sigma)$ is nonzero if the standard $L$-function $L^S(s, \sigma)$ is nonzero at $1$, and $\Theta_{n-1} (\sigma)$ is nonzero if $L^S(s,\sigma)$ 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)

Transactions of the AMS, 351, (1999), 781-811 (30 pages). pdf file (580 K).

In this paper we consider the theta correspondence between the sets $Irr(GSp(2,k))$ and $Irr(GO(X))$ when $k$ is a nonarchimedean local field and $\dim_k X =4$. 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 $Irr(Sp(2,k))$ 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.

On the modularity of some four dimensional Galois representations

Notes for a talk at CIRM-Luminy for the conference ``Representations du groupes p-adique Sp(4)'', June 15-18, 1998 (5 pages), pdf file (250 K).

Tempered representations and the theta correspondence

Canadian Journal of Mathematics, 50, (1998), 1105-1118 (13 pages). pdf file (450 K).

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 $\sigma \in Irr(O(V))$ and $\pi \in Irr(Sp(n,F))$ correspond under the theta correspondence. Assuming that $\sigma$ is tempered, we investigate the problem of determining the Langlands quotient data for $\pi$.

Poles of local L-functions and the theta correspondence

preprint, (1997), (19 pages), pdf file (500 K).

Let $V$ be a nondegenerate even dimensional symmetric bilinear space over a nonarchimedean local field $F$ of characteristic zero. Let $\sigma \in \Irr (\OO (V))$ be pre-unitary, and assume that $\sigma$ corresponds to a tempered element of $\Irr (\SSp(n_0,F))$ with respect to the theta correspondence for some $n_0$ with $2n_0 \geq \dim V$. We show that if $2n >2n_0$, and $\pi \in \Irr (\SSp(n,F))$ corresponds to $\sigma$, then the doubling $L$-function of $\pi$ twisted by the quadratic character $\chi_V$ of $F^\times$ associated to $V$ has $L(s,|\cdot |^{-(n-\dim V/2)})$ as a factor, and so has a pole at $n-\dim V /2$. 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).

Nonvanishing of global theta lifts

Proceedings of the AMS Summer Research Conference, Representation Theory of Real and p-adic Reductive Groups, July 1997, (6 pages). http://www.math.umd.edu/~jda/seattle_proceedings/.

The theta correspondence for similitudes

Israel Journal of Mathematics, 94, (1996), 285-317 (32 pages). pdf file (516 K).

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.

Expository publications:

The 1994 University of Maryland Conference on the Theta Correspondence, Dual Pairs and Automorphic Forms, College Park, Maryland

Lecture notes, (1994), (32 pages), http://www.math.umd.edu/~jda/preprints.html.

The Seminar on Galois Representations and L-Packets

Lecture notes, The Seminar on Galois Representations and L-Packets, University of Toronto, (1997), (102 pages), pdf file (1 MB).

Automorphic Representations of GL(n)

Lecture notes from a course taught at the University of Toronto, (1998).

Some expository talks:

Notes from some conferences:

Instructions for viewing the conference notes: These notes are pdf files which have been compressed. To expand the files after downloading to your computer, use Stuffit expander, available free from stuffit.com. Expanding on a Windows machine may require two steps: after expanding once, expand the created file.