$\def\Z{\mathbb Z} \def\R{\mathbb R} \def\C{\mathbb C} \def\bs{\backslash} \def\p{\mathfrak p} \def\OF{\mathfrak o} \def\GL{\rm GL} \def\PGL{\rm PGL} \def\SL{\rm SL} \def\SO{\rm SO} \def\Symp{\rm Sp} \def\GSp{\rm GSp} \def\PGSp{\rm PGSp} \def\meta{\widetilde{\rm SL}} \def\sgn{{\rm sgn}} \def\St{{\rm St}} \def\triv{1} \def\Mp{\rm Mp} \def\val{\rm val} \def\Irr{\rm Irr} \def\Norm{\rm N} \def\Mat{\rm M} \def\Gal{\rm Gal} \def\GSO{\rm GSO} \def\GO{\rm GO} \def\OO{\rm O} \def\Ind{\rm Ind} \def\ind{\rm ind} \def\cInd{\mathrm{c}\text{-}\mathrm{Ind}} \def\Trace{\rm T} \def\trace{\rm tr} \def\Real{\rm Real} \def\Hom{\rm Hom} \def\SSp{\rm Sp} \def\EO{\mathfrak{o}_E} \def\new{\rm new} \def\vl{\rm vol} \def\disc{\rm disc} \def\pisw{\pi_{\scriptscriptstyle SW}} \def\pis{\pi_{\scriptscriptstyle S}} \def\piw{\pi_{\scriptscriptstyle W}} \def\trip{\rm trip} \def\Left{\rm Left} \def\Right{\rm Right} \def\sroot{\Delta} \def\im{\rm im} \def\A{\mathbb A}$
Contact
brooksr@uidaho.edu
Brooks Roberts
875 Perimeter Drive, MS 1103
Department of Mathematics
Moscow, ID 83844-1103
USA
Research publications
• Johnson-Leung, J., & Roberts, B. (2015). Fourier coefficients for twists of Siegel paramodular forms (expanded version). Preprint.
In this paper, we calculate the Fourier coefficients of the paramodular twist of a Siegel modular form of paramodular level $N$ by a nontrivial quadratic Dirichlet character mod $p$ for $p$ a prime not dividing $N$. As an application, these formulas can be used to verify the nonvanishing of the twist for particular examples. We also deduce that the twists of Maass forms are identically zero.
• Roberts, B., & Schmidt, R. (2015). Some results on Bessel functionals for GSp(4). Preprint.
We prove that every irreducible, admissible representation $\pi$ of GSp(4,$F$), where $F$ is a non-archimedean local field of characteristic zero, admits a Bessel functional, provided $\pi$ is not one-dimensional. Given $\pi$, we explicitly determine the set of all split Bessel functionals admitted by $\pi$, and prove that these functionals are unique. If $\pi$ is not supercuspidal, or in an $L$-packet with a non-supercuspidal representation, we explicitly determine the set of all Bessel functionals admitted by $\pi$, and prove that these functionals are unique.
• Johnson-Leung, J., & Roberts, B. (2014). Twisting of Siegel paramodular forms. Preprint.
Let $S_k(\Gamma^{\mathrm{para}}(N))$ be the space of Siegel paramodular forms of level $N$ and weight $k$. Let $p\nmid N$ and let $\chi$ be a nontrivial quadratic Dirichlet character mod $p$. Based on our paper "Twisting of paramodular vectors", we define a linear twisting map $\mathcal{T}_\chi:S_k(\Gamma^{\mathrm{para}}(N))\rightarrow S_k(\Gamma^{\mathrm{para}}(Np^4))$. We calculate an explicit expression for this twist and give the commutation relations of this map with the Hecke operators and Atkin-Lehner involution for primes $\ell\neq p$.
• Johnson-Leung, J., & Roberts, B. (2014). Appendix to "Twisting of Siegel paramodular forms". Preprint.
In this appendix we present an expanded version of Section 4 of our paper "Twisting of Siegel paramodular forms", including the proofs of all of the technical lemmas.
• Johnson-Leung, J., & Roberts, B. (2014). Twisting of paramodular vectors. International Journal of Number Theory 10(4), 1043–1065.
Let $F$ be a non-archimedean local field of characteristic zero, let $(\pi,V)$ be an irreducible, admissible representation of $\GSp(4,F)$ with trivial central character, and let $\chi$ be a quadratic character of $F^\times$ with conductor $c(\chi)>0$. We define a twisting operator $T_\chi$ from paramodular vectors for $\pi$ of level $n$ to paramodular vectors for $\chi \otimes \pi$ of level $\max(n+2c(\chi),4c(\chi))$, and prove that this operator has properties analogous to the well-known $\GL(2)$ twisting operator.
• Johnson-Leung, J., & Roberts, B. (2012). Siegel modular forms of degree two attached to Hilbert modular forms. Journal of Number Theory, 132, 543–564.
Let $E/\mathbb Q$ be a real quadratic field and $\pi_0$ a cuspidal, irreducible, automorphic representation of $\GL(2,\A_E)$ with trivial central character and infinity type $(2,2n+2)$ for some non-negative integer $n$. We show that there exists a non-zero Siegel paramodular newform $F: \mathfrak H_2 \to \mathbb C$ with weight, level, Hecke eigenvalues, epsilon factor and $L$-function determined explicitly by $\pi_0$. We tabulate these invariants in terms of those of $\pi_0$ for every prime $p$ of $\mathbb Q$.
• Roberts, B., & Schmidt, R. (2011). On the number of local newforms in a metaplectic representation. In J. Cogdell, J. Funke, M. Rapoport, & T. Yang (Eds), Arithmetic Geometry and Automorphic Forms (pp. 505–530). Somerville, MA: International Press.
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.
• Roberts, B., & Schmidt, R. (2007). Local Newforms for GSp(4). (Springer Lecture Notes in Mathematics Vol. 1918). New York: Springer.
• Roberts, B., & Schmidt, R. (2006). A decomposition of the spaces $S_k(\Gamma _0(N))$ in degree $2$ and the construction of hypercuspidal modular forms. In Proceedings of the 9th Autumn Workshop on Number Theory, Hakuba, Japan.
• Roberts, B., & Schmidt, R. (2006). An alternative proof of a theorem about local newforms for GSp(4). In Proceedings of the 9th Autumn Workshop on Number Theory, Hakuba, Japan.
• Roberts, B., & Schmidt, R. (2006). On modular forms for the paramodular groups. In S. Bocherer, T. Ibukiyama, M. Kaneko, F. Sato (Eds), Automorphic Forms and Zeta Functions (pp. 334–364). Hackensack, NJ: World Scientific Publishing.
• Roberts, B., & Schmidt, R. (2003). New vectors for GSp(4): a conjecture and some evidence. Surikaisekikenkyusho Kokyuroku (Research Institute for Mathematical Sciences, Kyoto University), 1338, 107–121.
• Roberts, B. (2001). Global L-packets for GSp(2) and theta lifts. Documenta Mathematica, 6, 247–314. (Note: In this paper GSp(2) means 4 by 4 matrices, i.e., what is often called GSp(4).)
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 $F$.
• Roberts, B. (1999). Nonvanishing of global theta lifts from orthogonal groups. Journal of the Ramanujan Mathematical Society, 14, (1999), 153–216. (Note: In this paper Sp($n$) means $n$ by $n$ matrices, i.e., what is often called Sp($2n$).)
Let $X$ be an even dimensional symmetric bilinear space defined over a totally real number field $F$ with adeles $\A$, and let $\sigma=\otimes \sigma _v$ 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 are nonzero, then $\Theta_n(\sigma)$ is nonzero if the standard $L$-function $L^S(s,\sigma)$ is non-zero at $1$, and $\Theta_{n-1}(\sigma)$ is non-zero 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.
• Roberts, B. (1999). The nonarchimedean theta correspondence for GSp(2) and GO(4). Transactions of the AMS, 351, (1999), 781–811. (Note: In this paper GSp(2) means 4 by 4 matrices, i.e., what is often called GSp(4).)
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 $\dim X = 4$. 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 $\Irr(\OO(X))$ that occur in the theta correspondence between $\Irr(\SSp(2,k))$ and $\Irr(\OO(X))$. 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.
• Roberts, B. (1998). Tempered representations and the theta correspondence. Canadian Journal of Mathematics, 50, 1105-1118. (Note: In this paper Sp($n$) means $n$ by $n$ matrices, i.e., what is often called Sp($2n$).)
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 $\Irr(\OO(V))$ and $\pi$ in $\Irr(\SSp(n,F))$ correspond under the theta correspondence. Assuming that $\sigma$ is tempered, we investigate the problem of determining the Langlands quotient data for $\pi$.
• Roberts, B. (1998). Nonvanishing of GL(2) automorphic $L$-functions at $1/2$. Mathematische Annalen, 312, 575–598.
• Roberts, B. (1996) The theta correspondence for similitudes. Israel Journal of Mathematics, 94, 285–317.
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.
• Roberts, B. (1992). Lifting of Automorphic Forms on the Units of a Quaterion Algebra to Automorphic Forms on the Symplectic Groups (Doctoral dissertation). University of Chicago.
Other material
• Roberts, B., & Schmidt, R.. The GSp(4) wall poster.