brooksr@uidaho.edu
Brooks Roberts
875 Perimeter Drive, MS 1103
Department of Mathematics
Moscow, ID 838441103
USA
Brooks Roberts
875 Perimeter Drive, MS 1103
Department of Mathematics
Moscow, ID 838441103
USA

JohnsonLeung, J., Parker, J., & Roberts, B. (2024).
The paramodular Hecke algebra.
To appear in Research in Number Theory.
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We give a presentation via generators and relations of the local graded paramodular Hecke algebra of prime level. In particular, we prove that the paramodular Hecke algebra is isomorphic to the quotient of the free $\Z$algebra generated by four noncommuting variables by an ideal generated by seven relations. Using this description, we derive rationality results at the level of characters and give a characterization of the center of the Hecke algebra. Underlying our results are explicit formulas for the product of any generator with any double coset.

JohnsonLeung, J., Roberts, B., & Schmidt, R. (2023).
Stable Klingen Vectors and Paramodular
Newforms.
(Springer Lecture Notes in Mathematics Vol. 2342). New York: Springer.
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In this work we introduce the family of stable Klingen congruence subgroups of $\GSp(4)$. We use these subgroups to study both local paramodular vectors and Siegel modular forms of degree $2$ with paramodular level. In the first part, when $F$ is a nonarchimedean local field of characteristic zero and $(\pi,V)$ is an irreducible, admissible representation of $\GSp(4,F)$ with trivial central character, we establish a basic connection between the subspaces $V_s(n)$ of $V$ fixed by the stable Klingen congruence subgroups $\mathrm{K}_s(\p^n)$ and the spaces of paramodular vectors in $V$ and we derive a fundamental partition of the set of paramodular representations into two classes. We determine the spaces $V_s(n)$ for all $(\pi,V)$ and $n$. We relate the stable Klingen vectors in $V$ to the two paramodular Hecke eigenvalues of $\pi$ by introducing two stable Klingen Hecke operators and one level lowering operator. In contrast to the paramodular case, these three new operators are given by simple upper block formulas. We also prove additional results about stable Klingen vectors in $V$ especially when $\pi$ is generic. In the second part we apply these local results to a Siegel modular newform $F$ of degree $2$ with paramodular level $N$ that is an eigenform of the two paramodular Hecke operators at all primes $p$. We present new formulas relating the Hecke eigenvalues of $F$ at $p$ to the Fourier coefficients $a(S)$ of $F$ for $p^2 \mid N$. We verify that these formulas hold for the examples presented in \cite{PSY} and \cite{PSYW}, and we indicate how to use our formulas to generally compute Hecke eigenvalues at $p$ from Fourier coefficients of $F$ for $p^2 \mid N$. Finally, for $p^2 \mid N$ we express the formal power series in $p^{s}$ with coefficients given by the radial Fourier coefficients $a(p^t S)$, $t\geq 0$, as an explicit rational function in $p^{s}$ with denominator $L_p(s,F)^{1}$, where $L_p(s,F)$ is the spin $L$factor of $F$ at $p$.

JohnsonLeung, J., & Roberts, B. (2017).
Fourier coefficients for twists of Siegel paramodular forms.
Journal of the Ramanujan Mathematical Society, 32(2), 101–119.
Preprint.
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(published version),
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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. (2016).
Some results on Bessel functionals for GSp(4).
Documenta Mathematica, 21, 467–553.
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We prove that every irreducible, admissible representation $\pi$ of GSp(4,$F$), where $F$ is a nonarchimedean local field of characteristic zero, admits a Bessel functional, provided $\pi$ is not onedimensional. 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 nonsupercuspidal representation, we explicitly determine the set of all Bessel functionals admitted by $\pi$, and prove that these functionals are unique.

JohnsonLeung, J., & Roberts, B. (2017).
Twisting of Siegel paramodular forms.
International Journal of Number Theory, 13(7), 17553–1854.
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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 AtkinLehner involution for primes $\ell\neq p$.

JohnsonLeung, J., & Roberts, B. (2014).
Appendix to "Twisting of Siegel paramodular forms".
Preprint.
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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.

JohnsonLeung, J., & Roberts, B. (2014).
Twisting of paramodular vectors.
International Journal of Number Theory 10(4), 1043–1065.
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Let $F$ be a nonarchimedean 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 wellknown $\GL(2)$ twisting operator.

JohnsonLeung, J., & Roberts, B. (2012).
Siegel modular forms of degree two attached to Hilbert modular forms.
Journal of Number Theory, 132, 543–564.
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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 nonnegative integer $n$. We show that there exists a nonzero 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.
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The nonarchimedean local analogues of modular forms of halfintegral 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. PDF
 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. PDF
 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. PDF
 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. PDF
 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. PDF

Roberts, B. (2001).
Global Lpackets 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).)
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Let $F$ be a totally real number field. We define global Lpackets 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 LanglandsTunnell 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($2n$).)
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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 nonzero at $1$, and $\Theta_{n1}(\sigma)$ 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.

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).)
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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, 11051118.
(Note: In this paper Sp($n$) means $n$ by $n$ matrices, i.e., what is often called Sp($2n$).)
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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. PDF

Roberts, B. (1996)
The theta correspondence for similitudes.
Israel Journal of Mathematics, 94, 285–317.
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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.