arXiv subjects
Publications and source records attributed to Aaron Pollack.
We prove an automatic convergence theorem for holomorphic modular forms on tube domains. The argument works in some generality, and covers in particular the case of orthogonal groups, symplectic groups, unitary and quaternion unitary groups, and the exceptional group $E_7$.
Bruinier and Raum, building on work of Ibukiyama-Poor-Yuen, have studied a notion of ``formal Siegel modular forms". These objects are formal sums that have the symmetry properties of the Fourier expansion of a holomorphic Siegel modular form. These authors proved that formal Siegel modular forms necessarily converge absolutely on the Siegel half-space, and thus are the Fourier expansion of an honest Siegel modular form. The purpose of this note is to give a new proof of the cuspidal case of this ``automatic convergence" theorem of Bruinier-Raum. We use the same basic ideas in a separate paper to prove an automatic convergence theorem for cuspidal quaternionic modular forms on exceptional groups.
We prove that the space of cuspidal quaternionic modular forms on the groups of type $F_4$ and $E_n$ have a purely algebraic characterization. This characterization involves Fourier coefficients and Fourier-Jacobi expansions of the cuspidal modular forms. The main component of the proof of the algebraic characterization is to show that certain infinite sums, which are potentially the Fourier expansion of a cuspidal modular form, converge absolutely. As a consequence of the algebraic characterization, we deduce that the cuspidal quaternionic modular forms have a basis consisting of forms all of whose Fourier coefficients are algebraic numbers.
The classical Maass Spezialschar is a Hecke-stable subspace of the space of holomorphic Siegel modular forms of genus two and level one cut out by certain linear relations among Fourier coefficients. We define an analogous quaternionic Maass Spezialschar, which consists of the quaternionic modular forms of level one on split $\mathrm{SO}(8)$ whose Fourier coefficients satisfy certain linear relations. We characterize this space in terms of a theta lift from the space of holomorphic Siegel modular forms on $\mathrm{Sp}(4)$, and in terms of periods. We also give a conjecture for the Dirichlet series of the standard $L$-function of quaternionic modular eigenforms on $\mathrm{SO}(8)$ and verify our conjecture on the quaternionic Maass Spezialschar.
In a recent work, we found formulas for the Fourier coefficients of automorphic forms of type $G_2$: holomorphic Siegel modular forms on $\mathrm{Sp}_6$ that are theta lifts from $G_2^c$, and cuspidal quaternionic modular forms on split $G_2$. We have implemented these formulas in the mathematical software SAGE. In this paper, we explain the formulas of our recent paper and the SAGE implementation. We also deduce some theoretical consequences of our SAGE computations.
Let $E$ be a cubic \'etale extension of the rational numbers which is totally real, i.e., $E \otimes \mathbf{R} \simeq \mathbf{R} \times \mathbf{R} \times \mathbf{R}$. There is an algebraic $\mathbf{Q}$-group $S_E$ defined in terms of $E$, which is semisimple simply-connected of type $D_4$ and for which $S_E(\mathbf{R})$ is compact. We let $G_E$ denote a certain semisimple simply-connected algebraic $\mathbf{Q}$-group of type $D_4$, defined in terms of $E$, which is split over $\mathbf{R}$. Then $G_E \times S_E$ maps to quaternionic $E_8$. This latter group has an automorphic minimal representation, which can be used to lift automorhpic forms on $S_E$ to automorphic forms on $G_E$. We prove a Siegel-Weil theorem for this dual pair: I.e., we compute the lift of the trivial representation of $S_E$ to $G_E$, identifying the automorphic form on $G_E$ with a certain degenerate Eisenstein series. Along the way, we prove a few more "smaller" Siegel-Weil theorems, for dual pairs $M \times S_E$ with $M \subseteq G_E$. The main result of this paper is used in the companion paper "Exceptional theta functions and arithmeticity of modular forms on $G_2$" to prove that the cuspidal quaternionic modular forms on $G_2$ have an algebraic structure, defined in terms of Fourier coefficients.
Quaternionic modular forms on the split exceptional group $G_2 = G_2^s$ were defined by Gan-Gross-Savin. A remarkable property of these automorphic functions is that they have a robust notion of Fourier expansion and Fourier coefficients, similar to the classical holomorphic modular forms on Shimura varieties. In this paper we prove that in even weight $\ell$ at least $6$, there is a basis of the space of cuspidal modular forms of weight $\ell$ such that all the Fourier coefficients of elements of this basis are in the cyclotomic extension of $\mathbf{Q}$. Our main tool for proving this is to develop a notion of "exceptional theta functions" on $G_2$.
We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by $\pm 1$. We analyze the minimal modular form $\Theta_{F_4}$ on the double cover of $F_4$, following Loke--Savin and Ginzburg. Using $\Theta_{F_4}$, we define a modular form of weight $\frac{1}{2}$ on (the double cover of) $G_2$. We prove that the Fourier coefficients of this modular form on $G_2$ see the $2$-torsion in the narrow class groups of totally real cubic fields.
The goal of this paper is to provide a complete and refined study of the standard $L$-functions $L(\pi,\operatorname{Std},s)$ for certain non-generic cuspidal automorphic representations $\pi$ of $G_2(\mathbb{A})$. For a cuspidal automorphic representation $\pi$ of $G_2(\mathbb{A})$ that corresponds to a modular form $\varphi$ of level one and of even weight on $G_2$, we explicitly define the completed standard $L$-function, $\Lambda(\pi,\operatorname{Std},s)$. Assuming that a certain Fourier coefficient of $\varphi$ is nonzero, we prove the functional equation $\Lambda(\pi,\operatorname{Std},s) = \Lambda(\pi,\operatorname{Std},1-s)$. Our proof proceeds via a careful analysis of a Rankin-Selberg integral that is due to an earlier work of Gurevich and Segal.
We develop a theory of modular forms on the groups $\mathrm{SO}(3,n+1)$, $n \geq 3$. This is very similar to, but simpler, than the notion of modular forms on quaternionic exceptional groups, which was initiated by Gross-Wallach and Gan-Gross-Savin. We prove the results analogous to those of earlier papers of the author on modular forms on exceptional groups, except now in the familiar setting of classical groups. Moreover, in the setting of $\mathrm{SO}(3,n+1)$, there is a family of absolutely convergent Eisenstein series, which are modular forms. We prove that these Eisenstein series have algebraic Fourier coefficients, like the classical holomorphic Eisenstein series on $\mathrm{SO}(2,n)$. As an application, we prove that the so-called "next-to-minimal" modular form on quaternionic $E_8$ has rational Fourier expansion, under a mild local assumption.
We consider a special theta lift $\theta(f)$ from cuspidal Siegel modular forms $f$ on $\mathrm{Sp}_4$ to "modular forms" $\theta(f)$ on $\mathrm{SO}(4,4)$. This lift can be considered an analogue of the Saito-Kurokawa lift, where now the image of the lift is representations of $\mathrm{SO}(4,4)$ that are quaternionic at infinity. We relate the Fourier coefficients of $\theta(f)$ to those of $f$, and in particular prove that $\theta(f)$ is nonzero and has algebraic Fourier coefficients if $f$ does. Restricting the $\theta(f)$ to $G_2 \subseteq \mathrm{SO}(4,4)$, we obtain cuspidal modular forms on $G_2$ of arbitrarily large weight with all algebraic Fourier coefficients. In the case of level one, we obtain precise formulas for the Fourier coefficients of $\theta(f)$ in terms of those of $f$. In particular, we construct nonzero cuspidal modular forms on $G_2$ of level one with all integer Fourier coefficients.
By applying the residue method for period integrals and Langlands-Shahidi's theory for residues of Eisenstein series, we study the period integrals for six spherical varieties. For each spherical variety, we prove a relation between the period integrals and certain automorphic L-functions. In some cases, we also study the local multiplicity of the spherical varieties.
Suppose that $G$ is a simple reductive group over $\mathbf{Q}$, with an exceptional Dynkin type, and with $G(\mathbf{R})$ quaternionic (in the sense of Gross-Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $\theta_{Gan}$ on quaternionic $E_8$, and some applications. The $Sym^{8}(V_2)$-valued automorphic function $\theta_{Gan}$ is a weight four, level one modular form on $E_8$, which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic $E_7, E_6$ and $G_2$. We also discuss a family of degenerate Heisenberg Eisenstein series on the groups $G$, which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups $\mathrm{GSp}_{2n}$.
The purpose of this partly expository paper is to give an introduction to modular forms on $G_2$. We do this by focusing on two aspects of $G_2$ modular forms. First, we discuss the Fourier expansion of modular forms, following work of Gan-Gross-Savin and the author. Then, following Gurevich-Segal and Segal, we discuss a Rankin-Selberg integral yielding the standard $L$-function of modular forms on $G_2$. As a corollary of the analysis of this Rankin-Selberg integral, one obtains a Dirichlet series for the standard $L$-function of $G_2$ modular forms; this involves the arithmetic invariant theory of cubic rings. We end by analyzing the archimedean zeta integral that arises from the Rankin-Selberg integral when the cusp form is an even weight modular form.
We calculate a $\mathrm{G}_2$-period of a Fourier coefficient of a cuspidal Eisenstein series on the split simply-connected group $\mathrm{E}_6$, and relate this period to the Ginzburg-Rallis period of cusp forms on $\mathrm{GL}_6$. This gives us a relation between the Ginzburg-Rallis period and the central value of the exterior cube L-function of $\mathrm{GL}_6$
Suppose that $G$ is a simple adjoint reductive group over $\mathbf{Q}$, with an exceptional Dynkin type, and with $G(\mathbf{R})$ quaternionic (in the sense of Gross-Wallach). Then there is a notion of modular forms for $G$, anchored on the so-called quaternionic discrete series representations of $G(\mathbf{R})$. The purpose of this paper is to give an explicit form of the Fourier expansion of modular forms on $G$, along the unipotent radical $N$ of the Heisenberg parabolic $P = MN$ of $G$.
We investigate arithmetic aspects of the middle degree cohomology of compactified Picard modular surfaces $X$ attached to the unitary similitude group $\mathrm{GU}(2,1)$ for an imaginary quadratic extension $E/\mathbf{Q}$. We construct new Beilinson--Flach classes on $X$ and compute their Archimedean regulator. We obtain a special value formula involving a non-critical $L$-value of the degree six standard $L$-function, a Whittaker period, and the regulator. This provides evidence for Beilinson's conjecture in this setting.
Inspired by a construction of Bump, Friedberg, and Ginzburg of a two-variable integral representation on $\mathrm{GSp}_4$ for the product of the standard and spin $L$-functions, we give two similar multivariate integral representations. The first is a three-variable Rankin-Selberg integral for cusp forms on $\mathrm{PGL}_4$ representing the product of the $L$-functions attached to the three fundamental representations of the Langlands $L$-group $\mathrm{SL}_4(\mathbf{C})$. The second integral, which is closely related, is a two-variable Rankin-Selberg integral for cusp forms on $\mathrm{PGU}(2,2)$ representing the product of the degree 8 standard $L$-function and the degree 6 exterior square $L$-function.