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Publications (25)

math.NT2024

Automatic convergence and arithmeticity of modular forms on exceptional groups

Aaron Pollack

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 Fourie…

math.NT2019

On the residue method for period integrals

Aaron Pollack, Chen Wan, MichaÅ Zydor

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…

math.NT2018

Unramified Godement-Jacquet theory for the spin similitude group

Aaron Pollack

Suppose $F$ is a non-archimedean local field. The classical Godement-Jacquet theory is that one can use Schwartz-Bruhat functions on $n \times n$ matrices $M_{n} (F)$ to define the lo…

math.NT2018

Modular forms on $G_{2}$ and their standard $L$ -function

Aaron Pollack

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 th…

math.NT2022

Modular forms of half-integral weight on exceptional groups

Spencer Leslie, Aaron Pollack

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…

math.NT2017

The exterior square $L$ -function on $GU (2, 2)$

Aaron Pollack

In this paper we give Rankin-Selberg integrals for the quasisplit unitary group on four variables, $GU (2, 2)$ , and a closely-related quasisplit form of $GSpin_{6}$ .…

math.NT2017

The Spin $L$ -function on $GSp_{6}$ via a non-unique model

Aaron Pollack, Shrenik Shah

We give two global integrals that unfold to a non-unique model and represent the partial Spin $L$ -function on $GSp_{6}$ . We deduce that for a wide class of cuspidal automorp…

math.NT2017

A multivariate integral representation on $GL_{2} \times GSp_{4}$ inspired by the pullback formula

Aaron Pollack, Shrenik Shah

We give a two variable Rankin-Selberg integral inspired by consideration of Garrett's pullback formula. For a globally generic cusp form on $GL_{2} \times GSp_{4}$ , th…

math.NT2018

Multivariate Rankin-Selberg integrals on $GL_{4}$ and $GU (2, 2)$

Aaron Pollack, Shrenik Shah

Inspired by a construction of Bump, Friedberg, and Ginzburg of a two-variable integral representation on $GSp_{4}$ for the product of the standard and spin $L$ -functions, we…

math.NT2026

The quaternionic Maass Spezialschar on split $SO (8)$

Jennifer Johnson-Leung, Finn McGlade, Isabella Negrini +2

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 Fo…

math.NT2019

Modular forms on indefinite orthogonal groups of rank three

Aaron Pollack

We develop a theory of modular forms on the groups $SO (3, n + 1)$ , $n \geq 3$ . This is very similar to, but simpler, than the notion of modular forms on quaternionic exceptio…

math.NT2018

The minimal modular form on quaternionic $E_{8}$

Aaron Pollack

Suppose that $G$ is a simple reductive group over $Q$ , with an exceptional Dynkin type, and with $G (R)$ quaternionic (in the sense of Gross-Wallach). In a previou…

math.NT2018

A $G_{2}$ -period of a Fourier coefficient of an Eisenstein series on $E_{6}$

Aaron Pollack, Chen Wan, MichaÅ Zydor

We calculate a $G_{2}$ -period of a Fourier coefficient of a cuspidal Eisenstein series on the split simply-connected group $E_{6}$ , and relate this period to the Ginz…

math.NT2018

Lifting laws and arithmetic invariant theory

Aaron Pollack

In this paper we discuss lifting laws which, roughly, are ways of "lifting" elements of the open orbit of one prehomogeneous vector space to elements of the minimal nonzero orbit o…

math.NT2023

Exceptional Siegel-Weil theorems for compact $Spin_{8}$

Aaron Pollack

Let $E$ be a cubic Ã©tale extension of the rational numbers which is totally real, i.e., $E \otimes R ≃ R \times R \times R$ . There is an a…

math.NT2024

Automatic convergence for Siegel modular forms

Aaron Pollack

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…

math.NT2016

On the Rankin-Selberg integral of Kohnen and Skoruppa

Aaron Pollack, Shrenik Shah

The Rankin-Selberg integral of Kohnen and Skoruppa produces the Spin $L$ -function for holomorphic Siegel modular forms of genus two. In this paper, we reinterpret and extend their…

math.NT2024

Computation of Fourier coefficients of automorphic forms of type $G_{2}$

Aaron Pollack

In a recent work, we found formulas for the Fourier coefficients of automorphic forms of type $G_{2}$ : holomorphic Siegel modular forms on $Sp_{6}$ that are theta lifts from $…

math.NT2022

The completed standard $L$ -function of modular forms on $G_{2}$

Fatma ÃiÃ§ek, Giuliana Davidoff, Sarah Dijols +3

The goal of this paper is to provide a complete and refined study of the standard $L$ -functions $L (I ¨ , Std, s)$ for certain non-generic cuspidal automorphic representa…

math.NT2017

The Spin $L$ -function on $GSp_{6}$ for Siegel modular forms

Aaron Pollack

We give a Rankin-Selberg integral representation for the Spin (degree eight) $L$ -function on $PGSp_{6}$ . The integral applies to the cuspidal automorphic representations ass…

math.NT2018

A class number formula for Picard modular surfaces

Aaron Pollack, Shrenik Shah

We investigate arithmetic aspects of the middle degree cohomology of compactified Picard modular surfaces $X$ attached to the unitary similitude group $GU (2, 1)$ for an ima…

math.NT2023

Exceptional theta functions and arithmeticity of modular forms on $G_{2}$

Aaron Pollack

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 ro…

math.NT2019

A quaternionic Saito-Kurokawa lift and cusp forms on $G_{2}$

Aaron Pollack

We consider a special theta lift $I^¸ (f)$ from cuspidal Siegel modular forms $f$ on $Sp_{4}$ to "modular forms" $I^¸ (f)$ on $SO (4, 4)$ . This lift can be considered an…

math.NT2018

The Fourier expansion of modular forms on quaternionic exceptional groups

Aaron Pollack

Suppose that $G$ is a simple adjoint reductive group over $Q$ , with an exceptional Dynkin type, and with $G (R)$ quaternionic (in the sense of Gross-Wallach). Then…

math.NT2026

Automatic convergence for holomorphic modular forms

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 grou…

Publications (25)

Automatic convergence and arithmeticity of modular forms on exceptional groups

On the residue method for period integrals

Unramified Godement-Jacquet theory for the spin similitude group

Modular forms on G2​ and their standard L-function

Modular forms of half-integral weight on exceptional groups

The exterior square L-function on GU(2,2)

The Spin L-function on GSp6​ via a non-unique model

A multivariate integral representation on GL2​×GSp4​ inspired by the pullback formula

Multivariate Rankin-Selberg integrals on GL4​ and GU(2,2)

The quaternionic Maass Spezialschar on split SO(8)

Modular forms on indefinite orthogonal groups of rank three

The minimal modular form on quaternionic E8​

A G2​-period of a Fourier coefficient of an Eisenstein series on E6​

Lifting laws and arithmetic invariant theory

Exceptional Siegel-Weil theorems for compact Spin8​

Automatic convergence for Siegel modular forms

On the Rankin-Selberg integral of Kohnen and Skoruppa

Computation of Fourier coefficients of automorphic forms of type G2​

The completed standard L-function of modular forms on G2​

The Spin L-function on GSp6​ for Siegel modular forms

A class number formula for Picard modular surfaces

Exceptional theta functions and arithmeticity of modular forms on G2​

A quaternionic Saito-Kurokawa lift and cusp forms on G2​

The Fourier expansion of modular forms on quaternionic exceptional groups

Automatic convergence for holomorphic modular forms

Modular forms on $G_{2}$ and their standard $L$ -function

The exterior square $L$ -function on $GU (2, 2)$

The Spin $L$ -function on $GSp_{6}$ via a non-unique model

A multivariate integral representation on $GL_{2} \times GSp_{4}$ inspired by the pullback formula

Multivariate Rankin-Selberg integrals on $GL_{4}$ and $GU (2, 2)$

The quaternionic Maass Spezialschar on split $SO (8)$

The minimal modular form on quaternionic $E_{8}$

A $G_{2}$ -period of a Fourier coefficient of an Eisenstein series on $E_{6}$

Exceptional Siegel-Weil theorems for compact $Spin_{8}$

Computation of Fourier coefficients of automorphic forms of type $G_{2}$

The completed standard $L$ -function of modular forms on $G_{2}$

The Spin $L$ -function on $GSp_{6}$ for Siegel modular forms

Exceptional theta functions and arithmeticity of modular forms on $G_{2}$

A quaternionic Saito-Kurokawa lift and cusp forms on $G_{2}$