Modular forms on $G_2$ and their standard $L$-function

TL;DR

This paper introduces modular forms on the exceptional group G_2, exploring their Fourier expansions and deriving the standard L-function via Rankin-Selberg integrals, connecting automorphic forms with cubic rings.

Contribution

It provides an expository overview of G_2 modular forms, including Fourier expansions and the derivation of their standard L-function through integral representations.

Findings

Fourier expansion of G_2 modular forms analyzed

Standard L-function expressed as a Dirichlet series involving cubic rings

Archimedean zeta integral studied for even weight forms

Abstract

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.