Cusp forms for exceptional group of type $E_{7}$

TL;DR

This paper constructs holomorphic cusp forms for the exceptional group of type E7 over the rationals, extending classical modular forms through a novel lift inspired by Ikeda's method.

Contribution

It introduces a new lift from classical modular forms to cusp forms on an E7 symmetric domain, generalizing Ikeda's construction to an exceptional group setting.

Findings

Constructed cusp forms of weight 2k for E7 from classical modular forms.

Established a new lifting method analogous to Ikeda's construction.

Demonstrated the existence of non-zero cusp forms for all k ≥ 10.

Abstract

Let $\bf{G}$ be the connected reductive group of type $E_{7,3}$ over $\mathbb{Q}$ and $\mathfrak{T}$ be the corresponding symmetric domain in $\mathbb{C}^{27}$. Let $\Gamma=\bf{G}(\mathbb{Z})$ be the arithmetic subgroup defined by Baily. In this paper, for any positive integer $k\ge 10$, we will construct a (non-zero) holomorphic cusp form on $\mathfrak{T}$ of weight $2k$ with respect to $\Gamma$ from a Hecke cusp form in $S_{2k-8}(SL_2(\mathbb{Z}))$. This lift is an analogue of Ikeda's construction.