Automorphic Forms on Feit's Hermitian Lattices

TL;DR

This paper studies automorphic forms arising from Hermitian lattices over Eisenstein integers, identifying eigenforms, conjecturing automorphic parameters, and exploring congruences related to classical modular forms and Eisenstein series.

Contribution

It introduces a basis of Hecke eigenforms for a specific lattice genus and proposes global Arthur parameters, advancing understanding of automorphic forms on unitary groups.

Findings

Found a basis of Hecke eigenforms for the lattice genus.

Guessed global Arthur parameters matching eigenvalues.

Revealed congruences supporting conjectures for Eisenstein series.

Abstract

We consider the genus of $20$ classes of unimodular Hermitian lattices of rank $12$ over the Eisenstein integers. This set is the domain for a certain space of algebraic modular forms. We find a basis of Hecke eigenforms, and guess global Arthur parameters for the associated automorphic representations, which recover the computed Hecke eigenvalues. Congruences between Hecke eigenspaces, combined with the assumed parameters, recover known congruences for classical modular forms, and support new instances of conjectured Eisenstein congruences for $\mathbb{U}(2,2)$ automorphic forms.