Automorphic Forms on Feit’s Hermitian Lattices

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.

Keywords:

• automorphic representations of unitary groups
• unimodular Hermitian lattices
• algebraic modular forms
• congruences of automorphic forms

2010 AMS Subject Classification:

• 11F55
• 11E39

Acknowledgments

We thank G. Chenevier and G. Nebe for helpful communications. We thank also the anonymous referee, for a thorough reading and many helpful comments. We found neighbors, and conducted isometry tests, using Magma code written by Markus Kirschmer, now publicly available at http://www.math.rwth-aachen.de/∼Markus.Kirschmer/.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

Sebastian Schönnenbeck is supported by the DFG collaborative research center TRR 195.