Parameters of solvable automorphic forms

TL;DR

This paper classifies Maass wave forms of prime power level using class field theory and Galois representations, revealing distinct forms of tetrahedral type at specific primes.

Contribution

It provides an analogous classification to Tate's work on modular forms, specifically for Maass wave forms and their associated representations.

Findings

Existence of inequivalent Maass cusp forms of tetrahedral type at specific primes

Identification of parameters using class field theory and Galois representations

Smallest primes where inequivalent forms occur are 7687, 16363, and 20887

Abstract

In a letter from Tate to Serre dated March 26, 1974, Tate suggested a classification of weight one modular forms of prime level in terms of their associated odd Artin representations. This paper carries out an analogous classification of Maass wave forms of prime power level in terms of complex even representations. The parameters are identified with techniques from class field theory and Galois representations. The classification reveals that there exist distinct Maass cusp forms of tetrahedral type on $\Gamma_1(\ell)$ that remain inequivalent modulo $3$ for $\ell = 7687, 16363$ and $20887$, and that these $\ell$ are the three smallest such primes.