Parameters of solvable automorphic forms

TL;DR

The paper provides an explicit, cohomology‑driven classification of solvable Maass wave forms on $\Gamma_1(\ell^k)$ by attaching complex even Galois representations with solvable projective image ($A_4$ for tetrahedral and $S_4$ for octahedral). It constructs these forms via global class field theory, relating conductor, ramification, and Hecke eigenvalues to arithmetic of totally real fields and their unramified extensions, and gives exact counts $n_\ell^{(A_4)}=2^{k_\ell}-1$ and $n_\ell^{(S_4)}=\sum_L (2^{k_L}-1)$. The smallest primes realizing multiple inequivalent tetrahedral forms modulo $3$ are identified (e.g., $\ell=7687$), and the octahedral case is treated with analogous explicitness. This work deepens the Langlands paradigm by connecting automorphic Maass forms of solvable type to concrete number-field data and explicit class-field–theoretic constructions, highlighting a precise reciprocity phenomenon for even representations.

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 $Γ_1(\ell)$ that remain inequivalent modulo $3$ for $\ell = 7687, 16363$ and $20887$, and that these $\ell$ are the three smallest such primes.

Theorems & Definitions (54)

• Theorem 1: Classificaton of Maass forms of tetrahedral type on $\Gamma_1(\ell)$
• Corollary 2
• Theorem 3
• Corollary 4
• Theorem 5
• Theorem 6
• Remark 7
• Definition 8
• Definition 9
• Lemma 10