Mod-2 dihedral Galois representations of prime conductor

TL;DR

This paper investigates mod-2 dihedral Galois representations of prime conductor by computing Hecke operator actions for primes up to 500,000, revealing new nonexistence results and connections to class field theory.

Contribution

It provides extensive computational data on mod-2 Galois representations and extends existing nonexistence results for elliptic curves and modular forms.

Findings

Identified eigenvalues of Hecke operators mod 2 for primes up to 500,000.

Connected computational results to class field theory and Galois representations.

Extended nonexistence results for certain elliptic curves and modular forms.

Abstract

For all odd primes N up to 500000, we compute the action of the Hecke operator T_2 on the space S_2(Gamma_0(N), Q) and determine whether or not the reduction mod 2 (with respect to a suitable basis) has 0 and/or 1 as eigenvalues. We then partially explain the results in terms of class field theory and modular mod-2 Galois representations. As a byproduct, we obtain some nonexistence results on elliptic curves and modular forms with certain mod-2 reductions, extending prior results of Setzer, Hadano, and Kida.