Mod-2 dihedral Galois representations of prime conductor
Number Theory
2024-11-27 v2 Algebraic Geometry
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.
Keywords
Cite
@article{arxiv.1806.04653,
title = {Mod-2 dihedral Galois representations of prime conductor},
author = {Kiran S. Kedlaya and Anna Medvedovsky},
journal= {arXiv preprint arXiv:1806.04653},
year = {2024}
}
Comments
16 pages; v2: final submitted version