Vol. 2, 2019

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Mod-2 dihedral Galois representations of prime conductor

Kiran S. Kedlaya and Anna Medvedovsky

Vol. 2 (2019), No. 1, 325–342
Abstract

For all odd primes $N$ up to $500000$, we compute the action of the Hecke operator ${T}_{2}$ on the space ${S}_{2}\left({\Gamma }_{0}\left(N\right),ℚ\right)$ 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
modular forms, mod 2 Galois representations, elliptic curves, conductor
Primary: 11F33