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 T2 on the space S2(Γ0(N), ) 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
Mathematical Subject Classification 2010
Primary: 11F33
Milestones
Received: 2 March 2018
Revised: 26 August 2018
Accepted: 9 September 2018
Published: 13 February 2019
Authors
Kiran S. Kedlaya
Department of Mathematics
University of California, San Diego
La Jolla, CA
United States
Anna Medvedovsky
Department of Mathematics and Statistics
Boston University
Boston, MA
United States