Mod-2 dihedral Galois representations of prime conductor

Kedlaya, Kiran S.; Medvedovsky, Anna

doi:10.2140/obs.2019.2.325

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Mod-2 dihedral Galois representations of prime conductor

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Medvedovsky, Anna
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.2140/obs.2019.2.325
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Citation

Kedlaya, K. S., & Medvedovsky, A. (2019). Mod-2 dihedral Galois representations of prime conductor. In R. Scheidler, & J. Sorenson (Eds.), Proceedings of the Thirteenth Algorithmic Number Theory Symposium (pp. 325-342). Berkeley: Mathematical Sciences Publishers.

Cite as: https://hdl.handle.net/21.11116/0000-0004-8A7C-6

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.