Mod-2 dihedral Galois representations of prime conductor

Abstract

For all odd primes N up to 500000, we compute the action of the Hecke operator T 2 on the space S 2 (Γ 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.

On Seven Conjectures of Kedlaya and Medvedovsky

“…They then analyze all cases of N mod 8 to determine how many distinct mod 2 representations arise from this construction. Finally, they conjecture lower bounds for the number of Z 2 eigenforms whose mod 2 representations ρ f are isomorphic to each of the representations obtained above [KM19,Conjecture 13]. The purpose of the current paper is to prove this conjecture, reproduced below.…”

“…In [KM19], Kedlaya and Medvedovsky prove that if a mod 2 representation is dihedral, modular and ordinary of prime level N , then it must be the induction of a character of the class group Cl(K) of a quadratic extension K = Q( √ ±N)/Q to Q [KM19, Section 5.2]. They then analyze all cases of N mod 8 to determine how many distinct mod 2 representations arise from this construction.…”