Abstract

For all odd primes $N$ up to $500000$, we compute the action of the Hecke operator $T_2$ on the space $S_2\left({\Gamma }_0\left(N\right),\mathbb{Q}\right)$ 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

Mathematical Subject Classification 2010

Milestones

Authors