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.

We prove that the killing rate of certain degree-lowering "recursion operators" on a polynomial algebra over a finite field grows slower than linearly in the degree of the polynomial attacked. We also explain the motivating application: obtaining a lower bound for the Krull dimension of a local component of a big mod-p Hecke algebra in the genus-zero case. We sketch the application for p = 2 and p = 3 in level one. The case p = 2 was first established in by Nicolas and Serre in 2012 using different methods.

We propose an algebraic definition of the space of-new mod-p modular forms for Γ 0 (N) in the case that is prime to N , which naturally generalizes to a notion of newforms modulo p in squarefree level. We use this notion of newforms to interpret the Hecke algebras on the graded pieces of the space of mod-2 level-3 modular forms described by Paul Monsky. Along the way, we describe a renormalized version of the Atkin-Lehner involution: no longer an involution, it is an automorphism of the algebra of modular forms, even in characteristic p.

For an odd prime p, we study the image of a continuous 2-dimensional (pseudo)representation ρ of a profinite group with coefficients in a local pro-p domain A. Under mild conditions, Bellaïche has proved that the image of ρ contains a nontrivial congruence subgroup of SL2(B) for a certain subring B of A. We prove that the ring B can be slightly enlarged and then described in terms of the conjugate self-twists of ρ, symmetries that naturally constrain its image; hence this new B is optimal. We use this result to recover, and in some cases improve, the known large-image results for Galois representations arising from elliptic and Hilbert modular forms due to Serre, Ribet and Momose, and Nekovář, and p-adic Hida or Coleman families of elliptic modular forms due to Hida, Lang, and Conti-Iovita-Tilouine.