Comparison of integral structures on spaces of modular forms of weight two, and computation of spaces of forms mod 2 of weight one, with appendices by Jean-Francois Mestre and Gabor Wiese

2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/140532Two integral structures on the Q-vector space of modular forms of weight two on X_0(N) are compared at primes p exactly dividing N. When p=2 and N is divisible by a prime that is 3 mod 4, this comparison leads to an algorithm for computing the space of weight one forms mod 2 on X_0(N/2). For p arbitrary and N>4 prime to p, a way to compute the Hecke algebra of mod p modular forms of weight one on Gamma_1(N) is presented, using forms of weight p, and, for p=2, parabolic group cohomology with mod 2 coefficients. Appendix A is a letter from Mestre to Serre, of October 1987, where he reports on computations of weight one forms mod 2 of prime level. Appendix B reports on an implementation for p=2 in Magma, using Stein's modular symbols package, with which Mestre's computations are redone and slightly extended.39 pages, LatexNumber TheoryAlgebraic Geometry14G35 (Primary) 11F11, 11Y40 (Secondary)Comparison of integral structures on spaces of modular forms of weight two, and computation of spaces of forms mod 2 of weight one, with appendices by Jean-Francois Mestre and Gabor Wiesetext