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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-François Mestre and Gabor Wiese. (English) Zbl 1095.14019

Given integers \(N\) and \(k\) with \(N \geq 1\), let \(S_k (\Gamma_0 (N), \mathbb C)\) be the \(\mathbb C\)-vector space of holomorphic cusp forms for the congruence subgroup \(\Gamma_0 (N)\) of \(\text{SL}(2,\mathbb Z)\). One way of introducing a \(\mathbb Q\)-structure on \(S_k (\Gamma_0 (N), \mathbb C)\) is by using \(q\)-expansions of modular forms belonging to that space. Another way is by regarding \(S_k (\Gamma_0 (N), \mathbb Q)\) as the space of global sections of the invertible sheaf of modules \(\underline{\omega}^{\otimes k} (-\text{cusps})\) on the \(\mathbb Q\)-stack of generalized elliptic curves over \(\mathbb Q\)-schemes with a \(\Gamma_0 (N)\)-level structure. These two methods also determine \(\mathbb Z\)-structures on \(S_k (\Gamma_0 (N), \mathbb C)\) as well as on the modular curve \(X_0 (N)\).
In this paper the author compares two integral structures on \(X_0 (N)\) at primes \(p\) dividing \(N\) at most once. 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 \(p\), he describes a method of computing the Hecke algebra of mod \(p\) modular forms of weight one on \(\Gamma_1 (N)\) by using forms of weight \(p\) and, for \(p=2\), parabolic cohomology with mod 2 coefficients.
Appendix A is a letter from Mestre to Serre about computations of weight one forms mod 2 of prime level, and Appendix B by Wiese discusses implementation for \(p=2\) in Magma, using Stein’s modular symbols package, with which Mestre’s computations are redone and slightly extended.

MSC:

14G35 Modular and Shimura varieties
11F33 Congruences for modular and \(p\)-adic modular forms
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11Y40 Algebraic number theory computations

Software:

Hecke1; Magma; CommMatAlg
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