On modular symbols and the cohomology of Hecke triangle surfaces. (English) Zbl 1214.11066
The paper under review gives a clear, self-contained algebraic account of the formalism of modular symbols and uses this to derive the standard algorithms used in computer algebra packages such as Magma and Sage to compute modular forms for congruence subgroups. The approach given here avoids the difficult technical references such as V. V. Shokurov, [Izv. Akad. Nauk SSSR, Ser. Mat. 44, 670–718 (1980; Zbl 0444.14030)] normally used in the literature, instead using algebraic methods and the Eichler-Shimura isomorphism.
The author generalizes these algorithms from congruence subgroups of the full modular group to the Hecke triangle groups, which are certain non-arithmetic Fuchsian groups of the first kind. Results on the group cohomology of Hecke triangle groups are proved, including an explicit formula for the parabolic subspace of the group cohomology and a spectral sequence for Hecke triangle surfaces is derived and proved, and in an aside the author outlines a geometric interpretation of this spectral sequence in terms of analytic modular stacks.
The work also generalizes a result of L. Merel from [Ann. Inst. Fourier 41, No.3, 519–537 (1991; Zbl 0727.11020)] on the cohomology of modular curves from weight 2 to higher weight; this work was also motivated by the desire to compute modular forms, and the author shows how to derive the previous results as a special case.
Finally, the article concludes with a brief section explaining how the results of earlier sections can be used to compute modular forms over arbitrary commutative rings, which is a topic of current research interest as it allows the computation of modular forms which are not reductions of forms defined over the complex numbers; for instance, in the author’s paper [J. Reine Angew. Math. 606, 79–103 (2007; Zbl 1126.11028)] this is used to compute Katz modular forms over the algebraic closure of \(\mathbb{F}_p\).
The author generalizes these algorithms from congruence subgroups of the full modular group to the Hecke triangle groups, which are certain non-arithmetic Fuchsian groups of the first kind. Results on the group cohomology of Hecke triangle groups are proved, including an explicit formula for the parabolic subspace of the group cohomology and a spectral sequence for Hecke triangle surfaces is derived and proved, and in an aside the author outlines a geometric interpretation of this spectral sequence in terms of analytic modular stacks.
The work also generalizes a result of L. Merel from [Ann. Inst. Fourier 41, No.3, 519–537 (1991; Zbl 0727.11020)] on the cohomology of modular curves from weight 2 to higher weight; this work was also motivated by the desire to compute modular forms, and the author shows how to derive the previous results as a special case.
Finally, the article concludes with a brief section explaining how the results of earlier sections can be used to compute modular forms over arbitrary commutative rings, which is a topic of current research interest as it allows the computation of modular forms which are not reductions of forms defined over the complex numbers; for instance, in the author’s paper [J. Reine Angew. Math. 606, 79–103 (2007; Zbl 1126.11028)] this is used to compute Katz modular forms over the algebraic closure of \(\mathbb{F}_p\).
Reviewer: Lloyd Kilford (Bristol)
MSC:
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F75 | Cohomology of arithmetic groups |
| 11Y40 | Algebraic number theory computations |
| 20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |
References:
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