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Lectures on Computing Cohomology of Arithmetic Groups

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Computations with Modular Forms

Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 6))

Abstract

Let G be the reductive \(\mathbb {Q}\)-group \(R_{F/\mathbb {Q}} \mathrm {GL}_{n}\), where \(F/\mathbb {Q}\) is a number field. Let Γ⊂G be an arithmetic group. We discuss some techniques to compute explicitly the cohomology of Γ and the action of the Hecke operators on the cohomology.

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Notes

  1. 1.

    Throughout these lectures we only work with congruence subgroups. For \(\mathrm{SL}_{2} (\mathbb {Z})\) this means any group containing Γ(N) for some N.

  2. 2.

    This is Harish-Chandra’s “Philosophy of cusp forms” [Har70]; see also [Bum04, Chap. 49].

  3. 3.

    Like Molière’s Monsieur Jourdain: “Par ma foi! Il y a plus de quarante ans que je dis de la prose sans que j’en susse rien, et je vous suis le plus obligé du monde de m’avoir appris cela.”

  4. 4.

    This definition of arithmetic group suffices for our purposes because we have defined our algebraic groups as subgroups of GL n . If one works more abstractly, then the correct condition is that \(\varGamma \subset \mathbf {G} (\mathbb {Q})\) is arithmetic if for any \(\mathbb {Q}\)-embedding ι:G→GL n , the group ι(Γ) is commensurable with \(\iota(\mathbf {G})\cap\mathrm{GL}_{n} (\mathbb {Z})\).

  5. 5.

    In fact the cuspidal cohomology can itself come from groups of lower rank, through functorial liftings. The paper [AGMcC08] contains evidence of cohomological lifts of paramodular forms on \(\mathrm{Sp}_{4}/\mathbb {Q}\) to \(\mathrm{SL}_{4}/\mathbb {Q}\).

  6. 6.

    Note added in proof: After this paper was prepared, dramatic progress connecting cohomology of arithmetic groups with Galois representations was announced by Harris-Lan-Taylor-Thorne and Scholze (see [HLTT13, Scho13]).

  7. 7.

    Although this construction sounds strange, we shall see that it is a reasonable notion of forms over F. Not every quadratic form of interest comes from a matrix A that is the image of a matrix from F under the embeddings; in particular the perfect forms defined in this section usually do not come from a matrix over F.

  8. 8.

    According to R. MacPherson, the great geometers of old were perfectly comfortable with multi-valued functions and would have embraced such a perspective. It is only modern mathematicians who have the paucity of imagination to insist that functions be single-valued.

  9. 9.

    A bouquet of spheres is wedge sum of a set of spheres.

  10. 10.

    For general number fields, it is not expected that every elliptic curve should correspond to a cusp form in this way. For instance, suppose F is complex quadratic. Then if E is defined over F and has complex multiplication by an order in 𝒪 F , then E should correspond to an Eisenstein series, cf. [EGM82].

  11. 11.

    Similar phenomena happen over complex quadratic fields [Cre92], and can be expected to happen over any CM field.

  12. 12.

    The cohomology computations in the following are simplified by the fact that F 1 and F 2 each have class number 1. One can still perform these computations for fields with higher class numbers, although it is best to work adelically. In practice this means that one has to work with several copies of the locally symmetric spaces instead of one, each equipped with its own Koecher decomposition. However, such complications are not always necessary. For F imaginary quadratic with odd class number, for instance, Lingham [Lin05] developed a technique to work with a single connected component.

  13. 13.

    Actually, to avoid precision problems we work with the large finite field \(\mathbb {F}_{12379}\) instead of \(\mathbb {C}\).

  14. 14.

    One curve was found by Mark Watkins using the method of Cremona-Lingham [CL07]; see the appendix to [GHY13].

  15. 15.

    One cannot help but notice the contrast between the highly symmetric perfect cone for F 1 and the minimally symmetric perfect cones for F 2.

  16. 16.

    See footnote 13.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003, USA

    Paul E. Gunnells

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  1. Interdisciplinary Center for Scientific Computing, Universität Heidelberg, Heidelberg, Baden-Württemberg, Germany

    Gebhard Böckle

  2. UR Mathématiques, Faculté des Sciences, de la Technologie et de la Communication, Luxembourg, Luxembourg

    Gabor Wiese

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Gunnells, P.E. (2014). Lectures on Computing Cohomology of Arithmetic Groups. In: Böckle, G., Wiese, G. (eds) Computations with Modular Forms. Contributions in Mathematical and Computational Sciences, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-03847-6_1

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