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Arithmetic Aspects of Bianchi Groups

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Computations with Modular Forms

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

Abstract

We discuss several arithmetic aspects of Bianchi groups, especially from a computational point of view. In particular, we consider computing the homology of Bianchi groups together with the Hecke action, connections with automorphic forms, abelian varieties, Galois representations and the torsion in the homology of Bianchi groups. Along the way, we list several open problems and conjectures, survey the related literature, presenting concrete examples and numerical data.

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Notes

  1. 1.

    Note that a gap in Humbert’s proof was filled in by Grunewald and Kühnlein in [GK98].

  2. 2.

    These are the cusps that correspond to non-trivial elements of the class group.

  3. 3.

    In this case, Scarth exhibits in [Sca03] a finite-index subgroup which splits as an amalgamated product.

  4. 4.

    Recall that a subgroup Γ is of congruence type if it contains the kernel of the surjection \(\mathrm{PSL}_{2}(\mathcal{O}) \rightarrow \mathrm{PSL}_{2}(\mathcal{O} / I)\) for some ideal I of \(\mathcal{O}\).

  5. 5.

    In fact, the case of symmetric spaces of rank one was done first by Serre in [Serr70], to investigate the Congruence Subgroup Problem for Bianchi groups, and served as a prototype for the general construction.

  6. 6.

    Segal records the following in [Seg12] about these computations of Grunewald:

    “Fritz’s gentle manner concealed a steely determination when it came to serious computation. He was banned for a while from using the Bielefeld University mainframe after his program had monopolized the entire system: in order to carry out the heavy-duty computation required for this project, he had devised a routine that managed to bypass the automatic quota checks.”

  7. 7.

    Note that beyond degree two, the homology groups are exclusively torsion at the primes 2 and 3. Their explicit structure depends on the number of conjugacy classes of the finite subgroups of the Bianchi group, see Rahm [Rahb].

  8. 8.

    This is also known as the Eichler-Shimura-Harder Isomorphism. However Prof. Harder advised us not to use this name.

  9. 9.

    The term newform is used, both in elliptic and Bianchi setting, in its usual sense, that is, cuspidal, primitive and new eigenform.

  10. 10.

    The group \(\mathcal{K}_{0}(\mathfrak{n})\) is defined as . Notice that \(\mathrm{GL}_{2}(\mathcal{O}_{K}) \cap \mathcal{K}_{0}(\mathfrak{n})\) is the standard congruence subgroup \(\varGamma_{0}(\mathfrak{n})\).

  11. 11.

    Note that if p≤max{k,ℓ}, then \(V_{k,\ell}(\mathcal{O} / (p))\) is a reducible \(\mathrm{PSL}_{2}(\mathcal{O} / (p))\)-module and its coinvariants typically give rise to p-torsion in the homology.

  12. 12.

    As we mentioned in the Introduction, Scholze just released a paper [Scho] which proves this belief to be true!

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Acknowledgements

It is a pleasure to thank Lassina Dembélé for his help with the part about connections with elliptic curves. We also thank T. Berger, J.-P. Serre and A. Rahm for their helpful feedback. Figures 1 and 2 were reproduced from Maite Aranés’ thesis with her kind permission. Finally, we thank the referee for the constructive comments which gave the paper its more coherent final form.

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  1. Mehmet Haluk Şengün

    Present address: Mathematics Institute, University of Warwick, CV4 7AL, Coventry, UK

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  1. Max Planck Institute for Mathematics, Vivatsgasse 7, 53111, Bonn, Germany

    Mehmet Haluk Şengün

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    Gabor Wiese

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Şengün, M.H. (2014). Arithmetic Aspects of Bianchi 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_11

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