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L-Functions and Automorphic Forms pp 81–96Cite as

Computing Invariants of the Weil Representation

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Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 10))

Abstract

We propose an algorithm for computing bases and dimensions of spaces of invariants of Weil representations of \({\mathrm {SL}}_2(\mathbb {Z})\) associated to finite quadratic modules. We prove that these spaces are defined over \(\mathbb {Z}\), and that their dimension remains stable if we replace the base field by suitable finite prime fields.

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Notes

  1. 1.

    However, if (A, Q) possesses a self-dual isotropic subgroup U (i.e. a subgroup which equals its dual with respect to the bilinear form associated to Q and such that Q(x) = 0 for all x in U) then the characteristic function of U is quickly checked to be an invariant. Moreover, one can show that in this case the characteristic functions of the self-dual isotropic subgroups span in fact the space \(\mathbb {C}[A]^G\) (A proof of this will be given in [10]). An arbitrary finite quadratic module does not necessarily possess self-dual isotropic subgroups and still admits nonzero invariants if its order is big enough.

  2. 2.

    An implementation is available at [4].

  3. 3.

    However, in [1] a different and simpler formula is given, which expresses the traces of the Weil representations in terms of the natural invariants for the conjugacy classes of \({\mathrm {SL}}_2(\mathbb {Z})\).

  4. 4.

    We say that a subspace V  of \(\mathbb {C}[A]\) is defined over the ring R if it possesses a basis whose elements are in R[A].

  5. 5.

    For this one needs that \(e(-\operatorname {sig}(\mathfrak {A})/8)/\sqrt {{\mathrm {card}}\left (A\right )}\) is in K N , which can be read off from Milgram’s formula.

References

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Acknowledgements

We thank Jonathan Schürr for carefully reading the manuscript and correcting various little errors, and we thank the anonymous referee for his very detailed report.

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Authors and Affiliations

  1. Mathematisches Institut, University of Cologne, Weyertal 86-90, 50931, Cologne, Germany

    Stephan Ehlen

  2. Fachbereich Mathematik, Universität Siegen, Walter-Flex-Str. 3, 57072, Siegen, Germany

    Nils-Peter Skoruppa

Authors
  1. Stephan Ehlen
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  2. Nils-Peter Skoruppa
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Corresponding author

Correspondence to Stephan Ehlen .

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Editors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany

    Jan Hendrik Bruinier

  2. Mathematisches Institut, Universität Heidelberg, Heidelberg, Germany

    Winfried Kohnen

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Ehlen, S., Skoruppa, NP. (2017). Computing Invariants of the Weil Representation. In: Bruinier, J., Kohnen, W. (eds) L-Functions and Automorphic Forms. Contributions in Mathematical and Computational Sciences, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-69712-3_5

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