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Examples of abelian surfaces with everywhere good reduction

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Abstract

We describe several explicit examples of simple abelian surfaces over real quadratic fields with real multiplication and everywhere good reduction. These examples provide evidence for the Eichler–Shimura conjecture for Hilbert modular forms over a real quadratic field. Several of the examples also support a conjecture of Brumer and Kramer on abelian varieties associated to Siegel modular forms with paramodular level structures.

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Notes

  1. More precisely, this defines a Hilbert modular form of parallel weight k.

  2. Stroeker’s result is stated for imaginary quadratic fields. Elkies [16] remarks that the argument implies the statement above for real quadratic fields.

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Acknowledgments

We thank Fred Diamond and Haluk Şengün for several helpful email exchanges and discussions, and Noam Elkies, Neil Dummigan, Matthias Schütt and the anonymous referee for useful comments on earlier drafts of this paper. We are also thankful to Florian Bouyer and Marco Streng for many helpful exchanges, and for kindly allowing us to use their reduction package. In the early stages of this project, the first-named author spent some time at the Max-Planck Institute for Mathematics in Bonn. He would like to express his gratitude for their hospitality and support.

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Authors and Affiliations

  1. Warwick Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Lassina Dembélé

  2. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139, USA

    Abhinav Kumar

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  1. Lassina Dembélé
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  2. Abhinav Kumar
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Correspondence to Lassina Dembélé.

Additional information

AK was supported in part by National Science Foundation Grant DMS-0952486 and by a grant from the Solomon Buchsbaum Research Fund. LD was supported by the Grant EPSRC EP/J002658/1.

Appendix

Appendix

In Table 8 below we list Hilbert modular form data for all the examples considered in this paper.

Table 8 Hecke eigenvalues for the Hilbert modular forms in this paper
Full size table

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Dembélé, L., Kumar, A. Examples of abelian surfaces with everywhere good reduction. Math. Ann. 364, 1365–1392 (2016). https://doi.org/10.1007/s00208-015-1252-6

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