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On Rational Bianchi Newforms and Abelian Surfaces with Quaternionic Multiplication

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Arithmetic Geometry, Number Theory, and Computation

Part of the book series: Simons Symposia ((SISY))

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Abstract

We study the rational Bianchi newforms (weight 2, trivial character, with rational Hecke eigenvalues) in the LMFDB that are not associated to elliptic curves, but instead to abelian surfaces with quaternionic multiplication. Two of these examples exhibit a rather special kind of behaviour: we show they arise from twisted base change of a classical newform with nebentypus character of order 4 and eight inner twists.

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Notes

  1. 1.

    Here, 2.0.7.1 is the LMFDB label for the base field \(K=\mathbb {Q}(\sqrt {-7})\) and 30625.1 the label for the level ideal (175), which has norm 30625. The final c is the alphabetic label for this specific newform at that level. We use either full labels such as 2.0.7.1-30625.1-c for Bianchi newforms, or the shorter version 30625.1-c which omits the field when that is clear from the context.

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Acknowledgements

The authors would like to thank Frank Calegari for useful discussions and suggestions and the anonymous referees for helpful feedback. Many thanks also to David Loeffler for directing our attention to the sources in Remark 6.6. Cremona was supported by EPSRC Programme Grant EP/K034383/1 LMF: L-Functions and Modular Forms, and the Horizon 2020 European Research Infrastructures project OpenDreamKit (#676541). Pacetti was partially supported by grant PICT-2018-02073. Dembélé, Schembri, and Voight were supported by a Simons Collaboration Grant (550029; to Voight).

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Cremona, J.E., Dembélé, L., Pacetti, A., Schembri, C., Voight, J. (2021). On Rational Bianchi Newforms and Abelian Surfaces with Quaternionic Multiplication. In: Balakrishnan, J.S., Elkies, N., Hassett, B., Poonen, B., Sutherland, A.V., Voight, J. (eds) Arithmetic Geometry, Number Theory, and Computation. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-030-80914-0_11

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