Skip to main content
Book cover

Arithmetic Geometry, Number Theory, and Computation pp 343–363Cite as

On Rational Bianchi Newforms and Abelian Surfaces with Quaternionic Multiplication

  • 560 Accesses

Part of the Simons Symposia book series (SISY)

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.

2010 Mathematics Subject Classification

  • Primary: 11G05
  • Secondary: 11G40

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-80914-0_11
  • Chapter length: 21 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   129.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-80914-0
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   169.99
Price excludes VAT (USA)
Hardcover Book
USD   249.99
Price excludes VAT (USA)
Learn about institutional subscriptions

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.

References

  1. George Boxer, Frank Calegari, Toby Gee, and Vincent Pilloni. Abelian surfaces over totally real fields are potentially modular, 2018. Preprint: https://arxiv.org/abs/1812.09269.

  2. Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The user language. J. Symbolic Comput., 24(3-4):235–265, 1997. Computational algebra and number theory (London, 1993).

    Google Scholar 

  3. Tobias Berger, Lassina Dembélé, Ariel Pacetti, and Mehmet Haluk Şengün. Theta lifts of Bianchi modular forms and applications to paramodularity. J. Lond. Math. Soc. (2), 92(2):353–370, 2015.

    Google Scholar 

  4. Armand Brumer and Kenneth Kramer. Paramodular abelian varieties of odd conductor. Trans. Amer. Math. Soc., 366(5):2463–2516, 2014.

    CrossRef  MathSciNet  Google Scholar 

  5. Armand Brumer and Kenneth Kramer. Corrigendum to paramodular abelian varieties of odd conductor. Trans. Amer. Math. Soc., 2019. available from https://www.ams.org/journals/tran/2019-372-03/S0002-9947-2019-07792-9/home.html.

  6. Armand Brumer, Ariel Pacetti, Cris Poor, Gonzalo Tornaría, John Voight, and David Yuen. On the paramodularity of typical abelian surfaces. Algebra & Number Theory 13(5):1145–1195, 2019.

    CrossRef  MathSciNet  Google Scholar 

  7. Florian Bouyer and Marco Streng. Examples of CM curves of genus two defined over the reflex field. LMS J. Comput. Math., 18(1):507–538, 2015.

    CrossRef  MathSciNet  Google Scholar 

  8. Kevin Buzzard. Integral models of certain Shimura curves. Duke Math. J., 87(3):591–612, 1997.

    CrossRef  MathSciNet  Google Scholar 

  9. J. E. Cremona. Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields. Compositio Mathematica, 51:275–323, 1984.

    MathSciNet  MATH  Google Scholar 

  10. J. E. Cremona. Abelian varieties with extra twist, cusp forms, and elliptic curves over imaginary quadratic fields. J. London Math. Soc. (2), 45(3):404–416, 1992.

    CrossRef  MathSciNet  Google Scholar 

  11. Pierre Deligne. Travaux de Shimura. pages 123–165. Lecture Notes in Math., Vol. 244, 1971.

    Google Scholar 

  12. Luis Dieulefait, Lucio Guerberoff, and Ariel Pacetti. Proving modularity for a given elliptic curve over an imaginary quadratic field. Math. Comp., 79(270):1145–1170, 2010.

    CrossRef  MathSciNet  Google Scholar 

  13. Koji Doi and Hidehisa Naganuma. On the functional equation of certain Dirichlet series. Invent. Math., 9:1–14, 1969/1970.

    CrossRef  MathSciNet  Google Scholar 

  14. P. Deligne and M. Rapoport. Les schémas de modules de courbes elliptiques. pages 143–316. Lecture Notes in Math., Vol. 349, 1973.

    Google Scholar 

  15. Noam D. Elkies. Shimura curve computations. In Algorithmic number theory (Portland, OR, 1998), volume 1423 of Lecture Notes in Comput. Sci., pages 1–47. Springer, Berlin, 1998.

    Google Scholar 

  16. Solomon Friedberg. On the imaginary quadratic Doi-Naganuma lifting of modular forms of arbitrary level. Nagoya Math. J., 92:1–20, 1983.

    CrossRef  MathSciNet  Google Scholar 

  17. P. Gérardin and J.-P. Labesse. The solution of a base change problem for GL(2) (following Langlands, Saito, Shintani). In Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 115–133. Amer. Math. Soc., Providence, R.I., 1979.

    Google Scholar 

  18. Jia-Wei Guo and Yifan Yang. Equations of hyperelliptic Shimura curves. Compos. Math., 153(1):1–40, 2017.

    CrossRef  MathSciNet  Google Scholar 

  19. John Jones and David Roberts. A database of number fields. LMS Journal of Computation and Mathematics, 17(1):595–618, 2014.

    CrossRef  MathSciNet  Google Scholar 

  20. Bruce W. Jordan and Ron A. Livné. Local Diophantine properties of Shimura curves. Math. Ann., 270(2):235–248, 1985.

    CrossRef  MathSciNet  Google Scholar 

  21. Bruce Winchester Jordan. On The Diophantine Arithmetic of Shimura Curves. ProQuest LLC, Ann Arbor, MI, 1981. Thesis (Ph.D.)–Harvard University.

    Google Scholar 

  22. Masanari Kida. Galois descent and twists of an abelian variety. Acta Arith., 73(1):51–57, 1995.

    CrossRef  MathSciNet  Google Scholar 

  23. Robert P. Langlands. Base change for GL(2), volume 96 of Annals of Mathematics Studies. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980.

    Google Scholar 

  24. LMFDB Collaboration. The L-functions and modular forms database. http://www.lmfdb.org, 2020. [Online; accessed 3 August 2020].

  25. Erez Lapid and Jonathan Rogawski. On twists of cuspidal representations of GL(2). Forum Math., 10(2):175–197, 1998.

    CrossRef  MathSciNet  Google Scholar 

  26. The PARI Group, Univ. Bordeaux. PARI/GP version 2.13.2, 2021. available from http://pari.math.u-bordeaux.fr/.

  27. Elisabeth E. Pyle. Abelian varieties over \(\mathbb Q\) with large endomorphism algebras and their simple components over \(\overline {\mathbb Q}\). In Modular curves and abelian varieties, volume 224 of Progr. Math., pages 189–239. Birkhäuser, Basel, 2004.

    Google Scholar 

  28. Jordi Quer. Fields of definition of building blocks. Math. Comp., 78(265):537–554, 2009.

    CrossRef  MathSciNet  Google Scholar 

  29. Dinakar Ramakrishnan. Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2). Ann. of Math. (2), 152(1):45–111, 2000.

    CrossRef  MathSciNet  Google Scholar 

  30. Kenneth A. Ribet. Twists of modular forms and endomorphisms of abelian varieties. Math. Ann., 253(1):43–62, 1980.

    CrossRef  MathSciNet  Google Scholar 

  31. Kenneth A. Ribet. Abelian varieties over Q and modular forms. In Modular curves and abelian varieties, volume 224 of Progr. Math., pages 241–261. Birkhäuser, Basel, 2004.

    Google Scholar 

  32. Mattia Sanna. On the equivalence of 3-adic representations. University of Warwick PhD thesis, 2020. Available from http://wrap.warwick.ac.uk/156281/

  33. Michael Stoll and John E. Cremona. On the reduction theory of binary forms. J. Reine Angew. Math., 565:79–99, 2003.

    MathSciNet  MATH  Google Scholar 

  34. Ciaran Schembri. Examples of genuine QM abelian surfaces which are modular. Res. Number Theory, 5(1):Art. 11, 12, 2019.

    Google Scholar 

  35. Goro Shimura. Construction of class fields and zeta functions of algebraic curves. Ann. of Math. (2), 85:58–159, 1967.

    CrossRef  MathSciNet  Google Scholar 

  36. Goro Shimura. Introduction to the arithmetic theory of automorphic functions, volume 11 of Publications of the Mathematical Society of Japan. Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original, Kanô Memorial Lectures, 1.

    Google Scholar 

  37. Richard Taylor. l-adic representations associated to modular forms over imaginary quadratic fields. II. Invent. Math., 116(1-3):619–643, 1994.

    Google Scholar 

  38. Richard Taylor. Representations of Galois groups associated to modular forms. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 435–442. Birkhäuser, Basel, 1995.

    Google Scholar 

  39. George C. Ţurcaş. On Fermat’s equation over some quadratic imaginary number fields. Res. Number Theory, 4(2):Art. 24, 16, 2018.

    Google Scholar 

  40. John Voight. Quaternion algebras. May 26, 2019. available from http://quatalg.org/.

Download references

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).

Author information

Authors and Affiliations

  1. Warwick Mathematics Institute, Zeeman Building, University of Warwick, Coventry, UK

    J. E. Cremona

  2. Department of Mathematics, University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Lassina Dembélé

  3. Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro, Portugal

    Ariel Pacetti

  4. Department of Mathematics, Dartmouth College, Hanover, NH, USA

    Ciaran Schembri & John Voight

Authors
  1. J. E. Cremona
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lassina Dembélé
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ariel Pacetti
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ciaran Schembri
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. John Voight
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to John Voight .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, Boston University, Boston, MA, USA

    Jennifer S. Balakrishnan

  2. Department of Mathematics, Harvard University, Cambridge, MA, USA

    Noam Elkies

  3. Institute for Computational and Experimental Research in Mathematics, Brown University, Providence, RI, USA

    Brendan Hassett

  4. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Prof. Bjorn Poonen

  5. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Andrew V. Sutherland

  6. Department of Mathematics, Dartmouth College, Hanover, NH, USA

    John Voight

Rights and permissions

Reprints and Permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

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

Download citation