SpringerLink
Log in

Root Numbers of 5-adic Curves of Genus Two Having Maximal Ramification

  • Published:
Milan Journal of Mathematics Aims and scope Submit manuscript

Abstract

The formulas for local root numbers of abelian varieties of dimension one are known. In this paper we treat the simplest unknown case in dimension two by considering a curve of genus 2 defined over a 5-adic field such that the inertia acts on the first \(\ell \)-adic cohomology group through the largest possible finite quotient, isomorphic to \(C_5\rtimes C_8\). We give a few criteria to identify such curves and prove a formula for their local root numbers in terms of invariants associated to a Weierstrass equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The formula is stated for the \(\epsilon _0\)-factor of \(\chi \), which is equal to the \(\epsilon \)-factor when \(\chi \) is ramified.

References

  1. Abbes, A., Saito, T.: Local Fourier transform and epsilon factors. Compos. Math. 146(6), 1507–1551 (2010). https://doi.org/10.1112/S0010437X09004631

    Article  MathSciNet  MATH  Google Scholar 

  2. Berndt, B.C., Evans, R.J., Williams, K.S.: Gauss and Jacobi Sums (Canadian Mathematical Society Series of Monographs and Advanced Texts), pp. xii+583. Wiley, New York (1998)

  3. Bisatt, M.: Explicit root numbers of abelian varieties. Trans. Am. Math. Soc. 372(11), 7889–7920 (2019). https://doi.org/10.1090/tran/7926

    Article  MathSciNet  MATH  Google Scholar 

  4. Bisatt, M.: Root number of the jacobian of y\(^2 = x^p\) + a, (2021). ar**v: 2102.05720 [math.NT]

  5. Brumer, A., Kramer, K., Sabitova, M.: Explicit determination of root numbers of abelian varieties. Trans. Am. Math. Soc. 370(4), 2589–2604 (2018). https://doi.org/10.1090/tran/7116

    Article  MathSciNet  MATH  Google Scholar 

  6. Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models (Ergeb. Math. Grenzgeb. (3)), vol. 21, pp. x+325. Springer, Berlin (1990) https://doi.org/10.1007/978-3-642-51438-8

  7. Chai, C.-L.: Néron models for semiabelian varieties: Congruence and change of base field. Asian J. Math. 4(4), 715–736 (2000). https://doi.org/10.4310/AJM.2000.v4.n4.a1

    Article  MathSciNet  MATH  Google Scholar 

  8. Coppola, N.: Wild galois representations: A family of hyperelliptic curves with large inertia image (2020). ar**v:2001.08287 [math.NT]

  9. Dokchitser, T., Dokchitser, V.: Root numbers of elliptic curves in residue characteristic 2. Bull. Lond. Math. Soc. 40(3), 516–524 (2008). https://doi.org/10.1112/blms/bdn034

    Article  MathSciNet  MATH  Google Scholar 

  10. Deligne, P. Les constantes des équations fonctionnelles des fonctions L. In: Modular Functions of One Variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp,: ser. Lecture Notes in Math., vol. 349, pp. 501–597. Springer, Berlin (1972)

  11. Dokchitser, T.: GroupNames. (Feb. 2020), [Online]. http://groupnames.org

  12. Hasse, H.: Theorie der relativ-zyklischen algebraischen Funktionenkörper, insbesondere bei endlichem Konstantenkörper. J. Reine Angew. Math. 172, 37–54 (1935). https://doi.org/10.1515/crll.1935.172.37

    Article  MathSciNet  MATH  Google Scholar 

  13. Homma, M.: Automorphisms of prime order of curves. Manuscripta Math. 33(1), 99–109 (1980/81). https://doi.org/10.1007/BF01298341

  14. Kobayashi, S.: The local root number of elliptic curves with wild ramification. Math. Ann. 323(3), 609–623 (2002). https://doi.org/10.1007/s002080200318

    Article  MathSciNet  MATH  Google Scholar 

  15. Kraus, A.: Sur le défaut de semi-stabilité des courbes elliptiques á réduction additive. Manuscripta Math. 69(4), 353–385 (1990). https://doi.org/10.1007/BF02567933

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, Q.: Courbes stables de genre 2 et leur schéma de modules. Math. Ann. 295(2), 201–222 (1993). https://doi.org/10.1007/BF01444884

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu, Q.: Conducteur et discriminant minimal de courbes de genre 2. Compos. Math. 94(1), 51–79 (1994)

    MathSciNet  MATH  Google Scholar 

  18. Liu, Q.: Modéles minimaux des courbes de genre deux. J. Reine Angew. Math. 453, 137–164 (1994). https://doi.org/10.1515/crll.1994.453.137

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, Q.: Modéles entiers des courbes hyperelliptiques sur un corps de valuation discréte. Trans. Am. Math. Soc. 348(11), 4577–4610 (1996). https://doi.org/10.1090/S0002-9947-96-01684-4

    Article  MATH  Google Scholar 

  20. Liu, Q.: Algebraic Geometry and Arithmetic Curves (Oxford Graduate Texts in Mathematics), vol. 6, pp. xvi+576. Oxford University Press, Oxford (2002)

  21. The LMFDB Collaboration: The L-functions and modular forms database. http://www.lmfdb.org (2021)

  22. Liu, Q., Tong, J.: Néron models of algebraic curves. Trans. Am. Math. Soc. 368(10), 7019–7043 (2016). https://doi.org/10.1090/tran/6642

    Article  MATH  Google Scholar 

  23. Mumford, D.: Tata Lectures on Theta. II (Progress in Mathematics), vol. 43, pp. xiv+272. Birkhäuser Boston, Inc., Boston (1984). https://doi.org/10.1007/978-0-8176-4578-6

  24. Neukirch, J. Algebraic Number Theory (Grundlehren Math. Wiss.), vol. 322, pp. xviii+571. Springer, Berlin (1999). https://doi.org/10.1007/978-3-662-03983-0

  25. Namikawa, Y., Ueno, K.: The complete classification of fibres in pencils of curves of genus two. Manuscripta Math. 9, 143–186 (1973). https://doi.org/10.1007/BF01297652

    Article  MathSciNet  MATH  Google Scholar 

  26. Rohrlich, D. E.: Elliptic Curves and the Weil–Deligne Group, in Elliptic Curves and Related Topics, ser. CRM Proc. Lecture Notes, vol. 4, pp. 125–157. Amer. Math. Soc., Providence (1994)

  27. Rohrlich, D.E.: Galois theory, elliptic curves, and root numbers. Compositio Math. 100(3), 311–349 (1996)

    MathSciNet  MATH  Google Scholar 

  28. Roquette, P.: Abschätzung der Automorphismenanzahl von Funktionenkörpern bei Primzahlcharakteristik. Math. Z. 117, 157–163 (1970). https://doi.org/10.1007/BF01109838

    Article  MathSciNet  MATH  Google Scholar 

  29. Sabitova, M.: Root numbers of abelian varieties. Trans. Am. Math. Soc. 359(9), 4259–4284 (2007). https://doi.org/10.1090/S0002-9947-07-04148-7

    Article  MathSciNet  MATH  Google Scholar 

  30. Serre, J.-P.: Rigidité du foncteur de jacobi d’échelon n \(\geqslant \) 3, Appendix to A. Grothendieck, Techniques de construction en géométrie analytique, X. Construction de l’espace de Teichmüller, Séminaire Henri Cartan, 1960/61, no. 17

  31. Serre, J.-P.: Local Fields (Graduate Texts in Mathematics). Springer, New York (1977)

  32. Serre, J.-P., Tate, J.: Good reduction of abelian varieties. Ann. Math. 2(88), 492–517 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  33. Stichtenoth, H.: Algebraic Function Fields and Codes (Graduate Texts in Mathematics), Second, vol. 254, pp. xiv+355. Springer, Berlin (2009)

  34. Silverberg, A., Zarhin, Y.G.: Inertia groups and abelian surfaces. J. Number Theory 110(1), 178–198 (2005). https://doi.org/10.1016/j.jnt.2004.05.015

    Article  MathSciNet  MATH  Google Scholar 

  35. Yelton, J.: Images of 2-adic representations associated to hyperelliptic Jacobians. J. Number Theory 151, 7–17 (2015). https://doi.org/10.1016/j.jnt.2014.10.020

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I thank my doctoral thesis advisors Adriano Marmora and Rutger Noot as well as Kȩstutis Česnavičius and Takeshi Saito for their remarks and suggestions concerning the manuscript. I also thank Jeff Yelton for answering my questions about his results on the splitting fields of the 4-torsion of Jacobians. The study presented in this paper constitutes a part of my thesis.

Author information

Authors and Affiliations

  1. IRMA, UMR 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67000, Strasbourg, France

    Lukas Melninkas

Authors
  1. Lukas Melninkas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lukas Melninkas.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Melninkas, L. Root Numbers of 5-adic Curves of Genus Two Having Maximal Ramification. Milan J. Math. 91, 255–279 (2023). https://doi.org/10.1007/s00032-023-00380-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00032-023-00380-7

Mathematics Subject Classification

Search

Navigation