Skip to main content
SpringerLink
Log in

Counting and equidistribution for quaternion algebras

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We aim at studying automorphic forms of bounded analytic conductor in the totally definite quaternion algebra setting. We prove the equidistribution of the universal family with respect to an explicit and geometrically meaningful measure. It leads to answering the Sato–Tate conjectures in this case, and contains the counting law of the universal family, with a power savings error term.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adler, J.D., DeBacker, S., Sally, P.Jr., Spice, L.: Supercuspidal characters of SL(2) over a p-adic field. In: Harmonic Analysis on Reductive, p-adic Groups. Amer. Math. Soc. pp. 19–69 (2011)

  2. Arthur, J.: The characters of supercuspidal representations as weighted orbital integrals. Proc. Indian Acad. Sci. Math. Sci. 97, 3–19 (1988)

    Article  MathSciNet  Google Scholar 

  3. Arthur, J.: An introduction to the trace formula. In: Harmonic analysis, the trace formula, and Shimura varieties. Clay. Math. Proc. vol. 4, pp. 1–263 (2005)

  4. Batyrev, V.V., Manin, Y.I.: Sur le nombre des points rationnels de hauteur bornée des variétés algébriques. Math. Ann. 286, 27–43 (1990)

    Article  MathSciNet  Google Scholar 

  5. Bernstein, J., Deligne, P., Kazhdan, D.: Trace paley-wiener theorem for reductivep-adic groups. J. Anal. Math. 47, 180–192 (1986)

    Article  Google Scholar 

  6. Binder, J.: Fields of Rationality of Cusp Forms. Israel J. Math. 222, 973–1028 (2017)

    Article  MathSciNet  Google Scholar 

  7. Brooks, E.H., Petrow, I.: Counting automorphic forms on norm one tori. Acta Arith. 183, 117–143 (2018)

    Article  MathSciNet  Google Scholar 

  8. Brumley, F.: Effective Multiplicity One for GL(N) and narrow zero-free regions for Rankin–Selberg L-functions. Am. J. Math. 128, 1455–1474 (2006)

    Article  MathSciNet  Google Scholar 

  9. Brumley, F., Milićević, D.: Counting cusp forms with analytic conductor (2018). Preprint arXiv:1805.00633

  10. Casselman, W.: On Some Results by Atkin and Lehner. Math. Ann. 201, 301–314 (1973)

    Article  MathSciNet  Google Scholar 

  11. Chambert-Loir, A., Tschinkel, Y.: Igusa integrals and volume asymptotics in analytic and adelic geometry. Confluentes Math. 2, 351–429 (2010)

    Article  MathSciNet  Google Scholar 

  12. Clozel, L., Delorme, P.: Le théorème de Paley-Wiener invariant pour les groupes réductifs. II. Ann. Sci. ÉNS. 23, 193–228 (1990)

    MATH  Google Scholar 

  13. Conrey, J.B., Farmer, D.W., Keating, J.P., Rubinstein, M.O., Snaith, N.C.: Integral moments of L-functions. Proc. Lond. Math. Soc. 91, 33–104 (2005)

    Article  MathSciNet  Google Scholar 

  14. Diamond, F., Shurman, J.: A First Course in Modular Forms. Graduate Texts in Math 228. Springer (2005)

  15. Gelbart, S.: Automorphic Forms on Adele Groups. Ann. Math. Studies. 83, Princeton University Press, Princeton (1975)

  16. Godement, R., Jacquet, H.: Zeta Functions of Simple Algebras. Lecture Notes in Math 260. Springer, Berlin (1972)

  17. Hare, K.E.: The size of characters of compact Lie groups. Stud. Math. 129, 1–18 (1998)

    Article  MathSciNet  Google Scholar 

  18. Henniart, G.: On the local Langlands and Jacquet–Langlands correpsondence. International Congress of Mathematicians. II, pp. 1171–1182 (2006)

  19. Hull, R.: The maximal order of generalized quaternion division algebras. Bull. Am. Math. Soc. 22, 1–11 (1936)

    MathSciNet  MATH  Google Scholar 

  20. Iwaniec, H., Sarnak, P.: Perspectives on the Analytic Theory of L-functions. Geom. Funct. Anal. pp. 705–741 (2000)

  21. Kim, J.-L., Shin, S.W., Templier, N.: Asymptotics and Local Constancy of Characters of p-Adic Groups. In: Simons Symposia. Springer, pp. 259–295 (2016)

  22. Knapp, A.W.: Representation Theory of Semisimple Groups: An Overview based on Examples. Princeton Landmarks in Math 36 Princeton University Press, Princeton (1986)

  23. Knapp, A.W.: Local Langlands Correspondence The Archimedean Case. In: Proc. Symp. Pure Math 55, 393–410. Amer. Math. Soc. (1994)

  24. Knapp, A.W., Trappa, P.E.: Representation of semisimple lie groups. In: Representation Theory of Lie Groups. Amer. Math. Soc, pp. 5–88 (2000)

  25. Knightly, A., Li, C.: Traces of Hecke operators. Math. Surveys and Monographs 133. Amer. Math. Soc (2006)

  26. Landau, E.: Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale. B. G. Teubner, Leipzig (1918)

    MATH  Google Scholar 

  27. Lang, S.: SL(2, R), Graduate Texts in Mathematics 105. Springer, Berlin (1985)

    Google Scholar 

  28. Langlands, R.P., Labesse, J.-P.: L-indistinguishability for SL(2). Can. J. Math. 31, 726–785 (1979)

    Article  MathSciNet  Google Scholar 

  29. Lansky, J., Raghuram, A.: On the correspondence of representations between GL(n) and division algebras. Proc. Am. Math. Soc. 131, 1641–1648 (2004)

    Article  MathSciNet  Google Scholar 

  30. Lesesvre, D.: Arithmetic Statistics for Quaternion Algebras. PhD dissertation, Université Paris 13 (2018)

  31. Northcott, D.G.: Periodic points on an algebraic variety. Ann. Math. 51, 167–177 (1950)

    Article  MathSciNet  Google Scholar 

  32. Petrow, I.: The Weyl law on algebraic tori (2018). Preprint https://arxiv.org/pdf/1808.09991.pdf

  33. Peyre, E.: Hauteurs et mesures de Tamagawa sur les variétés de Fano. Duke J. Math. 79, 101–218 (1995)

    Article  MathSciNet  Google Scholar 

  34. Prasad, D., Raghuram, A.: Representation Theory of GL(n) over Non-Archimedean Local Fields. Lectures given at the School on Automorphic Forms on GL(n). pp. 1-48 (2000)

  35. Sally, P.J., Shalika, J.: Characters of the discrete series of representations of SL(2) over a local field. Proc. Natl. Acad. Sci. USA 61, 1231–1237 (1968)

    Article  MathSciNet  Google Scholar 

  36. Sarnak, P.: Statistical properties of eigenvalues of the hecke operators. In: Analytic Number Theory and Diophantine Problems. Birkhäuser, pp. 321–331 (1987)

  37. Sarnak, P., Shin, S.W., Templier, N.: Families of L-functions and their symmetries. In: Families of automorphic forms and the trace formula. Springer, pp. 531–578 (2016)

  38. Sauvageot, F.: Principe de densité pour les groupes réductifs. Compos. Math. 108, 151–184 (1997)

    Article  MathSciNet  Google Scholar 

  39. Schanuel, S.: On heights in number fields. Bull. Am. Math. Soc. 70, 262–263 (1964)

    Article  MathSciNet  Google Scholar 

  40. Serre, J.-P.: Répartition asymptotique des valeurs propres de l’opérateur de Hecke T(p). J. Am. Math. Soc. 10, 75–102 (1997)

    Article  Google Scholar 

  41. Shin, S.W., Templier, N.: Sato-Tate theorem for families and low-lying zeros of automorphic L-functions. Invent. Math. 1, 1–177 (2016)

    Article  MathSciNet  Google Scholar 

  42. Waldspurger, J.-L.: La formule de Plancherel pour les groupes p-adiques, d’après Harish-Chandra. J. Inst. Math Jussieu. 2, 235–333 (2003)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

I am infinitely indebted to my advisor, Farrell Brumley, for having trusted in me for handling this problem and for having been strongly present and implied in its resolution. I am grateful to the time Gergely Harcos and Philippe Michel granted me in carefully reading the thesis from which this article blossomed. I would like to thank Valentin Blomer, Ernest Hunter Brooks, Andrew Corbett, Mikolaj Fraczyk, Élie Goudout, Guy Henniart and Ian Petrow for many enlightening discussions. This work has been faithfully supported by the ANR 14-CE25 PerCoLaTor, the Fondation Sciences Mathématiques de Paris and the Deutscher Akademischer Austauschdienst. At last, nothing would have been done without the warm and peaceful environments of the institutions which hosted me during these years: Université Paris 13, École Polytechnique Fédérale de Lausanne, Georg-August Universität and Sun Yat-Sen University.

Author information

Authors and Affiliations

  1. School of Mathematics, Sun Yat-Sen University, Zhuhai Campus, Tangjiawan, Zhuhai, 519082, Guangdong, China

    Didier Lesesvre

Authors
  1. Didier Lesesvre
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Didier Lesesvre.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lesesvre, D. Counting and equidistribution for quaternion algebras. Math. Z. 295, 129–159 (2020). https://doi.org/10.1007/s00209-019-02401-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-019-02401-x

Keywords

Search

Navigation