Abstract
We use a combinatorial result relating the discriminant of the cycle pairing on a weighted finite graph to the eigenvalues of its Laplacian to deduce a formula for the orders of component groups of Jacobians of modular curves arising from quaternion algebras over \({{\mathbb F}_q(T)}\) or \({{\mathbb Q}}\). Our formula over \({{\mathbb Q}}\) recovers a result of Jordan and Livné.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Springer, Berlin, 1990.
J.-F. Boutot and H. Carayol, Uniformisation p-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfeld, Astérisque (1991), no. 196-197, 45–158, Courbes modulaires et courbes de Shimura (Orsay, 1987/1988).
Denert M., Van Geel J.: The class number of hereditary orders in non-Eichler algebras over global function fields. Math. Ann. 282, 379–393 (1988)
Gekeler E.-U., Reversat M.: Jacobians of Drinfeld modular curves. J. Reine Angew. Math. 476, 27–93 (1996)
B. Gross, Heights and the special values of L-series, Number theory (Montreal, Que., 1985), CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 115–187.
A. Grothendieck, Modèles de Néron et monodromie, SGA 7, Exposé IX, 1972.
Hausberger T.: Uniformisation des variétés de Laumon-Rapoport-Stuhler et conjecture de Drinfeld-Carayol. Ann. Inst. Fourier (Grenoble) 55, 1285–1371 (2005)
Jordan B., Livné R.: On the Néron model of Jacobians of Shimura curves. Compositio Math. 60, 227–236 (1986)
Kurihara A.: On some examples of equations defining Shimura curves and the Mumford uniformization. J. Fac. Sci. Univ. Tokyo 25, 277–300 (1979)
Laumon G., Rapoport M., Stuhler U.: \({{\mathscr D}}\)-elliptic sheaves and the Langlands correspondence. Invent. Math. 113, 217–338 (1993)
Q. Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002.
Nagoshi H.: The distribution of eigenvalues of arithmetic infinite graphs. Forum Math. 14, 807–829 (2002)
M. Papikian, Graph Laplacians, component groups and Drinfeld modular curves, Münster J. Math., to appear, available at http://www.math.uni-muenster.de/mjm/.
Papikian M.: On Jacquet-Langlands isogeny over function fields. J. Number Theory 131, 1149–1175 (2011)
Papikian M.: Local Diophantine properties of modular curves of \({{\mathscr D}}\)-elliptic sheaves. J. Reine Angew. Math. 664, 115–140 (2012)
Papikian M., Wei F.-T.: The Eisenstein ideal and Jacquet-Langlands isogeny over function fields. Doc. Math. 20, 551–629 (2015)
Schweizer A.: On the Drinfeld modular polynomial \({{\Phi_T(X, Y)}}\). J. Number Theory 52, 53– (1995)
Serre J.-P.: Répartition asymptotique des valeurs propres de l’opérateur de Hecke T p . J. Amer. Math. Soc. 10, 75–102 (1997)
J.-P. Serre, Trees, Springer Monographs in Mathematics, Springer, Berlin, 2003.
M.-F. Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980.
Wei F.-T., Yu J.: Theta series and function field analogue of Gross formula. Doc. Math. 16, 723–765 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author’s research was partially supported by grants from the Simons Foundation (245 676) and the National Security Agency (H98230-15-1-0008). Dedicated to Ernst-Ulrich Gekeler.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Papikian, M. On component groups of Jacobians of quaternionic modular curves. Arch. Math. 107, 315–328 (2016). https://doi.org/10.1007/s00013-016-0927-x
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-016-0927-x