Quadratic modular symbols on Shimura curves
Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 2, p. 261-283

We introduce the concept of quadratic modular symbol and study how these symbols are related to quadratic p-adic L-functions. These objects were introduced in [3] in the case of modular curves. In this paper, we discuss a method to attach quadratic modular symbols and quadratic p-adic L-functions to more general Shimura curves.

Nous introduisons le concept de symbole modulaire quadratique et nous étudions comment ces symboles sont liés aux fonctions L p-adiques quadratiques. Ces objets ont été introduits dans [3] dans le cas des courbes modulaires. Dans cet article, nous proposons une méthode pour attacher des symboles modulaires et fonctions L p-adiques quadratiques aux courbes de Shimura plus générales.

@article{JTNB_2013__25_2_261_0,
     author = {Bayer, Pilar and Blanco-Chac\'on, Iv\'an},
     title = {Quadratic modular symbols on Shimura curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {25},
     number = {2},
     year = {2013},
     pages = {261-283},
     doi = {10.5802/jtnb.835},
     mrnumber = {3228307},
     zbl = {1295.11052},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2013__25_2_261_0}
}
Bayer, Pilar; Blanco-Chacón, Iván. Quadratic modular symbols on Shimura curves. Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 2, pp. 261-283. doi : 10.5802/jtnb.835. http://www.numdam.org/item/JTNB_2013__25_2_261_0/

[1] M. Alsina, P. Bayer, Quaternion orders, quadratic forms and Shimura curves. CRM Monograph Series 22. American Mathematical Society, 2004. | MR 2038122 | Zbl 1073.11040

[2] M. A. Armstrong, On the fundamental group of an orbit space. Proc. Cambridge Philos. Soc. 61 (1965), 639–646. | MR 187244 | Zbl 0125.40002

[3] P. Bayer, I. Blanco-Chacon, Quadratic modular symbols. RACSAM 106, n. 2 (2012), 429–441. | MR 2978924 | Zbl pre06132876

[4] P. Bayer, A. Travesa (eds.), Corbes modulars: taules. Notes del Seminari de Teoria de Nombres (UB-UAB-UPC). Universitat de Barcelona 1 (1996).

[5] B. Birch, Heegner points: the beginnings. In Heegner points and Rankin L-series. Math. Sci. Res. Inst. Publ. 49. Cambridge Univ. Press, 2004. | MR 2083207 | Zbl 1073.11001

[6] I. Blanco-Chacon, Upper triangular operators and p-adic L-functions. p-adic numbers, p-adic analyisis and applications 3 (2011), 1–14. | MR 2802032 | Zbl pre06105066

[7] J. E. Cremona, Algorithms for modular elliptic curves. Cambridge University Press, 1997. | MR 1628193 | Zbl 0758.14042

[8] H. Jacquet, R. Langlands, Automorphic forms on GL(2). Lecture Notes in Mathematics 114. Springer-Verlag, 1970. | MR 401654 | Zbl 0236.12010

[9] Ju. Y. Manin, Parabolic points and zeta functions of modular curves. Mat. Sbornik 6 (1972), 19–64. | MR 314846 | Zbl 0248.14010

[10] B. Mazur, J. Tate, J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math. 84, n. 1 (1986), 1–48. | MR 830037 | Zbl 0699.14028

[11] R. Pollack, G. Stevens, Overconvergent modular symbols and p-adic L-functions. Ann. Sci. Éc. Norm. Sup. 44, n. 1 (2011), 1–42. | Numdam | MR 2760194 | Zbl 1268.11075

[12] A. M. Robert, A course in p-adic analysis. Graduate Texts in Mathematics 198, Springer, 2000. | MR 1760253 | Zbl 0947.11035

[13] D. E. Rohrlich, L-functions and division towers. Math. Ann. 281 (1988), 611–632. | MR 958262 | Zbl 0656.14013

[14] G. Shimura, Construction of class fields and zeta functions of algebraic curves. Ann. of Math. 85, n. 2 (1967), 58–159. | MR 204426 | Zbl 0204.07201

[15] G. Shimura, Introduction to the arithmetic theory of automorphic forms. Princeton University Press, 1971.

[16] K. Takeuchi, Arithmetic Fuchsian groups with signature (1;e). J. Math. Soc. Japan 35, n. 3 (1983), 381–407. | MR 702765 | Zbl 0517.20022

[17] M. F. Vignéras, Arithmétique des algèbres de quaternions. Lecture Notes in Mathematics 800. Springer, 1980. | MR 580949 | Zbl 0422.12008

[18] Y. Yang, Schwarzian differential equations and Hecke eigenforms on Shimura curves. arXiv: 1110.6284v1, 2011. | MR 3011876