Overconvergent modular symbols and $p$-adic $L$-functions

Pollack, Robert; Stevens, Glenn

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 44 (2011) no. 1, pp. 1-42

This paper is a constructive investigation of the relationship between classical modular symbols and overconvergent $p$-adic modular symbols. Specifically, we give a constructive proof of a control theorem (Theorem 1.1) due to the second author [19] proving existence and uniqueness of overconvergent eigenliftings of classical modular eigensymbols of non-critical slope. As an application we describe a polynomial-time algorithm for explicit computation of associated $p$-adic $L$-functions in this case. In the case of critical slope, the control theorem fails to always produce eigenliftings (see Theorem 5.14 and [16] for a salvage), but the algorithm still “succeeds” at producing $p$-adic $L$-functions. In the final two sections we present numerical data in several critical slope examples and examine the Newton polygons of the associated $p$-adic $L$-functions.

Cet article est une exploration constructive des rapports entre les symboles modulaires classiques et les symboles modulaires $p$-adiques surconvergents. Plus précisément, nous donnons une preuve constructive d’un théorème de contrôle (Théorème 1.1) du deuxième auteur [19] ; ce théorème démontre l’existence et l’unicité des « liftings propres » des symboles propres modulaires classiques de pente non-critique. Comme application, nous décrivons un algorithme en temps polynomial pour le calcul explicite des fonctions $L$ $p$-adiques associées dans ce cas-là. Dans le cas de pente critique, le théorème de contrôle échoue toujours à produire des « liftings propres » (voir Théorème 5.14 et [16] pour un succédané), mais l’algorithme « réussit » néanmoins à produire des fonctions $L$ $p$-adiques. Dans les deux dernières sections, nous présentons des données numériques pour plusieurs exemples de pente critique et examinons le polygone de Newton des fonctions $L$ $p$-adiques associées.

Classification : 11F67

[1] Y. Amice & J. Vélu, Distributions $p$-adiques associées aux séries de Hecke, Astérisque 24-25 (1975), 119-131. | Zbl | Numdam

[2] A. Ash & G. Stevens, Modular forms in characteristic $l$ and special values of their $L$-functions, Duke Math. J. 53 (1986), 849-868. | Zbl | MR

[3] W. Bosma, J. Cannon & C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 235-265. | Zbl | MR

[4] R. F. Coleman, Classical and overconvergent modular forms, Invent. Math. 124 (1996), 215-241. | Zbl | MR

[5] H. Darmon, Integration on ${\mathscr{H}}_p\times \mathscr{H}$ and arithmetic applications, Ann. of Math. 154 (2001), 589-639. | Zbl | MR

[6] H. Darmon & R. Pollack, Efficient calculation of Stark-Heegner points via overconvergent modular symbols, Israel J. Math. 153 (2006), 319-354. | Zbl | MR

[7] M. Greenberg, Lifting modular symbols of non-critical slope, Israel J. Math. 161 (2007), 141-155. | Zbl | MR

[8] R. Greenberg, Iwasawa theory for elliptic curves, in Arithmetic theory of elliptic curves (Cetraro, 1997), Lecture Notes in Math. 1716, Springer, 1999, 51-144. | Zbl | MR

[9] R. Greenberg & G. Stevens, On the conjecture of Mazur 1991), Contemp. Math. 165, Amer. Math. Soc., 1994, 183-211. | Zbl | MR

[10] R. Greenberg & V. Vatsal, On the Iwasawa invariants of elliptic curves, Invent. Math. 142 (2000), 17-63. | Zbl | MR

[11] M. Kurihara & R. Pollack, Two $p$-adic $L$-functions and rational points on elliptic curves with supersingular reduction, in $L$-functions and Galois representations, London Math. Soc. Lecture Note Ser. 320, Cambridge Univ. Press, 2007, 300-332. | Zbl | MR

[12] J. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19-66. | Zbl | MR

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

[14] D. Pollack & R. Pollack, A construction of rigid analytic cohomology classes for congruence subgroups of ${\mathrm{SL}}_3\left(\mathbb{Z}\right)$, Canad. J. Math. 61 (2009), 674-690. | Zbl | MR

[15] R. Pollack, Tables of Iwasawa invariants of elliptic curves, http://math.bu.edu/people/rpollack/Data/data.html.

[16] R. Pollack & G. Stevens, Critical slope $p$-adic $L$-function, preprint http://math.bu.edu/people/rpollack/Papers/Critical_slope_padic_Lfunctions.pdf. | MR

[17] D. E. Rohrlich, On $L$-functions of elliptic curves and cyclotomic towers, Invent. Math. 75 (1984), 409-423. | Zbl | MR

[18] J-P. Serre, Endomorphismes complètement continus des espaces de Banach $p$-adiques, Publ. Math. I.H.É.S. 12 (1962), 69-85. | Zbl | MR | Numdam

[19] G. Stevens, Rigid analytic modular symbols, preprint.

[20] M. Trifković, Stark-Heegner points on elliptic curves defined over imaginary quadratic fields, Duke Math. J. 135 (2006), 415-453. | Zbl | MR

[21] M. M. Višik, Nonarchimedean measures associated with Dirichlet series, Mat. Sb. (N.S.) 99 (141) (1976), 248-260, 296. | Zbl | MR