zbMATH — the first resource for mathematics

\(p\)-adic periods, \(p\)-adic \(L\)-functions, and the \(p\)-adic uniformization of Shimura curves. (English) Zbl 1037.11045
From the introduction: Let \(E/\mathbb Q\) be a modular elliptic curve of conductor \(N\), and let \(p\) be a prime of split multiplicative reduction for \(E\). Write \(\mathbb C_p\) for a fixed completion of an algebraic closure of \(\mathbb Q_p\). Tate’s theory of \(p\)-adic uniformization of elliptic curves yields a rigid-analytic, \(\text{Gal} (\mathbb C_p, \mathbb Q_p)\)-equivariant uniformization of the \(\mathbb C_p\)-points of \[ E: 0\to q^{\mathbb Z}\to \mathbb C_p^\times @>\Phi_{\text{Tate}}>> E(\mathbb C_p)\to 0, \tag{1} \] where \(q\in p\mathbb Z_p\) is the \(p\)-adic period of \(E\).
B. Mazur, J. Tate and J. Teitelbaum conjectured in [[MTT] Invent. Math. 84, 1–48 (1986; Zbl 0699.14028)] that the cyclotomic \(p\)-adic \(L\)-function \(E/\mathbb Q\) vanishes at the central point to order one greater than that of its classical counterpart. Furthermore, they proposed a formula for the leading coefficient of such a \(p\)-adic \(L\)-function. In the special case where the analytic rank of \(E(\mathbb Q)\) is zero, they predicted that the ratio of the special value of the first derivative of the cyclotomic \(p\)-adic \(L\)-function and the algebraic part of the special value of the complex \(L\)-function of \(E/\mathbb Q\) is equal to the quantity \(\frac {\log_p(q)} {\text{ord}_p(q)}\) (where \(\log_p\) is Iwasawa’s cyclotomic logarithm), which is defined purely in terms of the \(p\)-adic uniformization of \(E\). Greenberg and Stevens gave a proof of this special case.
The article [M. Bertolini and H. Darmon, Invent. Math. 126, 413–456 (1996; Zbl 0882.11034); ibid. 453–491 (1998; Zbl 0899.11029)] formulates an analogue of the conjectures of [MTT] in which the cyclotomic \(\mathbb Z_p\)-extension of \(\mathbb Q\) is replaced by the anticyclotomic \(\mathbb Z_p\)-extension of an imaginary quadratic field \(K\). When \(p\) is split in \(K\) and the sign of the functional equation of \(L(E/K,s)\) is \(+1\), this conjecture relates the first derivative of the anticyclotomic \(p\)-adic \(L\)-function of \(E\) to the anticyclotomic logarithm of the \(p\)-adic period of \(E\). The present paper supplies a proof of this conjecture. Our proof is based on the theory of \(p\)-adic uniformization of Shimura curves.
The main result is the following.
Theorem. Suppose that \(c=1\). The equality (up to sign) \[ {\mathcal L}_p' (E/K)= \frac {\text{rec}_p(q)} {\text{ord}_p(q)} \sqrt{ L(E/K,1) \Omega_f^{-1} \cdot d^{1/2} u^2 n_f} \] holds in \(I/I^2 \otimes \mathbb Q\).

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G18 Arithmetic aspects of modular and Shimura varieties
11F33 Congruences for modular and \(p\)-adic modular forms
Full Text: DOI
[1] Katia Barré-Sirieix, Guy Diaz, François Gramain, and Georges Philibert, Une preuve de la conjecture de Mahler-Manin , Invent. Math. 124 (1996), no. 1-3, 1-9. · Zbl 0853.11059
[2] M. Bertolini and H. Darmon, Heegner points on Mumford-Tate curves , Invent. Math. 126 (1996), no. 3, 413-456. · Zbl 0882.11034
[3] Massimo Bertolini and Henri Darmon, Heegner points, \(p\)-adic \(L\)-functions, and the Cerednik-Drinfeld uniformization , Invent. Math. 131 (1998), no. 3, 453-491. · Zbl 0899.11029
[4] X. Boichut, On the Mazur-Tate-Teitelbaum \(p\)-adic conjecture for elliptic curves with bad reduction at \(p\) , in preparation.
[5] 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, 7, 45-158 (1992), in Courbes modulaires et courbes de Shimura (Orsay, 1987/1988), Soc. Math. France, Marseille. · Zbl 0781.14010
[6] I. V. Čerednik, Uniformization of algebraic curves by discrete arithmetic subgroups of \(\mathrm PGL_2(k\sbw)\) with compact quotient spaces , Mat. Sb. (N.S.) 100(142) (1976), no. 1, 59-88, 165, (in Russian); English trans. in Math. USSR-Sb. 29 (1976), 55-78.
[7] H. Daghigh, Modular forms, quaternion algebras, and special values of \(L\)-functions , Ph.D. thesis, McGill University, 1997.
[8] V. G. Drinfeld, Coverings of \(p\)-adic symmetric domains , Funkcional. Anal. i Priložen. 10 (1976), no. 2, 29-40. · Zbl 0346.14010
[9] Lothar Gerritzen and Marius van der Put, Schottky groups and Mumford curves , Lecture Notes in Mathematics, vol. 817, Springer, Berlin, 1980. · Zbl 0442.14009
[10] Ralph Greenberg and Glenn Stevens, \(p\)-adic \(L\)-functions and \(p\)-adic periods of modular forms , Invent. Math. 111 (1993), no. 2, 407-447. · Zbl 0778.11034
[11] Benedict H. Gross, Heights and the special values of \(L\)-series , Number theory (Montreal, Que., 1985) eds. H. Kisilevsky and J. Labute, CMS Conf. Proc., vol. 7, Amer. Math. Soc., Providence, RI, 1987, pp. 115-187. · Zbl 0623.10019
[12] Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of \(L\)-series , Invent. Math. 84 (1986), no. 2, 225-320. · Zbl 0608.14019
[13] H. Jacquet and R. P. Langlands, Automorphic forms on \(\mathrm GL(2)\) , Lecture Notes in Math., vol. 114, Springer-Verlag, Berlin, 1970. · Zbl 0236.12010
[14] K. Kato, M. Kurihara, and Tsuji, forthcoming work.
[15] Christoph Klingenberg, On \(p\)-adic \(L\)-functions of Mumford curves , \(p\)-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991) eds. B. Mazur and G. Stevens, Contemp. Math., vol. 165, Amer. Math. Soc., Providence, RI, 1994, pp. 277-315. · Zbl 0863.14014
[16] Y. I. Manin, \(p\)-adic automorphic functions , J. Soviet Math. 5 (1976), 279-333. · Zbl 0375.14007
[17] B. Mazur, J. Tate, and J. Teitelbaum, On \(p\)-adic analogues of the conjectures of Birch and Swinnerton-Dyer , Invent. Math. 84 (1986), no. 1, 1-48. · Zbl 0699.14028
[18] Kenneth A. Ribet and Shuzo Takahashi, Parametrizations of elliptic curves by Shimura curves and by classical modular curves , Proc. Nat. Acad. Sci. U.S.A. 94 (1997), no. 21, 11110-11114. JSTOR: · Zbl 0897.11018
[19] P. Schneider, Rigid-analytic \(L\)-transforms , Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983), Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 216-230. · Zbl 0572.14014
[20] Jean-Pierre Serre, Arbres, amalgames, \(\mathrm SL_2\) , Astérisque, vol. 46, Société Mathématique de France, Paris, 1977. · Zbl 0369.20013
[21] Takuro Shintani, On construction of holomorphic cusp forms of half integral weight , Nagoya Math. J. 58 (1975), 83-126. · Zbl 0316.10016
[22] Jeremy T. Teitelbaum, Values of \(p\)-adic \(L\)-functions and a \(p\)-adic Poisson kernel , Invent. Math. 101 (1990), no. 2, 395-410. · Zbl 0731.11065
[23] Marie-France Vignéras, Arithmétique des algèbres de quaternions , Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980. · Zbl 0422.12008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.