\(p\)-adic \(L\)-functions and \(p\)-adic periods of modular forms. (English) Zbl 0778.11034

Let \(E\) be a modular elliptic curve over \(\mathbb{Q}\), \(p\) be a prime for which \(E\) has split multiplicative reduction, and \(L_ p(E,s)\) be the \(p\)-adic \(L\)-function of \(E\). From the interpolation property of \(L_ p(E,s)\), it is automatically the case that \(L_ p(E,1)=0\).
This paper proves the following formula conjectured by Mazur, Tate, and Teitelbaum: \[ L_ p'(E,1)= {{\log_ p(q_ E)} \over {\text{ord}_ p(q_ E)}} {{L_ \infty(E,1)} \over {\Omega_ E}}. \] Here, \(q_ E\) is the Tate period of \(E\) at \(p\), \(\log_ p\) is Iwasawa’s \(p\)-adic logarithm, \(\text{ord}_ p\) is the normalized valuation at \(p\), \(L_ \infty(E,s)\) is the Hasse-Weil \(L\)-function of \(E\), and \(\Omega_ E\) is the real period of \(E\). The paper actually works in the more general setting of a “split multiplicative” weight 2 newform, but the main motivation is the situation described above.
The proof studies a two variable \(p\)-adic \(L\)-function which specializes to \(L_ p(E,s)\). The authors are actually able to determine the constant term of the two-variable \(p\)-adic \(L\)-function, from which they derive their result.
Reviewer: J.Jones (Tempe)


11G05 Elliptic curves over global fields
11R23 Iwasawa theory
14G20 Local ground fields in algebraic geometry
11F85 \(p\)-adic theory, local fields
11S40 Zeta functions and \(L\)-functions
Full Text: DOI EuDML


[1] [A-S] Ash, A., Stevens, G.: Modular forms in characteristicl and special values of theirL-functions. Duke Math. J.53, 849-868 (1986) · Zbl 0618.10026 · doi:10.1215/S0012-7094-86-05346-9
[2] [A-L] Atkin, A.O.L., Li, W.: Twists of Newforms and Pseudo-Eigenvalues ofW-operators. Invent. Math.48, 221-243 (1978) · Zbl 0377.10017 · doi:10.1007/BF01390245
[3] [A-V] Amice, Y., Vélu, J.: Distributionsp-adiques associées aux séries de Hecke. Astérisque, No. 24/25. Soc. Math. Fr. Paris, 1975, pp. 119-131
[4] [D] Deligne, P.: Formes modulaires et représentationsl-adiques. Séminaire Bourbaki355 (1969)
[5] [D-R] Deligne, P., Rapoport, M.: Schémas de modules de courbes elliptiques. Lect. Notes Math.,349, 143-316 (1973) · doi:10.1007/978-3-540-37855-6_4
[6] [Gr] Greenberg, R.: Iwasawa theory forp-adic representations. Advance Studies in Pure Mathematics17, 97-137 (1989) · Zbl 0739.11045
[7] [H1] Hida, H.: Galois representations intoGL 2 (Z p[[X]]) attached to ordinary cusp forms. Invent. Math.85, 545-613 (1986) · Zbl 0612.10021 · doi:10.1007/BF01390329
[8] [H2] Hida, H.: Iwasawa modules attached to congruences of cusp forms. Ann. Sci. École Norm. Sup.19, 231-273 (1986) · Zbl 0607.10022
[9] [H3] Hida, H.: Onp-adic Hecke algebras forGL 2 over totally real fields. Ann. Math.128, 295-384 (1988) · Zbl 0658.10034 · doi:10.2307/1971444
[10] [Kz] Katz, N.:p-AdicL-Functions for CM Fields. Invent. Math.49, 199-297 (1978) · Zbl 0417.12003 · doi:10.1007/BF01390187
[11] [K] Kitagawa, K.: On standardp-adicL-functions of families of elliptic cusp forms. UCLA PhD Thesis, 1991
[12] [Kl] Klingenberg, C.: Onp-adicL-functions of Mumford curves. Preprint, March 1990
[13] [Man1] Manin, J.: Cyclotomic fields and modular curves. Russian Math. Surveys26(6), 7-78 (1971) · Zbl 0266.14012 · doi:10.1070/RM1971v026n06ABEH001272
[14] [Man2] Manin, J.: Parabolic points and zeta functions of modular curves. Izv. Akad. Nauk SSSR (AMS translation)6(1), 19-64 (1972) · Zbl 0243.14008
[15] [McC] McGabe, J.:p-adic Theta Functions. Harvard Ph.D. Thesis 1968
[16] [Mo] Morikawa, H.: On theta functions and abelian varieties over valuation fields of rank one, I and II. Nagoya Math. J.20, 1-27 and 231-250 (1962) · Zbl 0115.39001
[17] [Mz1] Mazur, B.: Modular Curves and the Eisenstein Ideal. Publ. Math. I.H.E.S.47, 33-189 (1977)
[18] [Mz2] Mazur, B.: Two-variablep-adicL-functions. Unpublished Harvard Lecture Notes (1985)
[19] [Mz-SwD] Mazur, B., Swinnerton-Dyer, P.: Arithmetic of Weil curves. Invent. Math.25, 1-61 (1974) · Zbl 0281.14016 · doi:10.1007/BF01389997
[20] [Mz-T-T] Mazur, B., Tate, J., Teitelbaum, J.: Onp-adic analogs of the conjectures of Birch and Swinnerton-Dyer. Invent. Math.84, 1 48 (1986) · Zbl 0699.14028 · doi:10.1007/BF01388731
[21] [Mz-W] Mazur, B., Wiles, A.: Onp-adic analytic families of Galois representations. Compos. Math.59, 231-264 (1986) · Zbl 0654.12008
[22] [N] Nagata, M.: Local Rings. Interscience Tracts in Pure and Applied Mathematics13 (1962). John Wiley & Sons, Inc., New York, London
[23] [O] Ohta, M.: Onl-adic representations attached to automorphic forms. Jpn. J. Math.8, No. 1, 1-47 (1982) · Zbl 0505.10012
[24] [Ser] Serre, J-P.: Corps Locaux. Publications de l’Institut de Mathématique de l’Université de Nancago VII; Herman, Paris 1968
[25] [Sh1] Shimura, G.: Anl-adic method in the theory of automorphic forms. (1968, unpublished)
[26] [Sh2] Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. Princeton University Press, 1971 · Zbl 0221.10029
[27] [Vi] Vishik, M.: Nonarchimedean measures connected with Dirichlet series. Math. USSSR Sb.28, 216-228 (1976) · Zbl 0369.14010 · doi:10.1070/SM1976v028n02ABEH001648
[28] [W] Wiles, A.: On ordinary ?-adic representations associated to modular forms. Invent. Math.94, 529-573 (1988) · Zbl 0664.10013 · doi:10.1007/BF01394275
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.