\(p\)-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.
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)
MSC:
| 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 |
Keywords:
Mazur formula; Tate formula; Teitelbaum formula; modular elliptic curve; interpolation; Tate period; two-variable \(p\)-adic \(L\)-functionReferences:
| [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) · 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 |
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