×

\(p\)-adic Eisenstein-Kronecker series and non-critical values of \(p\)-adic Hecke \(L\)-function of an imaginary quadratic field when the conductor is divisible by \(p\). (English) Zbl 1385.11045

Summary: We relate non-critical special values of \(p\)-adic \(L\)-functions associated to algebraic Hecke characters of an imaginary quadratic number field with class number one to \(p\)-adic Eisenstein-Kronecker series constructed as the Coleman function, when the conductors of the algebraic Hecke characters are divisible by \(p\).
For a video summary of this paper, please click here or visit http://youtu.be/AZemqgfp5pQ.

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F85 \(p\)-adic theory, local fields
11G15 Complex multiplication and moduli of abelian varieties
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11R42 Zeta functions and \(L\)-functions of number fields
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bannai, K.; Furusho, H.; Kobayashi, S., \(p\)-adic Eisenstein-Kronecker function and the elliptic polylogarithm for CM elliptic curves (2008), preprint
[2] Bannai, K.; Kings, G., \(p\)-adic Beilinson conjecture for ordinary Hecke motives associated to imaginary quadratic fields, (Ichikawa, T.; Kida, M.; Yamazaki, T., RIMS Kokyuroku Bessatsu B25: Algebraic Number Theory and Related Topics 2009 (June 2011)), 9-30 · Zbl 1248.14025
[3] Bannai, K.; Kobayashi, S., Algebraic theta functions and the \(p\)-adic interpolation of Eisenstein-Kronecker numbers, Duke Math. J., 153, 229-295 (2010) · Zbl 1205.11076
[4] Bannai, K.; Kobayashi, S.; Tsuji, T., On the de Rham and \(p\)-adic realizations of the elliptic polylogarithm for CM elliptic curves, Ann. Sci. Éc. Norm. Super., 43, 2, 185-234 (2010) · Zbl 1197.11073
[5] Beilinson, A. A., Higher regulators and values of \(L\)-functions, J. Sov. Math., 30, 2036-2070 (1985) · Zbl 0588.14013
[6] Berthelot, P., Cohomologie rigide et cohomologie rigide à supports propres, premiére partie (1996), Institut de recherche math. de Rennes
[7] Besser, A., Syntomic regulators and \(p\)-adic integration II: \(K2\) of curves, Israel J. Math., 120, 1, 335-359 (December 2000)
[8] Besser, A., A generalization of Colemanʼs \(p\)-adic integration theory, Invent. Math., 142, 397-434 (2000) · Zbl 1053.14020
[9] Bloch, S., Higher Regulators, Algebraic \(K\)-Theory, and Zeta Functions of Elliptic Curves, CRM Monogr. Ser., vol. 11 (2000), American Mathematical Society · Zbl 0958.19001
[10] Bosch, S.; Güntzer, U.; Remmert, R., Non-Archimedean Analysis, Grundlehren Math. Wiss., vol. 261 (1984), Springer-Verlag · Zbl 0539.14017
[11] Breuil, C., Intégration sur les variétés \(p\)-adiques [dʼaprès Coleman, Colmez], (Séminaire N. Bourbaki (1999)), 319-350 · Zbl 1015.11028
[12] Coleman, R., Dilogarithms, regulators, and \(p\)-adic \(L\)-functions, Invent. Math., 69, 171-208 (1982) · Zbl 0516.12017
[13] Coleman, R.; de Shalit, E., \(p\)-adic regulators on curves and special values on \(p\)-adic \(L\)-function, Invent. Math., 93, 239-266 (1988) · Zbl 0655.14010
[14] Deligne, P., Valeurs de Fonctions \(L\) et Périodes Dʼintégrales, (Proc. Sympos. Pure Math., vol. 33 (1979)), 313-346, Part 2
[15] de Shalit, E., Iwasawa Theory of Elliptic Curves with Complex Multiplication, Perspect. Math., vol. 3 (1986)
[16] Fresnel, J.; van der Put, M., Rigid Analytic Geometry and Its Applications (2003), Boston Birkhäuser
[17] Gros, M., Régulateurs syntomiques et valeurs de fonctions L p-adiques. I, Invent. Math., 99, 2, 293-320 (1990), with an appendix by Masato Kurihara · Zbl 0667.14006
[18] Gros, M., Régulateurs syntomiques et valeurs de fonctions L p-adiques. II, Invent. Math., 115, 1, 61-79 (1994) · Zbl 0799.14010
[19] Hirotsune, T., A study on the relation of special values of \(p\)-adic \(L\)-functions to \(p\)-adic Eisenstein-Kronecker series (March 2012), Keio University, Masterʼs thesis
[20] Iwasawa, K., Lectures on \(p\)-Adic \(L\)-Functions, Ann. of Math. Stud., vol. 74 (1972), Princeton University Press · Zbl 0236.12001
[21] Katz, N., \(p\)-adic interpolation of real analytic Eisenstein series, Ann. of Math., 104, 459-571 (1976) · Zbl 0354.14007
[22] Lang, S., Elliptic Functions, Grad. Texts in Math., vol. 112 (1973), Springer-Verlag
[23] Lubin, J., One-parameter formal Lie groups over \(p\)-adic integer rings, Ann. of Math., 80, 464-484 (1964) · Zbl 0135.07003
[24] Manin, Ju. I.; Višik, S., \(p\)-adic Hecke series for quadratic imaginary fields, Mat. Sb., 95(137), 3(11) (1974)
[25] Perrin-Riou, B., Fonctions L p-adiques des représentations \(p\)-adiques, Astérisque, 229 (1995) · Zbl 0845.11040
[26] Silverman, J., Advanced Topics in the Arithmetic of Elliptic Curves, Grad. Texts in Math., vol. 151 (1994), Springer · Zbl 0911.14015
[27] Silverman, J., The Arithmetic of Elliptic Curves, Grad. Texts in Math., vol. 106 (2008), Springer
[28] Tate, J., \(p\)-divisible groups, (Proceedings of a Conference on Local Fields (1967), Springer), 158-183 · Zbl 0157.27601
[29] Tate, J., Rigid analytic space, Invent. Math., 12, 257-289 (1971) · Zbl 0212.25601
[30] Weil, A., Elliptic Functions According to Eisenstein and Kronecker, Ergeb. Math. Grenzgeb., vol. 88 (1976) · Zbl 0318.33004
[31] Yager, R. I., On two variable \(p\)-adic \(L\)-functions, Ann. of Math., 115, 411-449 (1982) · Zbl 0496.12010
[32] Zagier, D., The Bloch-Wigner-Ramakrishnan polylogarithm function, Math. Ann., 286, 613-624 (1990) · Zbl 0698.33001
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.