On the de Rham and p-adic realizations of the elliptic polylogarithm for CM elliptic curves
[Sur les réalisations de de Rham et p-adiques du polylogarithme elliptique des courbes elliptiques à multiplication complexe]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 2, pp. 185-234.

Dans cet article, nous donnons une description explicite des réalisations de de Rham et p-adiques des polylogarithmes elliptiques en utilisant la fonction thêta de Kronecker. Considérons en particulier une courbe elliptique E définie sur un corps quadratique imaginaire 𝕂, à multiplication complexe par l’anneau des entiers 𝒪 𝕂 de 𝕂, et ayant bonne réduction en chaque place au-dessus d’un nombre premier p5 non ramifié dans 𝕂. On notera que le nombre de classe de 𝕂 est nécessairement égal à un. Nous montrons alors que les spécialisations des polylogarithmes p-adiques aux points de torsion de E d’ordre premier à p sont reliées aux nombres d’Eisenstein-Kronecker p-adiques. Ce résultat est valable même si E a une réduction supersingulière en p. C’est un analogue p-adique d’un cas spécial du résultat de Beilinson et Levin exprimant la réalisation de Hodge du polylogarithme elliptique en utilisant les séries d’Eisenstein-Kronecker-Lerch. Si p est quelconque, nous établissons un lien entre les nombres d’Eisenstein-Kronecker p-adiques et les valeurs spéciales des fonctions L associées aux caractères de Hecke de 𝕂.

In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. In particular, consider an elliptic curve E defined over an imaginary quadratic field 𝕂 with complex multiplication by the full ring of integers 𝒪 𝕂 of 𝕂. Note that our condition implies that 𝕂 has class number one. Assume in addition that E has good reduction above a prime p5 unramified in 𝒪 𝕂 . In this case, we prove that the specializations of the p-adic elliptic polylogarithm to torsion points of E of order prime to p are related to p-adic Eisenstein-Kronecker numbers. Our result is valid even if E has supersingular reduction at p. This is a p-adic analogue in a special case of the result of Beilinson and Levin, expressing the Hodge realization of the elliptic polylogarithm in terms of Eisenstein-Kronecker-Lerch series. When p is ordinary, then we relate the p-adic Eisenstein-Kronecker numbers to special values of p-adic L-functions associated to certain Hecke characters of 𝕂.

DOI : 10.24033/asens.2119
Classification : 11G55, 11G07, 11G15, 14F30, 14G10
Keywords: elliptic curves, complex multiplication, elliptic polylogarithms, $p$-adic $L$-functions
Mot clés : courbes elliptiques, multiplication complexe, polylogarithmes elliptiques, fonctions $L$ $p$-adiques
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     title = {On the de {Rham} and $p$-adic realizations of the elliptic polylogarithm for {CM} elliptic curves},
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Bannai, Kenichi; Kobayashi, Shinichi; Tsuji, Takeshi. On the de Rham and $p$-adic realizations of the elliptic polylogarithm for CM elliptic curves. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 43 (2010) no. 2, pp. 185-234. doi : 10.24033/asens.2119. http://archive.numdam.org/articles/10.24033/asens.2119/

[1] M. Abramowitz & I. A. Stegun (éds.), Weierstrass elliptic and related functions, Ch. 18, in Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover Publications Inc., 1992, p. 627-671. | MR | Zbl

[2] F. Baldassarri & B. Chiarellotto, Algebraic versus rigid cohomology with logarithmic coefficients, in Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), Perspect. Math. 15, Academic Press, 1994, 11-50. | MR | Zbl

[3] K. Bannai, Rigid syntomic cohomology and p-adic polylogarithms, J. reine angew. Math. 529 (2000), 205-237. | MR | Zbl

[4] K. Bannai, On the p-adic realization of elliptic polylogarithms for CM-elliptic curves, Duke Math. J. 113 (2002), 193-236. | MR | Zbl

[5] K. Bannai & G. Kings, p-adic elliptic polylogarithm, p-adic Eisenstein series and Katz measure, preprint arXiv:0707.3747, to appear in Amer. J. Math. | MR | Zbl

[6] K. Bannai & S. Kobayashi, Algebraic theta functions and p-adic interpolation of Eisenstein-Kronecker numbers, preprint arXiv:math.NT/0610163, to appear in Duke Math. J. | MR | Zbl

[7] A. Beĭlinson & A. Levin, The elliptic polylogarithm, in Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math. 55, Amer. Math. Soc., 1994, 123-190. | MR | Zbl

[8] P. Berthelot, Géométrie rigide et cohomologie des variétés algébriques de caractéristique p, Mém. Soc. Math. France (N.S.) 23 (1986), 7-32. | Numdam | MR | Zbl

[9] P. Berthelot, Cohomologie rigide et cohomologie rigide à support propre, première partie, preprint IRMAR 96-03, 1996.

[10] P. Berthelot, Finitude et pureté cohomologique en cohomologie rigide, Invent. Math. 128 (1997), 329-377. | MR | Zbl

[11] J. L. Boxall, A new construction of 𝔭-adic L-functions attached to certain elliptic curves with complex multiplication, Ann. Inst. Fourier (Grenoble) 36 (1986), 31-68. | Numdam | MR | Zbl

[12] J. L. Boxall, p-adic interpolation of logarithmic derivatives associated to certain Lubin-Tate formal groups, Ann. Inst. Fourier (Grenoble) 36 (1986), 1-27. | Numdam | MR | Zbl

[13] P. Colmez, Fonctions L p-adiques, Séminaire Bourbaki, vol. 1998/99, exposé no 851, Astérisque 266 (2000), 21-58. | Numdam | MR | Zbl

[14] R. M. Damerell, L-functions of elliptic curves with complex multiplication. I, Acta Arith. 17 (1970), 287-301. | Zbl

[15] R. M. Damerell, L-functions of elliptic curves with complex multiplication. II, Acta Arith. 19 (1971), 311-317. | Zbl

[16] J.-M. Fontaine, Modules galoisiens, modules filtrés et anneaux de Barsotti-Tate, in Journées de Géométrie Algébrique de Rennes. (Rennes, 1978), Vol. III, Astérisque 65, Soc. Math. France, 1979, 3-80. | Numdam | Zbl

[17] A. Huber & G. Kings, Degeneration of l-adic Eisenstein classes and of the elliptic polylog, Invent. Math. 135 (1999), 545-594. | Zbl

[18] A. Huber & J. Wildeshaus, Classical motivic polylogarithm according to Beilinson and Deligne, Doc. Math. 3 (1998), 27-133; correction: idem, 297-299. | Zbl

[19] N. M. Katz, p-adic interpolation of real analytic Eisenstein series, Ann. of Math. 104 (1976), 459-571. | Zbl

[20] A. Levin, Elliptic polylogarithms: an analytic theory, Compositio Math. 106 (1997), 267-282. | Zbl

[21] A. Levin & G. Racinet, Towards multiple elliptic polylogarithm, preprint arXiv:math/0703237.

[22] J. I. Manin & M. M. Višik, p-adic Hecke series of imaginary quadratic fields, Mat. Sb. (N.S.) 95 (1974), 357-383. | Zbl

[23] P. Monsky & G. Washnitzer, Formal cohomology. I, Ann. of Math. 88 (1968), 181-217. | Zbl

[24] B. Perrin-Riou, Fonctions L p-adiques des représentations p-adiques, Astérisque 229 (1995). | Numdam | Zbl

[25] P. Schneider & J. Teitelbaum, p-adic Fourier theory, Doc. Math. 6 (2001), 447-481. | MR | Zbl

[26] E. De Shalit, Iwasawa theory of elliptic curves with complex multiplication, Perspectives in Mathematics 3, Academic Press Inc., 1987. | MR | Zbl

[27] A. Shiho, Crystalline fundamental groups. II. Log convergent cohomology and rigid cohomology, J. Math. Sci. Univ. Tokyo 9 (2002), 1-163. | MR | Zbl

[28] N. Solomon, p-adic elliptic polylogarithms and arithmetic applications, Thèse, Ben-Gurion University, 2009.

[29] N. Tsuzuki, On base change theorem and coherence in rigid cohomology, Doc. Math. extra vol. (2003), 891-918. | MR | Zbl

[30] A. Weil, Elliptic functions according to Eisenstein and Kronecker, Ergebn. Math. Grenzg. 88, Springer, 1976. | MR | Zbl

[31] E. Weisstein, Weierstrass sigma function, http://mathworld.wolfram.com/WeierstrassSigmaFunction.html.

[32] J. Wildeshaus, Realizations of polylogarithms, Lecture Notes in Math. 1650, Springer, 1997. | MR | Zbl

[33] S. Yamamoto, On p-adic L-functions for CM elliptic curves at supersingular primes, Mémoire, University of Tokyo, 2002.

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