On the de Rham and p-adic realizations of the elliptic polylogarithm for CM elliptic curves
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 43 (2010) no. 2, p. 185-234

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.

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.

DOI : https://doi.org/10.24033/asens.2119
Classification:  11G55,  11G07,  11G15,  14F30,  14G10
Keywords: elliptic curves, complex multiplication, elliptic polylogarithms, p-adic L-functions
@article{ASENS_2010_4_43_2_185_0,
     author = {Bannai, Kenichi and Kobayashi, Shinichi and Tsuji, Takeshi},
     title = {On the de Rham and $p$-adic realizations of the elliptic polylogarithm for CM elliptic curves},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 43},
     number = {2},
     year = {2010},
     pages = {185-234},
     doi = {10.24033/asens.2119},
     zbl = {1197.11073},
     mrnumber = {2662664},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2010_4_43_2_185_0}
}
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, Serie 4, Volume 43 (2010) no. 2, pp. 185-234. doi : 10.24033/asens.2119. http://www.numdam.org/item/ASENS_2010_4_43_2_185_0/

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