On the de Rham and $p$-adic realizations of the elliptic polylogarithm for CM elliptic curves

Bannai, Kenichi; Kobayashi, Shinichi; Tsuji, Takeshi

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 $p\ge 5$ 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 $p\ge 5$ 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 $𝕂$.

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