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

Kenichi Bannai, Shinichi Kobayashi, Takeshi Tsuji

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9 Citations (Scopus)


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 OK 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 ≥ 5 unramified in OK. 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.

Original languageEnglish
Pages (from-to)185-234
Number of pages50
JournalAnnales Scientifiques de l'Ecole Normale Superieure
Issue number2
Publication statusPublished - 2010
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)


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