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

Kenichi Bannai; Shinichi Kobayashi; Takeshi Tsuji

Annales scientifiques de l'École Normale Supérieure (2010)

  • Volume: 43, Issue: 2, page 185-234
  • ISSN: 0012-9593

Abstract

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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 ≥ 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 .

How to cite

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Bannai, Kenichi, Kobayashi, Shinichi, and 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 43.2 (2010): 185-234. <http://eudml.org/doc/272129>.

@article{Bannai2010,
abstract = {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 $\mathbb \{K\}$ with complex multiplication by the full ring of integers $\mathcal \{O\}_\mathbb \{K\}$ of $\mathbb \{K\}$. Note that our condition implies that $\mathbb \{K\}$ has class number one. Assume in addition that $E$ has good reduction above a prime $p \ge 5$ unramified in $\mathcal \{O\}_\mathbb \{K\}$. 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 $\mathbb \{K\}$.},
author = {Bannai, Kenichi, Kobayashi, Shinichi, Tsuji, Takeshi},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {elliptic curves; complex multiplication; elliptic polylogarithms; $p$-adic $L$-functions},
language = {eng},
number = {2},
pages = {185-234},
publisher = {Société mathématique de France},
title = {On the de Rham and $p$-adic realizations of the elliptic polylogarithm for CM elliptic curves},
url = {http://eudml.org/doc/272129},
volume = {43},
year = {2010},
}

TY - JOUR
AU - Bannai, Kenichi
AU - Kobayashi, Shinichi
AU - Tsuji, Takeshi
TI - On the de Rham and $p$-adic realizations of the elliptic polylogarithm for CM elliptic curves
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2010
PB - Société mathématique de France
VL - 43
IS - 2
SP - 185
EP - 234
AB - 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 $\mathbb {K}$ with complex multiplication by the full ring of integers $\mathcal {O}_\mathbb {K}$ of $\mathbb {K}$. Note that our condition implies that $\mathbb {K}$ has class number one. Assume in addition that $E$ has good reduction above a prime $p \ge 5$ unramified in $\mathcal {O}_\mathbb {K}$. 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 $\mathbb {K}$.
LA - eng
KW - elliptic curves; complex multiplication; elliptic polylogarithms; $p$-adic $L$-functions
UR - http://eudml.org/doc/272129
ER -

References

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