$p$-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas
Number Theory
2020-09-11 v2
Abstract
Consider an elliptic curve defined over an imaginary quadratic field with good reduction at the primes above and has complex multiplication by the full ring of integers of . In this paper, we construct -adic analogues of the Eisenstein-Kronecker series for such elliptic curve as Coleman functions on the elliptic curve. We then prove -adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.
Cite
@article{arxiv.0807.4007,
title = {$p$-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas},
author = {Kenichi Bannai and Hidekazu Furusho and Shinichi Kobayashi},
journal= {arXiv preprint arXiv:0807.4007},
year = {2020}
}
Comments
v2. The current version is the synthesis of {\S}1-{\S}3 of the first version of this article with the content of arXiv:0807.4008 "The Kronecker limit formulas via the distribution relation." {\S}4,{\S}5 of the first version of this paper will be treated in a subsequent article