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p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry (Co)homology theory

Published online by Cambridge University Press:  11 January 2016

Kenichi Bannai ,
Hidekazu Furusho  and
Shinichi Kobayashi
Kenichi Bannai
Affiliation:
Department of Mathematics, Keio University, Yokohama 223-8522, Japan, [email protected]
Hidekazu Furusho
Affiliation:
Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan, [email protected]
Shinichi Kobayashi
Affiliation:
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan, [email protected]
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Abstract

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Consider an elliptic curve defined over an imaginary quadratic field K with good reduction at the primes above p ≥ 5 and with complex multiplication by the full ring of integers of K. In this paper, we construct p-adic analogues of the Eisenstein-Kronecker series for such an elliptic curve as Coleman functions on the elliptic curve. We then prove p-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.

MSC classification

Secondary: 11G55: Polylogarithms and relations with $K$-theory 11G07: Elliptic curves over local fields 11G15: Complex multiplication and moduli of abelian varieties 14F30: $p$-adic cohomology, crystalline cohomology 14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture)
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 219 , September 2015 , pp. 269 - 302
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

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