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

Published online by Cambridge University Press:  11 January 2016

Kenichi Bannai
Affiliation:
Department of Mathematics, Keio University, Yokohama 223-8522, Japan, bannai@math.keio.ac.jp
Hidekazu Furusho
Affiliation:
Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan, furusho@math.nagoya-u.ac.jp
Shinichi Kobayashi
Affiliation:
Mathematical Institute, Tohoku University, Sendai 980-8578, Japan, shinichi@math.tohoku.ac.jp
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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.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2015

References

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