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arXiv · 0807.4007

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

Kenichi Bannai·Hidekazu Furusho·Shinichi Kobayashi
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Abstract

Consider an elliptic curve defined over an imaginary quadratic field $K$ with good reduction at the primes above $p\geq 5$ and has complex multiplication by the full ring of integers $\mathcal{O}_K$ of $K$. In this paper, we construct $p$-adic analogues of the Eisenstein-Kronecker series for such 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.

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Kenichi Bannai, Hidekazu Furusho, Shinichi Kobayashi. 2008-07-25. $p$-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas. https://doi.org/10.1215/00277630-2891995

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