## Nagoya Mathematical Journal

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

#### Abstract

Consider an elliptic curve defined over an imaginary quadratic field $K$ with good reduction at the primes above $p\geq5$ and with 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 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.

#### Article information

Source
Nagoya Math. J., Volume 219 (2015), 269-302.

Dates
Revised: 18 June 2014
Accepted: 14 August 2014
First available in Project Euclid: 20 October 2015

https://projecteuclid.org/euclid.nmj/1445345522

Digital Object Identifier
doi:10.1215/00277630-2891995

Mathematical Reviews number (MathSciNet)
MR3413578

Zentralblatt MATH identifier
1336.11050

#### Citation

Bannai, Kenichi; Furusho, Hidekazu; Kobayashi, Shinichi. $p$ -adic Eisenstein–Kronecker series for CM elliptic curves and the Kronecker limit formulas. Nagoya Math. J. 219 (2015), 269--302. doi:10.1215/00277630-2891995. https://projecteuclid.org/euclid.nmj/1445345522

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