Nagoya Mathematical Journal
- Nagoya Math. J.
- Volume 219 (2015), 269-302.
-adic Eisenstein–Kronecker series for CM elliptic curves and the Kronecker limit formulas
Consider an elliptic curve defined over an imaginary quadratic field with good reduction at the primes above and with complex multiplication by the full ring of integers of . In this paper, we construct -adic analogues of the Eisenstein–Kronecker series for such an 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.
Nagoya Math. J., Volume 219 (2015), 269-302.
Received: 28 June 2012
Revised: 18 June 2014
Accepted: 14 August 2014
First available in Project Euclid: 20 October 2015
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 11G55: Polylogarithms and relations with $K$-theory
Secondary: 11G07: Elliptic curves over local fields [See also 14G20, 14H52] 11G15: Complex multiplication and moduli of abelian varieties [See also 14K22] 14F30: $p$-adic cohomology, crystalline cohomology 14G10: Zeta-functions and related questions [See also 11G40] (Birch- Swinnerton-Dyer conjecture)
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