On the de Rham and p-adic realizations of the Elliptic Polylogarithm for CM elliptic curves

Abstract

In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good reduction at p, then the specializations to torsion points of the p-adic elliptic polylogarithm are related to p-adic Eisenstein-Kronecker numbers, proving a p-adic analogue of the result of Beilinson and Levin expressing the complex elliptic polylogarithm in terms of Eisenstein-Kronecker-Lerch series. Our result is valid even if the elliptic curve has supersingular reduction at p.

61 pages, v2. Sections concerning the Hodge realization was moved to the appendix

References in corpus (2)

• Algebraic theta functions and p-adic interpolation of Eisenstein-Kronecker numbers
• Algebraic theta functions and Eisenstein-Kronecker numbers

Cited by in corpus (11)

• Algebraic theta functions and p-adic interpolation of Eisenstein-Kronecker numbers
• $p$-adic Polylogarithms and $p$-adic Hecke $L$-functions for Totally Real Fields
• $p$-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas
• Hyodo-Kato theory with syntomic coefficients
• Towards algebraic iterated integrals on elliptic curves via the universal vectorial extension
• The Hodge realization of the polylogarithm on the product of multiplicative groups
• The algebraic de Rham realization of the elliptic polylogarithm via the Poincaré bundle
• Eisenstein-Kronecker series via the Poincaré bundle
• The Kronecker limit formulas via the distribution relation
• Elliptic KZB connections via universal vector extensions
• The Hodge Realization of the Polylogarithm and the Shintani Generating Class for Totally Real Fields