p-adic elliptic polylogarithm, p-adic Eisenstein series and Katz measure

TL;DR

This paper establishes a connection between motivic Eisenstein classes and p-adic Eisenstein-Kronecker series, enabling explicit computation of their syntomic realizations on the ordinary locus of modular curves.

Contribution

It demonstrates that the syntomic realization of motivic Eisenstein classes can be expressed via p-adic Eisenstein-Kronecker series on the ordinary locus.

Findings

Syntomic realization expressed using p-adic Eisenstein-Kronecker series

Restriction to the ordinary locus is crucial for the expression

Uses Katz's two-variable p-adic measure for modular forms

Abstract

The specializations of the motivic elliptic polylog are called motivic Eisenstein classes. For applications to special values of L-Functions, it is important to compute the realizations of these classes. In this paper, we prove that the syntomic realization of the motivic Eisenstein classes, restricted to the ordinary locus of the modular curve, may be expressed using p-adic Eisenstein-Kronecker series. These p-adic modular forms are defined using the two-variable p-adic measure with values in p-adic modular forms constructed by Katz.