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$$p$$-adic elliptic polylogarithm, $$p$$-adic Eisenstein series and Katz measure. (English) Zbl 1225.11075
Authors’ abstract: The specializations of the motivic elliptic polylogarithm on the universal elliptic curve to the modular curve are referred to as Eisenstein classes. In this paper, we prove that the syntomic realization of the Eisenstein classes restricted to the ordinary locus of the modular curve may be expressed using $$p$$-adic Eisenstein series of negative weight, which are $$p$$-adic modular forms defined using the two-variable $$p$$-adic measure with values in $$p$$-adic modular forms constructed by Katz. The motivation of our research is the $$p$$-adic Beilinson conjecture formulated by Perrin-Riou.
In a formula: $\text{Eis}_{\text{syn}}^{k+2}(\varphi)=[\alpha_{\text{Eis}}^{k+2}(\varphi),\text{Eis}_{\text{dR}}^{k+2}(\varphi)].$

##### MSC:
 11F85 $$p$$-adic theory, local fields 11G55 Polylogarithms and relations with $$K$$-theory
##### Keywords:
$$p$$-adic modular forms
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