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A classical construction of Katz gives a purely algebraic construction ofEisenstein--Kronecker series using the Gau\ss--Manin connection on theuniversal elliptic curve. This approach gives a systematic way to studyalgebraic and $p$-adic properties of real-analytic Eisenstein series. In thefirst part of this paper we provide an alternative algebraic construction ofEisenstein--Kronecker series via the Poincar\'e bundle. Building on this, wegive in the second part a new conceptional construction of Katz' two-variable$p$-adic Eisenstein measure through $p$-adic theta functions of the Poincar\'ebundle.

Eisenstein-Kronecker series via the Poincar\'e bundle