On the $p$-adic values of the weight two Eisenstein series for supersingular primes

Abstract

The weight two Eisenstein series may be considered as the first example of a Katz $p$-adic modular form. Classically, its values are defined for the primes of ordinary reduction. We offer a modified definition which applies uniformly to all primes of good reduction, both ordinary and supersingular. We show that, in the case of complex multiplication, these $p$-adic values coincide with the algebraic value of this Eisenstein series.