Extension of Eisenstein-Kronecker classes and a new construction of Katz's p-adic Eisenstein measure

In this work, we give a construction of a p-adic measure which takes values in the ring of p-adic Hilbert modular forms which recovers the one constructed by Katz via the use of Eisenstein series. Our construction is based on the definition of Kings-Sprang of Eisenstein-Kronecker classes, equivariant cohomology classes associated to the completed Poincaré bundle of an abelian scheme with complex multiplication, and which specialize to higher-dimensional Eisenstein-Kronecker series over the complex numbers. We show that these classes for the universal abelian scheme over the Hilbert moduli space still recover these Eisenstein-Kronecker series as Hilbert modular forms. Working in the case of Γ₀₀ (p^∞) -level structure and using a p-adic trivialization of the completion of the Poincaré bundle, these classes define our desired p-adic measure, whose moments give back the Eisenstein-Kronecker classes. A direct comparison with the formulas of Katz then shows that we recover his measure.