An Eisenstein--Kronecker Modular Symbol and Beilinson--Kato Distributions

Abstract

We construct an Eisenstein-Kronecker modular symbol with values in a completed bigraded quotient of modular forms by Eisenstein series. Its homogeneous finite-level specializations recover the Borisov-Gunnells symbols. The main theorem identifies its standard distribution, after the explicit transposition, orientation, finite symplectic Fourier, projection, and period normalizations, with Colmez's Eisenstein realization of the standard Busuioc-Park-Patashnick-Stevens Beilinson-Kato distribution. The comparison holds on the full BPPS test-function space, including the zero-torsion degenerations. The cyclic Manin relation is obtained from the Frobenius-Stickelberger identity and the theta heat equation; at level one its coefficient identities generate all linear relations among unordered products of two Eisenstein series whose weights are at least four. We also record compatible real-regulator and Poincare-bundle coefficient realizations with the same Fourier normalization.

arXiv comment48 pages. Substantially revised and expanded version; title changed, and the comparison with the BPPS Beilinson-Kato distribution and its regulator realizations has been added