Variations of Hodge-de Rham structure and elliptic modular units

Abstract

This is a revised version of ANT-0049. Given an elliptic curve E --> B over a base B with zero section i, we denote, letting E':= E - i(B), by L(E) the Q-vector space with basis ({s}, s \in E'(B)). Assume that B is smooth and separated over a field of characteristic 0. On the lowest step, the weak version of the elliptic Zagier conjecture predicts the existence of a homomorphism ϕfrom the kernel of a certain differential d on L(E) to the vector space O*(B) \otimes Q of units on B. This homomorphism should behave functorially with respect to change of the base B, and it should satisfy a certain norm compatibility. Also, if B is the spectrum of a local field, then the absolute value of ϕshould be expressible in terms of the local Néron height function. In this paper, we give a proof of this. We also connect the values of ϕon specific elements of ker(d) to modular, and to elliptic units.