On Arithmetic Monodromy Representations of Eisenstein Type in Fundamental Groups of Once Punctured Elliptic Curves

Abstract

We discuss certain arithmetic invariants arising from the monodromy representation in fundamental groups of a family of once punctured elliptic curves of characteristic zero. An explicit formula in terms of Kummer properties of modular units is given to describe these invariants. In the complex analytic model, the formula turns out to feature generalized Dedekind–Rademacher functions as the main periodic part of the invariant.