Factorization of coefficients for exponential and logarithm in function fields

Abstract

Let X be an elliptic curve or a ramifying hyperelliptic curve over Fq. We will discuss how to factorize the coefficients of the exponential and logarithm series for a Hayes module over such a curve. This allows us to obtain v-adic convergence results for such exponential and logarithm series, for v a “finite” prime. As an application, we can show that the v-adic Goss L-value Lv(1,Ψ) is log-algebraic for suitable characters Ψ.