Factorization of coefficients for exponential and logarithm in function fields

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