RTG Seminar on Number Theory Seminar

Classically over the rational numbers, the exponential and logarithm series converge p-adically within some open disc of C_p. For function fields, exponential and logarithm series arise naturally from Drinfeld modules, which are objects constructed by Drinfeld in his thesis to prove the Langlands conjecture for GL_2 over function fields. For a "finite place" v on such a curve, one can ask if the exp and log possess similar v-adic convergence properties. For the most basic case, namely that of the Carlitz module over F_q[T], this question has been long understood. In this talk, we will show the v-adic convergence for Drinfeld-(Hayes) modules on elliptic curves and a certain class of hyperelliptic curves. As an application, we are then able to obtain a formula for the v-adic L-value L_v(1,Psi) for characters in these cases, analogous to Leopoldt's formula in the number field case. Speaker(s): Angus Chung (UM)