Tannakian Galois groups and transcendence in positive characteristic
Matt Papanikolas, Texas A&M

Abstract: Transcendence problems over function fields in characteristic p begin with periods of Drinfeld modules and go back to work of Carlitz and Wade in the 1930's and 1940's. Until recently theorems on transcendence and algebraic independence in characteristic p have mirrored what is known for their classical counterparts (even if their proofs were quite different!). In this talk we will present new algebraic independence results over function fields.

By introducing a Tannakian formalism for Drinfeld modules and relating it to the Galois theory of certain Frobenius semi-linear difference equations, we determine the transcendence degrees of fields generated by periods of Drinfeld modules and more generally Anderson t-modules. More precisely, we show that the transcendence degree of the period matrix of a Drinfeld module is equal to the dimension of its Galois group. As an application, we prove that Carlitz logarithms of algebraic numbers that are linearly independent over F_q(t) are algebraically independent.