Thursday, January 27
Milner 317, 1:00 PM
Title: Galois groups and transcendence in characteristic p
Abstract: In this talk we will present new results on algebraic independence over function fields. In many respects, conjectures about the transcendence and algebraic independence of periods and logarithms of Drinfeld modules have mirrored the conjectures for their classical counterparts. However, until recently, with notable exceptions, theorems proved in positive characteristic also mirrored results in characteristic 0 (even if their proofs were quite different!).
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 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.