Hyperderivatives of Periods and Logarithms of Anderson t-modules, and Algebraic Independence

Abstract

In this dissertation, we study algebraic relations among periods, quasi-periods, logarithms and quasi-logarithms of Drinfeld modules. This work is motivated by the Tannakian theory for t-motives especially the function field analogue, proved by Papanikolas, of Grothendieck’s conjecture for periods of abelian varieties. Papanikolas’ theorem shows that the dimension of the Galois group associated to a t-motive is equal to the transcendence degree of the entries of the period matrix of the t-motive. In recent work, Papanikolas and the author proved that the period matrix of the prolongation t-motives, introduced by Maurischat, of t-motives associated to t-modules entail hyperderivatives of periods and quasi-periods. Computing the Galois group of these prolongations, we prove that the algebraic relations among the hyperderivatives of periods and quasi-periods of a Drinfeld module are the ones induced by the endomorphisms of the Drinfeld module. Furthermore, we construct a new t-motive using these prolongations and compute its Galois group, using which we investigate hyperderivatives of Drinfeld logarithms and quasi-logarithms, and prove transcendence results about them.