Rigid Analytic Trivializations and Periods of Drinfeld Modules and Their Tensor Products
Abstract
The purpose of this research is to study Drinfeld modules, tensor product of Drinfeld modules, their rigid analytic trivializations, and their periods. A formula for rigid analytic trivializations for Drinfeld modules was originally given by Pellarin. In this research, we provide a new method to construct a rigid analytic trivialization for Drinfeld modules. Unlike Pellarin's formula, our method does not require periods of Drinfeld modules. Given a rank $r$ Drinfeld module, we provide a recursive process that produce a convergent $t$-division sequence. Consequently we use the $t$-division sequence to construct a sequence of matrices $(\Upsilon_n)_{n\geq1}$ and by computing the limit of $(\Upsilon_n)_{n\geq1}$, we obtain our rigid analytic trivialization for a Drinfeld module. Using the function $\cL_\phi(\xi;t)$ introduced by El-Guindy and Papanikolas, we are able to find an explicit formula for our rigid analytic trivialization. Furthermore, in the second part of our research, we investigate tensor products of two Drinfeld modules $\phi_1$ and $\phi_2$. Using the theory of $t$-motives, we define a $t$-action for $\phi_1\otimes\phi_2$. Inspired by a formula for periods of the tensor product of Carlitz module by Anderson and Thakur, we discover a formula for periods of the tensor product $\phi_1\otimes\phi_2$. Moreover, we provide a formula for Anderson generating functions for the tensor product $\phi_1\otimes\phi_2$.
Citation
Khaochim, Chalinee (2021). Rigid Analytic Trivializations and Periods of Drinfeld Modules and Their Tensor Products. Doctoral dissertation, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /193222.