Extending the Radius of Convergence of the Drinfeld Module Logarithm

Authors

Ethan Clapp, Louisiana Tech UniversityFollow

Document Type

Article

Publication Date

Spring 2023

Abstract

Drinfeld modules are, in essence, a way to create a function field analogue to complex multiplication. With these modules, we can define a logarithm function analogous to the natural logarithm by virtue of a power series. Similarly to how we extend the radius of convergence of the natural logarithm with complex numbers, we can also extend the radius of convergence for these Drinfeld module logarithms. While there are proofs for extending the radius of convergence of the Carlitz module, the simplest form of Drinfeld module, there is no proof for a generalized Drinfeld. In this paper, a method of extending the Carlitz logarithm’s radius of convergence using Newton polygons will be examined. Afterwards, a way to apply this proof to a general Drinfeld logarithm function will be walked through.

Recommended Citation

Clapp, Ethan, "Extending the Radius of Convergence of the Drinfeld Module Logarithm" (2023). Mathematics Senior Capstone Papers. 32.
https://digitalcommons.latech.edu/mathematics-senior-capstone-papers/32