Abelian Extensions of Function Fields with One or Two Variables

Abstract

This thesis investigates abelian extensions of function fields with one or two variables with the aid of elliptic modules and shtukas. We treat the one-variable case and the two-variable case in Chapter 2 and in Chapter 3, respectively.

Regarding the case of a function field with one variable, there exists an explicit construction of the reciprocity map due to Zywina, which he obtained building on the theory of elliptic modules developed by Drinfeld and Hayes. Chapter 2 of this thesis thoroughly explains this theory of elliptic modules and describes Zywina’s construction with an example.

For the two-variable case, this thesis focuses on a function field of the product of two copies of a curve. Building on Drinfeld’s works on the moduli of shtukas, we geometrically construct reasonably large abelian extensions of such a function field for different choices of “levels” on the curve. The main strategy for this construction is to closely analyze certain moduli of rank 1 shtukas, using Picard schemes. We also provide an interpretation of our geometric construction by relating it to ray class fields of the function field of the curve. Chapter 3 of this thesis starts with some background in algebraic geometry, then introduces shtukas and Picard schemes and finally shows how to construct these abelian extensions.