Towards a topological proof of Wright's Theorem

Abstract

Malle's Conjecture concerns the asymptotic behavior of the number of degree-n extensions of a number field with Galois group permutation-isomorphic to G. Using the function field analogy, a similar conjecture can be made for finite extensions of F_q(t), where q is a power of a prime. In fact, a similar conjecture has been proven by Wright (1989), in the case where G is abelian. In this work, we also prove an asymptotic bound on the number of field extensions of F_q(t) in the abelian case, using substantially different methods. This serves to highlight connections between different fields within mathematics and to test the feasibility of the particular method outlined in this paper.