On algebraic function fields and their associated L-functions

Abstract

We study counting functions of field extensions of the rational function field Fq(T). Moreover, we study the distribution of the low-lying zeros of certain L-functions associated to these extensions. Our first results concern S3-cubic extensions of Fq(T), ordered by discriminant, with q coprime to 2 and 3. We derive an asymptotic formula, with an error term of order ≪ϵ X2/3+ϵ, matching the current best result over number fields due to Bhargava, Taniguchi and Thorne. We also obtain an asymptotic formula for a refined counting function where one specifies the splitting type of finitely many primes. In addition to obtaining an upper bound for the error term, we also obtain a lower bound by studying the one-level density of certain Artin Lfunctions associated with these fields. This generalises conditional results over Q obtained by Cho, Fiorilli, Lee and Södergren. Next, we study a certain family of Artin L-functions associated with D4-quartic extensions of Fq(T). We prove for large q coprime to 2 that, when ordered by conductor, at least 77% of these L-functions are non-vanishing at the central point s = 1/2, improving results over Q by Durlanık. We also obtain an asymptotic formula for the counting function of these fields with a power-saving error term, generalising results due to Friedrichsen.