arXiv subjects

Victor Ahlquist

Publications and source records attributed to Victor Ahlquist.

2 recordsLinked to original sources

arXiv2025

Non-vanishing of Artin $L$-functions associated with $D_4$-quartic function fields ordered by conductor

We study the low-lying zeros of certain Artin $L$-functions associated with $D_4$-quartic function fields. Specifically, we prove that when ordered by conductor, at least $77\%$ of these $L$-functions are non-vanishing at the central point. This generalises and extends results over $\mathbb{Q}$ due to Durlanik, proving that an infinite number of these $L$-functions are non-vanishing. We obtain these results by examining the low-lying zeros of the $L$-functions using the one-level density. Specifically, we apply and extend a method used by Rudnick, who studied Dirichlet $L$-functions associated with quadratic function field extensions, to the $D_4$-case. The main difficulty is studying $L$-functions which are associated to $D_4$-fields whose quadratic subfield is of large discriminant. These $L$-functions are studied by utilising the so-called flipped field of a $D_4$ extension, combining a method introduced by Friedrichsen for counting $D_4$-fields, with explicit ramification theory in such fields provided by Altu\u{g}, Shankar, Varma and Wilson.

math.NT↗

arXiv2025

On the counting function of cubic function fields

We study the counting function of cubic function fields. Specifically, we derive an asymptotic formula for this counting function including a secondary term and an error term of order $\mathcal{O}\big(X^{2/3+\epsilon}\big)$, which matches the best-known result, due to Bhargava, Taniguchi and Thorne, over $\mathbb{Q}$. Furthermore, we obtain estimates for the refined counting function, where one specifies the splitting behaviour of finitely many primes. Also in this case, our error term matches what is known for number fields. However, in the function field setting, the secondary term becomes more difficult to write down explicitly. Our proof uses geometry of numbers methods, which are especially effective for function fields. In particular, we obtain an exact formula for the number of orbits of cubic forms with fixed absolute discriminant. Moreover, by studying the one-level density of a family of Artin $L$-functions associated to these cubic fields, we prove an unconditional lower bound on the error term in the estimate for the refined counting function. This generalises a conditional result over $\mathbb{Q}$, due to Cho, Fiorilli, Lee and S\"odergren.

math.NT↗