Large values of quadratic Dirichlet $L$-functions over monic irreducible polynomial in ${\mathbb{F}}_q[t]$

Abstract

We prove an $\Omega$-result for the quadratic Dirichlet $L$-function $|L(1/2,{\chi }_P)|$ over irreducible polynomials $P$ associated with the hyperelliptic curve of genus $g$ over a fixed finite field ${\mathbb{F}}_q$ in the large genus limit. In particular, we showed that for any $\epsilon \in (0,1/2)$, \[ \max_{\substack{P\in \mathcal{P}_{2g+1}}}|L(1/2, χ_P)|\gg \exp\left(\left(\sqrt{\left(1/2-ε\right)\ln q}+o(1)\right)\sqrt{\frac{g \ln_2 g}{\ln g}}\right), \] where ${\mathcal{P}}_{2g+1}$ is the set of all monic irreducible polynomial of degree $2g+1$. This matches with the order of magnitude of the Bondarenko--Seip bound.

12 Pages; To appear in Proceedings of American Mathematical Society