Extreme Values of Quadratic Dirichlet $L$-Functions over Monic Irreducible Polynomials in $\mathbb{F}_q[t]$

Abstract

In this paper, we establish a new $\Omega$-result for the central values $|L(1/2,\chi_P)|$ of quadratic Dirichlet $L$-functions, where $P$ ranges over monic irreducible polynomials associated with hyperelliptic curves of genus $g$ over a fixed finite field $\mathbb{F}_q$. We consider the asymptotic setting in which $q$ is fixed and $g\to\infty$. More precisely, for every $\epsilon \in (0,1/2)$, we prove that \[ \max_{P \in \mathcal{P}_{2g+1}} |L(1/2, \chi_P)| \gg \exp \left( \left( \sqrt{\frac{\sqrt{q}+1}{\sqrt{q}-1} (1/2-\epsilon)} \, \, \ln q + o(1) \right) \sqrt{\frac{g \ln_2 g}{\ln g}} \right), \] where $\mathcal{P}_{2g+1}$ is the set of all monic irreducible polynomials of degree $ 2g+1$ in $\mathbb{F}_q[t]$. Our result extends the recent work of Darbar and Maiti (2024) and yields an improved lower bound for the extreme values in this family. we also investigate the extreme values of these quadratic $L$-functions near the central line. In addition, for $1/2<\sigma<1$ and sufficiently large $n$, we study the extreme values of $L(\sigma,\chi_P)$, where $P\in\mathcal{P}_n$, and obtain an improved lower bound compared with the result of Lumley (2021).