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 values $|L(\sigma,\chi_P)|$ of quadratic Dirichlet $L$-functions at and near the central point, where \[ \sigma=\frac12+\frac{\kappa}{\ln g}, \qquad \kappa\geq 0, \] and $P$ 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 $\epsilon \in (0,1/2)$ andevery fixed $\kappa\geq 0$, we prove that \[ \max_{P \in \mathcal{P}_{2g+1}} |L(\sigma,\chi_P)| \gg \exp \left(\left(e^{-\kappa} \ln q \,\sqrt{\frac{\sqrt{q}+1}{\sqrt{q}-1}(1/2-\epsilon)} +o(1)\right)\sqrt{\frac{g\ln_2 g}{\ln g}} \right), \] where $\mathcal{P}_{2g+1}$ is the set of all monic irreducible polynomials of degree $2g+1$ in $\mathbb{F}_q[t]$. In particular, when $\kappa=0$, this gives the corresponding result at the central point $\sigma=1/2$. Our result extends the recent work of Darbar and Maiti (2024) and yields an improved lower bound for the extreme values in this family. 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).