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

In this paper, we establish a new $Ω$-result for the central values $|L(1/2,χ_P)|$ 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