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

We prove an $\mathrm{\Omega }$-result for the quadratic Dirichlet $L$-function $\mathrm{\mid }L(1\mathrm{/}2,{\chi }_P)\mathrm{\mid }$ 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 $\epsilon \in (0,1\mathrm{/}2)$, $$\underset{\begin{array}{c}P\in {\mathcal{P}}_{2g+1}\end{array}}{\max }\mathrm{\mid }L(1\mathrm{/}2,{\chi }_P)\mathrm{\mid }\gg \exp \left((\sqrt{(1\mathrm{/}2-\epsilon )\ln q}+o(1))\sqrt{\frac{g{\ln }_{2}g}{\ln g}}\right),$$ 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.