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Large values of quadratic Dirichlet $L$-functions over monic irreducible polynomial in $\mathbb {F}_q[t]$
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Proc. Amer. Math. Soc. 152 (2024), 3243-3254

DOI: https://doi.org/10.1090/proc/16828

Published electronically: June 14, 2024

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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 $\epsilon \in (0, 1/2)$, \[ \max _{\substack {P\in \mathcal {P}_{2g+1}}}|L(1/2, \chi _P)|\gg \exp \left (\left (\sqrt {\left (1/2-\epsilon \right )\ln q}+o(1)\right )\sqrt {\frac {g \ln _2 g}{\ln g}}\right ), \] where $\mathcal {P}_{2g+1}$ is the set of all monic irreducible polynomials of degree $2g+1$. This matches with the order of magnitude of the Bondarenko–Seip bound.

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