Large values of quadratic Dirichlet \(L\)-functions over monic irreducible polynomial in \(\mathbb{F}_q[t]\) (Q6566376)

Let \(P \in \mathbb{F}_q[t]\) be a monic irreducible polynomial. The quadratic character \(\chi_P\) attached to \(P\) is defined using quadratic residue symbol for \(\mathbb{F}_q[t]\) by \(\chi_P (f) =\left(\frac{f}{P}\right)\) and the corresponding Dirichlet \(L\)-function is denoted by \(L(s, \chi_P )\). Let \(\mathcal{P}_n\) be the family of all curves given in affine form by an equation \(C_P : y^2 = P(t)\), where \(P\) is an irreducible polynomial of degree \(n\). The main objective of this paper is to study large values of the subfamily \(\lbrace L(1/2, \chi_P) \rbrace_{P\in \mathcal{P}_{2g+1}}\). More precisely, it is shown that as \(g\to \infty\) and for any \(\varepsilon \in (0, 1/2)\), we have \N\[\N\max\limits_{P\in \mathcal{P}_{2g+1}}|L(1/2,\chi_P)|\gg\exp\bigg(\left (\sqrt{\left(1/2-\varepsilon\right)\ln q}+o(1)\right)\sqrt{\frac{g \ln_2 g}{\ln g}}\:\bigg).\N\]\NNote that it seems very difficult to find large values of the subfamily \(\lbrace L(1/2, \chi_P) \rbrace_{P\in \mathcal{P}_{2g+2}}\) as same strength of the above estimate due to lack of positivity to the extra \(\lambda\)-terms coming from the approximate functional equation for \(L(1/2, \chi_P)\) in Lemma \(7\).