Large values of quadratic Dirichlet 𝐿-functions over monic irreducible polynomial in 𝔽_{𝕢}[𝕥]

We prove an Ω \Omega -result for the quadratic Dirichlet L L -function | L ( 1 / 2 , χ P ) | |L(1/2, \chi _P)| over irreducible polynomials P P associated with the hyperelliptic curve of genus g g over a fixed finite field F q \mathbb {F}_q in the large genus limit. In particular, we showed that for any ϵ ∈ ( 0 , 1 / 2 ) \epsilon \in (0, 1/2) , \[ max P ∈ P 2 g + 1 | L ( 1 / 2 , χ P ) | ≫ exp ⁡ ( ( ( 1 / 2 − ϵ ) ln ⁡ q + o ( 1 ) ) g ln 2 ⁡ g ln ⁡ g ) , \max _{\substack {P\in \mathcal {P}_{2g+1}}}|L(1/2, \chi _P)|\gg \exp \left (\left (\sqrt {\left (1/2-\epsilon \right )\ln q}+o(1)\right )\sqrt {\frac {g \ln _2 g}{\ln g}}\right ), \] where P 2 g + 1 \mathcal {P}_{2g+1} is the set of all monic irreducible polynomials of degree 2 g + 1 2g+1 . This matches with the order of magnitude of the Bondarenko–Seip bound.