Large values of quadratic Dirichlet 𝐿-functions over monic irreducible polynomial in 𝔽🖁𝕥

We prove an Ω -result for the quadratic Dirichlet L L -function | L (1 / 2, χ P) | |L (1/2, P) | over irreducible polynomials P P associated with the hyperelliptic curve of genus g g over a fixed finite field F q Fq in the large genus limit. In particular, we showed that for any ϵ ∈ (0, 1 / 2) (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), ₏ {₏₂₆+₁}|L (1/2, P) | ( ( (1/2-) q+o (1) ) {g ₂ g g}), \ where P 2 g + 1 P₂₆+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.