Large values of quadratic Dirichlet 𝐿-functions over monic irreducible polynomial in 𝔽𝕢[𝕥]

We prove an Ω Ω -result for the quadratic Dirichlet L L -function | L ( 1 / 2 , χ P ) | |L(1/2, χ _P)| over irreducible polynomials P P associated with the hyperelliptic curve of genus g g over a fixed finite field F q F_q in the large genus limit. In particular, we showed that for any ϵ ∈ ( 0 , 1 / 2 ) ε ∈ (0, 1/2) , \[ max P ∈ P 2 g + 1 | L ( 1 / 2 , χ P ) | ≫ exp ⁡ ( ( ( 1 / 2 − ϵ ) ln ⁡ q + o ( 1 ) ) g ln 2 ⁡ g ln ⁡ g ) , max _{ {P∈ P2g+1}}|L(1/2, χ _P)| exp ( (√ (1/2-ε )ln q+o(1) )√g ln _2 g/ln g ), \] where P 2 g + 1 P2g+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.