Conjectures for the integral moments and ratios of \(L\)-functions over function fields (Q403285)

\textit{J. B. Conrey} et al. [Proc. Lond. Math. Soc. (3) 91, No. 1, 33--104 (2005; Zbl 1075.11058); J. Number Theory 128, No. 6, 1516--1554 (2008; Zbl 1141.11040)] made two conjectures for the integral moments and ratios of \(L\)-functions defined over number fields. In this paper, the authors develop the function field analogues of these conjectures for the family of quadratic Dirichlet \(L\)-functions associated with hyperelliptic curves of genus \(g\) over a fixed finite field \(\mathbb{F}_q\). Specifically, the authors give a heuristic for the moments and ratios of the family of quadratic Dirichlet \(L\)-functions associated with hyperelliptic curves of genus \(g\) over a fixed finite field \(\mathbb{F}_q\) in the limit as \(g\rightarrow \infty\). Like in the number field case, the author observe that there is a striking resemblance to the corresponding formulae for the characteristic polynomials of random matrices. As an application, the authors calculate the one-level density for the zeros of these \(L\)-functions.