The first moment of quadratic Dirichlet $L$-functions in the even hyperelliptic ensemble

Abstract

We establish an asymptotic formula for the first moment of the central values of quadratic Dirichlet $L$-functions in the even hyperelliptic ensemble $\mathcal{H}_{2g+2}$ over $\mathbb{F}_q[x]$, for every fixed odd prime power $q$. Working with the completed $L$-function yields an exact two-term central-value formula with truncation levels $g$ and $g-1$; combining the two truncations before the square-dual contour shifts leaves the three cubic points $z^3 = q^{-4}$ as the only secondary poles, and they contribute the secondary term $q^{2g/3+2}(a_g g + b_g)$ with an error $O_\varepsilon(q^{g(1+\varepsilon)/2})$. The coefficients are real and $m \mapsto (a_m, b_m)$ has minimal period exactly three, so the coefficient of $g$ at the order $q^{2g/3}$ depends on $g$ modulo 3, in contrast with the earlier even-degree formulas; the difference is confirmed numerically.