The mixed second moment of quadratic Dirichlet $$L$$-functions over function fields.(English)Zbl 1492.11131

The authors study moments of $$L$$-functions in the function field case. More precisely, they consider the family of quadratic Dirichlet $$L$$-functions associated to hyperelliptic curves over a fixed finite field $$\mathbb{F}_q$$ of odd cardinality $$q$$. For a monic and square-free polynomial $$D \in \mathbb{F}_q [t]$$ of degree $$2g + 1$$, the equation $$y^2 = D(x)$$ defines a hyperelliptic curve of genus g. The zeta function of this curve is a rational function whose numerator $$L(s, \chi_D)$$ is the quadratic Dirichlet $$L$$-function associated to the quadratic character $$\chi_D$$ modulo $$D$$. Therefore, the completed $$L$$-function $\Lambda(s,\chi_D)=q^{g(s-1/2)}L(s,\chi_D)$ satisfies the functional equation $$\Lambda(s,\chi_D)=\Lambda(1-s,\chi_D)$$.
The authors provide a mixed second moment formula for $$\Lambda(s,\chi)$$ when $$q \equiv 1$$ (mod $$4$$) and $$g\to \infty$$ as follows: $\sum_{D\in \mathcal{H}_{2g+1}}\frac{\Lambda''(1/2, \chi_D)\Lambda(1/2,\chi_D)}{(\log q)^2}=\frac{q^{2g+1}}{\zeta_q(2)}P(2g+1)+O(q^{g(1+\varepsilon)}),$ where $$P(x)$$ is an explicit polynomial of degree $$5$$ and the summation is over the set $$\mathcal{H}_{2g+1}\subset \mathbb{F}_q[t]$$ of all monic square-free polynomials of degree $$2g+1$$.
The above formula yields also a non-vanishing result regarding the number of $$D\in \mathcal{H}_{2g+1}$$ such that $\Lambda''(1/2, \chi_D)\Lambda(1/2,\chi_D)\neq 0$ which is asymptotically at least $$\frac{q^{2g+1}}{g^4}$$.

MSC:

 11M38 Zeta and $$L$$-functions in characteristic $$p$$ 11G20 Curves over finite and local fields 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11M50 Relations with random matrices 11R58 Arithmetic theory of algebraic function fields
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