The Ratios Conjecture and upper bounds for negative moments of $L$-functions over function fields

We prove special cases of the Ratios Conjecture for the family of quadratic Dirichlet $L$–functions over function fields. More specifically, we study the average of $L(1\mathrm{/}2+\alpha ,{\chi }_D)\mathrm{/}L(1\mathrm{/}2+\beta ,{\chi }_D)$, when $D$ varies over monic, square-free polynomials of degree $2g+1$ over ${\mathbb{F}}_q[x]$, as $g \to \infty$,andweobtainanasymptoticformulawhen$\Re \beta \gg g^{-1/2+\varepsilon}$\mathrm{.}Wealsostudyaveragesofproductsof$2$over$2$and$3$over$3$ $L$–functions, and obtain asymptotic formulas when the shifts in the denominator have real part bigger than $g^{-1\mathrm{/}4+\epsilon }$ and $g^{-1\mathrm{/}6+\epsilon }$ respectively. The main ingredient in the proof is obtaining upper bounds for negative moments of $L$–functions. The upper bounds we obtain are expected to be almost sharp in the ranges described above. As an application, we recover the asymptotic formula for the one-level density of zeros in the family with the support of the Fourier transform in $(-2,2)$.