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

Abstract: 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 $L(1/2+\alpha,\chi_D)/L(1/2+\beta,\chi_D)$, when $D$ varies over monic, square-free polynomials of degree $2g+1$ over $\mathbb{F}_q[x]$, as $g \to \infty$, and we obtain an asymptotic formula when $\Re \beta \gg g^{-1/2+\varepsilon}$. We also study averages of products of $2$ over $2$ and $3$ over $3$ $L$--functions, and obtain asymptotic formulas when the shifts in the denominator have real part bigger than $g^{-1/4+\varepsilon}$ and $g^{-1/6+\varepsilon}$ 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 $(-2,2)$.