Moments of quadratic twists of elliptic curve $L$-functions over function fields

Abstract

We calculate the first and second moments of $L$-functions in the family of quadratic twists of a fixed elliptic curve $E$ over ${\mathbb{𝔽}}_q\left[x\right]$, asymptotically in the limit as the degree of the twists tends to infinity. We also compute moments involving derivatives of $L$-functions over quadratic twists, enabling us to deduce lower bounds on the correlations between the analytic ranks of the twists of two distinct curves.