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

We calculate the first and second moments of L-functions in the family of quadratic twists of a fixed elliptic curve E over Fq[x], 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.