L-functions of twisted elliptic curves over function fields

Traditionally number theorists have studied, both theoretically and computationally, elliptic curves and their L-functions over number fields, in particular over the rational numbers. Much less work has been done over function fields, especially computationally, where the underlying geometry of the function field plays an intimate role in the arithmetic of elliptic curves. We make use of this underlying geometry to develop a method to compute the L-function of an elliptic curve and its twists over the function field of the projective line over a finite field. This method requires computing the number of points on an elliptic curve over a finite field, for which we present a novel algorithm. If the j-invariant of an elliptic curve over a function field is non-constant, its L-function is a polynomial, hence its analytic rank and value at a given point can be computed exactly. We present data in this direction for a family of quadratic twists of four fixed elliptic curves over a few function fields of differing characteristic. First we present analytic rank data that confirms a conjecture of Goldfeld, in stark contrast to the corresponding data in the number field setting. Second, we present data on the integral moments of the value of the L-function at the symmetry point, which on the surface appears to refute random matrix theory conjectures.