The Average Analytic Rank of a Family of Elliptic Curves Original Research Article

This paper studies the family of elliptic curves Em: X3 + Y3 = m where m is a cubefree integer. Assuming the Generalized Rieman Hypothesis, the average rank of Em′s with even analytic rank is proved to be ≤5/2, asymptotically. We also obtain some results for the case of Em′s with odd analytic rank and mean value theorems for higher moments of the analytic ranks.