The elliptic curves , where is a fourth-power-free integer, form a family of quartic twists. We study in this talk the average analytic rank over the family. Under the GRH, we show that the average analytic rank is bounded by , and by assuming a conjecture of Heath-Brown and Patterson about the distribution of quartic Gauss sums. Since the same result holds when we restricts to the subfamilies of curves where the root number is fixed (i.e. ), this shows that there is a positive proportion of curves with among the curves with even analytic rank, and a positive proportions of curves with among the curves with odd analytic rank.
Our results are similar to the results obtained by Heath-Brown for the analytic rank of the quadratic twists under the GRH. For the quadratic twists, it was shown in the recent ground-breaking work of Smith that half of the quadratic twists have algebraic rank 0 and half of the quadratic twists have algebraic rank 1, under the assumption that the Tate-Shafarevic group is finite. For the case of the quartic twists , no bound for the average algebraic rank is known.
This is joint work with L. Devin, A. Fazzari and E. Waxman.
