Date/heure
27 février 2025
14:30 - 15:30
Lieu
Salle Döblin
Oratrice ou orateur
Chantal David (Université Concordia, Montréal)
Catégorie d'évènement Séminaire de Théorie des Nombres de Nancy-Metz
Résumé
The elliptic curves $E_d : y^2 = x^3 – dx$, where $d$ is a fourth-power-free integer, form a family of quartic twists. We study in this talk the average analytic rank $r(d)$ over the family. Under the GRH, we show that the average analytic rank is bounded by $13/6$, and by $3/2$ 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 $E_d$ where the root number is fixed (i.e. $W(E_d) = \pm 1$), this shows that there is a positive proportion of curves with $r(E_d)=0$ among the curves with even analytic rank, and a positive proportions of curves with $r(E_d)=1$ 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 $dy^2 = x^3 + ax + b$ 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 $E_d : y^2 = x^3 – dx$, no bound for the average algebraic rank is known.
This is joint work with L. Devin, A. Fazzari and E. Waxman.