For an abelian variety A over a number field F , we prove that the average rank of the quadratic twists of A is bounded, under the assumption that the multiplication-by-3 isogeny on A factors as a composition of 3-isogenies over F . This is the first such boundedness result for an absolutely simple abelian variety A of dimension greater than one. In fact, we exhibit such twist families in arbitrarily large dimension and over any number field.In dimension one, we deduce that if E/F is an elliptic curve admitting a 3-isogeny, then the average rank of its quadratic twists is bounded. If F is totally real, we moreover show that a positive proportion of twists have rank 0 and a positive proportion have 3-Selmer rank 1. These results on bounded average ranks in families of quadratic twists represent new progress towards Goldfeld's conjecture -which states that the average rank in the quadratic twist family of an elliptic curve over Q should be 1/2 -and the first progress towards the analogous conjecture over number fields other than Q.Our results follow from a computation of the average size of the φ-Selmer group in the family of quadratic twists of an abelian variety admitting a 3-isogeny φ.

The elliptic curve Ek:y2=x3+k admits a natural 3‐isogeny ϕk:Ek→E−27k. We compute the average size of the ϕk‐Selmer group as k varies over the integers. Unlike previous results of Bhargava and Shankar on n‐Selmer groups of elliptic curves, we show that this average can be very sensitive to congruence conditions on k; this sensitivity can be precisely controlled by the Tamagawa numbers of Ek and E−27k. As a consequence, we prove that the average rank of the curves Ek, k∈Z, is less than 1.21 and over 23% (respectively, 41%) of the curves in this family have rank 0 (respectively, 3‐Selmer rank 1).

We show that a positive proportion of cubic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof involves the comparison of 2-descent and 3-descent in a certain family of Mordell curves E k : y 2 = x 3 + k. As a byproduct of our methods, we show that, for every r ≥ 0, a positive proportion of curves E k have Tate-Shafarevich group with 3-rank at least r.

Let F be the field of rational functions on a smooth projective curve over a finite field, and let π be an unramified cuspidal automorphic representation for PGL2 over F . We prove a variant of the formula of Yun and Zhang relating derivatives of the L-function of π to the self-intersections of Heegner-Drinfeld cycles on moduli spaces of shtukas.In our variant, instead of a self-intersection, we compute the intersection pairing of Heegner-Drinfeld cycles coming from two different quadratic extensions of F , and relate the intersection to the r-th derivative of a product of two toric period integrals.

Let F be a totally real number field and A/F a principally polarized abelian variety with real multiplication by the ring of integers O of a totally real field. Assuming A admits an O-linear 3-isogeny over F , we prove that a positive proportion of the quadratic twists A d have rank 0. We also prove that a positive proportion of A d have rank dim A, assuming the groups X(A d ) are finite. If A is the Jacobian of a hyperelliptic curve C, we deduce that a positive proportion of twists C d have no rational points other than those fixed by the hyperelliptic involution.1 This terminology is used in [27], but with a slightly different meaning. 2