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 multiplicationby- 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 1. In fact, we exhibit such twist families in arbitrarily large dimension and over any number field. In dimension 1, 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 toward 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 toward 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.
Bibliographical noteFunding Information:
The first author’s work was partially supported by a Simons Investigator Grant and by National Science Foundation (NSF) grant DMS-1001828. The third author was partially supported by NSF grant DMS-1601398.
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