3-isogeny selmer groups and ranks of abelian varieties in quadratic twist families over a number field

Manjul Bhargava, Zev Klagsbrun, Robert J.Lemke Oliver, Ari Shnidman

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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.

Original languageAmerican English
Pages (from-to)2951-2989
Number of pages39
JournalDuke Mathematical Journal
Issue number15
StatePublished - 2019

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