Duke Mathematical Journal
- Duke Math. J.
- Volume 168, Number 15 (2019), 2951-2989.
-Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field
Manjul Bhargava, Zev Klagsbrun, Robert J. Lemke Oliver, and Ari Shnidman
Abstract
For an abelian variety over a number field , we prove that the average rank of the quadratic twists of is bounded, under the assumption that the multiplication-by--isogeny on factors as a composition of -isogenies over . This is the first such boundedness result for an absolutely simple abelian variety of dimension greater than . In fact, we exhibit such twist families in arbitrarily large dimension and over any number field. In dimension , we deduce that if is an elliptic curve admitting a -isogeny, then the average rank of its quadratic twists is bounded. If is totally real, we moreover show that a positive proportion of twists have rank and a positive proportion have -Selmer rank . 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 should be —and the first progress toward the analogous conjecture over number fields other than . Our results follow from a computation of the average size of the -Selmer group in the family of quadratic twists of an abelian variety admitting a -isogeny .
Article information
Source
Duke Math. J., Volume 168, Number 15 (2019), 2951-2989.
Dates
Received: 25 April 2018
Revised: 16 April 2019
First available in Project Euclid: 24 September 2019
Permanent link to this document
https://projecteuclid.org/euclid.dmj/1569290597
Digital Object Identifier
doi:10.1215/00127094-2019-0031
Mathematical Reviews number (MathSciNet)
MR4017518
Subjects
Primary: 14G05: Rational points
Secondary: 11G10: Abelian varieties of dimension > 1 [See also 14Kxx]
Keywords
abelian varieties Selmer groups geometry of numbers rational points quadratic twists
Citation
Bhargava, Manjul; Klagsbrun, Zev; Lemke Oliver, Robert J.; Shnidman, Ari. $3$ -Isogeny Selmer groups and ranks of Abelian varieties in quadratic twist families over a number field. Duke Math. J. 168 (2019), no. 15, 2951--2989. doi:10.1215/00127094-2019-0031. https://projecteuclid.org/euclid.dmj/1569290597