TY - JOUR

T1 - Quadratic Twists of Abelian Varieties with Real Multiplication

AU - Shnidman, Ari

N1 - Publisher Copyright:
© 2021 Oxford University Press. All rights reserved.

PY - 2021/3/1

Y1 - 2021/3/1

N2 - Let F be a totally real number field and A/F an abelian variety with real multiplication (RM) 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 Ad have rank 0. If moreover A is principally polarized and III(Ad) is finite, then a positive proportion of Ad have O-rank 1. Our proofsmake use of the geometry-of-numbers methods from our previous work with Bhargava, Klagsbrun, and Lemke Oliver and develop them further in the case of RM. We quantify these results for A/Q of prime level, using Mazur s study of the Eisenstein ideal. For example, suppose p ? 10 or 19 (mod 27), and let A be the unique optimal quotient of J0(p) with a rational point P of order 3. We prove that at least 25% of twists Ad have rank 0 and the average O-rank of Ad(F) is at most 7/6. Using the presence of two different 3-isogenies in this case, we also prove that roughly 1/8 of twists of the quotient A/(P) have nontrivial 3-Torsion in their Tate Shafarevich groups.

AB - Let F be a totally real number field and A/F an abelian variety with real multiplication (RM) 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 Ad have rank 0. If moreover A is principally polarized and III(Ad) is finite, then a positive proportion of Ad have O-rank 1. Our proofsmake use of the geometry-of-numbers methods from our previous work with Bhargava, Klagsbrun, and Lemke Oliver and develop them further in the case of RM. We quantify these results for A/Q of prime level, using Mazur s study of the Eisenstein ideal. For example, suppose p ? 10 or 19 (mod 27), and let A be the unique optimal quotient of J0(p) with a rational point P of order 3. We prove that at least 25% of twists Ad have rank 0 and the average O-rank of Ad(F) is at most 7/6. Using the presence of two different 3-isogenies in this case, we also prove that roughly 1/8 of twists of the quotient A/(P) have nontrivial 3-Torsion in their Tate Shafarevich groups.

UR - http://www.scopus.com/inward/record.url?scp=85108669829&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnz185

DO - 10.1093/imrn/rnz185

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AN - SCOPUS:85108669829

SN - 1073-7928

VL - 2021

SP - 3267

EP - 3298

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 5

ER -