Quadratic Twists of Abelian Varieties with Real Multiplication

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.