Quadratic twists of abelian varieties with real multiplication

TL;DR

This paper investigates the ranks of quadratic twists of abelian varieties with real multiplication, showing that a positive proportion have minimal or maximal rank under certain conditions, with implications for rational points on related hyperelliptic curves.

Contribution

It establishes that a positive proportion of quadratic twists of certain abelian varieties have rank 0 or the full dimension, under specific algebraic assumptions, advancing understanding of their arithmetic properties.

Findings

A positive proportion of quadratic twists have rank 0.

A positive proportion of quadratic twists have rank equal to the dimension of the abelian variety.

For Jacobians of hyperelliptic curves, many twists have no rational points beyond the hyperelliptic involution.

Abstract

Let $F$ be a totally real number field and $A/F$ a principally polarized abelian variety with real multiplication by the ring of integers $\mathcal{O}$ of a totally real field. Assuming $A$ admits an $\mathcal{O}$-linear 3-isogeny over $F$, we prove that a positive proportion of the quadratic twists $A_d$ have rank 0. We also prove that a positive proportion of $A_d$ have rank $\dim A$, assuming the Tate-Shafarevich groups are finite. If $A$ is the Jacobian of a hyperelliptic curve $C$, we deduce that a positive proportion of twists $C_d$ have no rational points other than those fixed by the hyperelliptic involution.