Torsion and twists of abelian varieties

In this article, we investigate the possible torsion subgroups of twists of abelian varieties with good reduction. As an application, we prove a theorem concerning ramified primes over any quadratic extension where odd-order torsion growth is achieved. In particular, we show that for every rational elliptic curve and every imaginary quadratic field not equal to $\mathbb{Q}(\sqrt{-3})$ satisfying the Heegner hypothesis no odd-order torsion growth can occur.