# arithmetic geometry – When are marked points torsion

We consider a family of genus one curves defined by

$$displaystyle d_0 z^2 = F_{a,b}(u,v) text{ (1)},$$

where

$$F_{a,b}(u,v) = a(u^2 – v^2)^2 + 4bu^2 v^2$$

and $$d_0 = F_{a,b}(u_0, v_0)$$, where $$u_0, v_0$$ are fixed integers independent of $$a,b$$. Then, provided that $$a,b$$ are integers, the curve defined by (1) is a genus one curve with a marked rational point, namely $$(u_0, v_0, 1)$$.

How does one determine whether the marked point $$(u_0, v_0, 1)$$ is torsion? As $$a,b$$ vary over a box, say $$max{|a|, |b|} leq X$$, can one say anything about the distribution of pairs $$(a,b)$$ for which $$(u_0, v_0, 1)$$ is torsion/not torsion?