A way to study families of algebraic curves

Question

I was looking for a way to study rational points on a family of curves instead of only one at a time, is there any?

Answer 1

Yes, there is any. Let $E_m$ be the family of elliptic curves given by $$ y^2=x^3-x+m^2 $$ Then there are infinitely many rational numbers $n$ such that $E_n(\Bbb Q)$ has rank at least $3$. Another family is given by $$ y^2=x^3-m^2x+1, $$ for $m\in \Bbb N$. Here we have $$ rank (E_m)\ge 2 $$ for all $m\ge 2$.