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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?

Bernard
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Guilherme Gondin
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    What sort of family of curves, and what sort of 'study'? Can you be more specific about what you're trying to do? – Steven Stadnicki Mar 05 '20 at 20:38
  • I'm starting to study curves, I can't be much more specific. – Guilherme Gondin Mar 05 '20 at 20:47
  • But my objective is to find rational points in more than just a elliptic curve at once. – Guilherme Gondin Mar 05 '20 at 20:48
  • Do you mean 'I can't be much more specific' in the sense of 'I don't know enough about the subject to say more yet' or in the sense of 'I am working on something and don't want it poached'? If the former, then you should certainly start by learning more about how rational points on specific curves are studied; if the latter, then I suspect you have some misconceptions about mathematical academia... – Steven Stadnicki Mar 05 '20 at 21:07
  • Sure in the first, and you can't be more right on your answer, this was nothing more than a thought of mine and I should study more, just tought whould be interesting to read the comments. – Guilherme Gondin Mar 05 '20 at 21:40

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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 $m\ge 2$.

Dietrich Burde
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