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The motivation of this question can be found in

Is it possible to say that every point $P$ in $C(ℚ)$ other than the 'basis' is of finite order?

Given the elliptic curve: $$C:y²=x³+ax+b$$

for $a,b∈ℤ$.

We know that $C(ℚ)≠∅$, so the rank is $r≥0$. From the current literature we do know about the case $r≥2$ except some special cases. My question is then:

Given an arbitrary rank $r≥2$, is it possible to say that there is a curve $C$ such that its rank exactly $r$?

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Safwane
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    No, this is not known. – Álvaro Lozano-Robledo Apr 07 '13 at 11:48
  • @ÁlvaroLozano-Robledo: Thank you very much for your valauble comment. But still we can assume ((say) in a proof) that $r≥2$ if we want to exclud the cases $r=0,1$. – Safwane Apr 07 '13 at 12:00
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    Your comment makes no sense to me. I'll expand on Alvaro Lozano-Robledo's comment. It is still an open question as to whether or not the ranks of elliptic are arbitrarily large, see http://web.math.pmf.unizg.hr/~duje/tors/rankhist.html If for any $r\geq 2$ you could find a curve of that rank, then you would have solved this problem! – Matt Apr 07 '13 at 17:06
  • @Matt: I want to say that the given of an elliptic curve must give an integr $r≥0$. If I want to begin from $r≥2$, then there is no problem. – Safwane Apr 07 '13 at 18:47
  • "If I want to begin from $r\geq 2$" ... begin what? – Álvaro Lozano-Robledo Apr 07 '13 at 19:26
  • Yes. I'm sorry, but your comments continue to be extremely unclear. I have no idea what you are looking for and/or asking. – Matt Apr 07 '13 at 19:47
  • @ÁlvaroLozano-Robledo: I am sorry. I will reformulate the question. It is really very unclear. – Safwane Apr 07 '13 at 20:35
  • @Matt: I am sorry. Iill reformulate the question. It is really very unclear. – Safwane Apr 07 '13 at 20:36

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