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Recently, I have heard of some heuristics that would suggest that the rank of elliptic curves are bounded (specifically in the congruent number family). I always though that the best way to prove something about the rank of an elliptic curve is to look at the rank of the 2-Selmer group. Indeed, I was able to find a reference that says that the rank in the congruent number family of elliptic curves is bounded by the 2-Selmer rank.

Clearly it is not known that the 2-Selmer rank is bounded, but is it known that it is unbounded in the congruent number family? What about in general? Is this equivalent to asking if the rank of elliptic curves is unbounded?

I would appreciate any answers or references.

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Probably you mean "size" of the 2-Selmer group or "dimension" as $\mathbb{F}_2$-vector space, not "rank" since it is finite. The corank of the 2-primary Selmer group is equal to the rank of the curve unless Sha is infinite. – Chris Wuthrich Aug 5 '14 at 22:52
You want Heath-Brown's paper. The size of Selmer groups for the congruent number problem, II. Inv. Math. 118, no. 1 (1994), 331–370. which in fact computes the distribution of the 2-Selmer group size for this congruent number curve quadratic twist family. – NAME_IN_CAPS Aug 5 '14 at 23:27
on math.SE (but I guess it might make sense to keep it here). – user9072 Aug 5 '14 at 23:55
up vote 9 down vote accepted

Cassels showed in "Arithmetic on curves of genus 1 VI" that there is no bound on the size of the 3-Selmer group for the curves of $x^3+y^3+dz^3=0$. Well, he actually showed that the 3-torsion of Sha can be arbitrarily large.

Similar results are known for other primes. Tom Fisher found arbitrarily large 5-torsion for instance. He refers to Bölling for the fact that the 2-torsion can be arbitrarily large. Kloosterman and Schaefer show that the $p$-torsion can grow over certain number fields.

I have little hope that the $p$-Selmer group for a prime $p$ is bounded, because the Tate-Shafarevich part of it can grow.

I would not know about $p=2$ for the congruent number curves.

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