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It is a well-known problem on the theory of elliptic curves that the rank of an elliptic curve (the number of generators of the free part of the Mordell group of the elliptic curve) cannot be effectively computed given the coefficients defining the curve, and that curves with large rank are difficult to construct. However, it is expected that curves with arbitrarily large rank exist.

I am wondering if it is expected that the function $f : \mathcal{C} \rightarrow \mathbb{N}$ where $\mathcal{C}$ is the set of elliptic curves over $\mathbb{Q}$ defined as $f(C) = \text{Rank}(C)$ for a curve $C \in \mathcal{C}$, is surjective.

That is, is every natural number the rank of some elliptic curve over $\mathbb{Q}$? If not, what obstructions would lead one to believe otherwise?

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If the rank of elliptic curves over $\mathbb{Q}$ is unbounded, there is no known reason for this to be the case. It could very well be that the set of ranks has density $0$ on the naturals, for all we know.

But there is no known obstruction either.

Perhaps the proof of unboundedness sheds some light into which ranks occur, but not necessarily. Or perhaps a counterexample of the type "no elliptic curve can have rank x" can be proved, regardless of unboundedness.

On the other direction, the consensus seems to be now that the rank is actually bounded, see for example the very recent paper "A heuristic for boundedness of ranks of elliptic curves" (2016). If this was the case, it obviously solves the issue as well.

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