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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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... and someone should have answered there already that we now have poeple around that try to make us believe that the rank is bounded, based on random matrix theory and such. – Chris Wuthrich Jul 28 '13 at 11:09
The question here is a bit different, though. If the rank is unbounded then I would think that every natural number could appear. Otherwise probably every one up to some bound. But these will be wild speculations at best. – Chris Wuthrich Jul 28 '13 at 11:12

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