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For every positive integer $n$, is there some number field $K$ and elliptic curve $E/K$ such that $E(K)$ has rank $n$?

It's easy to show that the set of such $n$ is unbounded. But can one show that every positive integer is the rank of some elliptic curve over a number field?

The analogous question for a fixed number field is expected to have a negative answer (c.f. e.g., this question) but is still conjectural. But I wonder if one might be able to prove a positive answer to the question I asked above.

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    $\begingroup$ I am not sure whether it is fair to say that the fixed number field case should admit a negative answer; of course that comes down to a famous unsolved problem. For the existence of $E/K$, an iterated application of Silverman's specialization theorem reduces us to proving that the group of sections of the elliptic scheme $A_{1,n+1} \to A_{1,n}$ has rank $n$, generated by the $n$ 'point doubling' sections $(x_1, x_2,\ldots,x_n) \mapsto (x_1,x_1;x_2,\ldots,x_n)$, etc. (Here $A_{g,n}$ is the space of $n$-pointed abelian varieties of dimension $g$). This should be true. $\endgroup$ – Vesselin Dimitrov Jun 26 '19 at 15:02
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    $\begingroup$ @RP_ I would -1 your comment if I could. Since when does usefulness of the result dictate whether the question is good for this site? Which ranks are possible over a fixed number field is similarly useless, and yet it's a very interesting open problem. $\endgroup$ – Wojowu Jun 26 '19 at 15:22
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    $\begingroup$ @RP: What would be the use of knowing the answer of any question on this website? Unless you're only in it for cryptographic applications, this question is useful because of human curiosity. $\endgroup$ – David Corwin Jun 26 '19 at 15:23
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    $\begingroup$ One may ask more generally: given a representation of a finite group $\rho : G \to \mathrm{GL}_n(\mathbb{Q})$, does $\rho$ arise as the Galois module $E(L) \otimes \mathbb{Q}$ for some elliptic curve $E/K$ and some extension $L/K$ with Galois group $G$. We can also fix the number field $K$. If I recall correctly, if one replaces "arises" by "appears inside" then for function fields $K=\mathbb{F}_p(t)$ these kinds of things are known. $\endgroup$ – François Brunault Jun 26 '19 at 15:49
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    $\begingroup$ @VesselinDimitrov The Mordell-Weil group of $E$ is isomorphic to the relative Picard group of $E/\mathbb{Q}$, so this is a natural analogue. By the way it is known that every abelian group is the Picard group of some Dedekind domain (Claborn 1966). One may ask the same thing while restricting the class of abelian groups and/or Dedekind domains considered. $\endgroup$ – François Brunault Jun 27 '19 at 7:51

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