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One of the most interesting questions in Mathematics concerns the Mordell-Weil rank of the group of rational points on elliptic curves $E/\mathbb{Q}$, namely whether this quantity is bounded as one varies over all elliptic curves defined over the rationals (or some other number field $K$). It is known that there are infinitely many elliptic curves with small rank (say rank at most 2), and that there exist elliptic curves over $\mathbb{Q}$ with rank as large as 28 (due to Elkies).

What is the largest positive integer $r$ such that it is known that there are infinitely many elliptic curves over the rationals with rank at least $r$?

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    $\begingroup$ Have a look at Dujella's tables at web.math.pmf.unizg.hr/~duje/tors/generic.html . According to this, the record is 19 (the curves are parameterized by rational points on an elliptic curve of positive rank), due to Elkies in 2006. $\endgroup$ – Michael Stoll Jun 10 '18 at 10:57
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    $\begingroup$ @MichaelStoll It might be more fair to credit this to Elkies and Neron, since Elkies found an elliptic surface $E\to C$, where everything is defined over $\mathbb Q$ and $E(\mathbb Q(C))$ has rank (at least) 19 and $C(\mathbb Q)$ has rank 1, but one then needs to apply Neron's theorem to deduce that infinitely many of the specializations have rank $\ge19$. (I realize that you know this, but I think it's useful to mention for people who might read your post and not know.) Also,you should post your comment as an answer, since then it can be accepted. $\endgroup$ – Joe Silverman Jun 10 '18 at 11:45
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    $\begingroup$ @JoeSilverman Thanks for pointing this out. -- I was hoping that Noam would give a more detailed answer and didn't want to preempt him. I'll turn my comment into an answer eventually if he doesn't. $\endgroup$ – Michael Stoll Jun 10 '18 at 12:14
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    $\begingroup$ It may be worth noting the heuristics of Granville (described here by Watkins) and of Park, Poonen, Voight and Wood which suggest that the answer should be 21. $\endgroup$ – j.c. Jun 10 '18 at 17:54
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As suggested by @JoeSilverman, I am turning my comment into an answer.

According to the table maintained by Dujella at http://web.math.pmf.unizg.hr/~duje/tors/generic.html, the current record is 19, due to Noam Elkies in 2006. This is apparently obtained by a family of elliptic curves over an elliptic curve over $\mathbb Q$ with positive rank such that the generic fiber has Mordell-Weil group of rank 19. By a theorem of Neron (or a stronger version due to Silverman [Thm. C in "Heights and the specialization map for families of abelian varieties", J. Reine Angew. Math. 342 (1983), 197-211]), for all but finitely many of the infinitely many rational points of the base elliptic curve, the specialization map from the Mordell-Weil group of the generic fiber to that of the fiber above that point is injective, so that the rank of the latter is also at least 19.

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