# Largest rank assumed by infinitely many elliptic curves

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 $r$?

• 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. Jun 10, 2018 at 10:57
• @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. Jun 10, 2018 at 11:45
• @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. Jun 10, 2018 at 12:14
• 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.
– j.c.
Jun 10, 2018 at 17:54

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.