What heuristic evidence is there concerning the unboundedness or boundedness of Mordell-Weil ranks of elliptic curves over $\Bbb Q$?

Question

Some experts have a hunch that for any nonnegative integer $r$ there are infinitely many elliptic curves over $\Bbb Q$ with Mordell-Weil rank at least $r$.

The best empirical evidence for this hunch can be found in Andrej Dujella's tables here and here, the strongest of this evidence being provided by Elkies using constructions involving K3 surfaces.

The only heuristic evidence I know of supporting this hunch is the results of Tate and Shafarevich (strengthened by Ulmer) that the Mordell-Weil ranks of elliptic curves over the a fixed function field $\Bbb F_q(t)$ can be arbitrarily large.

What other heuristic evidence (if any) is there that the ranks of elliptic curves over $\Bbb Q$ should be unbounded?

Seeing as how some experts do no believe this conjecture, I'd also accept answer to the companion question:

What heuristic evidence (if any) is there that the ranks of elliptic curves over $\Bbb Q$ should be uniformly bounded?

Answer 1

Remke remarks: "I guess that the main evidence is the fact that every now and then someone comes up with a new record."

As a not-entirely-serious quantification of that remark, I graphed the following data: For each year starting in 1974 in which a new E(Q) rank record was found, graph the point (year,highest rank found that year). Here's the data: (1974,6),(1975,7),(1977,9),(1982,12),(1986,14),(1992,19), (1993,20),(1994,21),(1997,22),(1998,23),(2000,24),(2006,28),which I took from http://web.math.hr/~duje/tors/rankhist.html. The data looks remarkably linear. The linear correlation coefficient is 0.998. The slope is 0.686. Some of the linearity probably comes from Moore's law (log-linear increase in computing power), but the table also represents major algorithmic advances of Mestre, Elkies, ..., so it's not clear (to me) why theoretical advances should also give linear growth!

Addendum 8/24: Elkies and Klagsbrun have added a new (year,rank) point, (2024,29), to the list. With that addition, the correlation coefficient is down to 0.95, and the slope is reduced to 0.525. Still impressive, though!

Answer 2

An update from 2018: There has been some recent work on a heuristic suggesting that there are infinitely many elliptic curves of every rank $<21$ but only finitely many of rank $> 21$ (it's unclear to me what the model says about the case rank $=21.$)

I'm not sure what the full attribution should be, but one can read about these ideas in Bjorn Poonen's ICM article https://arxiv.org/abs/1711.10112v2 and the references within.

Update 2024: See the article Park, Jennifer; Poonen, Bjorn; Voight, John; Wood, Melanie Matchett, A heuristic for boundedness of ranks of elliptic curves, J. Eur. Math. Soc. (JEMS) 21, No. 9, 2859-2903 (2019). ZBL1469.11173.

Answer 3

I am not aware of much evidence for arbitrary high rank elliptic curves. Silverman in his book (Arithmetic of elliptic curves) gives as evidence the lack of evidence for the opposite statement. I guess that the main evidence is the fact that every now and then someone comes up with a new record.

The function field case is some sort of evidence, however, most constructions rely on supersingular elliptic surfaces or supersingular Fermat surfaces. These objects are quite far away from characteristic zero objects. (One exception to this is the construction by Bouw-Diem-Scholten.) Shioda tried to use Fermat surfaces in order to produce high rank elliptic curves over $K(t)$, with $\operatorname{char}(K)=0$, but the best you can get from that is rank 68 if $K$ is algebraically closed, for $K=\mathbb{Q}$ this maximum is much lower.

A second piece of evidence is the fact that we can construct big Selmer groups. We can find arbitrarily many elements of order m in the m-Selmer group, for $m=2,\dots,10,12,13,16,25$. However, every construction of large Selmer groups seems to allow a refinement that makes the Tate-Shafarevich group large. (For some reason nobody wrote this up for the composite $m$ mentioned above, but I am quite sure that this has been done.) Matsuno gave a construction s.t. for each prime number $p$ and each field $K$ containing a degree $p$ Galois extension of $\mathbb{Q}$ one can produce a series of elliptic curves $E/\mathbb{Q}$ the group $S^p(E/K)$ can be arbitrarily large, but also this construction allows a refinement to get big Tate-Shafarevich groups.

Answer 4

There are dozens of articles in the mathematical literature giving a rather strong correlation between the ranks of certain elliptic curves (for example $y^2 = k^3 + k$ for certain values of $k$) and the $3$-rank of the class groups of quadratic number fields (Quer, Top, DeLong), or the $2$-rank of the class groups of pure cubic fields (Frey et al, Schneiders). By Lenstra-Cohen, we certainly expect the former to be unbounded; I don't know whether Cohen-Martinet predicts that the second set is not bounded since $2$ is some kind of bad prime in this connection. I do wonder, however, whether there really are (not were!) experts who suggest that these ranks are bounded.

I have also seen attempts at transferring the Cohen-Lenstra heuristics to the Tate-Shafarevich group of elliptic curves (C. Delaunay); I do not know whether these heuristics are strong enough to predict that sha is small often enough to give a large rank of the Mordell-Weil group.

Answer 5

Manjul Bhargava and Arul Shankar have shown that the average rank is at most 1.17. But perhaps this is not evidence one way or the other for the boundedness/unboundedness.

Answer 6

Dear Felipe, as long as we restrict ourselves to elliptic curves over $\mathbb{Q}$, the only results known in this direction are that the 2, 3, 5, 7 and 13 primary torsion of Tate-Shafarevich groups can be arbitrarily large. These are due to Kramer, Cassels, Fisher, Matsuno and Matsuno, respectively. It is still unknown whether it is true that for any $p$, the $p$ primary torsion of sha can be arbitrarily large. It is not even known whether the $p$-torsion of sha can be non-trivial for arbitrary $p$.

As far as I know, almost everybody nowadays believes that ranks of elliptic curves over $\mathbb{Q}$ can be arbitrarily large, but when pressed for evidence, most people point to the function field case. In fact, it is not even known, whether the rank can get arbitrarily large over number fields of uniformly bounded degree.

Answer 7

It was proved a long time ago that you can make elliptic curves with either large rank or large Sha. But I'm not sure what to make of that as an heuristic.