The following are two facts about $\mathbb{Z}/9\mathbb{Z}$, $\mathbb{Z}/10\mathbb{Z}$, $\mathbb{Z}/12\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$.

(a) According to Andrej Dujella's web-site "High rank elliptic curves with prescribed torsion", nobody has yet produced a parametric family of elliptic curves, over $\mathbb{Q}(t)$, with these groups as the torsion subgroup and the rank of the curves at least $1$. All the other possible torsion subgroups have such families.

(b) According to Noam Elkies' preprint "Three lectures on elliptic surfaces and curves of high rank", these are the only groups, in the list of elliptic curve (over $\mathbb{Q}$) torsion possibilities, that do NOT appear as the torsion for elliptic $K3$ surfaces.

It seems that these observations ought to be related.

Is this wishful thinking, obvious, or somewhere between? Any observations, references etc. welcome.


Here's the short answer:

Observation (b) is, in my understanding, Elkies' explanation as to why the techniques he used to break most of the records on Dujella's list cannot be directly applied for these maximal torsion subgroups.

Observation (a) follows from observation (b) and the fact that Elkies techniques are the best available for finding elliptic curves with large rank over $\Bbb Q(t)$. This isn't a mathematically precise statement, but check the scoreboard.

End of Short Answer

I'm personally interested in trying to find elliptic curves over $\Bbb Q(t)$ with these torsion subgroups and positive rank. It would help with increasing the rank records over $\Bbb Q$ via specialization, and there might also be some applications to the elliptic curve method (ECM) of factorization (See Atkin and Morain's Paper).

I have a long standing personal obsession with the $\Bbb Z / 2 \times \Bbb Z / 8$ case. There the universal elliptic curve is:

$$E_{\Bbb Z / 2 \times \Bbb Z / 8} : y^2 = x(x + (2t)^4)(x + (t^2 - 1)^4)$$

which has discriminant degree is 48. Elkies pointed out to me that it's a quadratic base change from a K3 surface, namely the universal elliptic curve with an 8-torsion point.

A general result of Alice Silverberg's ensures that the only $\Bbb Q(t)$-rational points of this elliptic curve are the 16 torsion points. If you want to find elliptic curves over $\Bbb Q(t)$ with 16-torsion points and positive rank, then you need to find rational curves on the associated surface other than the singular fibers and the 16 curves coming from the torsion sections.

No one has been able to find such a curve on this surface. I've asked around wether or not such a curve should exist, I've gotten one no, one yes, and nothing else very definitive.

My guess is that such curves do exist, but I don't have a very good idea where to start looking other than staring at a lot of data concerning rank 2 curves over $\Bbb Q$ with 16 torsion points and hoping to find a pattern.

Most of the same information applies for the other 3 torsion subgroups mentioned, but I don't know the universal elliptic curves by heart, and I don't already have a massive amount of data computed regarding specializations. Of course one can find models for them in Kubert's table. Interestingly Dujella attributes the records over $\Bbb Q(t)$ for these torsion subgroups to Kubert. It might be more appropriate to attribute some of them Fricke, or Levi at the latest.


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