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Andrej Dujella's excellent web-site "High rank elliptic curves with prescribed torsion", at http://web.math.pmf.unizg.hr/~duje/tors/tors.html gives example of the (current) largest known rank of an elliptic curve over $\mathbb{Q}$ having each of the fifteen possible torsion subgroup structures.

The related web-site "Infinite families of elliptic curves with high rank and prescribed torsion", at http://web.math.pmf.unizg.hr/~duje/tors/generic.html, gives information on the existence of infinite families.

There are no known families with torsion

$\mathbb{Z} / 9\mathbb{Z}$, $\mathbb{Z} / 10\mathbb{Z}$, $\mathbb{Z} / 12\mathbb{Z}$, and $\mathbb{Z} / 2\mathbb{Z} \times \mathbb{Z} / 8\mathbb{Z}$.

I have already spent a reasonable amount of time hunting for such a beast. To save me wasting my time, I would like to ask experts whether they would expect such a family to exist, and if so, why? Alternatively, if they do not expect such a family to exist, could they also give a plausible reason?

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    $\begingroup$ There are families for those cases, because the torsion that occurs over the rationals correspond exactly to the cases when the corresponding modular curve has genus zero (and an automatic rational point at a cusp). I am sure the people who first thought about this (Beppo Levi, Andy Ogg,...) wrote down equations. I don't know a place where you can find the formulas, maybe they are a bit ugly. $\endgroup$ – Felipe Voloch Jul 31 '14 at 10:09
  • $\begingroup$ The following paper has all the formulas in the cyclic cases. sciencedirect.com/science/article/pii/S0022314X02927800 $\endgroup$ – Felipe Voloch Jul 31 '14 at 10:15
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    $\begingroup$ @FelipeVoloch: I think the question concerns elliptic curves over $\mathbb{Q}(t)$ (i.e., families) with given torsion subgroup $T$ and with rank as high as the largest known rank of an elliptic curve over $\mathbb{Q}$ having the same torsion group $T$. $\endgroup$ – Vesselin Dimitrov Jul 31 '14 at 12:04
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    $\begingroup$ @VesselinDimitrov I think Felipe's interpretation is consistent with the information in the second linked page in the question, which has some entries claiming zero maximal known rank. $\endgroup$ – S. Carnahan Jul 31 '14 at 14:33
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    $\begingroup$ Similar Mathoverflow question mathoverflow.net/questions/89645/… indicates that for these four torsion groups there are no examples of elliptic curves over Q(t) with rank >=1. $\endgroup$ – duje Jul 31 '14 at 21:14
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It might be worth noting that in the following paper, which is also in Dujella's list of references,

A. O. L. Atkin and F. Morain, Finding suitable curves for the elliptic curve method of factorization, Math. Comp. 60 (1993) (available here: http://www.ams.org/journals/mcom/1993-60-201/S0025-5718-1993-1140645-1/S0025-5718-1993-1140645-1.pdf)

A. O. L. Atkin and F. Morain exhibits families of rank $ \geq 1$ and torsion group $\mathbb Z/8\mathbb Z \times \mathbb Z/2\mathbb Z$, $\mathbb Z/5\mathbb Z$, $\mathbb Z/7\mathbb Z$, $\mathbb Z/9\mathbb Z$ and $\mathbb Z/10\mathbb Z$.

However, their families are not $1$-parameter families, i.e. curves over $\mathbb Q(T)$, but rather parametrized by points on elliptic curves. Whether or not that adds anything to the likelihood of finding $1$-parameter families of rank $\geq 1$ for those less frequent torsion groups, I do not know.

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    $\begingroup$ Information on current records for infinite families, which are not necessarily curves over Q(t), can be found in the second table at web.math.pmf.unizg.hr/~duje/tors/generic.html There might be some "regularities" (not big fluctuations) in products d*r, where r's are current rank records and d's are quantities from the NAME_IN_CAPS answer. $\endgroup$ – duje Aug 1 '14 at 13:39
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One reason might be that these families are the least dense.

See Table 1 of http://www-personal.umich.edu/~asnowden/papers/torsion-112013.pdf

Or maybe I will just copy it here. $T$ is the torsion group, $d$ is the their $d$ (reciprocal of power density), and $r$ is the rank record from Dujella. I don't know if one could argue that these 4 remaining families are so sparse that one should not expect an infinite family.

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    $\begingroup$ The first version looked fine (albeit less delicious). $\endgroup$ – S. Carnahan Aug 1 '14 at 4:59

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