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Mazur’s torsion theorem famously tells us exactly which finite groups can occur as the torsion subgroup of $E(\mathbb{Q})$ for an elliptic curve $E$ defined over $\mathbb{Q}$. In particular, it implies that only finitely many torsion subgroups are possible, which seems like a much weaker result.

My question: Is there any way to see that the weak version of bounded torsion (only finitely many groups occur, but nevermind what they are) is true, without recourse to the full proof of Mazur’s theorem? I saw some references to a paper of Demjanenko from 1971 (EDIT: originally said 1975) which claims to prove this, but it’s only available in Russian and other sources don’t seem to think its argument is correct.

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  • $\begingroup$ Do you mean this paper: mathscinet.ams.org/mathscinet-getitem?mr=437539 ? (subscription needed) If so, the English translation of the Russian original is: V. A. Dem'yanenko, Rational points of algebraic curves, Mathematical Notes of the Academy of Sciences of the USSR (1975) 18: 1124. $\endgroup$ – David Roberts Jul 23 at 2:48
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    $\begingroup$ Ah, so maybe this? mathscinet.ams.org/mathscinet-getitem?mr=285537 (Dem'janenko,V A, TORSION OF ELLIPTIC CURVES, Mathematics of the USSR-Izvestiya(1971),5(2):289, iopscience.iop.org/article/10.1070/IM1971v005n02ABEH001047) $\endgroup$ – David Roberts Jul 23 at 2:53
  • $\begingroup$ Yes, I think that’s it (I realized I meant 1971). Thanks for passing along! Do you happen to know whether people think it’s correct? $\endgroup$ – Spencer Dembner Jul 23 at 2:55
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    $\begingroup$ No idea. Cassels writes about the paper (for MR): Unfortunately, the exposition is so obscure that the reviewer has yet to meet someone who would vouch for the validity of the proof; on the other hand he has yet to be shown a mistake that unambiguously and irretrievably vitiates the argument. $\endgroup$ – David Roberts Jul 23 at 2:58
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    $\begingroup$ Even by 1996, the review by Jörg Jahnel of Demʹyanenko's On the torsion of elliptic curves points out: So the proof is mainly based on very complicated algebraic calculations. Furthermore it relies on a reference to an old paper of the same author [Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 280–307; MR0285537], which seems never to have been understood. Thus, unfortunately, it is impossible for the referee to certify that everything is correct. $\endgroup$ – David Roberts Jul 23 at 3:03
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By all means hold out hope, but I don't think that the ideas in Dem'janenko's papers are going to work. I spent a lot of time in grad school looking at them.

If I remember correctly, Dem'janenko also claimed to have proven that on $E:y^2=x^3+D$, if a $P\in E(\mathbb Q)$ is a non-torsion point, then $\hat h(P)\ge c\log|D|$ for an absolute constant $c$. But again, no one has managed to decipher his proof. Lang was intrigued enough to conjecture that $\hat h(P)\ge c\log|\Delta_E|$ for all elliptic curves, where $\Delta_E$ is the minimal discriminant. Using quite different techniques, I proved a weaker version of Lang's conjecture, and Hindry and I proved that Lang's full conjecture follows from $ABC$. However, for twists such as in Dem'janenko's paper, there is an alternative easier argument, and it's possible that that is what's lurking in his paper.

I mention all of this, because height arguments such as those in Dem'janenko, and in my work with Hindry, tend to lead to statements of the form: Let $P\in E(\mathbb Q)$. Then either $NP=0$ or $\hat h(P)$ isn't too small. So torsion bounds and height bounds come packaged together.

As for proofs that just handle torsion points, note that any such proof would be a strong uniform version of the Mordell conjecture for the modular curves $X_1(\mathbb Q)$, which is why it seems unlikely that purely elementary, albeit complicated, algebraic manipulations as in Dem'janenko's paper could give full uniformity.

On the other hand, Dem'janenko came up with a very nice way of proving the Mordell conjecture in certain situations. For example, if there are two independent maps $f_1,f_2:C\to{E}$ and if $E(\mathbb Q)$ has rank 1, then $C(\mathbb Q)$ is finite, essentially by comparing $\hat h(f_1(x))$ and $\hat h(f_2(x))$. Manin generalized this to the tower of modular curves $X_1(p^{n+1})\to X_1(p^n)$, and used this to give an elementary (at least, compared to Mazur's work) proof of $p$-power uniformity for torsion points:

Theorem (Manin) Fix $p$ and a number field $K$. There is a constant $C=C(p,K)$ such that for every elliptic curve $E/K$, there are no torsion points in $E(K)$ of exact order $p^C$.

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  • $\begingroup$ Thanks for the helpful explanation! $\endgroup$ – Spencer Dembner Jul 23 at 18:54
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    $\begingroup$ @Spencer Your welcome. I probably should have also mentioned that there is a very short proof that the $ABC$-conjecture implies uniform boundednesss of torsion. (This might be due to Frey?) Not that that would be easier than Mazur's proof, unless there's an easy proof of $ABC$! But it would be quite different. $\endgroup$ – Joe Silverman Jul 23 at 19:05
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The 1971 paper appears in English translation as

V. A. Dem'janenko, Torsion of elliptic curves, Mathematics of the USSR-Izvestiya 5 no 2 (1971) 289–318, doi:10.1070/IM1971v005n02ABEH001047

Cassels wrote a review for Math Reviews, saying

Unfortunately, the exposition is so obscure that the reviewer has yet to meet someone who would vouch for the validity of the proof; on the other hand he has yet to be shown a mistake that unambiguously and irretrievably vitiates the argument.

Even as recent as 1996, Jörg Jahnel in reviewing Demʹjanenko's On the torsion of elliptic curves writes

So the proof is mainly based on very complicated algebraic calculations. Furthermore it relies on a reference to an old paper of the same author [the 1971 paper above], which seems never to have been understood. Thus, unfortunately, it is impossible for the referee to certify that everything is correct.

Recent citations are very thin on the ground, and no one particularly seems to credit Dem'janenko with having proving what is claimed in the 1971 paper.

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    $\begingroup$ One weird aspect of Cassels' review is the first sentence: The author purports to prove the long-standing conjecture that the Mordell-Weil rank of an elliptic curve (= abelian variety of dimension one) over an algebraic number field is bounded by a constant depending only on the field. As the title suggests, the paper makes no claim about the Mordell--Weil rank, only abou the torsion points. $\endgroup$ – Bort Jul 23 at 13:59

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