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Mazur's theorem about elliptic curve claims that there are only finitely many possibilities for the torsion subgroup of a Modell-Weil group of an elliptic curve over $\mathbb{Q}$, and he also gives a list of possible torsion groups.

Now I'm trying to think of a possible proof of weak Mazur's theorem that there are only finitely many possibilities for the torsion group, not giving the complete list. My strategy is the following: it is known that the modular curve $Y_{1}(N) = \Gamma_{1}(N)\backslash \mathbb{H}$, which has a model over $\mathbb{Q}$, parametrizes elliptic curve with a torsion point of order $N$, and rational points on the curve $Y_{1}(N)_{\mathbb{Q}}(\mathbb{Q})$ parametrize elliptic curve over $\mathbb{Q}$ that has a rational point of order $N$. Now if $N$ is large enough, then the genus of $Y_{1}(N)$ (or the genus of $X_{1}(N)$) will be bigger than 1 (genus can be computed by the Riemann-Hurwitz formula) so there will be finitely many rational points on $Y_{1}(N)$ by Faltings' theorem, which means that there are finitely many elliptic curves over $\mathbb{Q}$ with a rational point of order $N$. We can even compute the smallest $N$ such that genus is bigger than 1.

1) Is this a right argument? If it is right, it proves that for a sufficiently large $N$ there exist finitely many elliptic curves over $\mathbb{Q}$ with a rational point of order $N$.

2) Can we show that there is no rational point on $Y_{1}(N)$ for sufficiently large $N$? I believe this will imply the weak form of Mazur's theorem.

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  • $\begingroup$ $Y_0(N)$ parametrizes elliptic curves equipped with a cyclic subgroup of order $N$, not a point of order $N$. Maybe you want $Y_1(N)$? $\endgroup$ Commented Jan 19, 2019 at 4:12
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    $\begingroup$ @André3000 Oh you are right - thanks! But I think both give almost the same results. $\endgroup$
    – Seewoo Lee
    Commented Jan 19, 2019 at 4:43

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1) Your argument in essence is correct but let me point out that "finitely many elliptic curves over $\mathbb Q$" can be misleading.

First of all notice that $X_{*}(N)/{\mathbb Q}$ parametrizes elliptic curves over $\mathbb Q$ with some level-N structure up to $\bar{\mathbb Q}$-isomorphism, or up to twist. Hence even if $X=X_*(N)$ has finitely many points there might be infinitely many non $\mathbb Q$-isomorphic elliptic curves corresponding to a unique point in $X$.

For example it is known (see Rational isogenies of Prime degree, Mazur) that $X_0(43)$ has a "unique point" corresponding to the elliptic curve $$ E: y^2+y=x^3−1590140x−771794326 $$ with Cremona's label 1849a2 but 1849a1, 16641e2, 16641e1, 29584h2, 29584h1, 46225a2, 46225a1... are rational elliptic curves $\bar{\mathbb Q}$-isomorphic to $E$.

2) The curve $X_*(N)$ has finitely many points for genus big enough but it is never empty. The reason is that $X_*(N) \setminus Y_*(N)$ has cuspidal rational points. See the introduction of Mazur's paper for a complete exposition.

I guess this implies that some arithmetic must be done on $Y_*(N)$ in order to prove the emptyness and that's what Mazur actually does.

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