# Proof of weak form of Mazur's theorem by Faltings' theorem

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.

• $Y_0(N)$ parametrizes elliptic curves equipped with a cyclic subgroup of order $N$, not a point of order $N$. Maybe you want $Y_1(N)$? – André 3000 Jan 19 at 4:12
• @André3000 Oh you are right - thanks! But I think both give almost the same results. – Seewoo Lee Jan 19 at 4:43

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.