The Torsion Subgroup

Theorem 2.1 (Mazur, April 16, 1976)   Let $E$ be an elliptic curve over  $\mathbb{Q}$. Then $E(\mathbb{Q})_{\tor}$ is isomorphic to one of the following 15 groups:

$$\mathbb{Z}/n\mathbb{Z} n\leq 10 n=12,$$

$$(\mathbb{Z}/2\mathbb{Z})\times (\mathbb{Z}/2n\mathbb{Z}) n \leq 4.$$

As we will see in the next section, all of these torsion subgroups really do occur. Mazur's theorem is very deep, and I can barely begin to hint at how he proved it. The basic idea is to define, for each positive integer $N$, a curve $Y_1(N)$ with the magnificient property that the points of $Y_1(N)$ with complex coordinates are in natural bijection with the (isomorphism classes of) pairs $(E,P)$, where $E$ is an elliptic curve and $P$ is a point of $E$ of order $N$. Moreover, $Y_1(N)$ is amazing in that it has a rational point if and only if there is an elliptic curve over $\mathbb{Q}$ with a rational point of order $N$. I won't define $Y_1(N)$, but here it is for the first few $N$:

$\quad N$	A curve that contains $Y_1(N)$
$1-10$, $12$	a straight line; these have lots of points!
$11$	$y^2 + y = x^3 - x^2$
$13$	$y^2 = x^6 + 2x^5 + x^4 + 2x^3 + 6x^2 + 4x + 1$
$14$	$y^2 +xy +y = x^3-x$
$15$	$y^2+xy+y = x^3+x^2$
$16$	$y^2 = (x - 1)(x + 1)(x^2 - 2x - 1)(x^2 + 1)$
$17$	The intersection of the hypersurfaces in $\P ^4$ defined by:
	$ac - b^2 + 5bd - 3be - c^2 - 4cd + 2ce - 4d^2 + 7de - 2e^2$,
	$ad - bc + bd - be + c^2 - 2cd - 2d^2 + 4de - e^2$, and
	$ae - be - cd + 2d^2 - 2de + e^2$.
$18$	$y^2 = x^6 + 4x^5 + 10x^4 + 10x^3 + 5x^2 + 2x + 1$

(Some of the curves in the right hand column have a few obvious rational points, but these points ``don't count''.)

Mazur proved that if $N=11$ or $N\geq 13$, then $Y_1(N)$ has no rational points. This result, together with the theory surrounding $Y_1(N)$, yields his theorem.