A Deeper Look at the Mordell-Weil Group — Epoche C1
The group, and why it is a group An elliptic curve over the rational numbers is the solution set of an equation $y^2 = x^3 + Ax + B$ with $A, B \in \mathbb{Q}$ and non-vanishing discriminant, together with one extra point 'at infinity', and the remarkable fact about it is that its rational solutions can be added. That addition is what this essay is about. The addition rule is geometric: a line meets a cubic curve in three points counted properly, so given rational points $P$ and $Q$ one draws the line through them, takes the third intersection point, and reflects it in the $x$-axis; the result is defined to be $P + Q$. When $P = Q$ one uses the tangent at $P$ instead of a secant. The point at infinity, written $O$, is where all vertical lines meet, and it serves as the identity element: reflecting a point in the $x$-axis produces its inverse, so $(x,y) + (x,-y) = O$. Two facts make this worth taking seriously. The construction never leaves the rationals, since the third intersection of a line through two rational points on a rational cubic is itself rational — solving a cubic two of whose roots are rational leaves a linear equation for the third. And the operation is associative, which is not obvious from the geometry and is the one point at which an elementary treatment has to do real work. The set of rational points, written $E(\mathbb{Q})$, is therefore an abelian group. The non-vanishing of the discriminant, which for the form above is $\Delta = -16\left(4A^3 + 27B^2\right)$, is exactly the condition that the cubic $x^3 + Ax + B$ has no repeated root, and hence that the curve has no singular point at which the tangent construction would break down. Mordell-Weil, and what the structure theorem does and does not give Louis Mordell proved in 1922 that $E(\mathbb{Q})$ is finitely generated: there is a finite list of rational points from which every other rational point can be obtained by repeated addition and subtraction. André Weil generalised this in 1929 to abelian varieties over number fields, whence the name. Once finite generation is known, the classification theorem for finitely generated abelian groups applies and yields $$E(\mathbb{Q}) \;\cong\; \mathbb{Z}^{r} \oplus E(\mathbb{Q})_{\text{tors}}$$ where $r$, the rank, is the number of independent points of infinite order, and $E(\mathbb{Q})_{\text{tors}}$ is the torsion subgroup, consisting of the points $P$ for which some multiple $NP$ equals $O$ — that is, the points of finite order. The torsion subgroup is finite, being a finitely generated torsion group. Here the earlier version of this essay slipped, and the slip is worth correcting because it obscures where the difficulty lies. It suggested that one might naively expect the group to be a direct sum of cyclic groups, and presented the appearance of a non-cyclic torsion subgroup as a surprise. But every finite abelian group is a direct sum of cyclic groups; that is the content of the classification theorem, and it is guaranteed, not surprising. The genuine question is a different one: which direct sums actually occur? Nothing in group theory answers that, because nothing in group theory knows anything about $\mathbb{Q}$. The answer is a theorem of arithmetic, and a deep one. Mazur's list, and two reasons for its shape Barry Mazur proved in 1977 that the torsion subgroup of an elliptic curve over $\mathbb{Q}$ is isomorphic to exactly one of fifteen groups: the cyclic groups $\mathbb{Z}/N\mathbb{Z}$ for $N = 1,2,3,4,5,6,7,8,9,10$ and $N=12$ — eleven groups, with $N=11$ and every $N \ge 13$ absent; the groups $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2N\mathbb{Z}$ for $N = 1,2,3,4$ — four groups, the largest being $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/8\mathbb{Z}$. Each half of the list has a reason, and the two reasons are of very different depths. The second half is elementary once one has the right tool, and it explains the parenthetical remark in the earlier version that $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ does not occur over $\mathbb{Q}$ — a correct remark that was left without justification. Over the complex numbers an elliptic curve is a torus $\mathbb{C}/\Lambda$ for a lattice $\Lambda$ of rank $2$, so its points of order dividing $m$ form $\tfrac{1}{m}\Lambda/\Lambda \cong \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$; write this group $E[m]$. Now invoke the Weil pairing, a map $$e_m : E[m] \times E[m] \longrightarrow \mu_m$$ into the group $\mu_m$ of $m$-th roots of unity, which is bilinear, non-degenerate (so it does not collapse), alternating, and compatible with the action of the Galois group of $\overline{\mathbb{Q}}$ over $\mathbb{Q}$ — meaning $e_m(\sigma P, \sigma Q) = \sigma\!\left(e_m(P,Q)\right)$ for every automorphism $\sigma$. Suppose all of $E[m]$ consisted of rational points. Then $\sigma P = P$ and $\sigma Q = Q$ for every $\sigma$, so $\sigma$ fixes $e_m(P,Q)$ for all $P,Q$; by non-degeneracy those values generate $\mu_m$, so every $m$-th root of unity is fixed by every element of the Galois group and therefore lies in $\mathbb{Q}$. But the only roots of unity in $\mathbb{Q}$ are $1$ and $-1$. Hence full $m$-torsion can be rational only for $m \le 2$, which rules out $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ immediately and explains why every non-cyclic entry on Mazur's list has $\mathbb{Z}/2\mathbb{Z}$ as its first factor. The first half of the list is where the depth is, and no such argument reaches it. To ask which $N$ occur is to ask, for each $N$, whether there exists a pair consisting of an elliptic curve over $\mathbb{Q}$ and a rational point of order $N$ on it. Such pairs are parametrised by an algebraic curve, the modular curve $X_1(N)$, and the question becomes whether $X_1(N)$ has rational points other than its cusps. For $N \le 10$ and $N = 12$ this curve has genus $0$ and, having a rational point, is a rational curve with infinitely many; that is why those values occur, and occur for infinitely ma