### Elliptic tales 2

In this second part, we will be concerned with elliptic curves specifically, rather than arbitrary polynomial curves. The main idea will be to introduce an abelian group structure on the points of an elliptic curve, then we will be able to study the curve algebraically. But, first a brief recap of group theory.

Suppose $S$ is a subset of the abelian group $(G,+)$. We define the subgroup of $G$ generated by $S$ to be the group:

$\langle S \rangle = \{n_1 x_1 + \ldots + n_k x_k| k\geq 0, n_i \in \mathbb{Z}, x_i \in S \}$

with the group operation $+$ borrowed from $G$ (you can check that $(\langle S \rangle, +)$ is indeed a group). $G$ is called a finitely generated abelian group if $G = \langle S \rangle$ for some finite subset $S$.

$x$ is called a torsion element of $G$ if $x$ has order $n$ for some integer $n \geq 1$, i.e. $nx=0$ for some $n \geq 1$, but $mx \neq 0$ for all smaller $m$. If $x$ is not a torsion element, we say that $x$ has infinite order. The set of torsion elements of $G$ make up the torsion subgroup of $G$, with the group operation again borrowed from $G$ (again, you check that this is indeed a group).

A theorem then tells us that each subgroup of a finitely generated group is also finitely generated and in particular, its torsion subgroup is also finitely generated, hence is finite.

Another important concept is rank. Suppose $G$ is a finitely generated abelian group and $H$ is torsion subgroup. The rank of $G$ is defined to be the smallest integer $r$ such that $G$ can be generated by $r$ elements along with all elements of $H$.

Now, we’re ready to take up elliptic curves and define an abelian group structure over them. We first define an elliptic curve over a field $K$, denoted $E(K)$, to be a nonsingular cubic curve containing at least one point with $K$-coordinates. We call this point $\mathcal{O}$. It isn’t a trivial assumption to assume that such a point exists. For example, it isn’t at all obvious that a given homogeneous cubic polynomial with rational coefficients should have a solution with rational coordinates.

We define the abelian group operation on $E(K)$ geometrically. Here’s how: let $P$ and $Q$ be two points on $E$. We define $P+Q$ as follows: first let $L$ be the line connecting $P$ and $Q$. This line intersects $E$ at a third point $R$. Then draw the line $L^\prime$ connecting  $R$ and $\mathcal{O}$. This line intersects the curve at a third point and that third point is defined to be the point $P+Q$. Pictorially, the construction looks like this (the curve $E$ is shown in red):

Some amount of work is needed to show that this construction does indeed define an abelian group (for example, you can check that $\mathcal{O}$ is the identity element of this group), but we will not do it here. Note that we’re also assuming that $E(K)$ is non-singular. The case of singular cubic equations requires special handling, but the group structure in this case turns out to be isomorphic to one of a small number of much simpler groups (again we will not do this reduction here), so in a sense, the non-singular case is the interesting case. Whether an elliptic curve $E$ is singular or not can be determined from its discriminant, $\Delta_E$, which is an easy-to-compute algebraic function of the coefficients describing the curve. The curve is singular if and only if $\Delta_E=0$.

We will later be interested in elliptic curves over finite fields such as $\mathbf{F}_p$, i.e. $E(\mathbf{F}_p)$, and especially in the size of $E(\mathbf{F}_p)$, which we denote by $N_p$. A theorem due to Hasse states that:

Theorem (Hasse): The number $N_p$ satisfies $-2\sqrt{p} \leq p+1-N_p \leq 2\sqrt{p}$.

The number $p+1-N_p$ turns out to be so important that it gets its own name, $a_p$, so Hasse’s theorem can be restated as saying that $|a_p| \leq 2\sqrt{p}$.

We will also be interested in  elliptic curves over rationals, $E(\mathbf{Q})$, and we can say a few things about the group structure of $E(\mathbf{Q})$:

1) A theorem due to Mordell says that $E(\mathbf{Q})$ is a finitely generated abelian group. This implies that the torsion subgroup $E(\mathbf{Q})_{\mathrm{tors}}$ is also finitely generated, hence is finite.

2) A theorem due to Mazur says that if $P$ is a point of order $n$ in $E(\mathbf{Q})_{\mathrm{tors}}$, then either $1 \leq n \leq 10$ or $n=12$. I fact, $E(\mathbf{Q})_{\mathrm{tors}}$ has a very simple structure: either $E(\mathbf{Q})_{\mathrm{tors}}$ is a cyclic group of order 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, or 12; or $E(\mathbf{Q})_{\mathrm{tors}}$ is generated by two elements $A_1$ and $A_2$, where the order of $A_1$ is 2, 4, 6, or 8, and the order of $A_2$ is 2.

3) A theorem due to Nagell and Lutz says that for an elliptic curve $E$ described by the equation $y^2 = x^3 + a_2 x^2 + a_4 x + a_6$, with $a_2, a_4, a_6$ integers, if $P$ is an element of $E(\mathbf{Q})_{\mathrm{tors}}$, then both $x_1$ and $y_1$ are integers and either $y_1=0$, in which case $2P=\mathcal{O}$, or else $y_1$ divides the discriminant $\Delta_E$. This theorem allows us to enumerate all torsion points of $E$ relatively easily.

4) It turns out to be much more difficult to say something about the rank of $E(\mathbf{Q})$. It is conjectured that there is no maximum rank for $E(\mathbf{Q})$, but this remains extremely difficult to prove. It is also believed that a random elliptic curve, i.e. an equation of the form $y^2 = x^3 + Ax + B$, where $A$ and $B$ are random integers, should have rank 0 or 1 with high probability.

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