How is the rank of an elliptic curve defined?

Question

It is my understanding that a rank of 1 in the context of the Birch and Swinnerton Dyer Conjecture means that only 1 copy of the set of integers is required to account for all the rational points on an elliptic curve. Does this mean that for example if there were two points on an elliptic curve with coordinates ( 2/3, 2/5 ) we would need two sets of integers to account for the one 2 in 2/3 and the one 2 in 2/5 and that the curve would have a rank of 2? Or does this relate to group theory e.g 2P + 3Q giving another rational point on a curve? Thanks for any help you can give.

Answer 1

A group generated by a single element is abelian and will be either isomorphic to $\mathbb{Z}$ or $\mathbb{Z}_k$ with $\mathbb{Z}_k$ the integers modulo $k$ for some $k \in \mathbb{N}$. Finitely generated abelian groups, like those in the conjecture, will be a direct sum of groups in this type. The parts that are isomorphic to $\mathbb{Z}_k$ are said to have torsion, which just means the are finite in size. This means they will be isomorphic to $\mathbb{Z}^n \bigoplus \mathbb{Z}_{k_1}\bigoplus \mathbb{Z}_{k_2}\bigoplus \cdots \bigoplus\mathbb{Z}_{k_i} $ for some $k_1,k_2,...k_i,n \in \mathbb{N}$.

The $\mathbb{Z}_{k_1}\bigoplus \mathbb{Z}_{k_2}\bigoplus \cdots \bigoplus\mathbb{Z}_{k_i}$ part is called the torsion subgroup since it is finite. When we compute the rank we ignore the torsion subgroups and only consider how many copies of $\mathbb{Z}$ are in the subgroup, which by looking at the $\mathbb{Z}^n$ component of the decomposition is $n$. Note that this is similar to the relationship between rank and dimension from linear algebra. If you know the torsion subgroups and the rank you know everything about the group.

Answer 2

Rank refers to the number of independent elements of infinite order.

The group of rational points is finitely generated, so in particular the set of elements of finite order is a finite abelian group that we'll call $T$ (it's called the torsion points). If the group of rational points has no points of infinite order, so $T$ is all the rational points, then we call this "rank $0$". If there are rational points of infinite order, then for some integer $r \geq 1$ there are rational points $P_1, \ldots, P_r$ of infinite order such that the $P_i$'s together with $T$ generate all the rational points: every rational point has the form $$ m_1P_1 + \cdots + m_rP_r + t $$ for some $m_i \in \mathbf Z$ and $t \in T$. (We can include the case of no rational points of infinite order as an instance of this using $r = 0$.)

The smallest $r$ for which we can find such points $P_i$ is called the rank of the elliptic curve. That's getting at the rank "from above". A way to get it "from below" is through the concept of additive independence: call $P_1, \ldots, P_r$ additively independent if $m_1P_1 + \cdots + m_rP_r = O$ for $m_i \in \mathbf Z$ (by "$O$" I mean the identity for the group law) only when each $m_i$ is $0$. Every subset of an additively independent set is additively independent. A finitely generated abelian group has an upper bound on the number of additively independent elements inside it, so there is a largest possible number of additively independent rational points on an elliptic curve. This number is also the rank from before.

So "rank 1" means both of the following (they are equivalent): (i) there is a rational point $P$ of infinite order such that every rational point is $mP + t$ for some integer $m$ and rational point of finite order $t$, and (ii) there is a rational point of infinite order and there is no additively independent set of two rational points of infinite order.