The BSD Rank Conjecture Implies that $E(\mathbb{Q})$ is Computable

Proposition 1.3   Let $E$ be an elliptic curve over $\mathbb{Q}$. If Conjecture 1.1 is true, then there is an algorithm to compute the rank of $E$.

Next we show that given the rank $r$, the full group $E(\mathbb{Q})$ is computable. The issue is that what we did above might have only computed a subgroup of finite index. The argument below follows [Cre97, §3.5] closely.

The naive height $h(P)$ of a point $P =(x,y) \in E(\mathbb{Q})$ is

$$h(P) = \log(\max(\numer (x),\denom (x))).$$

The Néron-Tate canonical height of $P$ is

$$\hat{h}(P) = \lim_{n\to\infty} \frac{h(2^n P)}{4^n}.$$

Note that if $P$ has finite order then $\hat{h}(P)=0$. Also, a standard result is that the height pairing

$$\langle P,Q\rangle = \frac{1}{2}\left( \hat{h}(P+Q) - \hat{h}(P) - \hat{h}(Q)\right)$$

defines a nondegenerate real-valued quadratic form on $E(\mathbb{Q})/_{\tor }$ with discrete image.

Lemma 1.4   Let $B>0$ be a positive real number such that

$$S = \{ P \in E(\mathbb{Q}) : \hat{h}(P) \leq B \}$$

contains a set of generators for $E(\mathbb{Q})/2 E(\mathbb{Q})$. Then $S$ generates $E(\mathbb{Q})$.

Proposition 1.5   Let $E$ be an elliptic curve over $\mathbb{Q}$. If Conjecture 1.1 is true, then there is an algorithm to compute $E(\mathbb{Q})$.