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Let $C$ be an elliptic curve over $ℚ$. The group $C(ℚ)$ is a finitely generated Abelian group and we have $C(ℚ)≃ℤ^{r}⊕C(ℚ)^\mathrm{tors}$, where $C(ℚ)^\mathrm{tors}$ is a finite abelian group (is the subgroup of elements of finite order in $C(ℚ)$).

Assume that the rank of $C(ℚ)$ is equal to $r$, then the cardinality of a maximal independent set in $C(ℚ)$ is $r$, thus there exist $r$ independent points ${P_1,P_2,\ldots,P_r}$ of infinite order in $C(ℚ)$, i.e., $P_k=(x_k,y_k)∈ℚ^2,k=1,\ldots,r$ such that if $∑_{k=1}^r α_k P_k=0$, then $α_k=0$ for all $k=1,\ldots,r$. (Here $α_k ∈ ℤ$.)

Now, taking $(x,y)∈G$. My question is: How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$.

Safwane
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1 Answers1

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Let $C$ be an elliptic curve over $\mathbb{Q}$. The Mordell-Weil theorem states that $C(\mathbb{Q})$, the set of rational points on $C$, is a finitely generated abelian group. Then, by the structure theorem for finite abelian groups, we have $$C(ℚ)≃ℤ^{r}⊕C(ℚ)^\mathrm{tors},$$ for some $r\geq 0$ (called the rank of $C/\mathbb{Q}$) and some finite subgroup $C(ℚ)^\mathrm{tors}$ formed by the points of finite order in $C(\mathbb{Q})$. In particular: $$C(ℚ)/C(ℚ)^\mathrm{tors}\cong \mathbb{Z}^r.$$ Let $P_1,\ldots, P_r$ be generators of the quotient $C(ℚ)/C(ℚ)^\mathrm{tors}$. This means that for any point $R\in C(\mathbb{Q})$ we have integers $n_1,\ldots,n_r$ such that $$R\equiv n_1P_1+\cdots +n_rP_r \bmod C(ℚ)^\mathrm{tors},$$ and, in turn, this means that there exists a torsion point $T\in C(ℚ)^\mathrm{tors}$ such that $$R=n_1P_1+\cdots +n_rP_r + T.$$

Álvaro Lozano-Robledo
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