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Let $C$ be an elliptic curve over $ℚ$ that has the form: $$y²=x³+ax+b...........................(1)$$

where $a,b$ are integers. 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 $C(ℚ)$). Here $r$ is the Mordell-Weil rank of $C(ℚ)$. Then $r$ is defined to be the cardinality of a maximal independent set in $C(ℚ)$, 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 $α_k ∈ ℤ$.)

We know that if $r=0$ if and only if $C(ℚ)$ is finite.

My question is about the existence of a result that determine the values or the shape of the integers $a,b$ in $(1)$ such that $r=0$, i.e., $C(ℚ)$ is finite.

Safwane
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1 Answers1

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There are, for example, certain twists $E_d$ of elliptic curves with rank $r=0$. Some conditions are given in Theorem $2$ of this article, which is about this topic. There are also other conditions, but in general they are not so easy in terms of the "shape of the integers $a,b$". For example, elliptic curves $E$ satisfying $L(E,1) \neq 0$ have rank $r=0$ by Kolyvagin's theorem.

Some examples of elliptic curves over $\Bbb Q$ of rank $r=0$:

$$y^2=x^3+1$$

$$y^2 = x^3 - 9122x + 106889$$

$$ y^2 = x^3 - x^2 - 42144x + 66420$$

$$ y^2 = x^3 - x^2 - 168615x + 21827700$$

$$ y^2 = x^3 - 210386x + 32627329$$

See also this post: Elliptic curves with finitely many rational points

Dietrich Burde
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  • what about assuming that $r=0$ without knowing the values of $a,b$. This assumption is needed in a proof. – Safwane Mar 22 '20 at 19:19
  • to distinguish between the cases $r=0$ and $r>0$ in the number of points of inifinite order described in the question. – Safwane Mar 22 '20 at 19:27