Infinitude of rational points on an elliptic curve

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $A,B\in\mathbb Q$ satisfy $4A^3+27B^2\ne0$, and let $E/\mathbb Q$ be the nonsingular curve $y^2=x^3+Ax+B$ together with its point at infinity $O$. Its rational points form a finitely generated abelian group \[ E(\mathbb Q)\cong E(\mathbb Q)_{\mathrm{tors}}\oplus\mathbb Z^r, \] where the torsion subgroup is finite and $r\in\mathbb Z_{\ge0}$ is the Mordell--Weil rank. Consequently $E(\mathbb Q)$ is infinite exactly when $r>0$.

Statement recorded in the supplied source.

Given $A,B$ as above, determine whether $E(\mathbb Q)$ is infinite or finite; equivalently, determine whether its Mordell--Weil rank is positive or zero. [S2]

Short English statement

Determine, for an arbitrary rational elliptic curve, whether it has infinitely many rational points.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8900012 (AMR-088-0012). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Christian Wuthrich, Open problems in number theory, lecture slides (December 2011) slides headed An elliptic curve and Question, with examples $y^2=x^3+x+1$ and $y^2=x^3+x+2$. https://www.maths.nottingham.ac.uk/plp/pmzcw/download/open.pdf.