Question 4.17 — Mazur

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 $V$ be a smooth algebraic variety over $\mathbb Q$. Give $V(\mathbb R)$ its usual real topology and set $Z=\overline{V(\mathbb Q)}^{\,V(\mathbb R)}$. The condition that $V(\mathbb Q)$ is Zariski dense means it is not contained in a proper Zariski-closed subset of $V$.

Statement recorded in the supplied source.

If $V(\mathbb Q)$ is Zariski dense in $V$, must $Z$ be a union of connected components of $V(\mathbb R)$? [S2] The cited survey records a negative answer to this general question: Colliot-Thélène, Skorobogatov and Swinnerton-Dyer constructed a smooth surface with Zariski-dense rational points whose real closure is not a union of components. The survey subsequently distinguishes the separate abelian-variety case. [S2]

Short English statement

If rational points are algebraically dense, must their real closure consist of whole real connected components?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700059 (AMR-086-0059). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Question 4.17 and following discussion, p.42. https://arxiv.org/abs/math/0312440v2.