Conjecture 4.19

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.

Fix a real embedding $K\hookrightarrow\mathbb R$ of a number field and a simple abelian variety $A/K$ of dimension $g\ge1$. Let $A(\mathbb R)^0$ be the identity component of its real Lie group, and let $\ell=\operatorname{rank}_{\mathbb Z}A(K)$. Fix a symmetric ample line bundle on $A$, its Néron--Tate canonical height $\widehat h$, and a projective embedding $A\hookrightarrow\mathbb P^N$ from a sufficiently high tensor power of that line bundle. For real projective points $z=(z_0:\cdots:z_N)$ and $w=(w_0:\cdots:w_N)$ set \[ \operatorname{dist}(z,w)=\frac{\max_{0\le i,j\le N}|z_iw_j-z_jw_i|}{\max_i|z_i|\max_j|w_j|}. \]

Statement recorded in the supplied source.

For every $\varepsilon>0$, does there exist $h_0>0$, depending on $A,K,\varepsilon$ (with the embedding and canonical height fixed), such that, for every real $h\ge h_0$ and every $\zeta\in A(\mathbb R)^0$, there is $\gamma\in A(K)$ satisfying \[ \widehat h(\gamma)\le h,\qquad \operatorname{dist}(\zeta,\gamma)\le h^{-\ell/(2g)+\varepsilon}? \] [S2]

Short English statement

Can every real point in the identity component be uniformly approximated by rational points with bounded canonical height at this rank-dependent rate?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700061 (AMR-086-0061). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 4.19 and projective-distance definition, p.43. https://arxiv.org/abs/math/0312440v2.