Conjecture 4.7 — David-Hindry

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 an abelian variety $A$ over a number field $K$ and a symmetric ample line bundle $\mathcal L$. Write $\deg_{\mathcal L}V$ for degree in this polarization and define $\delta(P)=\min_V(\deg_{\mathcal L}V)^{1/\operatorname{codim}_A V}$, with $V\subsetneq A$ a $K$-irreducible subvariety defined over $K$ containing $P$. The Néron--Tate height is $\widehat h_{\mathcal L}(P)=\lim_{r\to\infty}4^{-r}h_{\mathcal L}([2^r]P)$ for a Weil height of $\mathcal L$. Require that the image of $P$ in $A/B$ have infinite order for every proper abelian subvariety $B\subsetneq A$.

Statement recorded in the supplied source.

Does there exist $c>0$, depending only on $A$ and $\mathcal L$, such that all such $P\in A(\overline{\mathbb Q})$ satisfy \[\widehat h_{\mathcal L}(P)\ge c\,\delta(P)^{-1}?\] [S2]

Short English statement

Is the canonical height uniformly bounded below by the reciprocal of the geometric obstruction index for points remaining nontorsion in every proper quotient?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8700052 (AMR-086-0052). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Michel Waldschmidt, Open Diophantine Problems, arXiv:math/0312440v2 (24 January 2004), Conjecture 4.7 and preceding definition of delta. https://arxiv.org/abs/math/0312440.