Bombieri–Lang finiteness for rational points on deep torsion-level covers

Prove the Bombieri–Lang prediction that, whenever a totally real field F has degree n and an ideal η⊂𝒪_F satisfies |Nm(η)|>(2π)^{2n}, the torsion-level Hilbert modular variety X_1(η) has only finitely many K-rational points for every number field K.

Background

The paper proves that every positive-dimensional subvariety of the torsion-level Hilbert modular variety X_1(η) is of general type once the norm of η exceeds (2π){2n}. Applying the Bombieri–Lang conjecture to this geometric result predicts finiteness of rational points over every number field.

The predicted finiteness remains largely unresolved. The paper instead obtains an unconditional upper bound on the asymptotic growth rate of rational points of bounded height, showing that these points become sparser as the torsion level deepens, but it does not establish the conjectured finiteness.

References

The variety X_1(\eta) is defined over $Q$, and, given Corollary~\ref{general type int}, the Bombieri--Lang conjecture predicts that if $|Nm(\eta)|> (2\pi){2n},$ then for every number field $K,$

|X_1(\eta)(K)|<\infty.

Although the finiteness predicted by the Bombieri--Lang conjecture remains largely open, combining our effective estimate in Theorem~\ref{log deg} with the recent results of Brunebarbe--Maculan gives us an unconditional effective bound on the growth rate of $K$-rational points in the following sense:

— Sparsity of rational points on torsion level covers of Hilbert modular varieties  (2609.30033 - Memariansorkhabi, 24 Sep 2026) in Introduction, paragraph immediately following Corollary general type