Integral points on algebraic subvarieties of period domains: from number fields to finitely generated fields

Abstract

Contents1. Introduction 2. Arithmetic hyperbolicity and geometric hyperbolicity 3. Weakly bounded varieties and persistence of non-density 4. Hodge theory 5. Proof of Deligne-Schmid's theorem and Theorem 1.4 6. Locally symmetric varieties and Shimura varieties 7. The moduli of smooth hypersurfaces 8. Non-density of integral points on the Hilbert scheme References Abstract. We show that for a variety which admits a quasi-finite period map, finiteness (resp. non-Zariski-density) of S-integral points implies finiteness (resp. non-Zariski-density) of points over all Z-finitely-generated integral domains of characteristic zero. Our proofs rely on foundational results in Hodge theory due to Deligne, Griffiths, and Schmid, as well as the recent resolution of Griffiths's conjecture by Bakker-Brunebarbe-Tsimerman. We give straightforward applications to Shimura varieties, locally symmetric varieties, and the moduli space of smooth hypersurfaces in projective space. Using similar arguments and results of Viehweg-Zuo, we obtain similar arithmetic finiteness (resp. non-Zariski-density) statements for complete subvarieties of the moduli of canonically polarized varieties.

Arithmetic hyperbolicity: automorphisms and persistence

“…For instance, as a Brody hyperbolic projective variety has only finitely many automorphisms, Lang's conjecture predicts that the same should hold for arithmetically hyperbolic projective varieties. In this paper we verify this prediction, and also establish several other results predicted by related conjectures of Campana, Green-Griffiths, Hassett-Tschinkel, and Vojta. In subsequent papers we build on the results of this paper and verify several other predictions made by these conjectures (see [16,57,60]). A survey of all these results is presented in [52].…”

“…In the direction of this "reasonable" expectation, we show that arithmetic hyperbolicity persists, under a "mild boundedness" assumption; see Theorem 4.4. This more general result has shown to be useful in [16] and [57]; see also Section 4.4. 1.3.3.…”

“…In the next sections we will apply the above results. For other applications of the above results, we refer to [16,57].…”

On the Geometric Zilber-Pink Theorem and the Lawrence-Venkatesh method

“…Remark 2.3. Thanks to the main theorem of [5], the same conclusion of Theorem 2.1 holds true for over every finitely generated field K of characteristic zero (not necessarly a number field). This is indeed the level of generality we employ from now on.…”

Siegel's theorem via the Lawrence-Venkatesh method

“…e la er work moreover compares the Lawrence-Venkatesh method to the Kim-Chabauty approach to (effective) finiteness of rational points on curves. Finally, we mention the work of [JL19], who show that in the context of the Lawrence-Venkatesh method, one can o en extend results about finiteness or non-Zariski denseness of sets of points over number rings to the same results for points over more general finitely generated rings. e goal of the present paper is to show that Siegel's theorem admits a proof via the Lawrence-Venkatesh method as well.…”