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

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Ariyan Javanpeykar , Daniel Litt ·

YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.1007/s00229-023-01463-w · OA: W2965725474

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 $$\mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> -finitely generated integral domains of characteristic zero. Our proofs rely on foundational results in Hodge theory due to Deligne, Griffiths, and Schmid, and Bakker-Brunebarbe-Tsimerman. We give straightforward applications to arithmetic locally symmetric varieties, the moduli space of smooth hypersurfaces in projective space, and the moduli of smooth divisors in an abelian variety.