The inverse sieve problem for algebraic varieties over global fields

The inverse sieve problem for algebraic varieties over global fields

Rev. Mat. Iberoam. 37 (2021), 2245-2284 Let $K$ be a global field and let $Z$ be a geometrically irreducible algebraic variety defined over $K$. We show that if a big set $S\subseteq Z$ of rational points of bounded height occupies few residue classes modulo $\mathfrak{p}$ for many prime ideals $\ma...

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Bibliographic Details
Main Authors: Menconi, Juan Manuel, Paredes, Marcelo, Sasyk, Román
Format: Journal Article
Language:English
Published: 02-04-2020
Subjects:
Mathematics - Number Theory
Online Access:Get full text
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Summary:Rev. Mat. Iberoam. 37 (2021), 2245-2284 Let $K$ be a global field and let $Z$ be a geometrically irreducible algebraic variety defined over $K$. We show that if a big set $S\subseteq Z$ of rational points of bounded height occupies few residue classes modulo $\mathfrak{p}$ for many prime ideals $\mathfrak{p}$, then a positive proportion of $S$ must lie in the zero set of a polynomial of low degree that does not vanish at $Z$. This generalizes the main result of Walsh in [Duke Math. J., vol.161, (2012), 2001-2022].
DOI:10.48550/arxiv.1907.02049