Rational points and rational moduli spaces

TL;DR

This paper develops a geometric criterion for identifying non-degenerate varieties over $Q$ using moduli spaces of abelian varieties, leading to effective bounds and solutions for Diophantine problems on certain curves.

Contribution

It introduces a new non-degenerate criterion via moduli spaces and constructs explicit models to effectively bound rational points and solve classical Diophantine problems.

Findings

Non-degenerate varieties have finite rational points by Faltings.

Explicit height bounds are obtained for rational points on certain curves.

The approach verifies the effective Mordell conjecture for large classes of curves.

Abstract

Let $X$ be a variety over $\mathbb Q$. We introduce a geometric non-degenerate criterion for $X$ using moduli spaces $M$ over $\mathbb Q$ of abelian varieties. If $X$ is non-degenerate, then we construct via $M$ an open dense moduli space $U\subseteq X$ whose forgetful map defines a Parsin construction for $U(\mathbb Q)$. For example if $M$ is a Hilbert modular variety then $U$ is a coarse Hilbert moduli scheme and $X$ is non-degenerate iff a projective model $Y\subset \bar{M}$ of $X$ over $\mathbb Q$ contains no singular points of the minimal compactification $\bar{M}$. We motivate our constructions when $M$ is a rational variety over $\mathbb Q$ with $\dim(M)>\dim(X)$. We study various geometric aspects of the non-degenerate criterion and we deduce arithmetic applications: If $X$ is non-degenerate, then $U(\mathbb Q)$ is finite by Faltings. Moreover, our constructions are made for the effective strategy which combines the method of Faltings (Arakelov, Parsin, Szpiro) with modularity and Masser-Wustholz isogeny estimates. When $M$ is a coarse Hilbert moduli scheme, we use this strategy to explicitly bound the height and the number of $x\in U(\mathbb Q)$ if $X$ is non-degenerate. We illustrate our approach when $M$ is the Hilbert modular surface given by the classical icosahedron surface studied by Clebsch, Klein and Hirzebruch. For any curve $X$ over $\mathbb Q$, we construct and study explicit projective models $Y\subset\bar{M}$ called ico models. If $X$ is non-degenerate, then we give via $Y$ an effective Parsin construction and an explicit Weil height bound for $x\in U(\mathbb Q)$. As most ico models are non-degenerate and $X\setminus U$ is controlled, this establishes the effective Mordell conjecture for large classes of (explicit) curves over $\mathbb Q$. We also solve the ico analogue of the generalized Fermat problem by combining our height bounds with Diophantine approximations.