Rational points and rational moduli spaces

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.