Rational points and rational moduli spaces

TL;DR

This work introduces a geometric non-degenerate criterion for varieties $X$ over $\boldsymbol{Q}$ via moduli spaces $M$ of abelian varieties and shows that non-degenerate $X$ admit an open dense $U$ with a Paršin construction $\phi:U(\boldsymbol{Q})\to\underline{A}_g(T)$, yielding finiteness by Faltings. In the Hilbert/coarse Hilbert setting, explicit height bounds and point-count bounds are obtained for $U(\boldsymbol{Q})$; the approach leverages the GL$_2$-type Shafarevich finiteness and the modularity/isogeny estimates of Masser–Wüstholz. The paper develops ico models for curves, proving that every integral curve has an ico model and that non-degenerate curves (genus $\ge 2$) admit effective Paršin constructions and Mordell-type height bounds $h(x)\le c\nu_f^{24}$, with $\nu_f$ depending only on diagonal parts of defining equations. A geometric-framework for plane curves via a $(\tau)$-criterion yields explicit height bounds and constructive methods, leading to effective Mordell results for large families of non-degenerate plane curves and applications to the Fermat problem inside ico surfaces. The work also discusses a general Fermat conjecture inside projective varieties, connections to $abc$-type conjectures, and outlines potential algorithmic implications for computing $X(\boldsymbol{Q})$ in broad geometric contexts. Collectively, the results bridge moduli-theoretic geometry with explicit Diophantine bounds, offering a versatile toolkit for finiteness and effective height results across curves and higher-dimensional varieties.

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.

Theorems & Definitions (61)

• Theorem 4.1
• Corollary 4.2
• Theorem 5.1
• Proposition 5.2
• Corollary 5.3
• Lemma 5.4
• Lemma 5.5
• Lemma 5.6
• proof : Proof of Lemma \ref{['lem:coarsebasechange']}
• proof : Proof of Lemma \ref{['lem:extndmod']}