Effective Methods for Diophantine Finiteness

TL;DR

This paper develops effective methods to determine the size of algebraic monodromy groups in families of algebraic varieties, enabling a fully effective approach to Shafarevich-type problems in number theory.

Contribution

It introduces an effective technique for assessing the monodromy groups of subvarieties, advancing the strategy for proving the Shafarevich conjecture for algebraic families.

Findings

Provides an algorithm to decide if the monodromy group is sufficiently large.

Combines with existing $p$-adic methods for a comprehensive strategy.

Enables fully effective solutions to Shafarevich-type problems.

Abstract

Let $K \subset \mathbb{C}$ be a number field, and let $\mathcal{O}_{K,N} = \mathcal{O}_{K}[N^{-1}]$ be its ring of $N$-integers. Recently, Lawrence and Venkatesh proposed a general strategy for proving the Shafarevich conjecture for the fibres of a smooth projective family $f : X \to S$ defined over $\mathcal{O}_{K,N}$. To carry out their strategy, one needs to be able to decide whether the algebraic monodromy group $\mathbf{H}_{Z}$ of any positive-dimensional geometrically irreducible subvariety $Z \subset S_{\mathbb{C}}$ is "large enough", in the sense that a certain orbit of $\mathbf{H}_{Z}$ in a variety of Hodge flags has dimension bounded from below by a certain quantity. In this article we give an effective method for deciding this question. Combined with the effective methods of Lawrence-Venkatesh for understanding semisimplifications of global Galois representations using $p$-adic Hodge theory, this gives a fully effective strategy for solving Shafarevich-type problems for arbitrary families $f$.