Effective Methods for Diophantine Finiteness

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 $f$.

Preliminary version. The author plans an extended version of this article with explicit computations to appear sometime next year

References in corpus (1)

• On the Transcendence of Period Images