Effective Methods for Diophantine Finiteness

2021/10/28 by David Urbanik, Urbanik, David
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT

paper · pdf · doi:10.48550/arxiv.2110.14829

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

arxiv created 2021/10/28 · arxiv updated 2021/10/29

Abstract

Let K ⊂ ℂ be a number field, and let O_K,N = 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 → S defined over O_K,N. To carry out their strategy, one needs to be able to decide whether the algebraic monodromy group H_Z of any positive-dimensional geometrically irreducible subvariety Z ⊂ S_ℂ is "large enough", in the sense that a certain orbit of 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.

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