Hilbert's irreducibility theorem and the larger sieve

TL;DR

This paper presents an explicit form of Hilbert's irreducibility theorem utilizing a generalized larger sieve, with applications to Galois groups of random polynomials and adelic representations of elliptic curves.

Contribution

It introduces a new explicit version of Hilbert's irreducibility theorem based on a generalized larger sieve method.

Findings

Explicit bounds for Hilbert's irreducibility theorem

Applications to Galois groups of random polynomials

Insights into adelic representations of elliptic curves

Abstract

We describe an explicit version of Hilbert's irreducibility theorem using a generalization of Gallagher's larger sieve. We give applications to the Galois theory of random polynomials, and to the images of the adelic representation associated to elliptic curves varying in rational families.