Hilbert's irreducibility theorem and the larger sieve

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.

References in corpus (3)

• On the distribution of Galois groups
• Elliptic curves with maximal Galois action on their torsion points
• Variation of Periods Modulo p in Arithmetic Dynamics

Cited by in corpus (12)

• On the distribution of Galois groups
• Enumerative Galois theory for cubics and quartics
• Surjectivity of Galois Representations in Rational Families of Abelian Varieties
• Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants
• Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians
• Splitting Behavior of $S_n$-Polynomials
• The inverse sieve problem for algebraic varieties over global fields
• Elliptic Curves with Full 2-Torsion and Maximal Adelic Galois Representations
• Low-degree factors of random polynomials
• Polynomial values modulo primes on average and sharpness of the larger sieve
• The large sieve for self-dual Eisenstein series of varying levels
• A Lefschetz theorem for intersections with projective varieties