Galois groups over rational function fields and Explicit Hilbert Irreducibility

Abstract

x] be a polynomial in two variables with rational coefficients, and let G be the Galois group of P over the field Q(t). It follows from Hilbert's Irreducibility Theorem that for most rational numbers c the specialized polynomial P (c, x) has Galois group isomorphic to G and factors in the same way as P . In this paper we discuss methods for computing the group G and obtaining an explicit description of the exceptional numbers c, i.e., those for which P (c, x) has Galois group different from G or factors differently from P . To illustrate the methods we determine the exceptional specializations of three sample polynomials. In addition, we apply our techniques to prove a new result in arithmetic dynamics.

On degrees in family of maps constructed via modular forms

“…. , λ s−1 ) a γ , n ≥ 1, we see that the solution is given by (2)(3)(4)(5)(6)(7) a α = −a γ M α (λ 0 , . .…”

Hilbert's Irreducibility, Modular Forms, and Computation of Certain Galois Groups

“…The paper [22] constructs various models over C of X 0 (N) complementing previous works such as [1], [2], [10], [11], [13], [18], [20], [26] and [30]. In this paper we discuss applications of the theory developed in [21] and [22] in computing certain Galois groups and splitting fields of rational functions in Q (X 0 (N)) using famous Hilbert's irreducibility theorem ( [29], [12]) and modular forms via the implementation of the algorithm for computing Galois groups of polynomials with coefficients in the function field Q(T ) (an extension of the work [9]), and implemented in MAGMA (see [8]). Hilbert's irreducibility is ultimately related to the inverse Galois problem, and a relation to modular forms has been studied in many papers (see for example [3], [4]).…”

A finiteness theorem for specializations of dynatomic polynomials

“…Let X be the smooth projective curve with function field S, and for every index i, let X i be the quotient curve X /M i . It follows from the proof of Proposition 3.3.1 in [30] (see also [17,Thm. 1.1]) that there exist a finite set E ⊂ P 1 (Q) and morphisms π i : X i → P 1 such that…”

“…The proof of the lemma suggests that we may determine the elements of E n by finding a certain finite set E and determining all the rational points on the curves X /M i . The set E as well as affine models for these curves can be obtained using the methods of the article [17]; however, the rational points on X /M i seem impossible to determine due to the large genera of the curves. (For instance, when n = 5 one of the curves has genus 9526, as seen in the proof of Theorem 6.6.)…”