A Polynomial with Galois Groups SL₂(F₁₆)

Abstract

In this paper we display an explicit polynomial having Galois group SL 2 (F 16 ), filling in a gap in the tables of Jürgen Klüners and Gunter Malle. Furthermore, the polynomial has small Galois root discriminant; this fact answers a question of John Jones and David Roberts. The computation of this polynomial uses modular forms and their Galois representations.

Modular forms applied to the computational inverse Galois problem

“…In the present paper, we will display polynomials for several groups of the type PSL 2 (F q ) and PGL 2 (F q ) with q a perfect prime power. This can be seen as an extension of a previous result by the author for q = 16 (see [2]). Theoretical results stating that PSL 2 (F q ) appears as Galois group over Q for 'many' q can be found in [29] and [9].…”

“…Proposition 1. Let q ≥ 4 be a prime power and let θ : PGL 2 (F q ) → F q be as in (2). Let G be a subgroup of PSL 2 (F q ).…”

A database of number fields

“…(7), Ω], twenty are listed in [21] and three are new. The polynomial for the SL 2 (16) field was found by Bosman [8], starting from a classical modular form of weight two. We found polynomials for the new SL 3 (2) field and the three new PGL 2 (7) fields starting from Schaeffer's list [35, Appendix A] of ethereal modular forms of weight one.…”

“…The database gives many solvable fields satisfying the root discriminant bound. In this paper, for brevity, we restrict attention to nonsolvable fields, where, among other interesting things, modular forms [8,35] sometimes point the way to explicit polynomials.…”

An effective bound for the Huber constant for cofinite Fuchsian groups

“…A summary of the ideas from [ECJMB] is given in [Edi06], where the purpose is to focus attention to the weight twelve modular form associated to PSL(2, Z). The ideas from [ECJMB] have been used to achieve advances in many other computation problems; see, for example, [Bo07], [CL09], [Cou09], and [La06] for specific results, as well as the survey article [Chl08] for more general comments.…”