An approach for computing families of multi-branch-point covers and applications for symplectic Galois groups

Abstract

We propose an approach for the computation of multi-parameter families of Galois extensions with prescribed ramification type. More precisely, we combine existing deformation and interpolation techniques with recently developed strong tools for the computation of 3-point covers. To demonstrate the applicability of our method in relatively large degrees, we compute several families of polynomials with symplectic Galois groups, in particular obtaining the first totally real polynomials with Galois group PSp 6 (2).Note: Supplementary data are contained in an extra file, available at:https://arxiv.org/src/1803.08778/anc/anc.txt

Computation of Belyi maps with prescribed ramification and applications in Galois theory

“…Our first step consists of sketching the expected dessin: Now, if we walk counter-clockwise around the dessin and write down the edges we pass by along the way we obtain (1,4,5,5,4,3,3,2,2,1).…”

“…There are five primitive groups having these subdegrees, all between PSL (3,4) and N S 56 (PSL (3,4)). These five groups are also the only possibilities for the geometric monodromy group G from which only N S 56 (PSL (3,4)) contains a generating triple with the desired cycle structures. It follows A = G = N S 56 (PSL (3,4)).…”

“…Realization of all index-2 subgroups of N S 56 (PSL (3,4)). There are exactly three index-2 subgroups of N S 56 (PSL (3,4)) given by their primitive group identification [56,3], [56, 4] = A 56 ∩N S 56 (PSL (3,4)) and [56, 5] = PΣL (3,4). We treat all these groups separately: [56,4]: Note that N S 56 (PSL (3,4)) is an odd and [56,4] an even group.…”

“…In particular, we present the first (to the best of our knowledge) explicit realizations of PSp (4,4), PSp(4, 4):2, HS, Aut(HS), O + (8, 2) as Galois groups over Q(t) and J 1 as a Galois group over K(t) with K being a degree-7 number field.…”

“…• calculation of multi-branch-point covers with prescribed ramification (collaboration with J. König), Galois group verification for 2-transitive groups, explicit realization of PSp 6 (2) of degree 28 and 36, PSp 4 (3) of degree 27 and PSL 6 (2) of degree 63, see [4], [9], [11].…”