Computation of some moduli spaces of covers and explicit 𝒮_nand 𝒜_nregular ℚ(T)-extensions with totally real fibers

Abstract

We study and compute an infinite family of Hurwitz spaces parameterizing covers of P 1 C branched at four points and deduce explicit regular S n and A n -extensions over Q(T ) with totally real fibers.Introduction.

On computing Belyi maps

“…This also works in reverse, and provides another application of computing Belyȋ maps. Hallouin-Riboulet-Deyris [66] explicitly computed polynomials with Galois group A n and S n over Q(t) with four branch points for small values of n; starting from a relatively simple "degenerate" three-point branched cover, the four-point branched cover is obtained by complex approximation (using Puiseux expansions). These methods were considerably augmented by Hallouin in [67] to find another such family with group PSL 2 (F 8 ).…”

Parametric Galois extensions

“…Assume that n is even and n ≥ 6. From [20], there exists some k-regular Galois extension E 3 /k(T ) of group S n with four Q-rational branch points and inertia canonical invariant ([1 2 (n −2) 1 ], [1 n−3 3 1 ], [2 (n/2) ], [1 2 2 (n−2)/2 ]).…”

“…Apply the "double group trick" [37,Lemma 4.5.1] to the extension E 2 /k(T ) from Section 7.1.1(b) to obtain a three branch point k-regular Galois extension E 2 /k(T ) of group A n and, from the branch cycle lemma [15], [ Note that the branch point of E 3 /k(T ) associated with [1 2 ((n − 2)/2) 2 ] (in each case) is Q-rational from the branch cycle lemma (if n ≥ 8). 20 …”

“…(2) Assume that n = 6 and k is either a number field or a finite extension of a rational function field κ(X) with κ an arbitrary algebraically closed field of char- 20 In the case n = 6, the involved branch point might not be Q-rational as the two conjugacy classes…”

Study and computation of a Hurwitz space and totally real PSL2(F8)-extensions of Hallouin1 2005Journal of AlgebraSelf Cite4000View full textAdd to dashboardCite

“…Following methods of [Cou99], we then explicitly deform this degenerate element in order to compute an algebraic model of our family over Q((π )) the completion of our pointed Hurwitz space at one point in its boundary. Comparing with our previous work in the area (see [HRD03]), this deformation step requires new algorithms since the genus of the total space is no longer zero but one.…”