Abelian constraints in inverse Galois theory

Abstract

Abstract. We show that if a finite group G is the Galois group of a Galois cover of P 1 over Q, then the orders p n of the abelianization of its p-Sylow subgroups are bounded in terms of their index m, of the branch point number r and the smallest prime | |G| of good reduction of the branch divisor. This is a new constraint for the regular inverse Galois problem: if p n is suitably large compared to r and m, the branch points must coalesce modulo small primes. We further conjecture that p n should be bounded only in terms of r and m. We use a connection with some rationality question on the torsion of abelian varieties. For example, our conjecture follows from the so-called torsion conjectures. Our approach also yields a new viewpoint on Fried's Modular Tower program.

Moduli relations between l-adic representations and the regular inverse Galois problem

“…In cases the MTs, F G,C,ℓ, ℓ ψ⋆ , for proper quotients are variants on classical spaces. • [CaD08] • expands on this.…”

“…• [FrK97] • and • [CaD08] • have a version that applies to any (G, ℓ) with ℓ-perfect (Def. 1.8) G. Further, the latter gives good reasons to explicitly relate Hurwitz spaces and classical spaces as does Quest.…”

“…(1.9b) Classifying cusps and components on which they lie; geometrically separating them for G Q action (5.24) in • [Fr06] and [CaTa09] •. (1.9c) Using (1.9b) and §4.1 ℓ-adic representations, for labeling abelian variety collections a' la the Torsion Conjecture • [CaD08] •.…”

“…§ 3.4.3 continues Ex. 2.16 as a case included in • [CaD08] •. §3.5 starts the motivation from [Se68] for the eventually ℓ-Frattini Definition.…”

“…4.5 shows how MTs for each finite group G and ℓ-perfect prime of G generalizes what the same question applied to dihedral groups posed for hyerelliptic jacobians. This -expanded in • [CaD08] • -shows the MT project, through the RIGP, tying to classical considerations.…”