Abelian constraints in inverse Galois theory

Abstract

We show that if a finite group G is the Galois group of a Galois cover of ${\mathbb{P}^1}$ over ${\mathbb{Q}}$ , 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 ${\ell \hskip -2pt \not | \hskip 2pt |{G}|}$ 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 provides a new viewpoint on Fried’s Modular Tower program and a weak form of its main conjecture.