Descent varieties for algebraic covers

Abstract

This paper is about descent theory for algebraic covers. Typical questions concern fields of definition, models, moduli spaces, families, etc. of covers. Here we construct descent varieties. Associated to any given cover f , they have the property that whether they have rational points on a given field k is the obstruction to descending the field of definition of f to k. Our constructions have a global version above moduli spaces of covers (Hurwitz spaces): here descent varieties are parameter spaces for Hurwitz families with some versal property. Descent varieties provide a new diophantine viewpoint on descent theory by reducing the questions to that of finding points on varieties. There are concrete applications. We answer a question raised in [7] about totally p-adic models of covers. We also show that the subset of a given moduli space of covers of P 1 where the field of moduli is a field of definition is Zariski-dense.

Finiteness Results in Descent Theory

“…In this absolute situation where F = k, the Galois group Gal(k/k) (the absolute Galois group of k) is denoted by G k . The notion of model extends to that of model over a ring (and in fact over an arbitrary K-scheme [8]). Given a K-integral domain A, with quotient field k, an A-model of a F-curve X is an A-curve X such that the generic fibre X ⊗ A k becomes isomorphic to X after extending the scalars to F, that is, (X ⊗ A k) ⊗ k F X.…”

On the fields of definition of genus‐one covers of P1$\mathbb {P}^1$

“…Covers without automorphisms as well as regular covers are known to be defined over their field of moduli [7]. In fact, covers defined over the field of moduli form a Zariski‐dense subset of the corresponding Hurwitz space [9]. However, there are examples of covers (and even of Belyi pairs) not defined over the field of moduli [3, 5, 12].…”

On the fields of definition of genus-one covers of $\mathbb{P}^1$

“…Covers without automorphisms as well as regular covers are known to be defined over their field of moduli [7]. In fact, covers defined over the field of moduli form a Zariski-dense subset of the corresponding Hurwitz space [9]. However, there are examples of covers (and even of Belyi pairs) not defined over the field of moduli [4,5,15].…”