Prym varieties of a Galois covering. (Variétés de Prym d’un revêtement galoisien.) (French) Zbl 0814.14043

Mérindol, Jean-Yves

J. Reine Angew. Math. 461, 49-61 (1995).

The aim of this paper is to construct some generalisations of the Prym varieties. Let \(\pi : X \to X\) a Galoisian map between two Riemann surfaces. We denote by \(G\) the Galois group and we choose an absolutely irreducible \(\mathbb Q[G]\)-module \(V\). We construct, by a very simple way, some abelian variety in the Jacobian of \(X\) associated to \(V\). We note this variety, in fact well defined up to an isogeny, by \(\text{Prym}_ V(\pi)\) or \(\text{Prym}_ V(G)\).
An important case is when \(G = W\) is a Weyl group. In this case \(\text{Jac }X\) is isogenous to the product (taken over all the irreducible representations of \(G = W\)): \(\prod_{V \in \text{Irr }G} \text{Prym}_ V(G) \text{deg }V\). In particular, when \(V\) is the “usual” representation of \(W\) (as a reflection group), \(\text{Prym}_ V(G)\) is the Prym-Tyurin variety constructed by Kanev.
This variety appears also in the linearisation of some Lax equations. This fact, also observed by Kanev, can also be explained by an elementary computation in linear algebra related to some projectors.

Reviewer: J.-Y.Mérindol (Strasbourg)

MSC:

14K30	Picard schemes, higher Jacobians
14H30	Coverings of curves, fundamental group

Keywords:

Prym varieties; abelian variety; Jacobian; Weyl group; Lax equations