Galois closure of a 5-fold covering and decomposition of its Jacobian

TL;DR

This paper characterizes the Galois closure of 5-fold coverings of Riemann surfaces using ramification data and describes the Jacobian's decomposition into simpler abelian varieties, revealing detailed structural insights.

Contribution

It explicitly determines the Galois closure and decomposes the Jacobian into Prym and Jacobian varieties based on ramification data.

Findings

Galois closure determined by ramification divisor

Decomposition of Jacobian into Prym and Jacobian factors

Dimensions and polarizations computed from ramification data

Abstract

For an arbitrary 5-fold ramified covering between compact Riemann surfaces, every possible Galois closure is determined in terms of the ramification data of the map; namely, the ramification divisor of the covering map. Since the group that acts on the Galois closure also acts on the Jacobian variety of the covering surface, we describe its group algebra decomposition in terms of the Jacobian and Prym varieties of the intermediate coverings of the Galois closure. The dimension and induced polarization of each abelian variety in the decomposition is computed in terms of the ramification data of the covering map.