On etale covers of curves

Let K be a number field with ring of integers R. For each integer g > 1 we consider the collection of abelian, etale R-coverings f:Y --> X, where X and Y are connected proper curves over R and the genus of X is g. We ask the following question: is there a positive integer B = B(K, g) which bounds the degree of such coverings? In this note we provide partial results towards such a bound and study the relationship with bounds on torsion in abelian varieties.