"Curves dominated by an unramified cover of another curve"

Bjorn Poonen, University of California, Berkeley

Abstract:
Let X be a curve of genus at least 2 over a number field F. If Y is an unramified cover of X, the Chevalley-Weil descent method reduces the problem of determining X(F) to the problem of determining Y(F') for some finite extension F' of F. Also, if X dominates Y, where Y also has genus at least 2, then the problem of determining X(F) can trivially be reduced to the problem of determining Y(F). We survey what is known about the transitive relation (not an equivalence relation) on curves induced by these two relationships.