The Reducibility of Curves f(X)=g(Y) and Polynomial Monodromy

Curves of the form f(X)=g(Y), and in particular elliptic
curves Y²=X³+aX+b, play a major role in number theory, arithmetic
dynamics, functional equations, and beyond.
The classical Davenport–Lewis–Schinzel (DLS) problem concerns their
reducibility. Surprisingly, by studying  the  corresponding monodromy,
the problem was found to be related to symmetries of interesting
geometric objects such as the Fano plane. We shall describe this
relation and how the study of polynomial monodromy allows solving the
DLS and related problems.