Mappings of Curves
A meromorphic function on an algebraic curve is a mapping from this curve into the projective line. However, it is natural to consider also mappings into other complex curves, first of all, one-to-one mappings from a complex curve to itself, i.e., automorphisms of a curve. All automorphisms of a given curve form a group. For a curve of genus 0 (projective line), this group is three-dimensional. For any curve of genus 1 (elliptic curve), it is one-dimensional. For curves of higher genus it is finite, and for curves of genus g > 2 it usually consists only of the identity mapping. Curves with a large symmetry group are of special interest: like any symmetric object, they can be very beautiful.