On the inertia group of elliptic curves in the Cremona group of the plane

We study the group of birational transformations of the plane that fix (each point of) a curve of geometric genus 1. A precise description of the finite elements is given; it is shown in particular that the order is at most 6, and that the embedding of the fixed curve in P 2 need to be birationally equivalent to a plane smooth cubic. We show that for a smooth cubic, the group is generated by its elements of degree 3, and prove that it contains a free product of Z/2Z, indexed by the points of the curve. 1.