The number of conjugacy classes of elements of the Cremona group of some given finite order

This note presents the study of the conjugacy classes of elements of some given finite order n in the Cremona group of the plane. In particular, it is shown that the number of conjugacy classes is infinite if n is even, n=3 or n=5, and that it is equal to 3 (respectively 9) if n=9 (respectively 15), and is exactly 1 for all remaining odd orders. Some precise representative elements of the classes are given.