Let S be a finite set of places of K a number field or a function field over a field k. We prove finiteness up to conjugacy of the set of non isotrivial, separable, tame, self maps of $\mathbb{P}^1_K$ of a given degree and ramified at least in 3 points, with "critically excellent reduction" outside of S. The simple notion of good reduction (conservation of the degree in reduction) is not adequate as one can see by looking at monic polynomial maps. Critically excellent reduction says that the ramification and branch locus are etale.

Apply to the Lattes map associated to the multiplication by 2 in an elliptic curve this reproves Shafarevich original result on elliptic curves . We will recall the different "Shafarevich conjectures" already proved (Finite maps, Curves over function field, Abelian varieties over number fields, K3 surfaces,)

Joint work with Tom Tucker and Lloyd West.