A singular property of the supersingular elliptic curve in characteristic 2

Let E be the supersingular elliptic curve defined over k, the algebraic closure of the finite field with two elements, which is unique up to k-isomorphism. Denote by 0 its identity element and let C be the quotient of E-{0} under the action of the group Isom(E) (which is non-abelian, of order 24). The main result of this paper asserts that the set C(k) naturally parametrizes k-isomorphism classes of Lam\'e covers, which are tamely ramified covers of the projective line unramified outside three points having a particular ramification datum. This fact is surprising for two reasons: first of all, it is the first non-trivial example of a family of covers of the projective line unramified outside three points which is parametrized by the geometric points of a curve. Moreover, when considered in arbitrary characteristic, the explicit construction of Lam\'e covers is quite involved and their arithmetic properties still remain misterious. The simplicity of the problem in characteristic 2 has many deep consequences when combined with lifting techniques from positive characteristic to characteristic 0. As an illustration, we obtain some sharp statements concerning the (local) Galois action, as, for example, a bound on the number of isomorphism classes of Lam\'e covers defined over a fixed number field, only depending on the degree of the residual extension at 2. Finally, in the appendix we give a partial generalization of these results by showing that, for any positive integer g, the k-rational points of a suitable quotient of a genus g hyperelliptic curve parametrize k-isomorphism classes of tamely ramified covers of the projective line unramified outside three points.