A singular property of the supersingular elliptic curve in characteristic 2

TL;DR

This paper reveals a unique property of the supersingular elliptic curve in characteristic 2, showing it parametrizes Lamé covers, which has significant implications for understanding ramified covers and Galois actions in positive characteristic.

Contribution

It demonstrates that the set C(k) associated with the supersingular elliptic curve parametrizes Lamé covers, providing a new explicit example of such parametrization in characteristic 2.

Findings

C(k) parametrizes Lamé covers with specific ramification.

Explicit construction of Lamé covers is simpler in characteristic 2.

Results imply bounds on the number of Lamé covers over fixed number fields.

Abstract

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.