A formula for the supersingular polynomial

Abstract

1. Introduction. Let k be a perfect field of characteristic p > 0 and E/k an elliptic curve over k. Ifk denotes the algebraic closure of k, then E(k) is an Abelian group and its p-torsion, denoted by E[p], is either 0 or Z/pZ. (See, for instance, Theorem V.3.1 in [7].) E is then called supersingular if E[p] = 0, and ordinary otherwise. (As observed by Silverman in Remark V.3.2.2 of [7], there are other characterizations of supersingular elliptic curves relevant to various applications.)It is a known fact that, for a fixed characteristic p > 0, there are (up to isomorphism) finitely many supersingular elliptic curves. (See, for instance, Theorem V.4

Lifting the j-invariant: Questions of Mazur and Tate

“…as if we let r def = (p − 1)/2, r 1 def = r/3 , and r 2 def = r/2 , then, an easy computation (see [7]) shows that…”

An elementary proof for the number of supersingular elliptic curves

“…More on the supersingular polynomial, including different formulas, can be found in Kaneko and Zagier's [4], Brillhart and Morton's [1], and Morton's [5]. We also note that the published formula in [3] has a typo, but Eq. (1.2) is correct.…”

“…Formula (1.2) above, which was nearly deduced by Deuring in [2], was fully derived in [3] by using the fact the an elliptic curve is supersingular if, and only if, its Hasse invariant is zero. (This result is due to Deuring and Hasse.)…”

“…At the end of [3] an alternative proof is given, which is completely elementary. We first observe that ss p (X) has simple roots if, and only if, has simple roots, as it differs from ss p (X) only by a constant multiple and possible factors of X or (X − 1728) .…”

Computations with Witt vectors of length 3

“…(This is well known. See, for instance, [9].) Then, since in this case the canonical lifting of the elliptic curve given by j 0 = 0 is the curve with…”