In this paper we analyze the j-invariant of the canonical lifting of an elliptic curve as a Witt vector. We show that its coordinates are rational functions on the j-invariant of the elliptic curve in characteristic p. In particular, we prove that the second coordinate is always regular at j = 0 and j = 1728, even when those correspond to supersingular values. A proof is given which yields a new proof for some results of Kaneko and Zagier about the modular polynomial.
In this paper we analyze liftings of hyperelliptic curves over perfect fields in characteristic 2 to curves over rings of Witt vectors. This theory can be applied to construct error-correcting codes; lifts of points with minimal degrees are likely to yield the best codes, and these are the main focus of the paper. We find upper and lower bounds for their degrees, give conditions to achieve the lower bounds and analyze the existence of lifts of the Frobenius. Finally, we exhibit explicit computations for genus 2 and show codes obtained using this theory.
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
Abstract. The purpose of this paper is to find upper bounds for the degrees, or equivalently, for the order of the poles at O, of the coordinate functions of the elliptic Teichmüller lift of an ordinary elliptic curve over a perfect field of characteristic p. We prove the following bounds: ord0(xn) ≥ −(n + 2)p n + np n−1 , ord0(yn) ≥ −(n + 3)p n + np n−1 .Also, we prove that the bound for xn is not the exact order if, and only if, p divides (n + 1), and the bound for yn is not the exact order if, and only if, p divides (n + 1)(n + 2)/2. Finally, we give an algorithm to compute the reduction modulo p 3 of the canonical lift for p = 2, 3.
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