Degrees of the Elliptic Teichmüller Lift

Abstract

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.

Exponential sums over lifts of points

“…Thus we need to find sections whose coordinate functions have ''small'' pole orders: we show (Theorem 2.1) that if C k ¼ C#k has genus g; and if D 0 is the intersection divisor of C k with H#k in P n k ; then there exists a section with x ij having polar divisor less than E j :¼ p j ðjD þ ðj þ 1ÞD 0 Þ (in other words, x ij is in LðE j Þ; the space of global sections of the divisor E j ), where D is a divisor of degree 2g À 2 þ 2gÀ1 p l m with support contained in that of D 0 : Similar calculations have been made in the case of elliptic and hyperelliptic curves by Finotti (cf. [1]), and this result may be of independent interest from the rest of the article. Finally, we find local parameters for the curve C at the points of C-H; and we estimate the pole orders of the components of GðsÞðf Þ from the expansions of f in terms of these local parameters.…”

Algebraic Geometric Codes over Rings

“…Blache [4], [5] extended this work by considering other exponential sums along these and other curves defined over rings. Some of the results were also improved upon by Finotti [9], [10], [11], who used work of Mochizuki [33] to get better bounds on the degrees of the liftings of points.…”

Computations with Witt vectors of length 3

“…The algorithm described in [6] to compute the second and third coordinates of the canonical lifting of an ordinary elliptic curve starts by computing the Greenberg transform. But this requires a lot of computer power when the coefficients are left as unknowns, and that is exactly the problem we first encountered when trying to compute J 2 (X) for p ≥ 17.…”

“…At first, the author computed J 1 and J 2 by computing the canonical lifting of the elliptic curve E given by y 2 0 = x 3 0 +a 0 x 0 +b 0 over F p (a 0 , b 0 ), where a 0 and b 0 were variables, i.e., algebraically independent transcendental elements over F p , using the algorithm described in [6]. (Note that the algorithm gives more than just the canonical lifting E of E. It also gives a lifting of points from E(k) to E(W 3 (k)) called the elliptic Teichmüller lift.)…”

“…It should also be mentioned that the method from [6] used to obtain the initial examples of J 2 mentioned above is not the most efficient. There are better methods to compute the canonical lifting if we are not also interested in the elliptic Teichmüller lift.…”