Degrees of the Elliptic Teichmüller Lift Original Research Article

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)greater-or-equal, slanted−(n+2)pn+npn−1, ord0(yn)greater-or-equal, slanted−(n+3)pn+npn−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 p3 of the canonical lift for p≠2,3.