Denominators of the Weierstrass coefficients of the canonical lifting

Abstract

Given an ordinary elliptic curve E/k:y02=x03+a0x0+b0 over a field k of characteristic p≥5 with j-invariant j0, the j-invariant of its canonical lifting E/W(k):y2=x3+ax+b is j=(j0,J1(j0),J2(j0),…), for some Ji∈Fp(X). Thus the Weierstrass coefficients of E can be given by a=λ4⋅27j/(6912−4j), b=λ6⋅27j/(6912−4j), where λ=((b0/a0)1/2,0,0,…), and therefore can be seen as functions on (a0,b0). Here we study the denominators of the coordinates of these a and b. We show that the only possible factors for these denominators are powers of a0, b0, and the Hasse invariant h. Upper bounds for these powers are given for each one of them.