In this paper we study formulas for the coordinates of the Weierstrass coefficients $\mathit{a}=(a,A_1,A_2,\dots )$ and $\mathit{b}=(b,B_1,B_2,\dots )$ of the canonical lifting. More precisely we show that there are formulas for $A_1$ and $B_1$ that are given by modular functions, universal (meaning, single formulas that work in every possible case), and with no factor of Δ in their denominators. Two possible constructions are given. Moreover, a sufficient condition is given for the existence of $A_2$ and $B_2$ with these properties. Finally, we prove that the Hasse invariant, given as a polynomial on the Weierstrass coefficients of an elliptic curve of characteristic $p\ge 5$, has no repeated factors.

在本文中，我们研究了Weierstrass系数的坐标公式 $\mathit{一种}=（\mathrm{一种}，{\mathrm{一种}}_{\mathrm{1个}}，{\mathrm{一种}}_{\mathrm{2个}}，\dots ）$ 和 $\mathit{b}=（b，{乙}_{\mathrm{1个}}，{乙}_{\mathrm{2个}}，\dots ）$规范的提升。更确切地说，我们证明了存在以下公式${\mathrm{一种}}_{\mathrm{1个}}$ 和 ${乙}_{\mathrm{1个}}$由通用的模块化函数（意味着在每种可能的情况下均有效的单个公式）给出，并且其分母中没有Δ因子。给出了两种可能的构造。此外，给出了存在的充分条件${\mathrm{一种}}_{\mathrm{2个}}$ 和 ${乙}_{\mathrm{2个}}$具有这些特性。最后，我们证明了作为特征椭圆曲线的Weierstrass系数的多项式给出的Hasse不变量$p\ge 5$，没有重复因素。