超越超奇异性的两个判据：除多项式的系数
Beyond two criteria for supersingularity: coefficients of division polynomials
摘要：
设E：y^2 = x^3 + Ax + B是定义在特征为p \geq 3的有限域上的椭圆曲线.本文证明了E的p次除多项式n_p（x）在x ^{p（p-1）/2}处的系数等于（x^3 + Ax + B）^{（p-1）/2}在x^{p-1}处的系数。第一个系数为零当且仅当除法多项式没有根，这等价于E是超奇异的。Deuring（1941）证明了这种超奇异性也等价于第二个系数的消失。所以系数的零轨迹（作为A和B的函数）是相等的;本文的主要结果明显强于最后一个陈述。
英文摘要：
Let E: y^2 = x^3 + Ax + B be an elliptic curve defined over a finite field of characteristic p\geq 3. In this paper we prove that the coefficient at x^{p(p-1)/2} in the p-th division polynomial ψ_p(x) of E equals the coefficient at x^{p-1} in (x^3 + Ax + B)^{(p-1)/2}. The first coefficient is zero if and only if the division polynomial has no roots, which is equivalent to E being supersingular. Deuring (1941) proved that this supersingularity is also equivalent to the vanishing of the second coefficient. So the zero loci of the coefficients (as functions of A and B) are equal; the main result in this paper is clearly stronger than this last statement.
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