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| 摘要: |
| 设K是一个虚二次域,O为K中的一个order. 由定义,O的希尔伯特类多项式HO(x)是一个整系数的首一不可约多项式, 它的复根恰为所有具有O-复乘的椭圆曲线的j-不变量. 设p ∈ N 为一个在K中惯性的素数, 且p严格大于|disc(O)|. 若HO(x)(mod p)的Fp根的所组成的集合非空, 我们证明群Pic(O)[2]在该集合上有一个自由且传递的作用; 因此HO(x)(mod p)的Fp根的个数要么等于0, 要么等于|Pic(O)[2]|. 我们还给出了一个关于Fp根存在性的具体判别方法. 类似的结果首先由Xiao 等人在文献[25]中得到, 后又经李等人在文献[13]广泛推广. 本文结果已在李等人的工作中出现, 但方法与之完全不同. |
| 关键词: 希尔伯特类多项式 超奇异椭圆曲线 自同态环 四元数代数 理想类群 |
| DOI： |
| 分类号:0516.2 |
| 基金项目:Supported by NSF grants DMS-1844206, DMS-1802161. |
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| ON Fp-ROOTS OF THE HILBERT CLASS POLYNOMIAL MODULO p |
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CHEN Ming-jie,XUE Jiang-wei
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| Abstract: |
| The Hilbert class polynomial Ho(x) ∈ Z[x] attached to an order O in an imaginary quadratic field K is the monic polynomial whose roots are precisely the distinct j-invariants of elliptic curves over C with complex multiplication by O. Let p be a prime inert in K and strictly greater than |disc(O)|. We show that the number of Fp-roots of Ho(x)(mod p) is either zero or |Pic(O)[2]| by exhibiting a free and transitive action of Pic(O)[2] on the set of Fp-roots of Ho(x) (mod p) whenever it is nonempty. We also provide a concrete criterion for the existence of Fp-roots. A similar result was first obtained by Xiao et al. [25] and generalized much further by Li et al. [13] (that covers the current result) with a different approach. |
| Key words: Hilbert class polynomial supersingular elliptic curve endomorphism ring quaternion algebra Picard group |
