希尔伯特类多项式模p的Fp根的个数

ON F_(p)-ROOTS OF THE HILBERT CLASS POLYNOMIAL MODULOp

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H_(o)(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F_(p)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F_(p)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F_(p)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.

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 F_(p)-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 F_(p)-roots of Ho(x)(mod p)whenever it is nonempty.We also provide a concrete criterion for the existence of F_(p)-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.