Hilbert类多项式在素域F_(p)上的分解

Factorization of Hilbert class polynomials over prime nite elds F_(p)

设D为模4余0或1的负整数,令O=O_(D)为虚二次域K=q(√D)中判别式等于D的序.熟知Hilbert类多项式HD(x)是O的j不变量j_(D)=j(C/O)在K上的最小多项式.令n_(D)=(O_(Q(j_(D))):Z[j_(D)])表示Z[j_(D)]在Q(j_(D))的整数环O_(Q(j_(D)))中的指数.设p为任意素数.如果p|n_(D),或者者p|D在K中惯性且n_(D)的p-进离散赋值v_(p)(n_(D))≤3,则本文完全确定了H_(D)(x)在多项式环F_(p)[x]中的因式分解.

Let D be a negative integer congruent to 0 or 1(mod 4)and O=O_(D)be the corresponding order of K=q(√D).The Hilbert class polynomial HD(x)is the minimal polynomial of the j-invariant j_(D)=j(C/O)of O over K.Let n_(D)=(O_(Q(j_(D))):Z[j_(D)])denote the index of Z[j_(D)]in the ring of integers O_(Q(j_(D)))of Q_((j_(D))).Suppose p is any prime.We completely determine the factorization of HD(x)in Fp[x]if either p|n_(D)or p|D is inert in K and the p-adic valuation v_(p)(n_(D))≤3.