Trace of Frobenius endomorphism of an elliptic curve with complex multiplication
Trace of Frobenius endomorphism of an elliptic curve with complex multiplication
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DOI：
10.1017/s0004972700035875
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发表时间：
2004-01
影响因子：
0.7
通讯作者：
Noburo Ishii
中科院分区：
文献类型：
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作者：
Noburo Ishii
设E是一条复乘R的椭圆曲线，其中R是虚二次域K的一个判别式D < - 4的阶数。如果素数p在与R相关的环类域中被完全分解，则E在K除以p的素数理想p处具有良好的约简性，并且存在正整数u和υ使得4p = u2 - Du；众所周知，E模p约化的Frobenius自同态的迹ap的绝对值等于u。我们确定了当类数R为2或3，D可被3、4或5整除时，ap = u还是ap = - u。
Let E be an elliptic curve with complex multiplication by R, where R is an order of discriminant D < −4 of an imaginary quadratic field K. If a prime number p is decomposed completely in the ring class field associated with R, then E has good reduction at a prime ideal p of K dividing p and there exist positive integers u and υ such that 4p = u2 – Du;2. It is well known that the absolute value of the trace ap of the Frobenius endomorphism of the reduction of E modulo p is equal to u. We determine whether ap = u or ap = −u in the case where the class number of R is 2 or 3 and D is divisible by 3, 4 or 5.