Anomalous primes of the elliptic curve ED: y2*x3+D
Anomalous primes of the elliptic curve ED: y2*x3+D
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DOI：
10.1112/plms/pdv072
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发表时间：
2016-02
影响因子：
1.8
通讯作者：
H. Qin
中科院分区：
文献类型：
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作者：
H. Qin
设D∈Z是Q（−3）中既不是正方形也不是立方的整数，艾德是由y 2 = x 3 +D定义的椭圆曲线。Mazur指出，小于N的反常素数的个数应渐近地由cN/logN给出（c为正常数），特别是对于艾德，反常素数应有无穷多个.我们证明了Hardy-Littlewood猜想蕴涵Mazur猜想，除了D= 80 d 6，其中0 <$d∈Z[（1+−3）/2]，d 6 ∈Z。相反，如果Mazur猜想对某个D成立，则多项式12 x2 +18x+7表示无穷多个素数。所有的反常素数都属于二次级数q（h）=14（1+ 3 h ~ 2）。在Hardy-Littlewood猜想的假设下，我们得到了任意D的q（h）中素数的反常素数的密度.在某些情况下，密度是16，正如Mazur所指出的那样，但它对所有D都不成立。我们的结果更一般。事实上，我们将考虑所有属于q（h）的六种类型的素数，而不仅仅是反常素数。所有这些素数的密度结果。我们还讨论了艾德的Lang-Trotter猜想。
Let D∈Z be an integer that is neither a square nor a cube in Q(−3), and let ED be the elliptic curve defined by y2=x3+D. Mazur conjectured that the number of anomalous primes less than N should be given asymptotically by cN/logN ( c is a positive constant), and in particular there should be infinitely many anomalous primes for ED . We show that the Hardy–Littlewood conjecture implies the Mazur conjecture, except for D=80d6 , where 0≠d∈Z[(1+−3)/2] with d6∈Z. Conversely, if the Mazur conjecture holds for some D , then the polynomial 12x2+18x+7 represents infinitely many primes. All anomalous primes belong to the quadratic progression q(h)=14(1+3h2) . Assuming the Hardy–Littlewood conjecture, we obtain the density of the anomalous primes in the primes in q(h) for any D . The density is 16 in some cases, as Mazur had conjectured, but it fails to be true for all D . Our results are more general. In fact, we will consider all primes of six types which belong to q(h) , not just anomalous primes. The density results are established for all these primes. We also discuss the Lang–Trotter conjecture for ED .