Anomalous primes of the elliptic curve E-D : y(2) = x(3) + D

作者：Qin, Hourong^*

来源：Proceedings of the London Mathematical Society, 2016, 112(2): 415-453.

DOI：10.1112/plms/pdv072

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Let D is an element of Z be an integer that is neither a square nor a cube in Q(root-3), and let E-D be the elliptic curve defined by y(2) = x(3) + D. Mazur conjectured that the number of anomalous primes less than N should be given asymptotically by c root N/logN (c is a positive constant), and in particular there should be infinitely many anomalous primes for E-D. We show that the Hardy-Littlewood conjecture implies the Mazur conjecture, except for D = 80d(6), where 0 not equal d is an element of Z[(1 + root-3)/2] with d(6) is an element of Z. Conversely, if the Mazur conjecture holds for some D, then the polynomial 12x(2) + 18x + 7 represents infinitely many primes. All anomalous primes belong to the quadratic progression q(h) = 1/4 (1 + 3h(2)). 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 1/6 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 E-D.

• 出版日期2016-2
• 单位南京大学