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September 2014 CM elliptic curves and primes captured by quadratic polynomials
Qingzhong Ji, Hourong Qin
Asian J. Math. 18(4): 707-726 (September 2014).

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with complex multiplication. For a prime $p$, some formulas for $a_p = p + 1 \sharp E(\mathbb{F}_p)$ are given in terms of the binomial coefficients. We show that the equality $a_p = r$ holds for some fixed integer $r$ if and only if a certain quadratic polynomial represents the prime $p$. In particular, for $E \colon y^2 = x^3 + x, a_p = 2$ holding for an odd prime $p$ if and only if $p$ is of the form $n^2 + 1$ and for $E \colon y^2 = x^3 - 11x + 14, a_p = 2$ holding for an odd prime $p$ if and only if $p$ is of the form $(4n)^2 + 1; a_p = -2$ holding for an odd prime $p$ if and only if $p$ is of the form $(4n + 2)^2 + 1$. In some CM cases the Lang-Trotter conjecture and the Hardy-Littlewood conjecture are equivalent.

Citation

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Qingzhong Ji. Hourong Qin. "CM elliptic curves and primes captured by quadratic polynomials." Asian J. Math. 18 (4) 707 - 726, September 2014.

Information

Published: September 2014
First available in Project Euclid: 6 November 2014

zbMATH: 1315.11048
MathSciNet: MR3275725

Subjects:
Primary: 11G05 , 11G15 , 11N32

Keywords: anomalous prime , CM elliptic curve , Hardy-Littlewood conjecture

Rights: Copyright © 2014 International Press of Boston

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Vol.18 • No. 4 • September 2014
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