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**CM elliptic curves and primes captured by quadratic polynomials.**
*(English)*
Zbl 1315.11048

For an elliptic curve \(E\) defined over \(\mathbb Q\), let \(p\) be a prime number where \(E\) has good ordinary reduction, and \(a_p(E)\) the trace of Frobenius endomorphism of the reduction of \(E\) at \(p\). For an integer \(r\), let \(\Sigma_E^{(r)}\) be the set of prime numbers \(p\) such that \( a_p(E)\equiv r \mod p\). The authors study the existence of an elliptic curve such that the set \(\Sigma_E^{(r)}\) is an infinite set for a given \(r\). For elliptic curves \(E\) defined over \(\mathbb Q\) with complex multiplication \(R_E\), they determine the value of \(a_p(E) \mod p\). For some of them, they characterize anomalous primes by using a twist. Further, in other cases they obtain the value \(a_p(E) \mod p\) by using a binomial coefficient. Such a value determines \(a_p(E)\) by the Hasse inequality except in a few small \(p\). On the other hand, since the order \(R_E\) is of class number one and \(p\) is unramified, \(a_p(E)\) is the trace of a suitably chosen prime element \(\pi\) of \(R_E\) dividing \(p\) and \(p\) is the norm of \(\pi\). The choice of \(\pi\) is determined by some congruence condition, for example, the value of \(a_p(E)\mod p\). Using those facts, they show for certain \(E\) and \(r\) that the Lang-Trotter conjecture concerning the distribution of prime numbers in the set \(\Sigma_E^{(r)}\) [S. Lang and H. Trotter, Frobenius distributions in \(\mathrm{GL}_2\)-extensions. Distribution of Frobenius automorphisms in \(\mathrm{GL}_2\)-extensions of the rational numbers. York: Springer-Verlag (1976; Zbl 0329.12015)] is equivalent to the Hardy-Littlewood conjecture concerning that of prime numbers represented by a quadratic polynomial deduced from the norm of \(R_E\).

Reviewer: Noburo Ishii (Kyoto)

### MSC:

11G05 | Elliptic curves over global fields |

11G15 | Complex multiplication and moduli of abelian varieties |

11N32 | Primes represented by polynomials; other multiplicative structures of polynomial values |