## 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 $$R_E$$.

### 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

Zbl 0329.12015
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