On the distribution of supersingular primes. (English) Zbl 0864.11030

Fouvry, Etienne; Murty, M. Ram

Can. J. Math. 48, No. 1, 81-104 (1996).

Let \(E/ \mathbb{Q}\) be an elliptic curve which does not have complex multiplication, and let \(\pi_0(x)\) be the number of primes \(p\leq x\) such that \(E\) is supersingular at \(p\). It was shown by Elkies that \(\pi_0(x) \to\infty\) as \(x\to\infty\), but the asymptotic behaviour of \(\pi_0(x)\) is not known.
In this paper the authors show that \[ \pi_0(x)> {\log_3x \over (\log_4x)^{1+ \delta}} \] for any \(\delta>0\) and \(x\) sufficiently large \((\log_kx\) is the \(k\)-fold iterated logarithm). Lang and Trotter conjectured the asymptotic behaviour of \(\pi_0(x)\). In the second part of the paper the authors consider a family of elliptic curves and prove an “average version” of the Lang-Trotter conjecture.

Reviewer: H.-G.Rück (Essen)

MSC:

11G05	Elliptic curves over global fields
11N05	Distribution of primes

Keywords:

supersingular primes; elliptic curve; asymptotic behaviour; Lang-Trotter conjecture