Geodesic
The Square Sieve and the Lang–Trotter Conjecture
Canadian journal of mathematics, Tome 57 (2005) no. 6, pp. 1155-1177

Voir la notice de l'article provenant de la source Cambridge University Press

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and without complex multiplication. Let $K$ be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes $p\,\le \,x$ for which $\mathbb{Q}\left( {{\pi }_{p}} \right)\,=\,K$ , where ${{\pi }_{p}}$ denotes the Frobenius endomorphism of $E$ at $p$ . More precisely, under a generalized Riemann hypothesis we show that this number is ${{O}_{E}}\left( {{x}^{17/18}}\,\log x \right)$ , and unconditionally we show that this number is ${{O}_{E,K}}\left( \frac{x{{\left( \log \,\log x \right)}^{13/12}}}{{{\left( \log x \right)}^{25/24}}} \right)$ We also prove that the number of imaginary quadratic fields $K$ , with − disc $K\,\le \,x$ and of the form $K\,=\,\mathbb{Q}({{\pi }_{p}})$ , is ${{\gg }_{E}}\,\log \,\log \,\log \,x$ for $x\,\ge \,{{x}_{0}}\left( E \right)$ . These results represent progress towards a 1976 Lang–Trotter conjecture.
DOI : 10.4153/CJM-2005-045-7
Mots-clés : Primary: 11G05, secondary: 11N36, 11R45, Elliptic curves modulo p, Lang–Trotter conjecture, applications of sieve methods 
Cojocaru, Alina Carmen; Fouvry, Etienne; Murty, M. Ram. The Square Sieve and the Lang–Trotter Conjecture. Canadian journal of mathematics, Tome 57 (2005) no. 6, pp. 1155-1177. doi: 10.4153/CJM-2005-045-7
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