In this paper of 1988 N. Koblitz conjectured that given an elliptic curve over the rationals, the order of the group of points of its reduction modulo , , is a prime number for infinitely many primes . Since then a wide number of research articles has been dedicated to understand and solve this conjecture. In this paper we give the best result known nowadays. We can prove quantitatively that for infinitely many primes the reduction of the curve modulo has order which is eight times an almost prime number. The problem turns out to be the equivalent to the twin prime conjecture in the Gaussian domain. The result could be extended to any CM curve with certain considerations. We also point out the relation of the result with certain considerations. We also point out the relation of the result with the cyclicity of , and the Lang Trotter conjecture.
@article{ASNSP_2010_5_9_4_815_0, author = {Iwaniec, Henryk and Urroz, Jorge Jim\'enez}, title = {Orders of {CM} elliptic curves modulo $p$ with at most two primes}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {815--832}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 9}, number = {4}, year = {2010}, zbl = {1211.11106}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2010_5_9_4_815_0/} }
TY - JOUR AU - Iwaniec, Henryk AU - Urroz, Jorge Jiménez TI - Orders of CM elliptic curves modulo $p$ with at most two primes JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2010 SP - 815 EP - 832 VL - 9 IS - 4 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2010_5_9_4_815_0/ LA - en ID - ASNSP_2010_5_9_4_815_0 ER -
%0 Journal Article %A Iwaniec, Henryk %A Urroz, Jorge Jiménez %T Orders of CM elliptic curves modulo $p$ with at most two primes %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2010 %P 815-832 %V 9 %N 4 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2010_5_9_4_815_0/ %G en %F ASNSP_2010_5_9_4_815_0
Iwaniec, Henryk; Urroz, Jorge Jiménez. Orders of CM elliptic curves modulo $p$ with at most two primes. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 4, pp. 815-832. http://www.numdam.org/item/ASNSP_2010_5_9_4_815_0/
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