TY - JOUR
T1 - On group structures realized by elliptic curves over a fixed finite field
AU - Farashahi, Reza Rezaeian
AU - Shparlinski, Igor E.
PY - 2012
Y1 - 2012
N2 - We obtain explicit formulas for the number of nonisomorphic elliptic curves with a given group structure (considered as an abstract abelian group) and the number of distinct group structures of all elliptic curves over a finite field. We use these formulas to derive some asymptotic estimates and tight upper and lower bounds for various counting functions related to classification of elliptic curves according to their group structure. Finally, we present results of some numerical tests that exhibit several interesting phenomena in the distribution of group structures.
AB - We obtain explicit formulas for the number of nonisomorphic elliptic curves with a given group structure (considered as an abstract abelian group) and the number of distinct group structures of all elliptic curves over a finite field. We use these formulas to derive some asymptotic estimates and tight upper and lower bounds for various counting functions related to classification of elliptic curves according to their group structure. Finally, we present results of some numerical tests that exhibit several interesting phenomena in the distribution of group structures.
UR - http://www.scopus.com/inward/record.url?scp=84874318646&partnerID=8YFLogxK
U2 - 10.1080/10586458.2012.632754
DO - 10.1080/10586458.2012.632754
M3 - Article
AN - SCOPUS:84874318646
VL - 21
SP - 1
EP - 10
JO - Experimental Mathematics
JF - Experimental Mathematics
SN - 1058-6458
IS - 1
ER -