scholarly journals Counting points on elliptic curves over finite fields

1995 ◽  
Vol 7 (1) ◽  
pp. 219-254 ◽  
Author(s):  
René Schoof
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Curves Over Finite Fields ◽  
Counting Points

scholarly journals The characterization of elliptic curves over finite fields

1988 ◽  
Vol 45 (2) ◽  
pp. 275-286 ◽  
Author(s):  
J. W. P. Hirschfeld ◽  
J. F. Voloch
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Sufficient Conditions ◽  
Maximum Size ◽  
Desarguesian Plane ◽  
Cubic Curve ◽  
Odd Order ◽  
Even Order ◽  
Curves Over Finite Fields

AbstractIn a finite Desarguesian plane of odd order, it was shown by Segre thirty years ago that a set of maximum size with at most two points on a line is a conic. Here, in a plane of odd or even order, sufficient conditions are given for a set with at most three points on a line to be a cubic curve. The case of an elliptic curve is of particular interest.

scholarly journals THE AVERAGE NUMBER OF SUBGROUPS OF ELLIPTIC CURVES OVER FINITE FIELDS

2020 ◽  
Vol 71 (3) ◽  
pp. 781-822
Author(s):  
Corentin Perret-Gentil
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Short Interval ◽  
Divisor Function ◽  
Finite Abelian Groups ◽  
Cyclic Subgroups ◽  
Number Of Divisors ◽  
Curves Over Finite Fields ◽  
Euler Products

Abstract By adapting the technique of David, Koukoulopoulos and Smith for computing sums of Euler products, and using their interpretation of results of Schoof à la Gekeler, we determine the average number of subgroups (or cyclic subgroups) of an elliptic curve over a fixed finite field of prime size. This is in line with previous works computing the average number of (cyclic) subgroups of finite abelian groups of rank at most $2$. A required input is a good estimate for the divisor function in both short interval and arithmetic progressions, that we obtain by combining ideas of Ivić–Zhai and Blomer. With the same tools, an asymptotic for the average of the number of divisors of the number of rational points could also be given.

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