scholarly journals Counting Rational Points of an Algebraic Variety over Finite Fields

2019 ◽  
Vol 74 (1) ◽  
Author(s):  
Shuangnian Hu ◽  
Xiaoer Qin ◽  
Junyong Zhao
Keyword(s):  
Finite Fields ◽  
Algebraic Variety ◽  
Rational Points ◽  
Counting Rational Points

Introduction, main results, context

2011 ◽  
Author(s):  
Bas Edixhoven
Keyword(s):  
Finite Fields ◽  
Polynomial Time ◽  
Rational Points ◽  
Tau Function ◽  
Etale Cohomology ◽  
Polynomial Time Algorithms ◽  
Étale Cohomology ◽  
The Subject ◽  
Counting Rational Points

This chapter provides an introduction to the subject, precise statements of the main results, and places these in a somewhat wider context. Topics discussed include statement of the main results, Schoof's algorithm, Schoof's algorithm described in terms of ètale cohomology, other cases where ètale cohomology can be used to construct polynomial time algorithms for counting rational points of varieties over finite fields, congruences for Ramanujan's tau-function, and comparison with p-adic methods.

scholarly journals Construction of elliptic curves with cyclic groups over prime fields

2006 ◽  
Vol 73 (2) ◽  
pp. 245-254 ◽  
Author(s):  
Naoya Nakazawa
Keyword(s):  
Elliptic Curves ◽  
Finite Fields ◽  
Function Field ◽  
Modular Group ◽  
Modular Function ◽  
Invariant Function ◽  
Rational Points ◽  
Modular Invariant ◽  
Cyclic Groups ◽  
Prime Fields

The purpose of this article is to construct families of elliptic curves E over finite fields F so that the groups of F-rational points of E are cyclic, by using a representation of the modular invariant function by a generator of a modular function field associated with the modular group Γ0(N), where N = 5, 7 or 13.

On the curve Yn = Xℓ(Xm + 1) over finite fields

2019 ◽  
Vol 19 (2) ◽  
pp. 263-268 ◽  
Author(s):  
Saeed Tafazolian ◽  
Fernando Torres
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Plane Curve ◽  
Rational Points ◽  
Weil Bound

Abstract Let 𝓧 be the nonsingular model of a plane curve of type yn = f(x) over the finite field F of order q2, where f(x) is a separable polynomial of degree coprime to n. If the number of F-rational points of 𝓧 attains the Hasse–Weil bound, then the condition that n divides q+1 is equivalent to the solubility of f(x) in F; see [20]. In this paper, we investigate this condition for f(x) = xℓ(xm+1).

scholarly journals Counting rational points on quartic del Pezzo surfaces with a rational conic

2018 ◽  
Vol 373 (3-4) ◽  
pp. 977-1016
Author(s):  
T. D. Browning ◽  
E. Sofos
Keyword(s):  
Rational Points ◽  
Del Pezzo Surfaces ◽  
Del Pezzo ◽  
Counting Rational Points

scholarly journals Counting rational points on cubic curves

2010 ◽  
Vol 53 (9) ◽  
pp. 2259-2268 ◽  
Author(s):  
Roger Heath-Brown ◽  
Damiano Testa
Keyword(s):  
Rational Points ◽  
Cubic Curves ◽  
Counting Rational Points

scholarly journals Congruences for rational points on varieties over finite fields

2005 ◽  
Vol 333 (4) ◽  
pp. 797-809 ◽  
Author(s):  
N. Fakhruddin ◽  
C. S. Rajan
Keyword(s):  
Finite Fields ◽  
Rational Points
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