scholarly journals Estimates for the number of rational points on simple abelian varieties over finite fields

2020 ◽  
Author(s):  
Borys Kadets
Keyword(s):  
Finite Fields ◽  
Abelian Varieties ◽  
Rational Points

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.

Abelian Varieties over Finite Fields

2000 ◽  
pp. 63-65
Author(s):  
Serge Lang
Keyword(s):  
Finite Fields ◽  
Abelian Varieties

scholarly journals Abelian varieties over finite fields

1969 ◽  
Vol 2 (4) ◽  
pp. 521-560 ◽  
Author(s):  
William C. Waterhouse
Keyword(s):  
Finite Fields ◽  
Abelian Varieties

scholarly journals Galois criterion for torsion points of Drinfeld modules

2021 ◽  
pp. 1-12
Author(s):  
Chien-Hua Chen
Keyword(s):  
Maximal Ideal ◽  
Abelian Varieties ◽  
Number Fields ◽  
Rational Points ◽  
Drinfeld Module ◽  
Torsion Point ◽  
Drinfeld Modules ◽  
Torsion Points ◽  
Ideal Formula ◽  
Almost All

In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal [Formula: see text] of [Formula: see text], the question essentially asks whether, up to isogeny, a Drinfeld module [Formula: see text] over [Formula: see text] contains a rational [Formula: see text]-torsion point if the reduction of [Formula: see text] at almost all primes of [Formula: see text] contains a rational [Formula: see text]-torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2, but negative if the rank is 3. Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive.

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 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
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