scholarly journals Lefschetz classes of simple factors of the Jacobian variety of a Fermat curve of prime degree over finite fields

2013 ◽  
Vol 133 (4) ◽  
pp. 1194-1206 ◽  
Author(s):  
Rin Sugiyama
Keyword(s):  
Finite Fields ◽  
Jacobian Variety ◽  
Fermat Curve ◽  
Prime Degree

Rational points on Fermat curves over finite fields

2017 ◽  
Vol 16 (03) ◽  
pp. 1750046
Author(s):  
Wei Cao ◽  
Shanmeng Han ◽  
Ruyun Wang
Keyword(s):  
Finite Fields ◽  
Rational Point ◽  
Rational Points ◽  
Fermat Curve ◽  
Fermat Curves ◽  
Curves Over Finite Fields

Let [Formula: see text] be the [Formula: see text]-rational point on the Fermat curve [Formula: see text] with [Formula: see text]. It has recently been proved that if [Formula: see text] then each [Formula: see text] is a cube in [Formula: see text]. It is natural to wonder whether there is a generalization to [Formula: see text]. In this paper, we show that the result cannot be extended to [Formula: see text] in general and conjecture that each [Formula: see text] is a cube in [Formula: see text] if and only if [Formula: see text].

Fermat Jacobians of Prime Degree over Finite Fields

1999 ◽  
Vol 42 (1) ◽  
pp. 78-86 ◽  
Author(s):  
Josep González
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Algebraic Closure ◽  
Cyclotomic Field ◽  
Roots Of Unity ◽  
Characteristic P ◽  
Prime Degree ◽  
Numerical Criterion

AbstractWe study the splitting of Fermat Jacobians of prime degree l over an algebraic closure of a finite field of characteristic p not equal to l. We prove that their decomposition is determined by the residue degree of p in the cyclotomic field of the l-th roots of unity. We provide a numerical criterion that allows to compute the absolutely simple subvarieties and their multiplicity in the Fermat Jacobian.

Computing Vf modulo p

2011 ◽  
Author(s):  
Jean-Marc Couveignes
Keyword(s):  
Finite Fields ◽  
Algebraic Curves ◽  
Las Vegas ◽  
Jacobian Variety ◽  
Generating Sets ◽  
Monte Carlo Algorithms ◽  
Modulo P ◽  
Curves Over Finite Fields ◽  
Las Vegas Algorithm ◽  
Probabilistic Polynomial Time

This chapter addresses the problem of computing in the group of lsuperscript k-torsion rational points in the Jacobian variety of algebraic curves over finite fields, with an application to computing modular representations. An algorithm in this chapter usually means a probabilistic Las Vegas algorithm. In some places it gives deterministic or probabilistic Monte Carlo algorithms, but this will be stated explicitly. The main reason for using probabilistic Turing machines is that there is a need to construct generating sets for the Picard group of curves over finite fields. Solving such a problem in the deterministic world is out of reach at this time. The unique goal is to prove, as quickly as possible, that the problems studied in this chapter can be solved in probabilistic polynomial time.

ON THE NUMBER OF RATIONAL POINTS OF GENERALIZED FERMAT CURVES OVER FINITE FIELDS

2012 ◽  
Vol 08 (04) ◽  
pp. 1087-1097 ◽  
Author(s):  
STEFANIA FANALI ◽  
MASSIMO GIULIETTI
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Direct Product ◽  
Finite Fields ◽  
Classical Problem ◽  
Rational Points ◽  
Fermat Curve ◽  
Cyclic Groups ◽  
Fermat Curves ◽  
Curves Over Finite Fields

The Stöhr–Voloch approach has been largely used to deal with the classical problem of estimating the number of rational points of a Fermat curve over a finite field. The same method actually applies to any curve admitting as an automorphism group the direct product of two cyclic groups C1 and C2 of the same size k, and such that the quotient curves with respect to both C1 and C2 are rational. In this paper such a curve is called a generalized Fermat curve. Our main achievement is that of extending some known results on Fermat curves to generalized Fermat curves.

scholarly journals RATIONAL POINTS ON SOME FERMAT CURVES AND SURFACES OVER FINITE FIELDS

2014 ◽  
Vol 10 (02) ◽  
pp. 319-325 ◽  
Author(s):  
JOSÉ FELIPE VOLOCH ◽  
MICHAEL E. ZIEVE
Keyword(s):  
Finite Fields ◽  
Rational Points ◽  
Explicit Description ◽  
Fermat Curve ◽  
Curves And Surfaces ◽  
Fermat Curves

We give an explicit description of the 𝔽qi-rational points on the Fermat curve uq-1 + vq-1 + wq-1 = 0, for i ∈{1, 2, 3}. As a consequence, we observe that for any such point (u, v, w), the product uvw is a cube in 𝔽qi. We also describe the 𝔽q2-rational points on the Fermat surface uq-1 + vq-1 + wq-1 + xq-1 = 0, and show that the product of the coordinates of any such point is a square.

scholarly journals Generalized Jacobian Varieties and Separable Abelian Extensions of Function Fields

1957 ◽  
Vol 12 ◽  
pp. 231-254 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Field Theory ◽  
Finite Fields ◽  
Function Fields ◽  
Abelian Extension ◽  
Jacobian Variety ◽  
Generalized Jacobian ◽  
Abelian Extensions ◽  
Jacobian Varieties ◽  
Several Variables ◽  
Class Fields

Using Frobenius automorphisms ingeniouslly, S. Lang has established an elegant theory of unramified class fields of function fields in several variables over finite fields [2]. As an application of class field theory and theory of reduction he has proved that any separable unramified abelian extension of a function field of one variable comes from a pull back of a separable ingeny of its jacobian variety [3].

scholarly journals A remark on the decomposition of the Jacobian variety of Fermat curves of prime degree

2015 ◽  
Vol 105 (4) ◽  
pp. 333-341 ◽  
Author(s):  
Ruben A. Hidalgo ◽  
Rubí E. Rodríguez
Keyword(s):  
Jacobian Variety ◽  
Prime Degree ◽  
Fermat Curves

Finite Fields

1996 ◽  
Author(s):  
Rudolf Lidl ◽  
Harald Niederreiter
Keyword(s):  
Finite Fields
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