scholarly journals Computing Zeta Functions of Hyperelliptic Curves over Finite Fields of Characteristic 2

2002 ◽  
pp. 369-384 ◽  
Author(s):  
Frederik Vercauteren
Keyword(s):  
Finite Fields ◽  
Zeta Functions ◽  
Hyperelliptic Curves ◽  
Curves Over Finite Fields ◽  
Characteristic 2

scholarly journals Computing Zeta Functions of Artin–schreier Curves over Finite Fields

2002 ◽  
Vol 5 ◽  
pp. 34-55 ◽  
Author(s):  
Alan G. B. Lauder ◽  
Daqing Wan
Keyword(s):  
Finite Fields ◽  
Hyperelliptic Curve ◽  
Exponential Sums ◽  
Reduction Method ◽  
Zeta Functions ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Efficient Reduction ◽  
Curves Over Finite Fields ◽  
Characteristic 2

AbstractThe authors present a practical polynomial-time algorithm for computing the zeta function of certain Artin–Schreier curves over finite fields. This yields a method for computing the order of the Jacobian of an elliptic curve in characteristic 2, and more generally, any hyperelliptic curve in characteristic 2 whose affine equation is of a particular form. The algorithm is based upon an efficient reduction method for the Dwork cohomology of one-variable exponential sums.

scholarly journals Moments of zeta functions associated to hyperelliptic curves over finite fields

2015 ◽  
Vol 373 (2040) ◽  
pp. 20140307 ◽  
Author(s):  
Michael O. Rubinstein ◽  
Kaiyu Wu
Keyword(s):  
Asymptotic Formula ◽  
Finite Fields ◽  
Remainder Term ◽  
Prime Power ◽  
Zeta Functions ◽  
Number Fields ◽  
Hyperelliptic Curves ◽  
Central Point ◽  
Characteristic Polynomials ◽  
Curves Over Finite Fields

Let q be an odd prime power, and denote the set of square-free monic polynomials D ( x )∈ F q [ x ] of degree d . Katz and Sarnak showed that the moments, over , of the zeta functions associated to the curves y 2 = D ( x ), evaluated at the central point, tend, as , to the moments of characteristic polynomials, evaluated at the central point, of matrices in USp (2⌊( d −1)/2⌋). Using techniques that were originally developed for studying moments of L -functions over number fields, Andrade and Keating conjectured an asymptotic formula for the moments for q fixed and . We provide theoretical and numerical evidence in favour of their conjecture. In some cases, we are able to work out exact formulae for the moments and use these to precisely determine the size of the remainder term in the predicted moments.

A Note on Relations Between the Zeta-Functions of Galois Coverings of Curves Over Finite Fields

1990 ◽  
Vol 33 (3) ◽  
pp. 282-285 ◽  
Author(s):  
Amilcar Pacheco
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Group Ring ◽  
Algebraic Curve ◽  
Zeta Functions ◽  
Finite Subgroup ◽  
Rational Group ◽  
Galois Coverings ◽  
Curves Over Finite Fields ◽  
Special Case

AbstractLet C be a complete irreducible nonsingular algebraic curve defined over a finite field k. Let G be a finite subgroup of the group of automorphisms Aut(C) of C. We prove that certain idempotent relations in the rational group ring Q[G] imply other relations between the zeta-functions of the quotient curves C/H, where H is a subgroup of G. In particular we generalize some results of Kani in the special case of curves over finite fields.

Sign in / Sign up

Export Citation Format

Share Document