scholarly journals On the characteristic polynomials of the Frobenius endomorphism for projective curves over finite fields

2004 ◽  
Vol 10 (3) ◽  
pp. 412-431 ◽  
Author(s):  
Yves Aubry ◽  
Marc Perret
Keyword(s):  
Finite Fields ◽  
Characteristic Polynomials ◽  
Frobenius Endomorphism ◽  
Projective Curves ◽  
Curves Over Finite Fields

scholarly journals Derangements in cosets of primitive permutation groups

2015 ◽  
Vol 18 (1) ◽  
Author(s):  
Andrei Pavelescu
Keyword(s):  
Fixed Point ◽  
Finite Fields ◽  
Permutation Groups ◽  
Normal Subgroups ◽  
Primitive Groups ◽  
Branched Coverings ◽  
Affine Groups ◽  
Nonabelian Subgroup ◽  
Projective Curves ◽  
Curves Over Finite Fields

AbstractMotivated by questions arising in connection with branched coverings of connected smooth projective curves over finite fields, we study the proportion of fixed-point free elements (derangements) in cosets of normal subgroups of primitive permutations groups. Using the Aschbacher–O'Nan–Scott Theorem for primitive groups to partition the problem, we provide complete answers for affine groups and groups which contain a regular normal nonabelian subgroup.

scholarly journals ON THE MERTENS CONJECTURE FOR ELLIPTIC CURVES OVER FINITE FIELDS

2013 ◽  
Vol 89 (1) ◽  
pp. 19-32
Author(s):  
PETER HUMPHRIES
Keyword(s):  
Finite Field ◽  
Elliptic Curves ◽  
Finite Fields ◽  
Frobenius Endomorphism ◽  
Curves Over Finite Fields

AbstractWe introduce an analogue of the Mertens conjecture for elliptic curves over finite fields. Using a result of Waterhouse, we classify the isogeny classes of elliptic curves for which this conjecture holds in terms of the size of the finite field and the trace of the Frobenius endomorphism acting on the curve.

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.

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