 
Summary: Finite Fields and Their Applications 10 (2004) 412431
On the characteristic polynomials of the
Frobenius endomorphism for projective curves
over finite fields
Yves Aubrya,Ã,1
and Marc Perretb
a
Laboratoire de Mathe´matiques Nicolas Oresme, CNRSUMR 6139, Universite´ de Caen,
14032 Caen Cedex, France
b
GRIMM, Universite´ de Toulouse 2Le Mirail, 5, alle´es A. Machado, 31058 Toulouse, France
Received 13 November 2002; revised 8 September 2003
Communicated by Michael Tsfasman
Abstract
We give a formula for the number of rational points of projective algebraic curves defined
over a finite field, and a bound ``a` la Weil'' for connected ones. More precisely, we give the
characteristic polynomials of the Frobenius endomorphism on the e´ tale cadic cohomology
groups of the curve. Finally, as an analogue of Artin's holomorphy conjecture, we prove that,
if YÀ!X is a finite flat morphism between two varieties over a finite field, then the
characteristic polynomial of the Frobenius morphism on Hi
