Abstract
We provide a historical background to the congruence zeta function and the Weil conjectures for non-singular projective curves defined over finite fields. We state these conjectures, and also the more recent Weil theorem for singular curves defined over finite fields. We end by remarking on some explicit results we have obtained for the zeta functions of some concrete classes of curves (both non-singular and singular) defined over a certain class of finite fields.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
N. Anuradha and S. A. Katre, Number of points on the projective curves aYl = bXl + cZland aY2l = bX2l + cZ2ldefined over finite fields, l an odd prime, J. Number Theory 77 (1999), 288–313.
N. Anuradha, Zeta function of the projective curve aY21 = bX2l+cZ2lover a class of finite fields, for odd primes l, Indian J. Math. to appear.
N. Anuradha, Jacobsthal sums and explicit zeta functions for certain hyper-elliptic curves defined over a class of finite fields, preprint.
E. Artin, Quadratische Körper im Gebiet der höheren Kongruenzen I, II, Math. Z. 19 (1924), 153–206, 207–246.
Y. Aubry and M. Perret, A Weil theorem for singular curves, Proc. Conf. at Institut de Mathématique de Luminy, Marseille-Luminy (1993), de Gruyter, Berlin, 1996, 1–7.
E. Bombieri, Counting points on curves over finite fields (d’après S. A, Stepanov), Séminaire Bourbaki 1972/73, Exp. 430, Lecture Notes in Math., Vol. 383, pp. 234–241, Springer-Verlag, Berlin-Heidelberg-New York, 1974.
B. Dwork, On the rationality of the zeta function of an algebraic variety, Amer. J. Math. 82 (1960), 631–648.
H. Hasse, Zur Theorie der abstrakten elliptischen Funktionenkörper I, II, III, 3. Reine Angew. Math. 175 (1936), 55–62, 69–88, 193–208.
J. Igusa, On the theory of algebraic correspondences and its application to the Riemann hypothesis in function fields, J. Math. Soc. Japan 1 (1949), 147–197.
C. Moreno, Algebraic Curves over Finite Fields, Cambridge Tracts in Mathematics, Vol. 97, Cambridge Univ. Press, Cambridge, MA, 1991.
P. Roquette, Riemannsche Vermutung in Funktionenkörpern, Arch. Math. 4 (1953), 6–16.
P. Roquette, Arithmetischer Beweis der Riemannschen Vermutung in Kongruenzfunktionenkörpern beliebigen Geschlechts, 3. Reine Angew. Math. 191 (1953), 199–252.
F. K. Schmidt, Zur Zahlentheorie in Körpern von der Charakteristik p, Sitzungsber. Phys.-Med. Soz. Erlangen 58/59 (1926/27), 159–172.
F. K. Schmidt, Analytische Zahlentheorie in Körpern der Charakteristik p, Math. Z. 33 (1931), 1–32.
W. M. Schmidt, Equations over Finite Fields: An Elementary Approach, Lecture Notes in Math., Vol. 536, Springer-Verlag, Berlin-Heidelberg-New York, 1976.
A. Weil, Sur les courbes algébriques et les variétés qui s’en déduisent, Actualités Sci. Ind., No. 1041, Hermann, Paris, 1948.
A. Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 (1949), 497–508.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Hindustan Book Agency
About this chapter
Cite this chapter
Narasimhan, A. (2002). Zeta Functions for Curves Defined over Finite Fields. In: Adhikari, S.D., Katre, S.A., Ramakrishnan, B. (eds) Current Trends in Number Theory. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-09-5_14
Download citation
DOI: https://doi.org/10.1007/978-93-86279-09-5_14
Publisher Name: Hindustan Book Agency, Gurgaon
Print ISBN: 978-81-85931-33-3
Online ISBN: 978-93-86279-09-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)