Abstract
In the first section the Weil conjectures for non-singular primals are stated and several examples are given. Particularities for curves are described in section two. The remaining sections are devoted to elliptic cubic curves. In particular, the number of points that a cubic can have is precisely given, as well as the number of inequivalent curves with a fixed number of points.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H. Davenport, The Higher Arithmetic. 4th edition (Hutchinson, 1970).
R. De Groote and J.W.P. Hirschfeld, The number of points on an elliptic cubic curve over a finite field, Europ. J. Combin. 1 (1980), 327–333.
P. Deligne, La conjecture de Weil, I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307.
B.M. Dwork, On the rationality of the zeta function of an algebraic variety, Amer. J. Math. 82 (1960), 631–648.
H. Hasse, Abstrakte Begründung der komplexen Multiplikation und Riemannsche Vermutung in Funktionkörpern, Abh. Math. Sem. Univ. Hamburg 10 (1934), 325–348.
J.W.P. Hirschfeld, Classical configurations over finite fields: I. The double-six and the cubic surface with 27 lines, Rend. Mat. e Appl. 26 (1967), 115–152.
J.W.P. Hirschfeld, Projective Geometries over Finite Fields. (Oxford University Press, 1979).
N.M. Katz, An overview of Deligne's proof of the Riemann hypothesis for varieties over finite fields, Proc. Sympos. Pure Math. 28 (1976), 275–306.
N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta Functions. (Springer, 1977).
B. Mazur, Eigenvalues of Frobenius acting on algebraic varieties over finite fields, Proc. Sympos. Pure Math. 29 (1975), 231–261.
W.M. Schmidt, Equations over Finite Fields, an Elementary Approach. (Lecture Notes in Mathematics 536, Springer, 1976).
R.J. Schoof, Unpublished manuscript.
J.T. Tate, The arithmetic of elliptic curves, Invent. Math. 23 (1974), 179–206.
E. Ughi, On the number of points of elliptic curves over a finite field and a problem of B. Segre, Europ. J. Combin., to appear.
W.C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. École Norm. Sup. 2 (1969), 521–560.
A Weil, Sur les Courbes Algébriques et les Variétés qui s'en Déduisent. (Hermann, 1948).
A. Weil, Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55 (1949), 497–508.
Editor information
Rights and permissions
Copyright information
© 1983 Springer-Verlag
About this paper
Cite this paper
Hirschfeld, J.W.P. (1983). The weil conjectures in finite geometry. In: Casse, L.R.A. (eds) Combinatorial Mathematics X. Lecture Notes in Mathematics, vol 1036. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0071506
Download citation
DOI: https://doi.org/10.1007/BFb0071506
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-12708-6
Online ISBN: 978-3-540-38694-0
eBook Packages: Springer Book Archive