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Transactions of the American Mathematical Society

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A unified viewpoint for upper bounds for the number of points of curves over finite fields via euclidean geometry and semi-definite symmetric Toeplitz matrices


Authors: Emmanuel Hallouin and Marc Perret
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11G20, 14G05, 14G15, 14H99, 15B05, 11M38
DOI: https://doi.org/10.1090/tran/7813
Published electronically: June 17, 2019
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Abstract: We provide an infinite sequence of upper bounds for the number of rational points of absolutely irreducible smooth projective curves $ X$ over a finite field, starting from the Weil classical bound, continuing to the Ihara bound, passing through infinitely many $ n$-th order Weil bounds, and ending asymptotically at the Drinfeld-Vlăduţ bound. We relate this set of bounds to those of Oesterlé, proving that these are inverse functions in some sense. We explain how the Riemann hypothesis for the curve $ X$ can be merely seen as a euclidean property coming from the Toeplitz shape of some intersection matrix on the surface $ X\times X$ together with the general theory of symmetric Toeplitz matrices. We also give some interpretation for the defect of asymptotically exact towers.

This is achieved by pushing further the classical Weil proof in terms of euclidean relationships between classes in the euclidean part  $ \mathcal {F}_X$ of the numerical group  $ \operatorname {Num}(X\times X)$ generated by classes of graphs of iterations of the Frobenius morphism. The noteworthy Toeplitz shape of their intersection matrix takes a central place by implying a very strong cyclic structure on  $ \mathcal {F}_X$.


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  • [1] Mihály Bakonyi and Hugo J. Woerdeman, Matrix completions, moments, and sums of Hermitian squares, Princeton University Press, Princeton, NJ, 2011. MR 2807419
  • [2] Pierre Fatou, Sur les séries entières à coefficients entiers, C. R. Acad. Sci. Paris Sér. A 138 (1904), 342-344.
  • [3] Søren Have Hansen, Rational points on curves over finite fields, Lecture Notes Series, vol. 64, Aarhus Universitet, Matematisk Institut, Aarhus, 1995. MR 1467481
  • [4] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • [5] Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1990. Corrected reprint of the 1985 original. MR 1084815
  • [6] Yasutaka Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 721–724 (1982). MR 656048
  • [7] Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232
  • [8] Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz; Translated from the French by Andreas Hartmann. MR 1864396
  • [9] Jean-Pierre Serre, Rational points on curves over finite fields, with lectures given at Harvard University (notes by F. Q. Gouvea).
  • [10] Henning Stichtenoth, Algebraic function fields and codes, Universitext, Springer-Verlag, Berlin, 1993. MR 1251961
  • [11] Michael A. Tsfasman, Some remarks on the asymptotic number of points, Coding theory and algebraic geometry (Luminy, 1991) Lecture Notes in Math., vol. 1518, Springer, Berlin, 1992, pp. 178–192. MR 1186424, https://doi.org/10.1007/BFb0088001
  • [12] M. A. Tsfasman and S. G. Vlăduţ, Infinite global fields and the generalized Brauer-Siegel theorem, Mosc. Math. J. 2 (2002), no. 2, 329–402. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. MR 1944510, https://doi.org/10.17323/1609-4514-2002-2-2-329-402
  • [13] S. G. Vlèduts and V. G. Drinfel′d, The number of points of an algebraic curve, Funktsional. Anal. i Prilozhen. 17 (1983), no. 1, 68–69 (Russian). MR 695100
  • [14] André Weil, Variétés abéliennes et courbes algébriques, Actualités Sci. Ind., no. 1064 = Publ. Inst. Math. Univ. Strasbourg 8 (1946), Hermann & Cie., Paris, 1948 (French). MR 0029522
  • [15] Oscar Zariski, Algebraic surfaces, Classics in Mathematics, Springer-Verlag, Berlin, 1995. With appendices by S. S. Abhyankar, J. Lipman and D. Mumford; Preface to the appendices by Mumford; Reprint of the second (1971) edition. MR 1336146

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Additional Information

Emmanuel Hallouin
Affiliation: Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UT2J, F-31058 Toulouse, France
Email: hallouin@univ-tlse2.fr

Marc Perret
Affiliation: Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UT2J, F-31058 Toulouse, France
Email: perret@univ-tlse2.fr

DOI: https://doi.org/10.1090/tran/7813
Keywords: Curves over a finite field, rational point, Weil bound, Toeplitz matrices, zeta function
Received by editor(s): January 7, 2018
Received by editor(s) in revised form: September 26, 2018
Published electronically: June 17, 2019
Additional Notes: This work was funded by ANR grant ANR-15-CE39-0013-01 “manta”
Article copyright: © Copyright 2019 American Mathematical Society