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. **372** (2019), 5409-5451

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 over a finite field, starting from the Weil classical bound, continuing to the Ihara bound, passing through infinitely many *-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 can be merely seen as a euclidean property coming from the Toeplitz shape of some intersection matrix on the surface 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 of the numerical group 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 .

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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