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

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 $\mathcal {F}_X$.