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 X over a finite field, starting from the Weil classical bound, continuing to the Ihara bound, passing through infinitely many n 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 X can be merely seen as a euclidean property coming from the Toeplitz shape of some intersection matrix on the surface X × X 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 F X \mathcal {F}_X of the numerical group Num ⁡ ( X × X ) \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 F X \mathcal {F}_X .

Number of points of curves over finite fields in some relative situations from an euclidean point of vue

“…The following Proposition 3.2 is the relative form of a previous absolute bound [ HP19,Proposition 12]. Of course, although less nice, such upper bounds can be given for any size ♯X(F q n ).…”

“…in case there exists a covering X −→ Y . Twisting a little bit Weil's original proof [Wei48] of (1), the authors have given in a previous paper [HP19] proofs of Weil's, Drinfeld-Vladut's, Tsfasman's and some other new bounds from an euclidean point of vue. For instance, Weil bound (1) is only Schwarz inequality for two very natural vectors, namely γ 0 X coming from the class of the diagonal ∆ X inside X × X, and γ 1 X coming from the class of the graph Γ F X of the Frobenius morphism F X on X, lying in some euclidean subspace…”

“…1 Known absolute results [HP19] In this first section, we gather the notations and results of our previous work [HP19] that are needed in this paper.…”

Weil positivity and Trace formula, the archimedean place

“…The key observation in 2. is that, for q " 1, the operator K q has only one eigenvalue ą 1. The method used in 2. and 3. is closely related to the technique applied in [21] in the context of RH for curves over finite fields.…”

“…where F C : L 2 pCq Ñ H is the Fourier transform, and 1 P is the multiplication by the characteristic function of the set P " tu P C | |u| ě 1u. We take the case C " R with module exp : R Ñ R ˚considered in this paper and identify the dual p C " R using the bi-character νps, tq :" expp´istq which corresponds to (21) under the isomorphism given by the module. We give a "geometric" proof of the following (see [10] Chapter IV for the general theory, a compact operator has infinite order when its characteristic values form a sequence of rapid decay; this implies that it is of trace class).…”