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

Emmanuel Hallouin (1) , Marc Perret (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 GRIM - Groupe de Recherche en Informatique et Mathématiques

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 Weil classical bound, continuing to Ihara bound, passing through infinitely many n-th order Weil bounds and ending asymptotically to Drinfeld-Vl˘ adut¸boundadut¸adut¸bound. We relate this set of bounds to Oesterlé one, proving that these are inverse functions in some sense. We explain how Riemann hypothesis for the curve X can be merely seen as an euclidean property, coming from the Toeplitz shape of some intersection matrix on the surface X × 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 term of eu-clidean relationships between classes in the euclidean part F X of the numerical group Num(X × 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 .

Mots clés: Rational point; Curves over a finite field; Weil bound; Toeplitz matrices; Zeta function

Domaines: Théorie des nombres [math.NT]; Géométrie algébrique [math.AG]

Emmanuel Hallouin, Marc Perret. An unified viewpoint for upper bounds for the number of points of curves over finite fields via Euclidean geometry and semi-definite symmetric Toeplitz matrices. Transactions of the American Mathematical Society, In press. ⟨hal-01654406⟩