Julien Monaldi | Searcharxiv

arXiv2025

Maximal curves with respect to quadratic extensions over finite fields

We propose a detailed study of a canonical bound which relates the numbers of rational points of a curve over a finite field with that over its quadratic extension. Alternative proofs which make a connection with the variance enable to obtain optimal refinements. We focus on the curves reaching the bound, which we call Hallouin-Perret-maximal curves. We provide different characterizations and stress natural links with the curves which attain the Ihara bound. As consequences, we establish the list of such curves with low genus and we outline a maximality result which involves the Suzuki curves. At last we determine which polynomials correspond to the Jacobian of a Hallouin-Perret-maximal curve of genus 2.

math.AG↗

arXiv2023

Closed points on curves over finite fields

We are interested in the quantity $\rho$(q, g) defined as the smallest positive integer such that r $\ge$ $\rho$(q, g) implies that any absolutely irreducible smooth projective algebraic curve defined over F q of genus g has a closed point of degree r. We provide general upper bounds for this number and its exact value for g = 1, 2 and 3. We also improve the known upper bounds on the number of closed points of degree 2 on a curve.

math.AG↗