My Ph.D thesis defense took place on July 6, 2016 at 10:30 am inside the Aubert amphitheater of Polytech on the campus of Luminy (Marseille).
Titre : Number of rational points on singular curves over finite fields
Résumé : In this Ph.D thesis, we focus on some issues about the maximum number of rational points on a singular curve defined over a finite field. This topic has been extensively discussed in the smooth case since Weil's works. We have split our study into two stages. First, we provide a construction of singular curves of prescribed genera and base field and with many rational points: such a construction, based on some notions and tools from algebraic geometry and commutative algebra, yields a method for constructing, given a smooth curve $X$, another curve $X'$ with singularities, such that $X$ is the normalization of $X'$, and the added singularities are rational on the base field and with the prescribed singularity degree. Then, using a Euclidian approach, we prove a new bound for the number of closed points of degree two on a smooth curve defined over a finite field.Combining these two a priori independent results, we can study the following question: when is the Aubry-Perret bound (the analogue of the Weil bound in the singular case) reached? This leads naturally to the study of the properties of maximal curves and, when the cardinality of the base field is a square, to the analysis of the spectrum of their genera.
- Yves Aubry, Université de Toulon (Supervisor)
- Massimo Giulietti, Università degli Studi di Perugia (Examiner)
- Marc Hindry, Université Paris Diderot (Reviewer)
- James W. P. Hirschfeld, University of Sussex (Reviewer)
- David Kohel, Aix-Marseille Université (Examiner)
- Marc Perret, Université de Toulouse II (Examiner)
- Serge Vladuts, Aix-Marseille Université (Examiner)
Some memories of that wonderful day!