La discussione della mia tesi di dottorato si è svolta il 6 luglio 2016 alle 10:30 nell'amfiteatro Aubert di Polytech sul campus di Luminy (Marsiglia).


Titolo : Number of rational points on singular curves over finite fields

Abstract : 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.

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Qualche ricordo di questa magnifica giornata!