The goal of this thesis is to apply an approach due to Elkies to study the distribution of rational point counts for certain families of curves and surfaces over finite fields. A vector space of polynomials over a fixed finite field gives rise to a linear code, and the weight enumerator of this code gives information about point count distributions. The MacWilliams theorem gives a relation between the weight enumerator of a linear code and the weight enumerator of its dual code. For certain codes C coming from families of varieties where it is not known how to determine the distribution of point counts directly, we analyze low-weight codewords of the dual code and apply the MacWilliams theorem and its generalizations to gain information about the weight enumerator of C. These low-weight dual codes can be described in terms of point sets that fail to impose independent conditions on this family of varieties. Our main results concern rational point count distributions for del Pezzo surfaces of degree 2, and for certain families of genus 1 curves. These weight enumerators have interesting geometric and coding theoretic applications for small q. / Mathematics

Les courbes algébriques sont des objets centraux de la géométrie algébrique. Dans cette thèse, nous étudions ces objets sous différents angles de la géométrie algébrique tels que la géométrie algébrique effective et la géométrie arithmétique. Dans le premier chapitre, nous étudions les courbes non-hyperelliptiques de genre g et leurs jacobiennes liées par l’intermédiaire de diviseurs thêta caractéristiques. Ces derniers contiennent des propriétés géométriques extrinsèques qui permettent de calculer les constantes thêta. Dans le deuxième chapitre, nous nous concentrons sur les courbes hyperelliptiques de genre 2 et leur surface de Kummer associée avec une motivation cryptographique. Dans le troisième et dernier chapitre, nous étudions les revêtements doubles non-ramifiés des courbes non-hyperelliptiques de genre g pour obtenir des informations sur le p-rang. / Algebraic curves are central objects in algebraic geometry. In this thesis, we consider these objects from different angles of algebraic geometry such as computational algebraic geometry and arithmetic geometry. In the first chapter, we study non-hyperelliptic curves of genus g and their Jacobians linked via theta characteristic divisors. Such divisors provide extrinsic geometric properties which allow us to compute theta constants. In the second chapter, we focus on hyperelliptic curves of genus 2 and the associated Kummer surface with a cryptographic motivation. In the third and final chapter, we examine unramified double covers of non-hyperelliptic curves of genus g to obtain information about p-rank.

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