Thesis (M.A.)--California State University Channel Islands, 2008. / Submitted in partial fulfillment of the requirements for the degree of Masters Of Science in Mathematics. Title from PDF t.p. (viewed October 25, 2009).

We give algorithms to compute Coleman integrals on superelliptic curves over unramified extensions of the p-adics, and apply these via Chabauty methods to determine the set of rational points on such curves. We also determine the solution to an explicit instance of the Shafarevich conjecture by finding all elliptic curves with good reduction outside of the first 6 primes, subject to a heuristic. We use a combination of non-abelian Chabauty and the Mordell--Weil sieve to determine the rational points on several quotient modular curves, and therefore classify pairs of elliptic curves over the rationals with 67-, 73-, and 107-isogenies. We give methods to explicitly compute Coleman integrals on modular curves using a canonical lift of Frobenius and canonical local coordinates in each residue disk, and discuss the problem of computing the Weil pairing in finite rings.

del Pezzo surfaces are isomorphic to either P¹ x P¹ or P² blown up <i>a</i> times, where <i>a</i> ranges from 0 to 8. We will look at lines on del Pezzo surfaces isomorphic to P² blown up <i>a</i> times with <i>a</i> ranging from 0 to 6. We will show that when we count points of bounded height on one of these surfaces, the number of points on lines give us the primary growth order, but the secondary growth order calculates the number of points on the rest of the surface and hence is a better representation of the geometry of the surface.

del Pezzo surfaces are isomorphic to either P¹ x P¹ or P² blown up <i>a</i> times, where <i>a</i> ranges from 0 to 8. We will look at lines on del Pezzo surfaces isomorphic to P² blown up <i>a</i> times with <i>a</i> ranging from 0 to 6. We will show that when we count points of bounded height on one of these surfaces, the number of points on lines give us the primary growth order, but the secondary growth order calculates the number of points on the rest of the surface and hence is a better representation of the geometry of the surface.

Thesis (Ph. D.)--University of Missouri-Columbia, 2008. / The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on June 8, 2009) Vita. Includes bibliographical references.

Este trabalho é essencialmente sobre pontos racionais em curvas algébricas sobre corpos finitos ou, equivalentemente, lugares racionais em corpos de funções algébricas em uma variável sobre corpos finitos. O objetivo é a demonstração da existência de constantes aq e bq &isin; R&gt; 0 tais que se g &ge; aq. N + bq, então existe uma curva sobre Fq de gênero g com N pontos racionais. / This work is essentially about rational points on algebraic curves over finite fields or, equivalently, rational places on algebraic function fields of one variable over finite fields. The aim is the proof of the existence of constants aq and bq &isin; R&gt; 0 such that if g &ge; aq &isin; aq . N+bq then there exists a curve over Fq of genus g with N rational points.

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

<p>For the moduli stack $\mathcal{M}_{g,n/\mathbb{F}_p}$ of smooth curves of type $(g,n)$ over Spec $\mathbb{F}_p$ with the function field $K$, we show that if $g\geq3$, then the only $K$-rational points of the generic curve over $K$ are its $n$ tautological points. Furthermore, we show that if $g\geq 3$ and $n=0$, then Grothendieck's Section Conjecture holds for the generic curve over $K$. A primary tool used in this thesis is the theory of weighted completion developed by Richard Hain and Makoto Matsumoto.</p> / Dissertation

Depuis les 50 dernières années, de nombreux progrès ont été faits dans la compréhension du comportement asymptotique du nombre de points rationnels de hauteur bornée sur les variétés algébriques. Des conjectures précises ont été avancées par Baryrev, Manin et Peyre quant à la formule asymptotique attendue pour une variété générale.En 1962, à l'aide d'arguments issus de la méthode du cercle de Hardy et Littlewood, B. Birch a donné une estimation précise du nombre de points à coordonnées entières bornées dans une hypersurface définie par une équation homogène. Ceci revient à démontrer la conjecture de Batyrev-Manin-Peyre pour les hypersurfaces de l'espace projectif. Plus récemment, V. Blomer et J. Brüdern ont élaboré des techniques leur permettant d'établir une formule pour le comportement asymptotique du nombre de points de hauteur bornée pour des hypersurfaces d'espaces multiprojectifs définies par des équations multihomogènes diagonales. Parallèlement, D. Schindler a démontré la conjecture pour des hypersurfaces générales d'espaces biprojectifs, à l'aide de développements de la méthode de Birch.L'objet de cette thèse a été d'utiliser et de généraliser les techniques de Schindler, Blomer et Brüdern afin de démontrer la validité de la conjecture de Batyrev-Manin-Peyre pour le cas d'hypersurfaces de variétés toriques plus générales.Ce travail est composé de trois parties. La première partie concerne le cas particulier des hypersurfaces de tridegré (1,1,1) d'un espace triprojectif. Ce cas particulier constitue une première extension des techniques de Schindler à des variétés toriques dont le rang du groupe de Picard est 3. La deuxième partie est consacrée à l'étude des hypersurfaces d'une famille de variétés toriques dont le rang du groupe de Picard est 2 et contenant la famille des espaces biprojectifs. Il s'agit en effet d'étendre la méthode de Schindler afin d'obtenir une formule asymptotique pour le nombre de points de hauteur bornée sur ces variétés. Enfin, dans la dernière partie, nous généralisons les méthodes développées dans les deux parties précédentes à des hypersurfaces des variétés toriques complètes lisses de rang de groupe dont le cône effectif est supposé simplicial, ce qui nous permet de démontrer la conjecture de Batyrev-Manin-Peyre pour ces variétés. / For the last 50 years, many progresses have been made in the understanding of the asymptotic behaviour of the number of rational points of bouded height on algebraic varieties. Some precise conjectures have been advanced by Batyrev, Manin, and Peyre for the expected asymptotic formula for a general variety.In 1962, using some arguments of the Hardy-Littlewood circle method, B. Birch gave a precise estimate for the number of integral points whose coordinates are bounded on an hypersurface defined by an homogeneous equation. This amounts to demonstrating the Batyrev-Manin-Peyre conjecture for hypersurfaces of projective spaces. More recently, V. Blomer and J. Brüdern developed some methods permitting to establish a formula for the asymptotic growth of the number of points of bounded height on hypersurfaces of multiprojective spaces defined by multihomogeneous diagonal equations. In the same time, D. Schindler proved the conjecture for general hypersurfaces of biprojective spaces by using some developements of the method of Birch.The aim of this thesis was to use and generalize the methods of Schindler, blomer, and Brüdern in order to prove the Batyrev-Manin-Peyre conjecture in the case of hypersurfaces of some general toric varieties.This work contain three parts. The first one deals with the particular case of hypersurfaces of tridegree (1,1,1) of triprojective spaces. This particular case is a first extension of the method of Schindler to some toric varieties whose rank of the Picard group is 3. The second part deals with the study of hypersurfaces of a class of toric varieties whose rank of the Picard group is 2 and containing biprojective spaces. We establish a generalization of the method of Schindler method in order to find an asymptotic formula for the number of points of bounded height on these vrieties. Finally, in the last part, we generalize the methods developed in the last two part to treat the case of hypersurfaces of complete non-singular toric vareties whose effective cone is simplicial. This permits to prove the conjecture of batyrev-Manin-Peyre for these varieties.

L'étude de la distribution des points rationnels sur les variétés algébriques est un sujet classique de la géométrie diophantienne. Le programme proposé par V. Batyrev et Y. Manin dans des années 90 donne une prédiction sur l'ordre de croissance tandis que sa version ultérieure dûe à E. Peyre conjecture l'existence d'une distribution globale. Dans cette thèse nous nous proposons une étude de la distribution locale des points rationnels de hauteur bornée sur les variétés algébriques. Ceci envisage une description plus fine que celle globale en dénombrant les points le plus proche d'un point fixé. Nous nous plaçons sur le cadre récent du travail de D. McKinnon et M. Roth qui met en évidence que la géométrie de la variété gouverne l'approximation diophantienne sur elle et nous reprenons les résultats de S. Pagelot. L'ordre de croissance espéré et l'existence d'une mesure asymptotique sur certaines surfaces toriques sont démontrés, alors que démontrons-nous un résultat totalement différent pour une autre surface sur laquelle il n'y pas de mesure asymptotique et les meilleurs approximants génériques s'obtiennent sur des courbes rationnelles nodales. Ces deux phénomènes sont de nature radicalement différente au point de vu de l'approximation diophantienne. / The study of the distribution of rational points on algebraic varieties is a classic subject of Diophantine geometry. The program proposed by V. Batyrev and Y. Manin in the 1990s gives a prediction on the order of growth whereas its later version due to E. Peyre conjectures the existence of a global distribution. In this thesis we propose a study of the local distribution of rational points of bounded height on algebraic manifolds. This aims at giving a description finer than the global one by counting the points closest to a fixed point. We set ourselves on the recent framework of the work of D. McKinnon and M. Roth who prefers that the geometry of the variety governs the Diophantine approximation on it and we take up the results of S. Pagelot. The expected order of growth and the existence of an asymptotic measure on some toric surfaces are demonstrated, while we demonstrate a totally different result for another surface on which there is no asymptotic measure and the best generic approximates are obtained on nodal rational curves. These two phenomena are of a radically different nature from the point of view of the Diophantine approximation.