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 ∈ R> 0 tais que se g ≥ 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 ∈ R> 0 such that if g ≥ aq ∈ aq . N+bq then there exists a curve over Fq of genus g with N rational points.

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 ∈ R> 0 tais que se g ≥ 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 ∈ R> 0 such that if g ≥ aq ∈ aq . N+bq then there exists a curve over Fq of genus g with N rational points.

Let X be a Banach space and denote by SS₁(X) the set of all S₁-strictly singular operators from X to X. We prove that there is a Banach space X such that SS₁(X) is not a closed ideal. More specifically, we construct space X and operators T₁ and T₂ in SS₁(X) such that T₁+T₂ is not in SS₁(X). We show one example where the space X is reflexive and other where it is c₀-saturated. We also develop some results about S_alpha-strictly singular operators for alpha less than omega_1. / text

Soit k un corps algébriquement clos de caractéristique p>0. Soit C/k une courbe algébrique, propre, lisse et de genre g>1, munie d'un p-groupe G d'automorphismes tel que |G|/g> 2p/(p-1). Un tel couple (C,G) est appelé une "grosse action". Sous ces hypothèses, C--> C/G est un revêtement étale de la droite affine Spec k[X], complètement ramifié à l'infini. Après avoir précisé certaines propriétés du deuxième groupe de ramification G_2 de G à l'infini, on donne des exemples de telles actions avec G_2 abélien d'exposant quelconque. Ces exemples trouvent leur source dans la construction , via les corps de classes de rayon, de courbes algébriques sur un corps fini possédant beaucoup de points rationnels. On se concentre ensuite sur le cas où G_2 est un p-groupe abélien élémentaire. En considérant une filtration d'anneau de k[X] liée aux polynômes additifs, on obtient un théorème de structure pour les fonctions paramétrant le revêtement d'Artin-Schreier: C --> C/G_2. On exhibe alors des familles universelles et on discute l'espace de déformation correspondant lorsque p=5. On déduit de ces résultats une classification et une paramétrisation de telles actions lorsque |G|/g^2 est supérieur ou égal à 4/(p^2-1)^2. / Let k be an algebraically closed field of characteristic p>0 and C a connected nonsingular projective curve over k with genus g>1. We define a big action as a pair (C,G) where G is a p-subgroup of the k-automorphism group of C such that |G| /g > 2p / p-1. Then, C ---> C/G is an étale cover of the affine line Spec k[X] totally ramified at infinity. We first give necessary conditions on the second ramification G_2 of G at infinity for (C,G) to be a big action. We also display realizations of such actions with G_2 abelian of exponent as large as we want. Our main source of examples comes from the construction of curves with many rational points using ray class field theory for global function fields. Then we focus on the case where G_2 is p-elementary abelian. In particular, considering additive polynomials of k[X], we obtain a structure theorem for the functions parametrizing the Artin-Schreier cover C --> C/G_2. Then we display universal families and discuss the corresponding deformation space for p=5. All these results lead to the classification and the parametrization of big actions for |G|/g^2 greater or equal to 4/(p^2-1)^2.

Cette thèse traite de l'uniformisation, en caractéristique p>0, d'une valuation rationnelle, dans les cas particuliers où cette valuation est centrée en une singularité définie localement par des hypersurfaces d'équations :<br /><br />- soit z^p+f(x,y)=0, avec f non puissance p-ième et ord f>p,<br /><br />- soit z^p+e(x,y)z+f(x,y)=0, avec ord(ez+f)>p (cas d'Artin-Schreier).<br /><br />Historiquement c'est dans ces cas particuliers que s'est trouvé concentrée la difficulté de résoudre les surfaces en caractéristique positive.<br /><br />Les nouveautés ici consistent en une majoration du nombre minimum<br />d'éclatements de points fermés nécessaires pour uniformiser, et en une<br />description ``d'en bas'' de l'évolution du polygone de Newton ainsi que des<br />paramètres choisis pour les éclatés successifs le long de la valuation. <br /><br />Dans la première partie de la thèse, on revient sur l'obtention de la forme<br />normale de Giraud pour f dans l'anneau O_X(X), où X schéma régulier de<br />dimension deux et de caractéristique p. Le point de départ est une<br />décomposition polynomiale de f en les curvettes associées à la valuation. On<br />prévoit ensuite via une puissance p-ième d'en bas, le comportement du<br />polygone de Newton de f moins cette puissance p-ième, et on majore le nombre<br />minimum d'équerres du graphe dual de la valuation nécessaires à ce qu'il devienne droit de hauteur au plus 1, et minimal, cas correspondant à la forme normale.<br /><br /><br />Dans la deuxième partie de la thèse on utilise cette étude pour les cas particuliers ci-dessus mentionnés, on donne un algorithme permettant de prévoir les translations à faire à la sortie des équerres pour avoir un polygone de Newton minimal. On quantifie combien d'équerres sont suffisantes pour obtenir une singularité quasi-ordinaire.

This dissertation is devoted to groups generated by bounded automata and geometric objects related to these groups (limit spaces, Schreier graphs, etc.). It is shown that groups generated by bounded automata are contracting. We introduce the notion of a post-critical set of a finite automaton and prove that the limit space of a contracting self-similar group generated by a finite automaton is post-critically finite (finitely-ramified) if and only if the automaton is bounded. We show that the Schreier graphs on levels of automaton groups can be constructed by an iterative procedure of inflation of graphs. This was used to associate a piecewise linear map of the form fK(v) = minA∈KAv, where K is a finite set of nonnegative matrices, with every bounded automaton. We give an effective criterium for the existence of a strictly positive eigenvector of fK. The existence of nonnegative generalized eigenvectors of fK is proved and used to give an algorithmic way for finding the exponents λmax and λmin of the maximal and minimal growth of the components of f(n) K (v). We prove that the growth exponent of diameters of the Schreier graphs is equal to λmax and the orbital contracting coefficient of the group is equal to 1/λmin . We prove that the simple random walks on orbital Schreier graphs are recurrent. A number of examples are presented to illustrate the developed methods with special attention to iterated monodromy groups of quadratic polynomials. We present the first example of a group whose coefficients λmin and λmax have different values.

This dissertation is devoted to groups generated by bounded automata and geometric objects related to these groups (limit spaces, Schreier graphs, etc.). It is shown that groups generated by bounded automata are contracting. We introduce the notion of a post-critical set of a finite automaton and prove that the limit space of a contracting self-similar group generated by a finite automaton is post-critically finite (finitely-ramified) if and only if the automaton is bounded. We show that the Schreier graphs on levels of automaton groups can be constructed by an iterative procedure of inflation of graphs. This was used to associate a piecewise linear map of the form fK(v) = minA[set]KAv, where K is a finite set of nonnegative matrices, with every bounded automaton. We give an effective criterium for the existence of a strictly positive eigenvector of fK. The existence of nonnegative generalized eigenvectors of fK is proved and used to give an algorithmic way for finding the exponents λmax and λmin of the maximal and minimal growth of the components of fK(n)(v). We prove that the growth exponent of diameters of the Schreier graphs is equal to λmax and the orbital contracting coefficient of the group is equal to 1/λmin . We prove that the simple random walks on orbital Schreier graphs are recurrent. A number of examples are presented to illustrate the developed methods with special attention to iterated monodromy groups of quadratic polynomials. We present the first example of a group whose coefficients λmin and λmax have different values.

Let p be an odd prime, and define f(x) as follows: f(x) as the sum from 1 to k of a_i times x raised to the power of (p to the power of (alpha_i+1)) in F_(p to the power of n)[x] where 0 is less than or equal to alpha_1 < alpha_2 < ... < alpha_k where alpha_k is equal to alpha. We consider the exponential sum S(f, n) equal to the sum_(x as x runs over the finite field with (p to the n elements) of zeta_(p to the power of Tr_n (f(x))), where zeta_p equals e to the power of (2i times pi divided by p) and Tr_n is the trace from the finite field with p to the n elements to the finite field with p elements.We provide necessary background from number theory and review the basic facts about quadratic forms over a finite field with p elements through both the multivariable and single variable approach. Our main objective is to compute S(f, n) explicitly. The sum S(f, n) is determined by two quantities: the nullity and the type of the quadratic form Tr_n (f(x)). We give an effective algorithm for the computation of the nullity. Tables of numerical values of the nullity are included. However, the type is more subtle and more difficult to determine. Most of our investigation concerns the type. We obtain "relative formulas" for S(f, mn) in terms of S(f, n) when the p-adic order of m is less than or equal to the minimum p-adic order of the alphas. The formulas are obtained in three separate cases, using different methods: (i) m is q to the s power, where q is a prime different from 2 and p; (ii) m is 2 to the s power; and (iii) m is p. In case (i), we use a congruence relation resulting from a suitable Galios action. For case (ii), in addition to the congruence in case (i), a special partition of the finite field with p to the 2n elements is needed. In case (iii), the congruence method does not work. However, the Artin-Schreier Theorem allows us to compute the trace of the extension from the finite field with p to the pn elements to the fi nite field with p to the n elements rather explicitly.When the 2-adic order of each of the alphas is equal and it is less than the 2-adic order of n, we are able to determine S(f, n) explicitly. As a special case, we have explicit formulas for the sum of the monomial, S(ax to the power of (1+ (p to the power of alpha)).Most of the results of the thesis are new and generalize previous results by Carlitz, Baumert, McEliece, and Hou.

JSIAM Letters Vol. 6 (2014) p.33-36 / 京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18639号 / 理博第4018号 / 新制||理||1579(附属図書館) / 31553 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 池田 保, 教授 雪江 明彦, 教授 上田 哲生 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Cette thèse est consacrée à l'étude de la topologie à l'infini d'espaces généralisant les graphes de Schreier. Plus précisément, on considère le quotient X/H d'un espace métrique géodésique propre hyperbolique X par un groupe quasi-convexe-cocompact H d'isométries de X. On montre que ce quotient est un espace hyperbolique. Le résultat principal de cette thèse indique que le nombre de bouts de l'espace quotient X/H est déterminé par les classes d'équivalence sur une sphère de rayon explicitement calculable. Dans le cadre de la théorie des groupes, on montre que l'on peut construire explicitement des groupes et des sous-groupes pour lesquels il n'existe pas d'algorithme permettant de déterminer le nombre de bouts relatifs. Si le sous-groupe est quasi-convexe, on donne un algorithme permettant de calculer le nombre de bouts relatifs. / This thesis is devoted to the study of the topology at infinity of spaces generalizing Schreier graphs. More precisely, we consider the quotient X/H of a geodesic proper hyperbolic metric space X by a quasiconvex-cocompact group H of isometries of X. We show that this quotient is a hyperbolic space. The main result of the thesis indicates that the number of ends of the quotient space X/H is determined by equivalence classes on a sphere of computable radius. In the context of group theory, we show that one can construct explicitly groups and subgroups for which there are no algorithm to determine the number of relative ends. If the subgroup is quasiconvex, we give an algorithm to compute the number of relative ends.