Soient k un corps algébriquement clos de caractéristique p > 0 et C/k une courbe projective, lisse, intègre de genre g > 1 munie d’un p-groupe d’automorphismes G tel que |G| > 2p/(p-1)g. Le couple (C,G) est appelé grosse action. Si (C,G) est une grosse action, alors |G| <=4p/(p-1)^2g^2 (*). Dans cette thèse, nous étudions les répercussions arithmétiques des propriétés géométriques de grosses actions. Nous étudions d’abord l’arithmétique de l’extension de monodromie sauvage maximale de courbes sur un corps local K d’inégale caractéristique p à corps résiduel algébriquement clos, de genre arbitrairement grand ayant pour potentielle bonne réduction une grosse action satisfaisant le cas d’égalité de (*). On étudie en particulier les conducteurs de Swan attachés à ces courbes. Nous donnons ensuite les premiers exemples, à notre connaissance, de grosses actions (C,G) telles que le groupe dérivé D(G) soit non abélien. Ces courbes sont obtenues comme revêtements de S-corps de classes de rayons de P1(Fq) pour S non vide un sous-ensemble fini de P1(Fq). Enfin, on donne une méthode de calcul des S-corps de classes de Hilbert de revêtements abéliens de la droite projective d’exposant p et supersinguliers que l’on illustre pour des courbes de Deligne-Lusztig. / Let k be an algebraically closed field of characteristic p > 0 and C/k be a projective,smooth, integral curve of genus g > 1 endowed with a p-group of automorphisms G such that |G| > 2p/(p-1)g. The pair (C,G) is called big action. If (C,G) is a big action, then |G|<=4p/(p-1)^2g^2 (*). In this thesis, one studies arithmetical repercussions of geometric properties of big actions. One studies the arithmetic of the maximal wild monodromy extension of curves over a local field K of mixed characteristic p with algebraically closed residue field, with arbitrarily high genus having for potential good reduction a big action achieving equality in (*). One studies the associated Swan conductors. Then, one gives the first examples, to our knowledge, of big actions (C,G) with non abelian derived group D(G). These curves are obtained as coverings of S-ray class fields of P1(Fq) where S is a finite non empty subset of P1(Fq). Finally, one describes a method to compute S-Hilbert class fields of supersingular abelian covers of the projective line having exponent p and one illustrates it for some Deligne-Lusztig curves.

<p>The main focus of this dissertation is to find and explain the periodic points of certain algebraic functions that are related to some modular functions, which themselves can be represented by continued fractions. Some of these continued fractions are first explored by Srinivasa Ramanujan in early 20th century. Later on, much work has been done in terms of studying the continued fractions, and proving several relations, identities, and giving different representations for them.</p> <p><br></p> <p>The layout of this report is as follows. Chapter 1 has all the basic background knowledge and ingredients about algebraic number theory, class field theory, Ramanujan’s theta functions, etc. In Chapter 2, we look at the Ramanujan-Göllnitz-Gordon continued fraction that we call v(τ) and evaluate it at certain arguments in the field K = Q(√−d), with −d ≡ 1 (mod 8), in which the ideal (2) = ℘₂℘′₂  is a product of two prime ideals. We prove several identities related to itself and with other modular functions. Some of these are new, while some of them are known but with different proofs. These values of v(τ) are shown to generate the inertia field of ℘₂ or ℘′₂ in an extended ring class field over the field K. The conjugates over Q of these same values, together with 0, −1 ± √2, are shown to form the exact set of periodic points of a fixed algebraic function ˆF(x), independent of d. These are analogues of similar results for the Rogers-Ramanujan continued fraction. See [1] and [2]. This joint work with my advisor Dr. Morton, is submitted for publication to the New York Journal.</p> <p><br> In Chapters 3 and 4, we take a similar approach in studying two more continued fractions c(τ) and u(τ), the first of which is more commonly known as the Ramanujan’s cubic continued fraction. We show what fields a value of this continued fraction generates over Q, and we describe how the periodic points for described functions arise as values of these continued fractions. Then in the last chapter, we summarise all these results, give some possible directions for future research as well as mentioning some conjectures.</p>

Let K be an imaginary quadratic field with class number one and let [Special characters omitted.] be a degree one prime ideal of norm p not dividing 6 d K . In this thesis we generalize an algorithm of Schoof to compute the class number of ray class fields [Special characters omitted.] heuristically. We achieve this by using elliptic units analytically constructed by Stark and the Galois action on them given by Shimura's reciprocity law. We have discovered a very interesting phenomena where p divides the class number of [Special characters omitted.] . This is a counterexample to the elliptic analogue of a well-known conjecture, namely the Vandiver's conjecture.