abstract: In Iwasawa theory, one studies how an arithmetic or geometric object grows as its field of definition varies over certain sequences of number fields. For example, let $F/\mathbb{Q}$ be a finite extension of fields, and let $E:y^2 = x^3 + Ax + B$ with $A,B \in F$ be an elliptic curve. If $F = F_0 \subseteq F_1 \subseteq F_2 \subseteq \cdots F_\infty = \bigcup_{i=0}^\infty F_i$, one may be interested in properties like the ranks and torsion subgroups of the increasing family of curves $E(F_0) \subseteq E(F_1) \subseteq \cdots \subseteq E(F_\infty)$. The main technique for studying this sequence of curves when $\Gal(F_\infty/F)$ has a $p$-adic analytic structure is to use the action of $\Gal(F_n/F)$ on $E(F_n)$ and the Galois cohomology groups attached to $E$, i.e. the Selmer and Tate-Shafarevich groups. As $n$ varies, these Galois actions fit into a coherent family, and taking a direct limit one obtains a short exact sequence of modules $$0 \longrightarrow E(F_\infty) \otimes(\mathbb{Q}_p/\mathbb{Z}_p) \longrightarrow \Sel_E(F_\infty)_p \longrightarrow \Sha_E(F_\infty)_p \longrightarrow 0 $$ over the profinite group algebra $\mathbb{Z}_p[[\Gal(F_\infty/F)]]$. When $\Gal(F_\infty/F) \cong \mathbb{Z}_p$, this ring is isomorphic to $\Lambda = \mathbb{Z}_p[[T]]$, and the $\Lambda$-module structure of $\Sel_E(F_\infty)_p$ and $\Sha_E(F_\infty)_p$ encode all the information about the curves $E(F_n)$ as $n$ varies. In this dissertation, it will be shown how one can classify certain finitely generated $\Lambda$-modules with fixed characteristic polynomial $f(T) \in \mathbb{Z}_p[T]$ up to isomorphism. The results yield explicit generators for each module up to isomorphism. As an application, it is shown how to identify the isomorphism class of $\Sel_E(\mathbb{Q_\infty})_p$ in this explicit form, where $\mathbb{Q}_\infty$ is the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, and $E$ is an elliptic curve over $\mathbb{Q}$ with good ordinary reduction at $p$, and possessing the property that $E(\mathbb{Q})$ has no $p$-torsion. / Dissertation/Thesis / Ph.D. Mathematics 2011

Let E be an elliptic curve defined over the rationals Q, and p be a prime at least 5 where E has multiplicative reduction. This thesis studies the Iwasawa theory of E over certain false Tate curve extensions F[infinity], with Galois groupG = Gal(F[infinity]/Q). I show how the p[infinity]-Selmer group of E over F[infinity] controls the p[infinity]-Selmer rank growth within the false Tate curve extension, and how it is connected to the root numbers of E twisted by absolutely irreducible orthogonal Artin representations of G, and investigate the parity conjecture for twisted modules.

abstract: In 1984, Sinnott used $p$-adic measures on $\mathbb{Z}_p$ to give a new proof of the Ferrero-Washington Theorem for abelian number fields by realizing $p$-adic $L$-functions as (essentially) the $Gamma$-transform of certain $p$-adic rational function measures. Shortly afterward, Gillard and Schneps independently adapted Sinnott's techniques to the case of $p$-adic $L$-functions associated to elliptic curves with complex multiplication (CM) by realizing these $p$-adic $L$-functions as $Gamma$-transforms of certain $p$-adic rational function measures. The results in the CM case give the vanishing of the Iwasawa $mu$-invariant for certain $mathbb{Z}_p$-extensions of imaginary quadratic fields constructed from torsion points of CM elliptic curves. In this thesis, I develop the theory of $p$-adic measures on $mathbb{Z}_p^d$, with particular interest given to the case of $d>1$. Although I introduce these measures within the context of $p$-adic integration, this study includes a strong emphasis on the interpretation of $p$-adic measures as $p$-adic power series. With this dual perspective, I describe $p$-adic analytic operations as maps on power series; the most important of these operations is the multivariate $Gamma$-transform on $p$-adic measures. This thesis gives new significance to product measures, and in particular to the use of product measures to construct measures on $mathbb{Z}_p^2$ from measures on $mathbb{Z}_p$. I introduce a subring of pseudo-polynomial measures on $mathbb{Z}_p^2$ which is closed under the standard operations on measures, including the $Gamma$-transform. I obtain results on the Iwasawa-invariants of such pseudo-polynomial measures, and use these results to deduce certain continuity results for the $Gamma$-transform. As an application, I establish the vanishing of the Iwasawa $mu$-invariant of Yager's two-variable $p$-adic $L$-function from measure theoretic considerations. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2015

Thesis (Ph. D.)--University of Washington, 2003. / Vita. Includes bibliographical references (p. 79-80).

Heidelberg, Univ., Diss., 2001.

abstract: In 1959, Iwasawa proved that the size of the $p$-part of the class groups of a $\mathbb{Z}_p$-extension grows as a power of $p$ with exponent ${\mu}p^m+{\lambda}\,m+\nu$ for $m$ sufficiently large. Broadly, I construct conditions to verify if a given $m$ is indeed sufficiently large. More precisely, let $CG_m^i$ (class group) be the $\epsilon_i$-eigenspace component of the $p$-Sylow subgroup of the class group of the field at the $m$-th level in a $\mathbb{Z}_p$-extension; and let $IACG^i_m$ (Iwasawa analytic class group) be ${\mathbb{Z}_p[[T]]/((1+T)^{p^m}-1,f(T,\omega^{1-i}))}$, where $f$ is the associated Iwasawa power series. It is expected that $CG_m^i$ and $IACG^i_m$ be isomorphic, providing us with a powerful connection between algebraic and analytic techniques; however, as of yet, this isomorphism is unestablished in general. I consider the existence and the properties of an exact sequence $$0\longrightarrow\ker{\longrightarrow}CG_m^i{\longrightarrow}IACG_m^i{\longrightarrow}\textrm{coker}\longrightarrow0.$$ In the case of a $\mathbb{Z}_p$-extension where the Main Conjecture is established, there exists a pseudo-isomorphism between the respective inverse limits of $CG_m^i$ and $IACG_m^i$. I consider conditions for when such a pseudo-isomorphism immediately gives the existence of the desired exact sequence, and I also consider work-around methods that preserve cardinality for otherwise. However, I primarily focus on constructing conditions to verify if a given $m$ is sufficiently large that the kernel and cokernel of the above exact sequence have become well-behaved, providing similarity of growth both in the size and in the structure of $CG_m^i$ and $IACG_m^i$; as well as conditions to determine if any such $m$ exists. The primary motivating idea is that if $IACG_m^i$ is relatively easy to work with, and if the relationship between $CG_m^i$ and $IACG_m^i$ is understood; then $CG_m^i$ becomes easier to work with. Moreover, while the motivating framework is stated concretely in terms of the cyclotomic $\mathbb{Z}_p$-extension of $p$-power roots of unity, all results are generally applicable to arbitrary $\mathbb{Z}_p$-extensions as they are developed in terms of Iwasawa-Theory-inspired, yet abstracted, algebraic results on maps between inverse limits. / Dissertation/Thesis / Ph.D. Mathematics 2013

Le but de cette thèse est l'étude des invariants d'Iwasawa attachés aux p-groupes des classes généralisés dans les extensions de Lie p-adiques de corps de nombres.Ces invariants ont été introduits par Iwasawa pour les Zp-extensions. Les travaux de Venjakob sur la structure des modules sur l'algèbre d'Iwasawa d'un groupe de Lie p-adique ont permis d'en généraliser la définition à la théorie non-commutative. Par des techniques de descente et une étude algébrique fine de la structure des modules d'Iwasawa sur un groupe non-commutatif, on dégage des formules asymptotiques pour les p-groupes des classes généralisés le long d'une extension de corps de nombres de groupe de Galois p-valué. Ces formules ont pour paramètres les invariants d'Iwasawa de l'extension. Elles sont rendues plus précises dans le cas des Zp-extensions, où on remarque qu'un défaut de descente doit être pris en compte et est d'impact non négligeable sur le résultat final. Ces résultats asymptotiques sont ensuite exploités à l'aide de la théorie du miroir. Ceci conduit à des formules de dualité entre ramification et décomposition concernant les invariants d'Iwasawa / This thesis aim at exploring Iwasawa invariants attached to generalized p-class groups in p-adic Lie extensions of number fields. These invariants where introduced by Iwasawa for Zp-extensions. In his work on the structure of modules over the Iwasawa algebra of a p-adic Lie group, Venjakob extends the definition to the non commutative theory. Using descent techniques, along with a fine algebraic study of Iwasawa's modules structure over a non commutative group, we obtain asymptotic formulas for generalized p-class groups in a tower of number fields, with a p-valued group as Galois group. These formulas have Iwasawa invariants as parameters. They become more precise for Zp-extensions, where a significant descent default is involved. These asymptotic results are exploited thanks to reflexion theory. This leads to duality formulas between ramification and decomposition for Iwasawa invariants

Un outil clé dans la théorie des représentations p-adiques est l'algèbre d'Iwasawa, construit par Iwasawa pour étudier les nombres de classes d'une tour de corps de nombres. Pour un nombre premier p, l'algèbre d'Iwasawa d'un groupe de Lie p-adique G, est l'algèbre de groupe G complétée non-commutative. C'est aussi l'algèbre des mesures p-adiques sur G. Les objets provenant de groupes semi-simples, simplement connectés ont des présentations explicites comme la présentation par Serre des algèbres semi-simples et la présentation de groupe de Chevalley par Steinberg. Dans la partie I, nous donnons une description explicite des certaines algèbres d'Iwasawa. Nous trouvons une présentation explicite (par générateurs et relations) de l'algèbre d'Iwasawa pour le sous-groupe de congruence principal de tout groupe de Chevalley semi-simple, scindé et simplement connexe sur Z_p. Nous étendons également la méthode pour l'algèbre d'Iwasawa du sous-groupe pro-p Iwahori de GL (n, Z_p). Motivé par le changement de base entre les algèbres d'Iwasawa sur une extension de Q_p nous étudions les représentations p-adiques globalement analytiques au sens d'Emerton. Nous fournissons également des résultats concernant la représentation de série principale globalement analytique sous l'action du sous-groupe pro-p Iwahori de GL (n, Z_p) et déterminons la condition d'irréductibilité. Dans la partie II, nous faisons des expériences numériques en utilisant SAGE pour confirmer heuristiquement la conjecture de Greenberg sur la p-rationalité affirmant l'existence de corps de nombres "p-rationnels" ayant des groupes de Galois (Z/2Z)^t. Les corps p-rationnels sont des corps de nombres algébriques dont la cohomologie galoisienne est particulièrement simple. Ils sont utilisés pour construire des représentations galoisiennes ayant des images ouvertes. En généralisant le travail de Greenberg, nous construisons de nouvelles représentations galoisiennes du groupe de Galois absolu de Q ayant des images ouvertes dans des groupes réductifs sur Z_p (ex GL (n, Z_p), SL (n, Z_p ), SO (n, Z_p), Sp (2n, Z_p)). Nous prouvons des résultats qui montrent l'existence d'extensions de Lie p-adiques de Q où le groupe de Galois correspond à une certaine algèbre de Lie p-adique (par exemple sl(n), so(n), sp(2n)). Cela répond au problème classique de Galois inverse pour l'algèbre de Lie simple p-adique. / A key tool in p-adic representation theory is the Iwasawa algebra, originally constructed by Iwasawa in 1960's to study the class groups of number fields. Since then, it appeared in varied settings such as Lazard's work on p-adic Lie groups and Fontaine's work on local Galois representations. For a prime p, the Iwasawa algebra of a p-adic Lie group G, is a non-commutative completed group algebra of G which is also the algebra of p-adic measures on G. It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups as noticed by Clozel. In Part I, we lay the foundation by giving an explicit description of certain Iwasawa algebras. We first find an explicit presentation (by generators and relations) of the Iwasawa algebra for the principal congruence subgroup of any semi-simple, simply connected Chevalley group over Z_p. Furthermore, we extend the method to give a set of generators and relations for the Iwasawa algebra of the pro-p Iwahori subgroup of GL(n,Z_p). The base change map between the Iwasawa algebras over an extension of Q_p motivates us to study the globally analytic p-adic representations following Emerton's work. We also provide results concerning the globally analytic induced principal series representation under the action of the pro-p Iwahori subgroup of GL(n,Z_p) and determine its condition of irreducibility. In Part II, we do numerical experiments using a computer algebra system SAGE which give heuristic support to Greenberg's p-rationality conjecture affirming the existence of "p-rational" number fields with Galois groups (Z/2Z)^t. The p-rational fields are algebraic number fields whose Galois cohomology is particularly simple and they offer ways of constructing Galois representations with big open images. We go beyond Greenberg's work and construct new Galois representations of the absolute Galois group of Q with big open images in reductive groups over Z_p (ex. GL(n, Z_p), SL(n, Z_p), SO(n, Z_p), Sp(2n, Z_p)). We are proving results which show the existence of p-adic Lie extensions of Q where the Galois group corresponds to a certain specific p-adic Lie algebra (ex. sl(n), so(n), sp(2n)). This relates our work with a more general and classical inverse Galois problem for p-adic Lie extensions.

Dans le premier chapitre de cette thèse on rappelle l'énoncé ainsi que des équivalents de la conjecture de Leopoldt puis l'on donne un algorithme permettant de vérifier cette conjecture pour un corps de nombre et premier donnés. Pour la suite on suppose cette conjecture vraie pour le premier p fixé Et on étudie la torsion du groupe de Galois de l'extension abélienne maximale p-ramifiée. On présente une méthode qui détermine effectivement les facteurs invariants de ce groupe fini. Dans le troisième chapitre on donne des résultats numériques que l'on interpréte via des heuristiques à la Cohen-Lenstra. Dans le quatrième chapitre, à l'aide de l'algorithme qui permet le calcul de ce module, on donne des exemples de corps et de premiers vérifiant la conjecture de Greenberg. / In the first chapter of this thesis we explain Leopoldt's conjecture and some equivalent formulations. Then we give an algorithm that checks this conjecture for a given prime p and a number field. Next we assume that this conjecture is true, and we study the torsion part of the Galois group of the maximal abelian p-ramified p-extension of a given number field. We present a method to compute the invariant factors of this finite group. In the third chapter we give an interpretation of our numrical result by heuristics “à la” Cohen-Lenstra. In the fourth and last chapter, using our algorithm which computes this torsion submodule, we give new examples of numbers fields which satisfy Greenberg's conjecture.

We generalize the work of Gelbart, Miller, Pantchichkine, and Shahidi on constructing p-adic measures to the case of totally real fields K. This measure is the Mellin transform of the reciprocal of the p-adic L-function which interpolates the special values at negative integers of the Hecke L-function of K. To define this measure as a distribution, we study the non-constant terms in the Fourier expansion of a particular Eisenstein series of the Hilbert modular group of K. Proving the distribution is a measure requires studying the structure of the Iwasawa algebra.