This thesis is divided into two parts. In the first, we generalize results of Greenberg-Vatsal on the behavior of algebraic lambda-invariants of p-ordinary modular forms under congruence. In the second, we generalize a result of Emerton on maps between locally algebraic parabolically induced representations and unitary Banach space representations of GL₂ over a p-adic field. / text

Thesis (Ph. D.)--University of Washington, 2004. / Vita. Includes bibliographical references (p. 55-56).

No description available.

Thesis (Ph. D.)--University of Washington, 2003. / Vita. Includes bibliographical references (p. 65-68).

No description available.

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.

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

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

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