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- All Subjects: Theoretical mathematics
- Creators: Childress, Nancy
Description
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.
ContributorsElledge, Shawn Michael (Author) / Childress, Nancy (Thesis advisor) / Bremner, Andrew (Committee member) / Fishel, Susanna (Committee member) / Jones, John (Committee member) / Paupert, Julien (Committee member) / Arizona State University (Publisher)
Created2013
Description
Let $E$ be an elliptic curve defined over a number field $K$, $p$ a rational prime, and $\Lambda(\Gamma)$ the Iwasawa module of the cyclotomic extension of $K$. A famous conjecture by Mazur states that the $p$-primary component of the Selmer group of $E$ is $\Lambda(\Gamma)$-cotorsion when $E$ has good ordinary reduction at all primes of $K$ lying over $p$. The conjecture was proven in the case that $K$ is the field of rationals by Kato, but is known to be false when $E$ has supersingular reduction type. To salvage this result, Kobayashi introduced the signed Selmer groups, which impose stronger local conditions than their classical counterparts. Part of the construction of the signed Selmer groups involves using Honda's theory of commutative formal groups to define a canonical system of points. In this paper I offer an alternate construction that appeals to the Functional Equation Lemma, and explore a possible way of generalizing this method to elliptic curves defined over $p$-adic fields by passing from formal group laws to formal modules.
ContributorsReamy, Alexander (Author) / Sprung, Florian (Thesis advisor) / Childress, Nancy (Thesis advisor) / Paupert, Julien (Committee member) / Montaño, Jonathan (Committee member) / Kaliszewski, Steven (Committee member) / Arizona State University (Publisher)
Created2023