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On Criteria for Large Cyclotomic $\lambda$-invariants for Imaginary Quadratic Fields

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Iwasawa theory is a branch of number theory that studies the behavior of certain objects associated to a $\mathbb{Z}_p$-extension. We will focus our attention to the cyclotomic $\mathbb{Z}_p$-extensions of imaginary quadratic fields for varying primes p, and will give some conditions for when the corresponding lambda-invariants are greater than…

Iwasawa theory is a branch of number theory that studies the behavior of certain objects associated to a $\mathbb{Z}_p$-extension. We will focus our attention to the cyclotomic $\mathbb{Z}_p$-extensions of imaginary quadratic fields for varying primes p, and will give some conditions for when the corresponding lambda-invariants are greater than 1.
ContributorsStokes, Christopher Mathewson (Author) / Childress, Nancy (Thesis advisor) / Sprung, Florian (Committee member) / Montaño, Johnathan (Committee member) / Paupert, Julian (Committee member) / Kaliszewski, Steven (Committee member) / Arizona State University (Publisher)
Created2023
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Generalizations of the Signed Selmer Groups for Cyclotomic Extensions

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…

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