Heidelberg, Univ., Diss., 2001. / Computerdatei im Fernzugriff.

Heidelberg, Univ., Diss., 2001. / Computerdatei im Fernzugriff.

Heidelberg, University, Diss., 2001.

No description available.

The well-known Riemann-Hurwitz formula for Riemann surfaces (or the corresponding formulas of the same name for curves/function fields) is used in genus computations. In 1979, Yûji Kida proved a strikingly analogous formula in [Kid80] for p-extensions of CM-fields (p an odd prime) which is similarly used to compute Iwasawa λ -invariants. However, the relationship between Kida’s formula and the statement for surfaces is not entirely clear since the proofs are of a very different flavor. Also, there were a few hypotheses for Kida’s result which were not fully satisfying; for example, Kida’s formula requires CM-fields rather than more general number fields and excludes the prime p = 2. Around a year after Kida’s result was published, Kenkichi Iwasawa used Galois cohomology in [Iwa81] to establish a more general formula (about representations) that did not exclude the prime p = 2 nor need the CM-field assumption. Moreover, Kida’s formula follows as a corollary from Iwasawa’s formula. We’ll prove a slight generalization of Iwasawa’s formula and use this to give a new proof of a result of Kida in [Kid79] and Ferrero in [Fer80] which computes λ-invariants in imaginary quadratic extensions for the prime p = 2. We go on to produce special generalizations of Iwasawa’s formula in the case of cyclic p-extensions; these formulas can be realized as statements about Q(p)-representations, and, in the cases of degree p or p², about p-adic integral representations. One upshot of these formulas is a vanishing criterion for λ-invariants which generalizes a result of Takashi Fukuda et al. in [FKOT97]. Other applications include new congruences and inequalities for λ-invariants that cannot be gleaned from Iwasawa’s formula. Lastly, we give a scheme theoretic approach to produce a general formula for finite, separable morphisms of Dedekind schemes which simultaneously encompasses the classical Riemann-Hurwitz formula and Iwasawa’s formula.

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).

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