<p>The classical Main Conjecture (MC) in Iwasawa Theory relates values of p-adic L-function associated to 1-dimensional Artin characters over a totally real number field F to values of characteristic polynomials attached to certain Iwasawa modules. Wiles [47] proved the MC for odd primes p over arbitrary totally real base fields F and for the prime 2 over abelian totally real fields F.</p> <p>An equivariant version of the MC, which combines the information for all characters of the Galois group of a relative abelian extension E/F of number fields with F totally real, was formulated and proven for odd primes p by Ritter and Weiss in [33] under the assumption that the corresponding Iwasawa module is finitely generated over ℤ_p ("µ=0"). This assumption is satisfied for abelian fields and conjectured to be true in general.</p> <p>In this thesis we formulate an Equivariant Main Conjecture (EMC) for all prime numbers p, which coincides with the version of Ritter and Weiss for odd p, and we provide a unified proof of the EMC for all primes p under the assumptions µ=0 and the validity of the 2-adic MC. The proof combines the approach of Ritter and Weiss with ideas and techniques used recently by Greither and Popescu [13] to give a proof of a slightly different formulation of an EMC under the same assumptions (p odd and µ=0) as in [33].</p> <p>As an application of the EMC we prove the Coates-Sinnott Conjecture, again assuming µ=0. We also show that the p-adic version of the Coates-Sinnott Conjecture holds without the assumption µ=0 for abelian Galois extensions E/F of degree prime to p. These generalize previous results for odd primes due to Nguyen Quang Do in [27], Greither-Popescu [13], and Popescu in [30].</p> / Doctor of Philosophy (PhD)

Thesis (Ph. D.)--Oregon State University, 1972. / Typescript (photocopy). Includes bibliographical references. Also available on the World Wide Web.

For every pair of vertices $u,v$ in an oriented graph, a $u$-$v$ $geodesic$ is a shortest directed path from $u$ to $v$. For an oriented graph $D$, let $I_{D}[u,v]$ denoted the set of all vertices lying on a $u$-$v$ geodesic or a $v$-$u$ geodesic. And for $Ssubseteq V(D)$, let $I_{D}[S]$ denoted the union of all $I_{D}[u,v]$ for all $u,vin S$. If $S$ is a $convex$ set then $I_{D}[S]=S$. Let $[S]_{D}$ denoted the smallest convex set containing $S$. The $geodetic$ $number$ $g(D)$ of an oriented graph $D$ is the minimum cardinality of a set $S$ with $I_{D}[S]=V(D)$. The $hull$ $number$ $h(D)$ of an oriented graph $D$ is the minimum cardinality of a set $S$ with $[S]_{D}=V(D)$. For a connected graph $G$, let $g^{-}(G)=$min${g(D)$:$D$ is an orientation of $G$ $}$ and $g^{+}(G)=$max${g(D)$:$D$ is an orientation of $G$ $}$. And let $h^{-}(G)=$min${h(D)$:$D$ is an orientation of $G$ $}$ and $h^{+}(G)=$max${h(D)$:$D$ is an orientation of $G$ $}$. We show that $h^{+}(G)>h^{-}(G)$ and $g^{+}(G)>g^{-}(G)$ for every connected graph $G$ with $|V(G)|geq 3$. Then we show that $h^{+}(G)=h^{-}(G)+1$ if and only if $G$ is isomorphic to $K_{3}$ or $K_{1,r}$ for $rgeq 2$ and prove that for every connected graph $G$, $h^{+}(G)geq 5$ if and only if $|V(G)|geq 5$ and $G cong C_{5}$. Let $Sh^{*}(G)={h(D)$:$D$ is a strongly connected orientation of $G$ $}$ and we have $Sh^{*}(K_{n})={2}$. Let a graph $C(n,t)$ with $V(C(n,t))={1,2,...,n,x,y}$ and $E(C(n,t))={i(i+1):i=1,2,...,n-1}cup {1n}cup {1x}cup {ty}$. We also have $h^{-} (C(n,t))<g^{-}(C(n,t))<h^{+}(C(n,t)) =g^{+}(C(n,t))$ if $n geq 5$, $t eq frac{n}{2}$ and $3leq tleq n-1$. The last result answers a problem of Farrugia in [7].

Thesis (doctoral)--Georg-August-Universität zu Göttingen, 1919. / Vita.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1884. / Vita. Includes bibliographical references.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1899. / On t.p. "[Summe]" appears as the summation symbol, "k=1" appears under the summation symbol, and "[unendlich]" appears as the infinity symbol above the summation. Vita. Includes bibliographical references.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1881. / Vita. Includes bibliographical references.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1922. / Vita. Includes bibliographical references.

Academisch proefschrift--Amsterdam. / At head of title: Vrije Universiteit Amsterdam. Summary in English. "Stellingen": [4] p. inserted.

Thesis (Ph. D.)--University of Chicago, 1932. / Vita. Lithoprinted. "Private edition, distributed by the University of Chicago libraries, Chicago, Illinois."