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. 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 ℤp ("µ=0"). This assumption is satisfied for abelian fields and conjectured to be true in general. 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]. 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].

Doctor of Philosophy (PhD)