An Equivariant Main Conjecture in Iwasawa Theory and the Coates-Sinnott Conjecture

Abstract

<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>