We introduce non-abelian generalizations of Brumer’s conjecture, the Brumer-Stark conjecture and the strong Brumer-Stark property attached to a Galois CM-extension of number fields. Moreover, we discuss how they are related to the equivariant Tamagawa number conjecture, the strong Stark conjecture and a non-abelian generalization of Rubin’s conjecture due to D. Burns.
This article confirms a consequence of the non-abelian Iwasawa main conjecture. It is proved that under a technical condition the étale cohomology groups , where is a smooth, projective scheme, are generated by twists of norm compatible units in a tower of number fields associated to . Using the “Bloch-Kato-conjecture” a similar result is proven for motivic cohomology with finite coefficients.
We give a necessary condition for a surjective representation Gal to arise from the -torsion of a -curve. We pay a special attention to the case of quadratic -curves.
Much recent progress in the 2-class field tower problem revolves around demonstrating infinite such towers for fields – in particular, quadratic fields – whose class groups have large 4-ranks. Generalizing to all primes, we use Golod-Safarevic-type inequalities to analyse the source of the -rank of the class group as a quantity of relevance in the -class field tower problem. We also make significant partial progress toward demonstrating that all real quadratic number fields whose class groups...