Stark units and the main conjectures for totally real fields

The main theorem of the author’s thesis suggests that it should be possible to lift the Kolyvagin systems of Stark units, constructed by the author in an earlier paper, to a Kolyvagin system over the cyclotomic Iwasawa algebra. In this paper, we verify that this is indeed the case. This construction of Kolyvagin systems over the cyclotomic Iwasawa algebra from Stark units provides the first example towards a more systematic study of Kolyvagin system theory over an Iwasawa algebra when the core Selmer rank (in the sense of Mazur and Rubin) is greater than one. As a result of this construction, we reduce the main conjectures of Iwasawa theory for totally real fields to a statement in the context of local Iwasawa theory, assuming the truth of the Rubin–Stark conjecture and Leopoldt’s conjecture. This statement in the local Iwasawa theory context turns out to be interesting in its own right, as it suggests a relation between the solutions to p-adic and complex Stark conjectures.

K-groups with finite coefficients and arithmetic

In this paper we prove rank formulas for the even K-groups of number rings and relate Leopoldt's conjecture to K-theory. These results follow from a computation of the higher K-groups with finite coefficients.

Discrete logarithms and local units

Let K be a number field and (9 K its ring of integers. Let l be a prime number and e a positive integer. We give a method to construct l e th powers in (9 K using smooth algebraic integers. This method makes use of approximations of the l -adic logarithm to identify l e th powers. One version we give is successful if the class number of K is not divisible by l and if the units in C K which are congruent to 1 modulo l e +1 are l e th powers. A second version only depends on Leopoldt’s conjecture. We use the technique of constructing l e th powers to find discrete logarithms in a finite field of prime order. Our method for computing discrete logarithms is closely modelled after Gordon’s adaptation of the number field sieve to this problem. We conjecture th at the expected running time of our algorithm is L p [1/3; (64/9) 1/3 + o(1)] for p-> oo, where L p [ s; c ] = exp ( c (log q )s (log log q ) 1-8 ). This is the same running time as is conjectured for the number field sieve factoring algorithm.