Computation of 2-groups of positive classes of exceptional number fields

We present an algorithm for computing the 2-group of the positive divisor classes of a number field F in case F has exceptional dyadic places. As an application, we compute the 2-rank of the wild kernel WK2(F) in K2(F) for such number fields

On the pp-divisibility of even KK-groups of the ring of integers of a cyclotomic field

Let $k$ be a given positive odd integer and $p$ an odd prime. In this paper, we shall give a sufficient condition when a prime $p$ divides the order of the groups $K_{2k}(\mathbb{Z}[\zeta_m+\zeta_m^{-1}])$ and $K_{2k}(\mathbb{Z}[\zeta_m])$, where $\zeta_m$ is a primitive $m$th root of unity. When $F$ is a $p$-extension contained in $\mathbb{Q}(\zeta_l)$ for some prime $l$, we also establish a necessary and sufficient condition for the order of $K_{2(p-2)}(\mathcal{O}_F)$ to be divisible by $p$. This generalizes a previous result of Browkin which in turn has applications towards establishing the existence of infinitely many cyclic extensions of degree $p$ which are $(p, p-1)$-rational

Mod pq Galois representations and Serre's conjecture

Motives and automorphic forms of arithmetic type give rise to Galois representations that occur in {\it compatible families}. These compatible families are of p-adic representations with p varying. By reducing such a family mod p one obtains compatible families of mod p representations. While the representations that occur in such a p-adic or mod p family are strongly correlated, in a sense each member of the family reveals a new face of the motive. In recent celebrated work of Wiles playing off a pair of Galois representations in different characteristics has been crucial. In this paper we investigate when a pair of mod p and mod q representations of the absolute Galois group of a number field K simultaneously arises from an {\it automorphic motive}: we do this in the 1-dimensional (Section 2) and 2-dimensional (Section 3: this time assuming $K={\mathbb Q}$) cases. In Section 3 we formulate a mod pq version of Serre's conjecture refining in part a question of Barry Mazur and Ken Ribet.Comment: This is an older preprint that was made available elsewhere on Sep. 19, 200