Galois cohomology of ambiguous ideals

Galois Module Structure of Galois Cohomology

Let F be a field containing a primitive pth root of unity, and let U be an open normal subgroup of index p of the absolute Galois group GF of F . We determine the structure of the cohomology group H(U, Fp) as an Fp[GF /U ]-module for all n ∈ N. Previously this structure was known only for n = 1, and until recently the structure even of H(U, Fp) was determined only for F a local field, a case se...

Palindromy and Ambiguous Ideals Revisited

The purpose of this article is to revisit the relationship between ambiguous ideals and palindromy in the simple continued fraction expansions of quadratic irrationals begun by the first author and A. J. van der Poorten (1995, Bull. Austral. Math. Soc. 51, 215 233). We present simpler proofs of known results, new interrelationships, and correct some misinterpretations. We do this via the infras...

Étale cohomology and Galois Representations

In this essay we briefly introduce the main ideas behind the theory of studying algebraic varieties over a number field by constructing associated Galois representations, and see how this can be understood naturally in the context of an extension of monodromy theory from geometry. We then, following Deligne’s original method, use some of these ideas to prove the Riemann Hypothesis for varieties...

Local Cohomology at Monomial Ideals

Factoring of Prime Ideals in Galois Extensions

We return to the general AKLB setup: A is a Dedekind domain with fraction field K, L is a finite separable extension of K, and B is the integral closure of A in L. But now we add the condition that the extension L/K is normal, hence Galois. We will see shortly that the Galois assumption imposes a severe constraint on the numbers ei and fi in the ram-rel identity (4.1.6). Throughout this chapter...