If G is a non-cyclic finite group, non-isomorphic G-sets X, Y may give rise to isomorphic permutation representations CEquivalently, the map from the Burnside ring to the rational representation ring of G has a kernel. Its elements are called Brauer relations, and the purpose of this paper is to classify them in all finite groups, extending the Tornehave-Bouc classification in the case of p-groups.
Let F/k be a Galois extension of number fields with dihedral Galois group of order 2q, where q is an odd integer. We express a certain quotient of S -class numbers of intermediate fields, arising from Brauer-Kuroda relations, as a unit index. Our formula is valid for arbitrary extensions with Galois group D 2q and for arbitrary Galois-stable sets of primes S , containing the Archimedean ones. Our results have curious applications to determining the Galois module structure of the units modulo the roots of unity of a D 2q -extension from class numbers and S -class numbers. The techniques we use are mainly representation theoretic and we consider the representation theoretic results we obtain to be of independent interest.
Alex Bartel -Relations between class numbers in dihedral extensions
Let p be a prime number and M a quadratic number field, M Q( √ p) if p ≡ 1 mod 4. We will prove that for any positive integer d there exists a Galois extension F/Q with Galois group D 2p and an elliptic curve E/Q such that F contains M and the p-Selmer group of E/F has size at least p d .
We develop a theory of commensurability of groups, of rings, and of modules.
It allows us, in certain cases, to compare sizes of automorphism groups of
modules, even when those are infinite. This work is motivated by the
Cohen-Lenstra heuristics on class groups. The number theoretic implications
will be addressed in a separate paper.Comment: 26 pages; improved the introduction and the exposition in the body of
the paper. The final version is to appear in Compositio Mat
Abstract. Given a finite group G, a G-covering of closed Riemannian manifolds, and a so-called G-relation, a construction of Sunada produces a pair of manifolds M1 and M2 that are strongly isospectral. Such manifolds have the same dimension and the same volume, and their rational homology groups are isomorphic. Here, we investigate the relationship between their integral homology. The Cheeger-Müller Theorem implies that a certain product of orders of torsion homology and of regulators for M1 agrees with that for M2. We exhibit a connection between the torsion in the integral homology of M1 and M2 on the one hand, and the G-module structure of integral homology of the covering manifold on the other, by interpreting the quotients Reg i (M1)/ Reg i (M2) representation theoretically. Further, we prove that the p ∞ -torsion in the homology of M1 is isomorphic to that of M2 for all primes p #G. For p ≤ 71, we give examples of pairs of strongly isospectral hyperbolic 3-manifolds for which the p-torsion homology differs, and we conjecture such examples to exist for all primes p.
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