Factor equivalence of Galois modules and regulator constants

We compare two approaches to the study of Galois module structures: on the one hand factor equivalence, a technique that has been used by Fr\"ohlich and others to investigate the Galois module structure of rings of integers of number fields and of their unit groups, and on the other hand regulator constants, a set of invariants attached to integral group representations by Dokchitser and Dokchitser, and used by the author, among others, to study Galois module structures. We show that the two approaches are in fact closely related, and interpret results arising from these two approaches in terms of each other. We also use this comparison to derive a factorisability result on higher $K$-groups of rings of integers, which is a direct analogue of a theorem of de Smit on $S$-units.Comment: Minor corrections and some more details added in proofs; 11 pages. Final version to appear in Int. J. Number Theor

Brauer relations in finite groups

If G is a non-cyclic finite group, non-isomorphic G-sets X, Y may give rise to isomorphic permutation representations C[X] and C[Y]. Equivalently, the map from the Burnside ring to the 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.Comment: 39 pages; final versio

Relations between permutation representations in positive characteristic

Given a finite group G and a field F, a G-set X gives rise to an F[G]-permutation module F[X]. This defines a map from the Burnside ring of G to its representation ring over F. It is an old problem in representation theory, with wide-ranging applications in algebra, number theory, and geometry, to give explicit generators of the kernel K_F(G) of this map, i.e. to classify pairs of G-sets X, Y such that F[X] is isomorphic to F[Y]. When F has characteristic 0, a complete description of K_F(G) is now known. In this paper, we give a similar description of K_F(G) when F is a field of characteristic p>0 in all but the most complicated case, which is when G has a subquotient that is a non-p-hypo-elementary (p,p)-Dress group.Comment: 18 pages; minor corrections and improvements. Final version to appear in Bull. London Math. So

Torsion homology and regulators of isospectral manifolds

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 M_1 and M_2 that are strongly isospectral. Such manifolds have the same dimension and the same volume, and their rational homology groups are isomorphic. We investigate the relationship between their integral homology. The Cheeger-Mueller Theorem implies that a certain product of orders of torsion homology and of regulators for M_1 agrees with that for M_2. We exhibit a connection between the torsion in the integral homology of M_1 and M_2 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(M_1)/Reg_i(M_2) representation theoretically. Further, we prove that the p-primary torsion in the homology of M_1 is isomorphic to that of M_2 for all primes p not dividing #G. For p <= 71, we give examples of pairs of isospectral hyperbolic 3-manifolds for which the p-torsion homology differs, and we conjecture such examples to exist for all primes p.Comment: 21 pages; minor changes; included a data file; to appear in J. Topolog

On Brauer–Kuroda type relations of S-class numbers in dihedral extensions

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 D2q 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 D2q-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

Elliptic curves with p-Selmer growth for all p

It is known that, for every elliptic curve over ℚ, there exists a quadratic extension in which the rank does not go up. For a large class of elliptic curves, the same is known with the rank replaced by the size of the 2-Selmer group. We show, however, that there exists a large supply of semistable elliptic curves E/ℚ whose 2-Selmer group grows in size in every bi-quadratic extension, and such that, moreover, for any odd prime p, the size of the p-Selmer group grows in every D2p-extension and every elementary abelian p-extension of rank at least 2. We provide a simple criterion for an elliptic curve over an arbitrary number field to exhibit this behaviour. We also discuss generalizations to other Galois groups

Galois module structure of oriented Arakelov class groups

We show that Chinburg's Omega(3) conjecture implies tight restrictions on the Galois module structure of oriented Arakelov class groups of number fields. We apply our findings to formulating a probabilistic model for Arakelov class groups in families, offering a correction of the Cohen--Lenstra--Martinet heuristics on ideal class groups.Comment: 14 pages; comments welcom

Brauer relations in finite groups II : quasi-elementary groups of order paq

This is the second in a series of papers investigating the space of Brauer relations of a finite group, the kernel of the natural map from its Burnside ring to the rational representation ring. The first paper classified all primitive Brauer relations, that is those that do not come from proper subquotients. In the case of quasi-elementary groups the description is intricate, and it does not specify groups that have primitive relations in terms of generators and relations. In this paper we provide such a classification in terms of generators and relations for quasi-elementary groups of order paq

Index formulae for integral Galois modules

We prove very general index formulae for integral Galois modules, specifically for units in rings of integers of number fields, for higher K-groups of rings of integers, and for Mordell-Weil groups of elliptic curves over number fields. These formulae link the respective Galois module structure to other arithmetic invariants, such as class numbers, or Tamagawa numbers and Tate-Shafarevich groups. This is a generalisation of known results on units to other Galois modules and to many more Galois groups, and at the same time a unification of the approaches hitherto developed in the case of units.Comment: 14 pages; final versio