Alex Bartel

Brauer relations in finite groups

If GG is a non-cyclic finite group, non-isomorphic GG-sets X,YX,Y may give rise to isomorphic permutation representations C[X]≅C[Y]C[X]≅C[Y]. Equivalently, the map from the Burnside ring to the rational representation ring of GG 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 pp-groups.

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

Abstract 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.

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.

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.

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

Abstract. 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.

Large Selmer groups over number fields

Let p be a prime number and M a quadratic number field, M ≠ () if p ≡ 1 mod 4. We will prove that for any positive integer d there exists a Galois extension F/ with Galois group D2p and an elliptic curve E/ such that F contains M and the p-Selmer group of E/F has size at least pd.

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.

A note on Green functors with inflation

Abstract This note is motivated by the problem to understand, given a commutative ring F , which G -sets X , Y give rise to isomorphic F [ G ] -representations F [ X ] ≅ F [ Y ] . A typical step in such investigations is an argument that uses induction theorems to give very general sufficient conditions for all such relations to come from proper subquotients of G . In the present paper we axiomatise the situation, and prove such a result in the generality of Mackey functors and Green functors with inflation. Our result includes, as special cases, a result of Deligne on monomial relations, a result of the first author and Tim Dokchitser on Brauer relations in characteristic 0, and a new result on Brauer relations in characteristic p > 0 . We will need the new result in a forthcoming paper on Brauer relations in positive characteristic.

Commensurability of automorphism groups

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.

Rational representations and permutation representations of finite groups

We investigate the question which