A classification of exceptional components in group algebras over abelian number fields

Abstract

When considering the unit group of O F G (O F the ring of integers of an abelian number field F and a finite group G) certain components in the Wedderburn decomposition of F G cause problems for known generic constructions of units; these components are called exceptional. Exceptional components are divided into two types: type 1 are division rings, type 2 are 2 × 2-matrix rings. For exceptional components of type 1 we provide infinite classes of division rings by describing the seven cases of minimal groups (w.r.t. quotients) having those division rings in their Wedderburn decomposition over F . We also classify the exceptional components of type 2 appearing in group algebras of a finite group over number fields F by describing all 58 finite groups G having a faithful exceptional Wedderburn component of this type in F G.

A classification of the automorphism groups of polarized abelian threefolds over finite fields

“…(3) T * × G m,r where T * is the binary tetrahedral group of order 24, and G m,r is either cyclic of order m with gcd(m, 6) = 1, or of type (2) with gcd(|G m,r |, 6) = 1 (where |G m,r | denotes the order of the group G m,r ). In both cases, for all primes p | m, the smallest integer γ p satisfying 2 γp ≡ 1 (mod p) is odd.…”

“…We note that the answer for Problem 1.1 (2) gives a (partial) answer for Problem 1.1 (1). For the case when k = C, Birkenhake, González and Lange [3] computed all finite automorphism groups of complex tori of dimension 3, which are maximal in the isogeny class.…”

“…For the case when k = C, Birkenhake, González and Lange [3] computed all finite automorphism groups of complex tori of dimension 3, which are maximal in the isogeny class. The goal of this paper is to give an almost complete answer for Problem 1.1 (2) for the case when k is a finite field, by classifying all finite groups G that can be realized as the automorphism group Aut k (X, L) of a polarized abelian threefold (X, L) over a finite field k, which are maximal in the following sense: there is no finite group H such that G is isomorphic to a proper subgroup of H and H = Aut k (Y, M) for some abelian threefold Y over k that is k-isogenous to X with a polarization M.…”

Automorphism groups of simple polarized abelian varieties of odd prime dimension over finite fields

“…(3) T * × G m,r where T * is the binary tetrahedral group of order 24, and G m,r is either cyclic of order m with gcd(m, 6) = 1, or of the type (2) with gcd(|G m,r |, 6) = 1. In both cases, for all primes p | m, the smallest integer γ p satisfying 2 γp ≡ 1 (mod p) is odd.…”

“…Let X be a simple abelian variety of dimension g over a finite field k. Then it is well known that End 0 k (X) is a division algebra over Q with 2g ≤ dim Q End 0 k (X) < (2g) 2 . Before giving our first result, we also recall Albert's classification.…”