A relative cohomological invariant for group pairs

1994 ◽  
Vol 83 (1) ◽  
pp. 1-18 ◽  
Author(s):  
M. G. C. Andrade ◽  
E. L. C. Fanti
Keyword(s):  
Cohomological Invariant

scholarly journals Decomposition of Splitting Invariants in Split Real Groups

2011 ◽  
Vol 63 (5) ◽  
pp. 1083-1106 ◽  
Author(s):  
Tasho Kaletha
Keyword(s):  
Local Field ◽  
Maximal Torus ◽  
Decomposition Theorem ◽  
Reduction Formula ◽  
Transfer Factors ◽  
Connected Group ◽  
Real Group ◽  
Cohomological Invariant ◽  
Simply Connected ◽  
The Given

Abstract For a maximal torus in a quasi-split semi-simple simply-connected group over a local field of characteristic 0, Langlands and Shelstad constructed a cohomological invariant called the splitting invariant, which is an important component of their endoscopic transfer factors. We study this invari- ant in the case of a split real group and prove a decomposition theorem which expresses this invariant for a general torus as a product of the corresponding invariants for simple tori. We also show how this reduction formula allows for the comparison of splitting invariants between different tori in the given real group.

scholarly journals Regulator constants of integral representations of finite groups

2018 ◽  
Vol 168 (1) ◽  
pp. 75-117 ◽  
Author(s):  
ALEX TORZEWSKI
Keyword(s):  
Finite Group ◽  
Elliptic Curves ◽  
Finite Groups ◽  
Isomorphism Class ◽  
Integral Representations ◽  
Dihedral Groups ◽  
Representation Ring ◽  
Representations Of Finite Groups ◽  
Cohomological Invariant ◽  
A Finite Group

AbstractLet G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$[G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$[G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.

scholarly journals Motives for perfect PAC fields with pro-cyclic Galois group

2008 ◽  
Vol 73 (3) ◽  
pp. 1036-1050
Author(s):  
Immanuel Halupczok
Keyword(s):  
Galois Group ◽  
Finite Fields ◽  
Grothendieck Ring ◽  
Additional Information ◽  
Chow Motives ◽  
Closed Fields ◽  
Cohomological Invariant ◽  
Definable Sets ◽  
Grothendieck Rings ◽  
Algebraically Closed Fields

AbstractDenef and Loeser denned a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group.In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of Denef-Loeser is not injective.

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