9 research outputs found
On the genera of semisimple groups defined over an integral domain of a global function field
Let be the global function field of rational functions
over a smooth and projective curve defined over a finite field
. The ring of regular functions on where
is any finite set of closed points on is a Dedekind domain
of . For a semisimple -group with a smooth
fundamental group , we aim to describe both the set of genera of
and its principal genus (the latter if is isotropic at ) in terms of abelian groups
depending on and only. This leads to a
necessary and sufficient condition for the Hasse local-global principle to hold
for certain . We also use it to express the Tamagawa number
of a semisimple -group by the Euler Poincar\'e invariant. This
facilitates the computation of for twisted -groups.Comment: 18 page
On the flat cohomology of binary norm forms
Let be an order of index in the maximal order of a
quadratic number field . Let
be the orthogonal -group of the
associated norm form . We describe the structure of the pointed set
, which classifies
quadratic forms isomorphic (properly or improperly) to in the flat
topology. Gauss classified quadratic forms of fundamental discriminant and
showed that the composition of any binary -form of discriminant
with itself belongs to the principal genus. Using cohomological
language, we extend these results to forms of certain non-fundamental
discriminants.Comment: 24 pages, submitted. Comments are welcom
The Discriminant of an Algebraic Torus
For a torus T defined over a global field K, we revisit an analytic class
number formula obtained by Shyr in the 1970's as a generalization of
Dirichlet's class number formula. We prove a local-global presentation of the
quasi-discriminant of T, which enters into this formula, in terms of
cocharacters of T. This presentation can serve as a more natural definition of
this invariant.Comment: 17 page