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Tame class field theory for arithmetic schemes
We extend the unramified class field theory for arithmetic schemes of K. Kato
and S. Saito to the tame case. Let be a regular proper arithmetic scheme
and let be a divisor on whose vertical irreducible components are
normal schemes.
Theorem: There exists a natural reciprocity isomorphism \rec_{X,D}:
\CH_0(X,D) \liso \tilde \pi_1^t(X,D)^\ab\. Both groups are finite.
This paper corrects and generalizes my paper "Relative K-theory and class
field theory for arithmetic surfaces" (math.NT/0204330
On quadratic progression sequences on smooth plane curves
We study the arithmetic (geometric) progressions in the -coordinates of
quadratic points on smooth projective planar curves defined over a number field
. Unless the curve is hyperelliptic, we prove that these progressions must
be finite. We, moreover, show that the arithmetic gonality of the curve
determines the infinitude of these progressions in the set of
-points with field of definition of degree at most ,
The basic geometry of Witt vectors, II: Spaces
This is an account of the algebraic geometry of Witt vectors and arithmetic
jet spaces. The usual, "p-typical" Witt vectors of p-adic schemes of finite
type are already reasonably well understood. The main point here is to
generalize this theory in two ways. We allow not just p-typical Witt vectors
but those taken with respect to any set of primes in any ring of integers in
any global field, for example. This includes the "big" Witt vectors. We also
allow not just p-adic schemes of finite type but arbitrary algebraic spaces
over the ring of integers in the global field. We give similar generalizations
of Buium's formal arithmetic jet functor, which is dual to the Witt functor. We
also give concrete geometric descriptions of Witt spaces and arithmetic jet
spaces and investigate whether a number of standard geometric properties are
preserved by these functors.Comment: Final versio
A lattice in more than two Kac--Moody groups is arithmetic
Let be an irreducible lattice in a product of n infinite irreducible
complete Kac-Moody groups of simply laced type over finite fields. We show that
if n is at least 3, then each Kac-Moody groups is in fact a simple algebraic
group over a local field and is an arithmetic lattice. This relies on
the following alternative which is satisfied by any irreducible lattice
provided n is at least 2: either is an S-arithmetic (hence linear)
group, or it is not residually finite. In that case, it is even virtually
simple when the ground field is large enough.
More general CAT(0) groups are also considered throughout.Comment: Subsection 2.B was modified and an example was added ther
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