2,179 research outputs found
A note on as a direct summand of nonstandard models of weak systems of arithmetic
There are nonstandard models of normal open induction () for which
is a direct summand of their additive group. We show that this is
impossible for nonstandard models of
F-rational rings and the integral closures of ideals
We show in this paper that the Briancon-Skoda theorem holds for all ideals in
F-rational rings of positive prime characteristic, and also in rings with
rational singularities which are of finite type over a field of characteristic
0.
Moreover, in Gorenstein F-rational rings of characteristic p we show that in
many cases the bound given in the Briancon-Skoda theorem may be improved by at
least one.Comment: This is a pre-galleys version of the paper, but theorem numbers
should be the same. 9 page
A Theorem of Legendre in
We prove a classical theorem due to Legendre, about the existence of non
trivial solutions of quadratic diophantine equations of the form
, in the weak fragment of Peano Arithmetic
Hierarchies of Subsystems of Weak Arithmetic
We completely characterize the logical hierarchy of various subsystems of
weak arithmetic, namely: ZR, ZR + N, ZR + GCD, ZR + Bez, OI + N, OI + GCD, OI +
Bez.Comment: To appear in Set theory, Arithmetic, Philosophy: Essays in Memory of
Stanley Tennenbaum (edited by J. Kennedy and R. Kossak), Cambridge University
Press
Counting points of schemes over finite rings and counting representations of arithmetic lattices
We relate the singularities of a scheme to the asymptotics of the number
of points of over finite rings. This gives a partial answer to a question
of Mustata. We use this result to count representations of arithmetic lattices.
More precisely, if is an arithmetic lattice whose -rank is
greater than one, let be the number of irreducible
-dimensional representations of up to isomorphism. We prove that
there is a constant (for example, suffices) such that
for every such . This answers a question of Larsen
and Lubotzky.Comment: version 2: The prove of the main theorem was simplified and a result
on arithmetic algebraic geometry was added. 29 page
Transcendental l-adic Galois representations
We consider continuous representations of the Galois group G of a number
field K taking values in the completion C of an algebraic closure A of the
field of l-adic numbers. We give a construction of irreducible representations
of G in GL(2,C) which cannot be conjugated into GL(2,A) but for which there
exist density-one sets of primes with Frobenius traces in A. We also show that
any such representation must be ramified at infinitely many primes of K.Comment: 21 page
Geometric proofs of theorems of Ax-Kochen and Ersov
We give an algebraic geometric proof of the Theorem of Ax and Kochen on
p-adic diophantine equations in many variables. Unlike Ax-Kochen's proof, ours
does not use any notions from mathematical logic and is based on weak
toroidalization of morphisms. We also show how this geometric approach yields
new proofs of the Ax-Kochen-Ersov transfer principle for local fields, and of
quantifier elimination theorems of Basarab and Pas.Comment: To appear in a special volume of the American Journal of Mathematics
dedicated to the memory of Professor Jun-ichi Igus
The local Langlands correspondence for GL_n in families
Let E be a nonarchimedean local field with residue characteristic l, and
suppose we have an n-dimensional representation of the absolute Galois group
G_E of E over a reduced complete Noetherian local ring A with finite residue
field k of characteristic p different from l. We consider the problem of
associating to any such representation an admissible A[GL_n(E)]-module in a
manner compatible with the local Langlands correspondence at characteristic
zero points of Spec A. In particular we give a set of conditions that uniquely
characterise such an A[GL_n(E)]-module if it exists, and show that such an
A[GL_n(E)]-module always exists when A is the ring of integers of a finite
extension of Q_p. We also use these results to define a "modified mod p local
Langlands correspondence" that is more compatible with specialization of Galois
representations than the mod p local Langlands correspondence of Vigneras.Comment: 61 paper
Macaulayfication of Noetherian schemes
To reduce to resolving Cohen-Macaulay singularities, Faltings initiated the
program of "Macaulayfying" a given Noetherian scheme . For a wide class of
, Kawasaki built the sought Cohen-Macaulay modifications, with a crucial
drawback that his blowups did not preserve the locus
where is already Cohen-Macaulay. We extend Kawasaki's methods to show that
every quasi-excellent, Noetherian scheme has a Cohen-Macaulay
with a proper map that is an
isomorphism over . This completes Faltings' program, reduces
the conjectural resolution of singularities to the Cohen-Macaulay case, and
implies that every proper, smooth scheme over a number field has a proper,
flat, Cohen-Macaulay model over the ring of integers.Comment: 24 pages; final version, to appear in Duke Mathematical Journa
As Easy as : Hilbert's Tenth Problem for Subrings of the Rationals and Number Fields
Hilbert's Tenth Problem over the field of rational numbers is one
of the biggest open problems in the area of undecidability in number theory. In
this paper we construct new, computably presentable subrings of
having the property that Hilbert's Tenth Problem for , denoted , is
Turing equivalent to .
We are able to put several additional constraints on the rings that we
construct. Given any computable nonnegative real number we construct
such a ring with a set of primes of lower
density . We also construct examples of rings for which deciding
membership in is Turing equivalent to deciding and also equivalent
to deciding . Alternatively, we can make have
arbitrary computably enumerable degree above . Finally, we show
that the same can be done for subrings of number fields and their prime ideals
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