671 research outputs found
Purity for families of Galois representations
We formulate a notion of purity for -adic big Galois representations and
pseudorepresentations of Weil groups of -adic number fields for . This is obtained by showing that all powers of the monodromy of any big
Galois representation stay "as large as possible" under pure specializations.
The role of purity for families in the study of the variation of local Euler
factors, local automorphic types along irreducible components, the intersection
points of irreducible components of -adic families of automorphic Galois
representations is illustrated using the examples of Hida families and
eigenvarieties. Moreover, using purity for families, we improve a part of the
local Langlands correspondence for in families formulated by
Emerton and Helm.Comment: 25 pages, theorem 4.1 and 4.2 of version 1 are removed, simpler proof
of parts of theorem 4.3 of version 1 are give
On Frobenius semisimplicity in Hida families
Let be a prime and be a prime not dividing the tame
level of a -ordinary Hida family. We prove that the actions of the Frobenius
element at on the Galois representations attached to almost all
arithmetic specializations are semisimple and non-scalar. If denotes the
weight of a cusp form , then the
inequality predicted by the Ramanujan
conjecture, is a strict inequality for almost all members of the family
The density of ramified primes
Let be a number field, be a domain with fraction field
of characteristic zero and be a representation such that
is semisimple. If admits a
finite monomorphism from a power series ring with coefficients in a -adic
integer ring (resp. is an affinoid algebra over a -adic number
field) and is continuous with respect to the maximal ideal adic topology
(resp. the Banach algebra topology), then we prove that the set of ramified
primes of is of density zero. If is a complete local
Noetherian ring over with finite residue field of characteristic
, is continuous with respect to the maximal ideal adic topology and
the kernels of pure specializations of form a Zariski-dense subset of
, then we show that the set of ramified primes of
is of density zero. These results are analogues, in the context of big
Galois representations, of a result of Khare and Rajan, and are proved relying
on their result
Variation of Weyl modules in -adic families
Given a Weil-Deligne representation with coefficients in a domain, we prove
the rigidity of the structures of the Frobenius-semisimplifications of the Weyl
modules associated to its pure specializations. Moreover, we show that the
structures of the Frobenius-semisimplifications of the Weyl modules attached to
a collection of pure representations are rigid if these pure representations
lift to Weil-Deligne representations over domains containing a domain
and a pseudorepresentation over parametrizes the
traces of these lifts.Comment: 6 pages, comments welcom
Conductors in -adic families
Given a Weil-Deligne representation of the Weil group of an -adic
number field with coefficients in a domain , we show that its pure
specializations have the same conductor. More generally, we prove that the
conductors of a collection of pure representations are equal if they lift to
Weil-Deligne representations over domains containing and the
traces of these lifts are parametrized by a pseudorepresentation over
.Comment: arXiv admin note: text overlap with arXiv:1410.384
Lorentz Invariant CPT Violation for a Class of Nonlocal Thirring Model
It is possible to construct Lorentz invariant CPT violating models for
Nonlocal Quantum Field Theory. In this article, we present a class of Nonlocal
Thirring Models, in which the CPT invariance is violated while the Lorentz
invariance is present. As a result, in certain cases the mass-splitting between
particle and antiparticle are identified.Comment: 6 pages, 0 figures, Comments are welcom
Congruences of algebraic -functions of motives
We develop a framework to investigate conjectures on congruences between the
algebraic part of special values of -functions of congruent motives. We show
that algebraic local Euler factors satisfy precise interpolation properties in
-adic families of motives and that algebraic -adic -functions exist in
quite large generality for -adic families of automorphic motives. We
formulate two conjectures refining (and correcting) the currently existing
formulation of the Equivariant Tamagawa Number Conjecture with coefficients in
Hecke algebras and pointing out the links between conjectures on special values
and completed cohomology.Comment: 34 page
On minimal complements in groups
Let be nonempty subsets in an arbitrary group . The set
is said to be a complement to if and it is minimal if no
proper subset of is a complement to . We show that, if is finite
then every complement of has a minimal complement, answering a problem of
Nathanson. This also shows the existence of minimal -nets for every
in finitely generated groups. Further, we give necessary and
sufficient conditions for the existence of minimal complements of a certain
class of infinite subsets in finitely generated abelian groups, partially
answering another problem of Nathanson. Finally, we provide infinitely many
examples of infinite subsets of abelian groups of arbitrary finite rank
admitting minimal complements.Comment: Minor correction
Infinite co-minimal pairs in the integers and integral lattices
Given two nonempty subsets of a group , they are said to form a
co-minimal pair if , and for any
and for any
. In this article, we show several new results
on co-minimal pairs in the integers and the integral lattices. We prove that
for any , the group admits infinitely many
automorphisms such that for each such automorphism , there exists a
subset of such that and form a co-minimal
pair. The existence and construction of co-minimal pairs in the integers with
both the subsets and () of infinite cardinality was unknown.
We show that such pairs exist and explicitly construct these pairs satisfying a
number of algebraic properties
Quantum Restoration of Broken Symmetry in one Dimensional Loop Space
For 1 Dimensional loop space, a nonlinear nonlocal transformation of fields
is given to make the action of the self-interacting quantum field to the free
one. A specific type of Classically broken symmetry is restored in Quantum
theory. 1-D Sine Gordon system and Sech interactions are treated as the
explicit example.Comment: 5 pages, 0 figures, Comments are welcom
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