2,590 research outputs found
Rigid G2-Representations and motives of Type G2
We prove an effective Hilbert Irreducibility result for residual realizations
of a family of motives with motivic Galois group G2
On Serre's uniformity conjecture for semistable elliptic curves over totally real fields
Let be a totally real field, and let be a finite set of
non-archimedean places of . It follows from the work of Merel, Momose and
David that there is a constant so that if is an elliptic curve
defined over , semistable outside , then for all , the
representation is irreducible. We combine this with
modularity and level lowering to show the existence of an effectively
computable constant , and an effectively computable set of elliptic
curves over with CM such that the following holds. If
is an elliptic curve over semistable outside , and is prime,
then either is surjective, or for some .Comment: 7 pages. Improved version incorporating referee's comment
A multi-Frey approach to Fermat equations of signature
In this paper, we give a resolution of the generalized Fermat equations for all integers , and all integers which are not a multiple of , respectively,
using the modular method with Frey elliptic curves over totally real fields.
The results require a refined application of the multi-Frey technique, which we
show to be effective in new ways to reduce the bounds on the exponents .
We also give a number of results for the equations , where
, under additional local conditions on the solutions. This includes a
result which is reminiscent of the second case of Fermat's Last Theorem, and
which uses a new application of level raising at modulo .Comment: Includes more details regarding the connection of this paper with its
sequel 'Some extensions of the modular method and Fermat-equations of
signature (13,13,n)'. More precisely: extended Remark 7.4; added details on
the computational parts of the proofs of Proposition 9 and Theorem 2;
included new comments and polished the auxiliary Magma files for Proposition
9 and Theorem
Decompositions of Generalized Wavelet Representations
Let be a simply connected, connected nilpotent Lie group which admits a
uniform subgroup Let be an automorphism of defined by
We assume that the linear action of
is diagonalizable and we do not assume that is commutative. Let be a
unitary wavelet representation of the semi-direct product group defined by and We obtain a decomposition of
into a direct integral of unitary representations. Moreover, we provide an
explicit unitary operator intertwining the representations, a precise
description of the representations occurring, the measure used in the direct
integral decomposition and the support of the measure. We also study the
irreducibility of the fiber representations occurring in the direct integral
decomposition in various settings. We prove that in the case where is an
expansive automorphism then the decomposition of is in fact a direct
integral of unitary irreducible representations each occurring with infinite
multiplicities if and only if is non-commutative. This work naturally
extends results obtained by H. Lim, J. Packer and K. Taylor who obtained a
direct integral decomposition of in the case where is commutative and
the matrix is expansive, i.e. all eigenvalues have absolute values larger
than one
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