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1. CJM 2011 (vol 64 pp. 282)
Level Lowering Modulo Prime Powers and Twisted Fermat Equations We discuss a clean level lowering theorem modulo prime powers
for weight $2$ cusp forms.
Furthermore, we illustrate how this can be used to completely
solve certain twisted Fermat equations
$ax^n+by^n+cz^n=0$.
Keywords:modular forms, level lowering, Diophantine equations Categories:11D41, 11F33, 11F11, 11F80, 11G05 |
2. CJM 2010 (vol 63 pp. 277)
Locally Indecomposable Galois Representations
In a previous paper
the authors showed that, under some technical
conditions,
the local Galois representations attached to the members of
a non-CM family of ordinary cusp forms are indecomposable for all
except possibly finitely many
members of the family. In this paper we use deformation theoretic
methods to give examples of non-CM families for
which every classical member of weight at least two has a locally
indecomposable Galois representation.
Category:11F80 |
3. CJM 2008 (vol 60 pp. 1028)
Lifting $n$-Dimensional Galois Representations We investigate the problem of deforming $n$-dimensional mod $p$ Galois
representations to characteristic zero. The existence of 2-dimensional
deformations has been proven under certain conditions
by allowing ramification at additional primes in order to
annihilate a dual Selmer group. We use the same general methods to
prove the existence of $n$-dimensional deformations.
We then examine under which conditions we may place restrictions on
the shape of our deformations at $p$, with the goal of showing that
under the correct conditions, the deformations may have locally
geometric shape. We also use the existence of these deformations to
prove the existence as Galois groups over $\Q$ of certain infinite
subgroups of $p$-adic general linear groups.
Category:11F80 |
4. CJM 2008 (vol 60 pp. 491)
A Multi-Frey Approach to Some Multi-Parameter Families of Diophantine Equations We solve several multi-parameter families of binomial Thue equations of arbitrary
degree; for example, we solve the equation
\[
5^u x^n-2^r 3^s y^n= \pm 1,
\]
in non-zero integers $x$, $y$ and positive integers $u$, $r$, $s$ and $n \geq 3$.
Our approach uses several Frey curves simultaneously, Galois representations
and level-lowering, new lower bounds for linear
forms in $3$ logarithms due to Mignotte and a famous theorem of Bennett on binomial
Thue equations.
Keywords:Diophantine equations, Frey curves, level-lowering, linear forms in logarithms, Thue equation Categories:11F80, 11D61, 11D59, 11J86, 11Y50 |
5. CJM 2006 (vol 58 pp. 1203)
Orbites unipotentes et pÃ´les d'ordre maximal de la fonction $\mu $ de Harish-Chandra Dans un travail ant\'erieur, nous
avions montr\'e que l'induite parabolique (normalis\'ee) d'une
repr\'esentation irr\'eductible cuspidale $\sigma $ d'un
sous-groupe de Levi $M$ d'un groupe $p$-adique contient un
sous-quotient de carr\'e int\'egrable, si et seulement si la
fonction $\mu $ de Harish-Chandra a un p\^ole en $\sigma $ d'ordre
\'egal au rang parabolique de $M$. L'objet de cet article est
d'interpr\'eter ce r\'esultat en termes de fonctorialit\'e de
Langlands.
Categories:11F70, 11F80, 22E50 |
6. CJM 2005 (vol 57 pp. 1215)
Reciprocity Law for Compatible Systems of Abelian $\bmod p$ Galois Representations The main result of the paper
is a {\em reciprocity law} which proves that
compatible systems of semisimple, abelian mod $p$ representations
(of arbitrary dimension)
of absolute Galois groups of number fields, arise from Hecke characters.
In the last section analogs for Galois groups of function fields of these
results are explored, and a question is raised whose answer seems to
require developments in transcendence theory in characteristic $p$.
Category:11F80 |