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1. CJM Online first
Unitary Eigenvarieties at Isobaric Points In this article we
study the geometry of the eigenvarieties of unitary groups at points
corresponding to tempered non-stable representations with an
anti-ordinary (a.k.a evil) refinement. We prove that, except in the
case the Galois representation attached to the automorphic form is a
sum of characters, the eigenvariety is non-smooth at such a point,
and that (under some additional hypotheses) its tangent space is big
enough to account for all the relevant Selmer group. We also study the
local reducibility locus
at those points, proving that in general, in contrast with the case of
the eigencurve, it is a proper subscheme of the fiber of the
eigenvariety over the weight space.
Keywords:eigenvarieties, Galois representations, Selmer groups |
2. CJM 2013 (vol 66 pp. 1167)
Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves The purpose of this note is introducing a method for proving the
existence of no rational points on a coarse moduli space $X$ of abelian varieties
over a given number field $K$, in cases where the moduli problem is not fine and
points in $X(K)$ may not be represented by an abelian variety (with additional structure)
admitting a model over the field $K$. This is typically the case when the abelian
varieties that are being classified have even dimension. The main idea, inspired on
the work of Ellenberg and Skinner on the modularity of $\mathbb{Q}$-curves, is that to a
point $P=[A]\in X(K)$ represented by an abelian variety $A/\bar K$ one may still
attach a Galois representation of $\operatorname{Gal}(\bar K/K)$ with values in the quotient
group $\operatorname{GL}(T_\ell(A))/\operatorname{Aut}(A)$, provided
$\operatorname{Aut}(A)$ lies in the centre of $\operatorname{GL}(T_\ell(A))$.
We exemplify our method in the cases where $X$ is a Shimura curve over an imaginary
quadratic field or an Atkin-Lehner quotient over $\mathbb{Q}$.
Keywords:Shimura curves, rational points, Galois representations, Hasse principle, Brauer-Manin obstruction Categories:11G18, 14G35, 14G05 |
3. CJM 2011 (vol 63 pp. 616)
A Modular Quintic Calabi-Yau Threefold of Level 55 In this note we search the parameter space of Horrocks-Mumford quintic
threefolds and locate a Calabi-Yau threefold that is modular, in the
sense that the $L$-function of its middle-dimensional cohomology is
associated with a classical modular form of weight 4 and level 55.
Keywords: Calabi-Yau threefold, non-rigid Calabi-Yau threefold, two-dimensional Galois representation, modular variety, Horrocks-Mumford vector bundle Categories:14J15, 11F23, 14J32, 11G40 |
4. CJM 2008 (vol 60 pp. 208)
Constructing Galois Representations with Very Large Image Starting with a 2-dimensional mod $p$ Galois representation, we
construct a deformation to a power series ring in infinitely many
variables over the $p$-adics. The image of this representation is full
in the sense that it contains $\SL_2$ of this power series
ring. Furthermore, all ${\mathbb Z}_p$ specializations of this
deformation are potentially semistable at $p$.
Keywords:Galois representation, deformation Category:11f80 |