352 research outputs found
Serre's Modularity Conjecture
These are the lecture notes from a five-hour mini-course given at the Winter
School on Galois Theory held at the University of Luxembourg in February 2012.
Their aim is to give an overview of Serre's modularity conjecture and of its
proof by Khare, Wintenberger, and Kisin, as well as of the results of other
mathematicians that played an important role in the proof. Along the way we
remark on some recent (as of 2012) work concerning generalizations of the
conjecture
Essential dimension of abelian varieties over number fields
We affirmatively answer a conjecture in the preprint ``Essential dimension
and algebraic stacks,'' proving that the essential dimension of an abelian
variety over a number field is infinite.Comment: 4 pages. To appear in C. R. Math. Acad. Sci. Paris. Preprint posted
earlier to http://www.mathematik.uni-bielefeld.de/LAG
On the Selmer groups of abelian varieties over function fields of characteristic p>0
In this paper, we study a (p-adic) geometric analogue for abelian varieties
over a function field of characteristic p of the cyclotomic Iwasawa theory and
the non-commutative Iwasawa theory for abelian varieties over a number field
initiated by Mazur and Coates respectively. We will prove some analogue of the
principal results obtained in the case over a number field and we study new
phenomena which did not happen in the case of number field case. We propose
also a conjecture which might be considered as a counterpart of the principal
conjecture in the case over a number field. \par This is a preprint which is
distributed since 2005 which is still in the process of submision. Following a
recent modification of some technical mistakes in the previous version of the
paper as well as an amelioration of the presentation of the paper, we decide
wider distribution via the archive.Comment: 21 page
Rank 2 Local Systems, Barsotti-Tate Groups, and Shimura Curves
We develop a descent criterion for -linear abelian categories. Using
recent advances in the Langlands correspondence due to Abe, we build a
correspondence between certain rank 2 local systems and certain Barsotti-Tate
groups on complete curves over a finite field. We conjecture that such
Barsotti-Tate groups "come from" a family of fake elliptic curves. As an
application of these ideas, we provide a criterion for being a Shimura curve
over . Along the way, we formulate a conjecture on the
field-of-coefficients of certain compatible systems.Comment: 30 pages. Part of author's PhD thesis. Comments welcome
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