Lecture Series
Freydoon Shahidi - An Overview of Functoriality
It was almost 40 years ago when Langlands formulated his
striking ideas on the theory of automorphic forms under, by now, the
catchphrase "Functoriality Conjecture". This remarkably deep
conjecture has many consequences in number theory and arithmetic
geometry and in its simplest form implies very deep results such as
Artin's reciprocity law for abelian classfield theory. There has been
quite a bit of activity in establishing special cases of the conjecture
recently which nevertheless have far reaching consequences that cannot be
established by other means. The purpose of these lectures is to explain
the conjecture and the progress made on it to the experts in neighbouring
fields who are not experts in automorphic forms.
Here is a tentative outline of the lectures:
1. Generalities:Reductive groups over number fields and their
representations, Satake isomorphism.
2. L-groups and L-functions, statement of functoriality.
3. Functoriality through examples:Different transfers and endoscopy.
4. Methods (L-functions and converse theorems, trace formula) and
consequences. What's next?!
Akio Tamagawa - Recent progress in anabelian geometry
The `yoga' of anabelian geometry was initiated by A. Grothendieck
in the early 1980s. A fundamental idea is that for an `anabelian'
scheme over a field (finitely generated over the prime field),
the geometry is completely determined by its \'etale fundamental group,
equipped with the structure homomorphism to the absolute Galois group
of the base field.
In this lecture series, I will first review briefly
-- theory of \'etale fundamental groups of schemes,
-- basic formulation of anabelian geometry,
-- `classical' results (up to 1990s),
and then survey some recent progress concerning the following topics:
-- section conjecture,
-- anabelian properties of moduli spaces of curves,
-- absolute p-adic anabelian geometry,
-- theory of cuspidalizations,
-- algebraically closed base fields in positive characteristic,
-- relation with diophantine geometry,
-- etc.
Jean-Pierre Wintenberger - On Serre's modularity conjecture
The conjecture states that every odd 2-dimensional absolutely irreducible
representation of the Galois group of Q with values in a finite
field arises from a modular form. I will discuss
a proof of the conjecture obtained in collaboration
with Chandrashekhar Khare and previous works of many people
on which our work relies. I think the different parts will be :
- The statement of the conjecture and some consequences
- Non-existence of low ramified Galois representations
- "Lifting Modularity Theorems" and "Lifting Galois
representations with control on ramification"
- Potential Modularity Theorem.
- The induction.
Other Talks
John Coates - Non-commutative Iwasawa theory
My lecture will discuss some curious links between the p-adic arithmetic
world and complex L-functions which are suggested by non-commutative
Iwasawa theory, and some frgementary results one can prove in this
direction.
Makoto Matsumoto - Relative pro-l completion of fundamental groups
(joint work with Richard Hain).
The fundamental group of an algebraic variety over Q
is acted by the absolute Galois group of Q. This Galois
representation is (highly) nonabelian. To obtain similarities
to the abelian representation, pro-l completion and/or
prounipotent completion of the fundamental group is used.
When we consider the moduli stack M of (g,n) curves, one difficulty
is that its fundamental group (i.e. the mapping class group)
becomes trivial if we take pro-l completion or pro-unipotent completion
for g>2.
To avoid this, we need to consider relative pro-l completions
and/or relative prounipotent completion of the fundamental group
of M. We shall explain reasons why these completions fit in the
study of the universal representation on the fundamental group
of the curves. Using these notion, we prove that the pro-l
completion of the Torelli subgroup of the mapping class group
does not inject to the automorphism of the pro-l completion
of the fundamental groups of curves.
Jeehoon Park - The Eisenstein-Siegel distribution
Shuji Saito - A counterexample of a variant of the Bloch-Kato conjecture
and an application to algebraic cycles
This is a joint work with M. Asakura.
In this talk I would like to demonstrate how Hodge theory can play a crucial
role in an arithmetic question. The issue is to disprove a variant of the
Block-Kato conjecture which characterizes the image of an $\ell$-adic
regulator map from higher Chow groups to continuous \'etale cohomology of
a projective smooth variety over a $p$-adic field by using $p$-adic
Hodge theory (the original conjecture is stated for a variety over a number
field).
By aid of theory of mixed Hodge modules due to M. Saito,
the problem is reduced to showing the exactness of de Rham complex
associated to a certain variation of Hodge structure, which is shown by
the infinitesimal method in Hodge theory.
As an application, we construct an example of a projective smooth surface
$X$ over a $p$-adic field such that for any prime $\ell\not=p$, the $\ell$-primary
torsion subgroup of $CH_0(X)$, the Chow group of $0$-cycles on $X$, is
infinite.
Sugwoo Shin - Endoscopy and Shimura Varieties
The global Langlands correspondence is constructed in many cases using the cohomology of certain simple PEL-type Shimura varieties. To carry this out in a slightly more general case, we have to deal with endoscopy, which is perhaps the main technical difficulty. In this talk we briefly review some aspects of endoscopy theory and explain how it helps us construct Galois representations from automorphic representations of GL(n) satisfying appropriate conditions.
Ramadorai Sujatha - Iwasawa theory and modular forms
We shall discuss some recent results on
noncommutative Iwasawa theory for the Galois representations
associated to modular forms and Hida families.
Daqing Wan - Moment zeta functions
I shall discuss the basic properties and examples
of moment zeta functions, which measure the variation of
zeta functions attached to a family of varieties.
Andrei Yafaev - Manin-Mumford and Andre-Oort conjectures : a unified approach
Chia Fu Yu - Subvarieties of Hilbert-Blumenthal varieties
In this talk we will outline the progress on geometry of
Hilbert-Blumenthal varieties modulo a prime p. Then we will present new
results on global properties of subvarieties such as Lie-alpha strata and
Newton polygon strata of a Hilbert-Blumenthal variety where p is maximally
ramified in the totally real field.
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