Nonabelian Fundamental Groups in Arithmetic Geometry  Introductory Workshop
Monday 27th July 2009 to Friday 31st July 2009
09:00 to 09:55  Registration  
09:55 to 10:00  Welcome  David Wallace (INI Director)  
10:00 to 11:00 
P Deligne ([IAS]) Counting ladic representations, in the function field case
We will explain some countings similar to the one Drinfeld did in 1981, and wonder what they mean.

INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30 
On the noncommutative Iwasawa Main Conjecture for CMelliptic curves
We discuss under which assumptions the (commutative) 2variable Main Conjecture for CMelliptic curves (due to Rubin, Yager, Katz etc.) implies the noncommutative Main Conjecture as formulated together with Coates, Fukaya, Kato and Sujatha.
Related Links
•http://www.mathi.uniheidelberg.de/~otmar/  personal homepage

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
Grothendieck's Section Conjecture and zerocycles on varieties
After some background material on Grothendieck's Section Conjecture, we discuss an obstruction for the existence of splittings of the abelianized homotopy exact sequence for the étale fundamental group. As an application, we explain how to find examples for smooth projective curves over Q that have points everywhere locally but the homotopy exact sequence does not split. This is joint work with David Harari, with explicit examples contributed by Victor Flynn.

INI 1  
15:00 to 15:30  Tea  
15:30 to 16:30  On the padic local Langlands correspondence for $GL_2({\bf Q}_p)$  INI 1  
16:30 to 17:30  Discussion  
17:30 to 18:30  Welcome Wine Reception  
18:45 to 19:30  Dinner at Wolfson Court (Residents Only) 
10:00 to 11:00 
J Coates ([Cambridge]) Iwasawa theory of elliptic curves with complex multiplication
We shall discuss, and prove in the very simplest case, one of the conjectures made in a previous joint paper with Fukaya, Kato, Sujatha, and Venjakob about the dual Selmer group of elliptic curves over those padic Lie extensions of the base field F which contain the cyclotomic Zpextension of F. The results discussed are joint work with Sujatha.

INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30 
Reciprocity maps and Selmer groups
This talk concerns certain homomorphisms that arise in the study of Galois cohomology with restricted ramification. Given a set S of primes of a number field containing all those above a given prime p, the Sreciprocity map is a homomorphism on Sunits that interpolates values of a cup product with those Sunits. We will discuss the properties of and connections between this and related homomorphisms, and study their application to Selmer groups of reducible representations. Finally, we will explore a connection with a conjecture of the author on the relationship between these maps for cyclotomic fields and a modular twovariable padic Lfunction, taken modulo an Eisenstein ideal.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
Obstructions to homotopy sections of curves over number fields
Grothendieck's section conjecture is analogous to equivalences between fixed points and homotopy fixed points of Galois actions on related topological spaces. We use cohomological obstructions of Jordan Ellenberg coming from nilpotent approximations to the curve to study the sections of etale pi_1 of the structure map. We will relate Ellenberg's obstructions to Massey products, and explicitly compute mod 2 versions of the first and second for P^1{0,1,infty} over Q. Over R, we show the first obstruction alone determines the connected components of real points of the curve from those of the Jacobian.

INI 1  
15:00 to 15:30  Tea  
15:30 to 16:30 
Potential automorphy of n dimensional Galois representations
I will discuss recent improvements in the potential automorphy theorems available for Galois representations of any dimension. In particular I will discuss the case of ordinary Galois representations and applications to elliptic modular forms, in particular the proof of the SatoTate conjecture for all elliptic modular new forms.

INI 1  
16:30 to 17:30  Discussion  
18:45 to 19:30  Dinner at Wolfson Court (Residents Only) 
10:00 to 11:00 
On the birational padic section conjecture
I plan to explain Grothendieck's section conjecture, which relates rational points of (completions of) hyperbolic curves to conjugacy classes of sections of the canonical projection between fundamental groups. I will explain a few variants of this conjecture (birational, padic), and finally discuss the status of the art of the conjecture.

INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30 
On the section conjecture for universal curves over function fields
In this talk I will discuss a version of Grothendieck's Section Conjecture for the universal curve over the function field of the moduli space of curves type (g,n) with a level m structure.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 17:00  Excursion  
18:45 to 19:30  Dinner at Wolfson Court (Residents Only) 
10:00 to 11:00 
M Emerton ([Northwestern]) padically completed cohomology and the padic Langlands program
Speaking at a general level, a major goal of the padic Langlands program (from a global, rather than local, perspective) is to find a padic generalization of the notion of automorphic eigenform, the hope being that every padic global Galois representation will correspond to such an object. (Recall that only those Galois representations that are motivic, i.e. that come from geometry, are expected to correspond to classical automorphic eigenforms).
In certain contexts (namely, when one has Shimura varieties at hand), one can begin with a geometric definition of automorphic forms, and generalize it to obtain a geometric definition of padic automorphic forms. However, in the nonShimura variety context, such an approach is not available. Furthermore, this approach is somewhat remote from the representationtheoretic point of view on automorphic forms, which plays such an important role in the classical Langlands program.
In this talk I will explain a different, and very general, approach to the problem of padic interpolation, via the theory of padically completed cohomology. This approach has close ties to the padic and mod p representation theory of padic groups, and to noncommutative= Iwasawa theory.
After introducing the basic objects (namely, the padically completed cohomology spaces attached to a given reductive group), I will explain several key conjectures that we expect to hold, including the conjectural relationship to Galois deformation spaces. Although these conjectures seem out of reach at present in general, some progress has been made towards them in particular cases. I will describe some of this progress, and along the way will introduce some of the tools that we have developed for studying padically completed cohomology, the most important of these being the Poincare duality spectral sequence.
This is joint work with Frank Calegari.

INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30 
The cuspidalization of sections of arithmetic fundamental groups
All results presented in this talk are part of a joint work with Akio Tamagawa. We introduce the problem of cuspidalization of sections of arithmetic fundamental groups which relates the Grothendieck section conjecture to its birational analog. We exhibit a necessary condition for a section of the arithmetic fundamental group of a hyperbolic curve to arise from a rational point which we call the goodness condition.
We prove that good sections of arithmetic fundamental groups of hyperbolic curves can be lifted to sections of the maximal cuspidally abelian Galois group of the function field of the curve (under quite general assumptions). As an application we prove a (geometrically prop) padic local version of the Grothendieck section conjecture under the assumption that the existence of sections of cuspidally (prop) abelian arithmetic fundamental groups implies the existence of tame points. We also prove, using cuspidalization techniques, that for a hyperbolic curve X over a padic local field and a set of points S of X which is dense in the padic topology every section of the arithmetic fundamental group of U=X\S arises from a rational point. As a corollary we deduce that the existence of a section of the absolute Galois group of a function field of a curve over a number field implies that the set of adelic points of the curve is nonempty.

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
The structure of Galois groups and birational algebraic geometry
In the talk I will discuss the relation between recent results on the structure of Galois groups for functional fields and birational geometry of algebraic varieties. Most of the results discussed in the talk are obtained in joint work with Yuri Tschinkel

INI 1  
15:00 to 15:30  Tea  
15:30 to 16:30 
Annihilating TateShafarevic groups
We describe how main conjectures in noncommutative Iwasawa theory lead naturally to the (conjectural) construction of a family of explicit annihilators of the BlochKatoTateShafarevic Groups that are attached to a wide class of padic representations over nonabelian extensions of number fields. Concrete examples to be discussed include a natural nonabelian analogue of Stickelberger's Theorem (which is proved) and of the refinement of the Birch and SwinnertonDyer Conjecture due to Mazur and Tate. Parts of this talk represent joint work with James Barrett and Henri Johnston.

INI 1  
16:30 to 17:30  Discussion  
18:45 to 23:00  Conference Dinner at Emmanual College (Old Library) 
10:00 to 11:00 
Patching and a localglobal principle for curves (joint with Julia Hartmann and Daniel Krashen)
Using patching, we establish a localglobal principle for actions of algebraic groups that are defined over the function field of a curve over a complete discretely valued field. This result has applications to quadratic forms and to Brauer groups.
In the case of quadratic forms, we obtain a result on the uinvariant of function fields, in particular reproving the theorem of Parimala and Suresh that the uinvariant of a onevariable padic function field is 8. Concerning Brauer groups, we obtain results on the periodindex problem for such fields, in particular reproving a result of Lieblich. We also obtain localglobal principles for quadratic forms and for Brauer groups.
Our localglobal principle for group actions holds in general for connected rational groups. In the disconnected case, the validity of the principle depends on the topology of a graph associated to the closed fiber of a model of the curve. The fundamental group of this graph is isomorphic to a certain quotient of the etale fundamental group; and the localglobal principle holds even for disconnected rational groups if and only if the graph is a tree. In that case, the localglobal principle for quadratic forms can be strengthened. In general, the cohomology of the graph determines the kernel of the localglobal map on Witt groups.

INI 1  
11:00 to 11:30  Coffee  
11:30 to 12:30 
TateShafarevich groups over anticyclotomic Z p extensions
Let E be an elliptic curve over Q with supersingular reduction at p and K an imaginary quadratic extension of Q. We analyze the structure of the pprimary part of the TateShafarevich group of E over the anticyclotomic Z_pextension K_\infty/K by viewing it as a module over Z_p[Gal(K_\infty/K)].

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
A Beilinson ([Chicago]) Epsilonfactors for the period determinants
I will explain why the determinants of the period matrices of holonomic Dmodules on curves admit a natural epsilonfactorization, as was conjectured by Deligne back in 1984. See http://xxx.lanl.gov/abs/0903.2674

INI 1  
15:00 to 15:30  Tea  
15:30 to 16:30  Galois theory and Diophantine geometry  INI 1  
16:30 to 17:30  Discussion  
18:45 to 19:30  Dinner at Wolfson Court (Residents Only) 