# Workshop Programme

## for period 27 - 31 July 2009

### Non-abelian Fundamental Groups in Arithmetic Geometry - Introductory Workshop

27 - 31 July 2009

Timetable

Monday 27 July | ||||

09:00-09:55 | Registration | |||

09:55-10:00 | Welcome - David Wallace (INI Director) | |||

10:00-11:00 | Deligne, P (IAS) |
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Counting l-adic representations, in the function field case | Sem 1 | |||

We will explain some countings similar to the one Drinfeld did in 1981, and wonder what they mean. |
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11:00-11:30 | Coffee | |||

11:30-12:30 | Venjakob, O (Heidelberg) |
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On the noncommutative Iwasawa Main Conjecture for CM-elliptic curves | Sem 1 | |||

We discuss under which assumptions the (commutative) 2-variable Main Conjecture for CM-elliptic curves (due to Rubin, Yager, Katz etc.) implies the non-commutative Main Conjecture as formulated together with Coates, Fukaya, Kato and Sujatha. Related Links •http://www.mathi.uni-heidelberg.de/~otmar/ - personal homepage |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Szamuely, T (Renyi Institute) |
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Grothendieck's Section Conjecture and zero-cycles on varieties | Sem 1 | |||

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. |
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15:00-15:30 | Tea | |||

15:30-16:30 | Colmez, P (Paris) |
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On the p-adic local Langlands correspondence for $GL_2({\bf Q}_p)$ | Sem 1 | |||

16:30-17:30 | Discussion | |||

17:30-18:30 | Welcome Wine Reception | |||

18:45-19:30 | Dinner at Wolfson Court (Residents Only) |

Tuesday 28 July | ||||

10:00-11:00 | Coates, J (Cambridge) |
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Iwasawa theory of elliptic curves with complex multiplication | Sem 1 | |||

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 p-adic Lie extensions of the base field F which contain the cyclotomic Zp-extension of F. The results discussed are joint work with Sujatha. |
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11:00-11:30 | Coffee | |||

11:30-12:30 | Sharifi, R (Arizona) |
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Reciprocity maps and Selmer groups | Sem 1 | |||

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 S-reciprocity map is a homomorphism on S-units that interpolates values of a cup product with those S-units. 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 two-variable p-adic L-function, taken modulo an Eisenstein ideal. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Wickelgren, K (Stanford) |
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Obstructions to homotopy sections of curves over number fields | Sem 1 | |||

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. |
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15:00-15:30 | Tea | |||

15:30-16:30 | Taylor, R (Harvard) |
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Potential automorphy of n dimensional Galois representations | Sem 1 | |||

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 Sato-Tate conjecture for all elliptic modular new forms. |
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16:30-17:30 | Discussion | |||

18:45-19:30 | Dinner at Wolfson Court (Residents Only) |

Wednesday 29 July | ||||

10:00-11:00 | Pop, F (Penn) |
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On the birational p-adic section conjecture | Sem 1 | |||

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, p-adic), and finally discuss the status of the art of the conjecture. |
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11:00-11:30 | Coffee | |||

11:30-12:30 | Hain, R (Duke) |
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On the section conjecture for universal curves over function fields | Sem 1 | |||

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. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-17:00 | Excursion | |||

18:45-19:30 | Dinner at Wolfson Court (Residents Only) |

Thursday 30 July | ||||

10:00-11:00 | Emerton, M (Northwestern) |
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p-adically completed cohomology and the p-adic Langlands program | Sem 1 | |||

Speaking at a general level, a major goal of the p-adic Langlands program (from a global, rather than local, perspective) is to find a p-adic generalization of the notion of automorphic eigenform, the hope being that every p-adic 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 p-adic automorphic forms. However, in the non-Shimura variety context, such an approach is not available. Furthermore, this approach is somewhat remote from the representation-theoretic 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 p-adic interpolation, via the theory of p-adically completed cohomology. This approach has close ties to the p-adic and mod p representation theory of p-adic groups, and to non-commutative= Iwasawa theory. After introducing the basic objects (namely, the p-adically 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 p-adically completed cohomology, the most important of these being the Poincare duality spectral sequence. This is joint work with Frank Calegari. |
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11:00-11:30 | Coffee | |||

11:30-12:30 | Saidi, M (Exeter) |
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The cuspidalization of sections of arithmetic fundamental groups | Sem 1 | |||

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 pro-p) p-adic local version of the Grothendieck section conjecture under the assumption that the existence of sections of cuspidally (pro-p) abelian arithmetic fundamental groups implies the existence of tame points. We also prove, using cuspidalization techniques, that for a hyperbolic curve X over a p-adic local field and a set of points S of X which is dense in the p-adic 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 non-empty. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Bogomolov, F (NYU) |
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The structure of Galois groups and birational algebraic geometry | Sem 1 | |||

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 |
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15:00-15:30 | Tea | |||

15:30-16:30 | Burns, D (KCL) |
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Annihilating Tate-Shafarevic groups | Sem 1 | |||

We describe how main conjectures in non-commutative Iwasawa theory lead naturally to the (conjectural) construction of a family of explicit annihilators of the Bloch-Kato-Tate-Shafarevic Groups that are attached to a wide class of p-adic representations over non-abelian extensions of number fields. Concrete examples to be discussed include a natural non-abelian analogue of Stickelberger's Theorem (which is proved) and of the refinement of the Birch and Swinnerton-Dyer Conjecture due to Mazur and Tate. Parts of this talk represent joint work with James Barrett and Henri Johnston. |
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16:30-17:30 | Discussion | |||

18:45-23:00 | Conference Dinner at Emmanual College (Old Library) |

Friday 31 July | ||||

10:00-11:00 | Harbater, D (Penn) |
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Patching and a local-global principle for curves (joint with Julia Hartmann and Daniel Krashen) | Sem 1 | |||

Using patching, we establish a local-global 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 u-invariant of function fields, in particular reproving the theorem of Parimala and Suresh that the u-invariant of a one-variable p-adic function field is 8. Concerning Brauer groups, we obtain results on the period-index problem for such fields, in particular reproving a result of Lieblich. We also obtain local-global principles for quadratic forms and for Brauer groups. Our local-global 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 local-global principle holds even for disconnected rational groups if and only if the graph is a tree. In that case, the local-global principle for quadratic forms can be strengthened. In general, the cohomology of the graph determines the kernel of the local-global map on Witt groups. |
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11:00-11:30 | Coffee | |||

11:30-12:30 | Ciperiani, M (Columbia) |
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Tate-Shafarevich groups over anticyclotomic Z p extensions | Sem 1 | |||

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 p-primary part of the Tate-Shafarevich group of E over the anticyclotomic Z_p-extension K_\infty/K by viewing it as a module over Z_p[Gal(K_\infty/K)]. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Beilinson, A (Chicago) |
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Epsilon-factors for the period determinants | Sem 1 | |||

I will explain why the determinants of the period matrices of holonomic D-modules on curves admit a natural epsilon-factorization, as was conjectured by Deligne back in 1984. See http://xxx.lanl.gov/abs/0903.2674 |
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15:00-15:30 | Tea | |||

15:30-16:30 | Kim, M (UCL) |
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Galois theory and Diophantine geometry | Sem 1 | |||

16:30-17:30 | Discussion | |||

18:45-19:30 | Dinner at Wolfson Court (Residents Only) |