# 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) 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. 11:00-11:30 Coffee 11:30-12:30 Venjakob, O (Heidelberg) 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 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Szamuely, T (Renyi Institute) 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. 15:00-15:30 Tea 15:30-16:30 Colmez, P (Paris) 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) 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. 11:00-11:30 Coffee 11:30-12:30 Sharifi, R (Arizona) 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. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Wickelgren, K (Stanford) 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. 15:00-15:30 Tea 15:30-16:30 Taylor, R (Harvard) 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. 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) 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. 11:00-11:30 Coffee 11:30-12:30 Hain, R (Duke) 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. 12:30-13:30 Lunch at Wolfson Court 14:00-17:00 Excursion 18:45-19:30 Dinner at Wolfson Court (Residents Only)