# Workshop Programme

## for period 14 - 18 December 2009

### Final Workshop

14 - 18 December 2009

Timetable

Monday 14 December | ||||

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

09:55-10:00 | Welcome from Sir David | |||

10:00-11:00 | Henniart, G (Paris-Sud 11) |
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Simple cuspidals and the local Langlands correspondence for GL(n) | Sem 1 | |||

Class field theory is the case n=1 of the Langlands correspondence. The case of general n may be considered as a kind of non-abelian class field theory. However the Langlands correspondence relates two equally mysterious sides. On the example of simple cuspidal representations for GL(n) over a non-Archimedean local field (pointed out by Gross and Reeder), we shall see that it is not so easy, but still possible, to determine the corresponding Galois representations (joint work in progress with Bushnell). |
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11:00-11:30 | Coffee and Tea | |||

11:30-12:30 | Ardakov, K (Nottingham) |
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Iwasawa algebras and enveloping algebras | Sem 1 | |||

I will compare and contrast the representation theory of Iwasawa algebras and of universal enveloping algebras. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Witte, M (Regensburg) |
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A Noncommutative Iwasawa Main Conjecture for Varieties over Finite Fields | Sem 1 | |||

We discuss a noncommutative Iwasawa Main Conjecture for $\ell$-adic Lie coverings of separated schemes of finite type over a finite field of characteristic $p$ different from $\ell$. |
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15:00-15:30 | Coffee and Tea | |||

15:30-16:30 | Nakamura, H (Okayama) |
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Arithmetic invariants of Eisenstein type arising from fundamental groups of once punctured elliptic curves | Sem 1 | |||

We discuss behaviors of certain arithmetic invariants (introduced by Bloch, Tsunogai) for fundamental groups of once-punctured elliptic curves. Anabelian geometry of tangential basepoints on M(1,1) and M(1,2) will also be brought into the view. |
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16:30-17:15 | Discussion | |||

17:15-18:15 | Welcome Wine Reception | |||

18:15-18:45 | Dinner at Wolfson Court |

Tuesday 15 December | ||||

10:00-11:00 | Jannsen, U (Regensburg) |
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Weights and Hasse principles for higher-dimensional fields | Sem 1 | |||

We present a Hasse principle for higher-dimensional fields which proves a conjecture of K. Kato. In addition to earlier results we also treat the case of p-torsion in positive characteristic p, asssuming resolution of singularities. Due to recent results on resolution we obtain unconditional results for low dimension. The principal tool is the consideration of weights on cohomology, as initiated in Deligne's proof of the Weil conjectures. The consideration of these weights is less standard for p-torsion in characteistic p. |
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11:00-11:30 | Coffee and Tea | |||

11:30-12:30 | Clozel, L (Paris-Sud 11) |
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Arthur's theory of automorphic forms on classical groups (a survey) | Sem 1 | |||

As requested by the organizers, I will try to give the flavour of Arthur's results describing completely the automorphic forms on orthogonal or symplectic groups. In view of the exhaustive results on Galois representations associated to forms on GL(n), I will try to explain the consequences for temperedness (or not) of cohomological representations of these groups, and possibly on the Galois representations appearing in the related Shimura varieties. This is a purely expository lecture. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Dokchitser, V (Cambridge) |
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Parity of ranks of elliptic curves I | Sem 1 | |||

I will explain why both the Birch-Swinnerton-Dyer conjecture and the Shafarevich-Tate conjecture imply that the parity of the rank of an elliptic curve over a number field can be expressed as a sum of (computable) local invariants, and describe some arithmetic consequences. |
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15:00-15:30 | Coffee and Tea | |||

15:30-16:30 | Rössler, D (Paris-Sud 11) |
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Conjectures on the logarithmic derivatives of Artin L-functions | Sem 1 | |||

We shall present a general conjecture relating the logarithmic derivatives of Artin L-functions to arithmetic intersection numbers on certain Shimura varieties. This is joint work with V. Maillot. |
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16:30-17:30 | Discussion | |||

18:00-19:15 | Wine Reception at CUP Bookshop |

Wednesday 16 December | ||||

09:00-10:00 | Pop, F (Pennsylvania) |
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On the pro-l abelian-by-central I/OM | Sem 1 | |||

After presenting the classical I/OM (Ihara problem / Oda-Matsumoto conjecture), I will present its abelian-by-central variant, and show its connection with a Programme initiated by Bogomolov. I will finally sketch a proof of the abelian-by-central I/OM, which is a much stronger assertion than the classical I/OM. |
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10:00-10:15 | Break | |||

10:15-11:15 | Kakde, M (UCL) |
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K 1 of some noncommutative p adic group rings | Sem 1 | |||

In this talk we will compute K_1 of some p-adic group rings which reduces, by a strategy of Burns and Kato, the noncommutative main conjecture for totally real number fields to commutative main conjecture for totally real number fields (a theorem of Wiles assuming Iwasawa invariant mu to be 0) and certain congruences between abelian p-adic zeta functions. |
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11:15-11:45 | Coffee and Tea | |||

11:45-12:45 | Kim, M (UCL) |
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Diophantine geometry and Galois theory 9 | Sem 1 | |||

12:45-13:30 | Lunch at Wolfson Court | |||

14:00-19:00 | Free time | |||

19:30-22:00 | Conference Dinner at Christ's College |

Thursday 17 December | ||||

10:00-11:00 | Stix, J (Heidelberg) |
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On the p-adic section conjecture | Sem 1 | |||

The talk presents results from joint work with Florian Pop, in which we found that sections of the fundamental group of a hyperbolic curve over a p-adic field all meet certain geometric requirements. |
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11:00-11:30 | Coffee and Tea | |||

11:30-12:30 | Rodriguez-Villegas, F (Texas at Austin) |
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On the mixed Hodge polynomials of character varieties of Riemann surfaces | Sem 1 | |||

I will describe a calculation of the number of points over finite fields of the varieties of the title (parameterizing generic representations of the fundamental group of a Riemann surface into GL_n). Since the varieties are polynomial count this calculation yields their E-polynomial (a geometric invariant). The results are best expressed in terms of a generating function involving the Macdonald polynomials of combinatorics. We conjecture that a natural deformation of these formulas in fact gives the full mixed Hodge polynomial of the varieties. This is joint work with T. Hausel and E. Letellier. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Popescu, C (UC, San Diego) |
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The Galois module structure of p-adic realisations of Picard 1-motives and applications | Sem 1 | |||

15:00-15:30 | Coffee and Tea | |||

15:30-16:30 | Flicker, Y (Ohio State) |
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Counting local systems with local principal unipotent monodromy | Sem 1 | |||

We compute, jointly with P. Deligne, the number of equivalence classes of irreducible rank n ell-adic local systems on the geometric X-S, namely n-dimensional ell-adic representations of pi_1(geometrix(X-S)), invariant under the Frobenius, whose local monodromy at each point of S is a single Jordan block of rank n. Here X is a smooth projective absolutely irreducible curve over the finite field of cardinality q, S a finite set of closed points of X of cardinality N>1, ell a prime with (ell,q)=1, and n>1 an integer. |
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16:30-18:00 | Discussion | |||

18:15-18:45 | Dinner at Wolfson Court |

Friday 18 December | ||||

09:00-10:00 | Esnault, H (Duisburg-Essen) |
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Relation between the \'etale and the algebraic fundamental groups | Sem 1 | |||

Joint work with V. Mehta. We show the relation between those two groups. Over the complex numbers, we know that if the 'etale fundamental group is trivial, so is the proalgebraic one. Among other things, we show the corresponding statement in char. p>0 over X projective smooth (Gieseker conjecture). |
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10:00-10:15 | Break | |||

10:15-11:15 | Burns, D (KCL) |
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Congruences between derivatives of Artin L-functions | Sem 1 | |||

11:15-11:45 | Coffee and Tea | |||

11:45-12:45 | Dokchitser, T (Cambridge) |
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Parity of ranks of elliptic curves II | Sem 1 | |||

In a follow-up to Vladimir's talk, I will discuss local formulae relating root numbers of elliptic curves to their Tamagawa numbers and periods. In particular, I will explain the proof of the parity conjecture for p-infinity Selmer ranks of elliptic curves with a p-isogeny and related results. |
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12:45-13:30 | Lunch at Wolfson Court |