ICMS

Sep 28, 2009 - Oct 2, 2009

14 India Street, Edinburgh EH3 6EZ

Organisers

Name	Institution
Brown, Ken A	University of Glasgow
Burns, David	King's College London
Coates, John	University of Cambridge
Schneider, Peter	Universität Münster

The aim of the workshop will be to discuss related recent developments in non-commutative algebra and Iwasawa theory. Iwasawa theory, via its main conjectures, remains the only systematic method known today for studying the mysterious connexions between purely arithmetic problems and special values of complex L-functions, typified by the conjecture of Birch and Swinnerton-Dyer. There is ever growing evidence that these main conjectures hold in vast generality, in particular, for all motives over p-adic Lie extension of number fields. At the same time, the Iwasawa algebras of compact p-adic Lie groups provide one of the most interesting examples of non-commutative rings. It is now clear that, intertwined with the arithmetic problems of Iwasawa theory, many intriguing questions of non-commutative algebra arise, which are interesting in their own right, and whose solution could eventually be very useful for understanding some of the classical problems of number theory. The Workshop will have lectures by both number-theorists and experts in non-commutative algebra, aimed at explaining to each other their results, conjectures, and hopes for future work.

This workshop is organised by ICMS as a satellite meeting of the Isaac Newton Institute, Cambridge.

Arrangements

Participation
Participation is by invitation only.

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and, if so, how to apply for one. If you do require a visa, ICMS can provide a signed invitation letter.

Venue
The workshop will take place at ICMS, 14 India Street, Edinburgh, EH3 6EZ. This house is the birthplace of James Clerk Maxwell and is situated in the historic New Town of Edinburgh, near the city centre.

The seminar room at 14 India Street has whiteboards, 2 overhead projectors, a data projector and laptop. We ask that you use our ICMS laptop for your presentation which you may bring on a memory stick or CD.

Wireless access is available throughout the building. There are also 4 public PCs which may be used at any time for internet access and to check email.

Travel
The ICMS travel pages contain advice on how to travel to Edinburgh. For local information, the finding ICMS page shows the location of ICMS and contains useful maps of the city centre. If you asked ICMS to book hotel accommodation for you, this map shows the location of The Ballantrae Albany Hotel in relation to the workshop venue at ICMS.

There are major roadworks taking place in Edinburgh at present due to the installation of tram tracks throughout the city centre. You are advised to travel by public transport and not by car as parking is very difficult to find and may prove expensive.

Accommodation
If you requested that ICMS arrange your accommodation, you will have been notified by email of the Ballantrae Albany Hotel details and the nights booked on your behalf. If you have not received an email with the details, please contact Audrey Brown at ICMS. The hotel will invoice ICMS directly so you will not be asked to pay the bed and breakfast costs. For information, this map shows The Ballantrae Albany Hotel.

Alternatively, participants are free to make their own arrangements and may claim back the cost, with receipts, up to a maximum of £65.00 per night bed and breakfast for a maximum of 6 nights. A list of Edinburgh accommodation of various sorts and prices is available here . Sections 1-3 are particularly relevant.

Meals and Refreshments
Morning and afternoon refreshments will be provided on each day of the workshop.

On Monday 28 September and Friday 2 October, a light buffet lunch will be provided free of charge to participants in the Exhibition Room at 14 India Street. On Tuesday, Wednesday and Thursday participants are free to explore the many cafés, sandwich shops, restaurants and bars nearby.

Following the lectures on Monday 28 September, there will be an informal wine reception in the Exhibition Room at 14 India Street.

After the wine reception on Monday 28 September, at 18.00, all participants are invited to an evening meal in Ristorante Librizzi (Sicilian/Italian) Restaurant, 69 North Castle Street, Edinburgh. A further informal meal will take place on Tuesday 29 September at 19.00 in The Nargile (Turkish) Restaurant, 73 Hanover Street, Edinburgh. A Workshop Dinner will be held on the evening of Thursday 1 October at 19.00 in The Magnum (Scottish) Restaurant, 1 Albany Street, Edinburgh (map showing location of restaurants) The workshop grant will cover the cost of these meals.

Registration
Registration will take place between 09.00 and 09.45 on the morning of Monday 28 September in the Exhibition Room at 14 India Street. There is no registration fee.

Financial Arrangements
Unless otherwise specified in your invitation letter, the workshop grant will cover the cost of your bed and breakfast accommodation, tea/coffee Monday to Friday, lunch on Monday and Friday only, a short wine reception on Monday following the talks, the informal meals on Monday and Tuesday evening and the Workshop Dinner on Thursday evening.

If we have agreed to pay some of your travel costs, you will be informed by email. Reimbursement will take place after the workshop directly through the Isaac Newton Institute (INI). At Registration, you will be issued with an INI travel claim form - please note that receipts or other supporting documentation are required for the travel costs claimed.

Programme

This workshop will start at 09.00 on Monday 28 September and finish with lunch 12.00-14.00 on Friday 2 October 2009.

Monday 28 September

09.00 - 09.45	Registration and coffee
09.45 - 10.00	Introduction and welcome
10.00 - 11.00	Peter Schneider (Universität Münster) K_1 and generalized Coleman maps
11.00 - 11.30	Coffee/tea break
11.30 - 12.30	Simon Wadsley (University of Cambridge) Euler characteristics and Iwasawa algebras
12.30 - 14.00	Lunch provided in the Exhibition Room
14.00 - 15.00	Karl Rubin (University of California, Irvine) Refined class number formulas and Kolyvagin systems
15.00 - 15.30	Coffee/tea break
15.30 - 16.30	Sarah Zerbes (University of Exeter) The higher Hilbert pairing via (phi,G)-modules
16.30 - 17.45	Wine Reception in the Exhibition Room at ICMS
18.00	Informal meal at Ristorante (Sicilian/Italian) Librizzi, 69 North Castle Street, Edinburgh

 

Tuesday 29 September

09.30 - 10.30	Romyar Sharifi (University of Arizona) Selmer groups and reducible representations
10.30 - 11.00	Coffee/tea break
11.00 - 12.00	Annette Huber-Klawitter (University of Freiburg) p-adic Borel regulators and Tamagawa numbers
12.00 - 14.00	Lunch break
14.00 - 15.00	Konstantin Ardakov (University of Nottingham) Roseblade's Theorem D for Iwasawa algebras
15.00 - 15.30	Coffee/tea break
15.30 - 16.30	Christopher Brookes (University of Cambridge) Deformation spaces for pro-p groups and Iwasawa algebras
19.00	Informal evening meal at Nargile (Turkish) Restaurant, 73 Hanover Street, Edinburgh

 

Wednesday 30 September

09.30 - 10.30	Irene Lau (Universität Augsburg) On the structure of the Iwasawa algebra
10.30 - 11.00	Coffee/tea break
11.00 - 12.00	Mahesh Kakde (University College London) Computation of K_1 of non-commutative groups rings
12.00	Lunch break and free afternoon

 

Thursday 1 October

09.30 - 10.30	Ramdorai Sujatha (Tata Institute of Fundamental Research) Non-commutative Iwasawa theory in Hida families
10.30 - 11.00	Coffee/tea break
11.00 - 12.00	Takashi Hara (University of Tokyo) Inductive construction of the p-adic zeta functions for non-commutative p-extensions of totally real fields with exponent p
12.00 - 14.00	Lunch break
14.00 - 15.00	Otmar Venjakob (University of Heidelberg) On localisations of skew power series rings
15.00 - 15.30	Coffee/tea break
15.30 - 16.30	Jürgen Ritter (University of Augsburg) Toward a proof of the 'main conjecture' of equivariant Iwasawa theory
19.00	Workshop dinner at The Magnum (Scottish) Restaurant, 1 Albany Street, Edinburgh (dress informal)

 

Friday 2 October

09.30 - 10.30	Fabien Trihan (University of Nottingham) On an analogue of the Iwasawa Main Conjecture for abelian varieties over function fields of characteristic p>0
10.30 - 11.00	Coffee/tea break
11.00 - 12.00	John Coates (University of Cambridge) On studying the Tate-Shafarevich group of elliptic curves with complex multiplication via Iwasawa theory
12.00 - 14.00	Lunch provided in the Exhibition Room, 14 India Street, Edinburgh

Presentations:

Presentation Details
Ardakov, Konstantin
Roseblade's Theorem D for Iwasawa algebras
View Abstract Hide Abstract
Let G be a free abelian pro-p group of finite rank and let kG be its completed group algebra over a field k of characteristic p. An ideal P of kG is said to be controlled by a subgroup H if it can be generated by some elements of the subalgebra kH. The ideal P is said to be faithful if the group G embeds faithfully into the group of units of the factor ring kG/P. We will show that if P is a faithful prime ideal of kG which is not controlled by any proper subgroup of G, then there are only finitely many k-linear automorphisms of kG that stabilise both P and G. This is a direct analogue in the world of Iwasawa algebras of Theorem D of Roseblade's 1978 paper on prime ideals in polycyclic group rings.
Brookes, Christopher
Deformation spaces for pro-p groups and Iwasawa algebras
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My talk will focus on spaces of group actions on trees and some associated completions and deformations of group algebras. Such spaces have been helpful in various parts of group theory in the study of automorphisms.
Coates, John
On studying the Tate-Shafarevich group of elliptic curves with complex multiplication via Iwasawa theory
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My lecture will discuss some recent work, both theoretical and numerical, with Liang and Sujatha on the on the Tate-Shafarevich of elliptic curves with complex multiplication.
Hara, Takashi
Inductive construction of the p-adic zeta functions for non-commutative p-extensions of totally real fields with exponent p
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Let F be a totally real number field and p an odd prime number. Let us consider the p-adic Lie extension the finite part of whose Galois group is isomorphic to a p-group with exponent p. In this talk, inspired by the method of Ritter-Weiss and Kazuya Kato, we will construct the p-adic zeta function associated to such an extension by using algebraic K-theory, Deligne-Ribet's q-expansion principle for Hilbert modular forms and certain inductive technique.
Huber-Klawitter, Annette
p-adic Borel regulators and Tamagawa numbers
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We define a p-adic analogue of the Borel regulator. The main ingredient is Lazard's isomorphism between Lie algebra cohomology and locally analytic group cohomology of p-adic Lie groups. We can prove on the one hand that it computes the Soule regulator to etale cohomology, ie, a term coming up in the Bloch-Kato conjecture. On the other hand it can be used to define cohomological versions of the local Tamagawa measure on algebraic groups. They correct Borel's famous formula for special values of Dedekind zeta-functions to give an integral identity. (Joint work with G Kings)
Kakde, Mahesh
Computation of K_1 of non-commutative groups rings
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In this talk I will present a description of the Whitehead group of the group ring of a finite p-group G with coefficients in Z_p in terms of units of group rings of abelian sub-quotients of G. The description will use Snaith's explicit Brauer induciton map. I will mention the applications of these this description in Non-commutative Iwasawa theory.
Lau, Irene
On the structure of the Iwasawa algebra
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In Iwasawa theory the Iwasawa algebra QG plays a crucial role. Yet, very little about its structure is known. In our talk we resolve the structure of the Iwasawa algebra for pro-p groups G=H.Z_p (with finite H).
Ritter, Jürgen
Toward a proof of the 'main conjecture' of equivariant Iwasawa theory
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The talk reports on recent progress made towards proving the `main conjecture´ in joint work with A Weiss.
Rubin, Karl
Refined class number formulas and Kolyvagin systems
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In 1995, Darmon formulated a "refined class number formula" for real quadratic fields, an algebraic Stark-type conjecture inspired by work of Mazur & Tate and of Gross. This talk (which describes joint work with Barry Mazur) will discuss a proof of Darmon's conjecture using the theory of Kolyvagin systems, and a proposed substantial generalization.
Schneider, Peter
K_1 and generalized Coleman maps
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I will report on joint work in progress with O Venjakob. We are developing a general strategy to compute K_1 of the completed group ring of pro-p Lie groups over the field F_p.
Sharifi, Romyar
Selmer groups and reducible representations
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We will consider certain Selmer complexes attached to residual representations. We will describe a seemingly non-obvious relationship between the connecting homomorphism arising from one of these and the S-reciprocity map for a number field F, where S is a set of primes of F containing those over a fixed prime p. These are both essentially maps from the S-units of F to a tensor product of pro-p abelian Galois groups with restricted ramification. For certain cyclotomic fields F, we will use this to suggest a connection with reductions of the p-adic L-functions of cuspidal eigenforms that satisfy mod p congruences with Eisenstein series.
Sujatha, Ramdorai
Non-commutative Iwasawa theory in Hida families
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We shall discuss the behaviour of various invariants that occur in non-commutative Iwasawa theory, along Hida families.
Trihan, Fabien
On an analogue of the Iwasawa Main Conjecture for abelian varieties over function fields of characteristic p>0
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We study a (p-adic) geometric analogue for abelian varieties with semistable reduction over a function field of characteristic p of the Iwasawa Main Conjecture for abelian varieties over a number field initiated by Mazur.
Venjakob, Otmar
On localisations of skew power series rings
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Inspired by the canonical Ore set for Iwasawa-algebras in my joint work with Coates, Fukaya, Kato and Sujatha it was a natural question to which extend similar localisations exist for general skew power series rings. In this talk we shall report on joint work with Peter Schneider which answers this question fairly generally.
Wadsley, Simon
Euler characteristics and Iwasawa algebras
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Whenever we have a finitely generated module for an Iwasawa algebra with finitely many homology groups all of which are finite we may define the Euler characteristic of the module to be the alternating product of the orders of these homology groups. We will discuss when these Euler characteristics are well-defined and how to calculate them, beginning with what is known and then suggesting possible strategies for dealing with more general cases.
Zerbes, Sarah
The higher Hilbert pairing via (phi,G)-modules
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I give a new cohomological proof of the formulae of Brueckner-Vostokov for the higher Hilbert pairing using the theory of (phi,G)-modules for higher dimensional local fields of mixed characteristic (0,p) as developed by Abrashkin/Andreatta/Scholl.

Participants

Name	Institution
Ardakov, Konstantin	University of Nottingham
Bouganis, Athanasios	University of Heidelberg
Brookes, Christopher	University of Cambridge
Brown, Ken A	University of Glasgow
Coates, John	University of Cambridge
Hara, Takashi	University of Tokyo
Huber-Klawitter, Annette	University of Freiburg
Kakde, Mahesh	University College London
Kang, Yunling	Nanjing University
Lau, Irene	Universität Augsburg
Lechner, Sabine	University of Freiburg
Liang, Zhibin	Capital Normal University
Ritter, Jürgen	University of Augsburg
Rubin, Karl	University of California, Irvine
Schneider, Peter	Universität Münster
Sharifi, Romyar	University of Arizona
Sujatha, Ramdorai	Tata Institute of Fundamental Research
Trihan, Fabien	University of Nottingham
Venjakob, Otmar	University of Heidelberg
Wadsley, Simon	University of Cambridge
Witte, Malte	Universität Regensburg
Zerbes, Sarah	University of Exeter