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MATHEMATISCHES FORSCHUNGSINSTITUT OBERWOLFACH

T a g u n g s b e r i c h t

Algebraic Number Theory

MATHEMATISCHES FORSCHUNGSINSTITUT OBERWOLFACH

T a g u n g s b e r i c h t

Algebraic Number Theory

The meeting has b een organized by Christopher Deninger M unster Peter Schneider M unster

and Anthony Scholl Durham

The sub ject of the conference was Algebraic Number Theory and Arithmetic Algebraic Ge

ometry In particular the following asp ects have b een treated in the talks

Iwasawa Theory

Galois group actions

Mo dular Forms and Galois representations

Langlands corresp ondences

padic Ho dge Theory

rigid analytic metho ds in Arithmetic Geometry

classical and padic p olylogarithms

Higher dimensional class eld theory

Cycles and Chow groups

Some sp eakers rep orted on their pro ofs of outstanding conjectures Henniart Laorgue on

lo cal resp ectively global Langlands corresp ondences Fontaine on the weakly admiss

admissconjecture joint work with Colmez

VORTRAGSAUSZUGE

Amnon Besser BeerSheva Israel

padic regulators for K theory of elds and padic p olylogarithms

joint work with Rob de Jeu

Goncharov conjectured the existence of complexes for a eld F

F n B B F B F F

n n n

Q Q Q

in degrees n

where B is generated by symbols x x mathbbP F f g and an isomorphism

n n

r r

H F n H F Q n Zagier further conjectured that when F is a number eld the

comp osition

H F nH F Q n H F R n R

M D

F C

where the eigenspace with resp ect to the action of complex conjugation on the embeddings is

given by x P x where P is a version of the complex p olylogarithm de Jeu

n Z ag n Z ag n

constructed a complex with similar prop erties and a map H H F Q n

M M

n n

and proved the analogue of Zagiers conjecture

In this work we treat the padic analogue when K Q we have a syntomic regulator

H K Q n K Gros Niziol Nekovar Besser We state the following conjecture

Conjecture If F K where F is a number eld then the comp osed

r eg H H F Q n H K Q n K

p n

M M

is given by x L x where L is a version of Colemans padic p olylogarithm

n mo dn mo dn

We prove this conjecture when jxj jx j and derive consequences for syntomic regu

lators of cyclotomic elements in K theory

Don Blasius UCLA

Motives for Hilb ert Mo dular Forms

According to a basic theorem of Shimura there is attached to each holomorphic newform f

of weight with rational Hecke eigenvalues an elliptic curve E dened over Q such that

LE s Lf s It is exp ected that this result will generalize to totally real elds F the

result is known if F Q is o dd or f b elongs to the discrete series at some nite prime v We

show that if the Ho dge Conjecture is true then the sought elliptic curve always exists More

generally if the Hecke eld of f is T there exists under the same hypothesis an ab elian

variety A dened over F with dim A T Q having a natural action of T such that

f f f f

LA s Lf s Thus Shimuras full result over Q will generalize to F

For metho ds we employ base change and the JacquetLanglands corresp ondence to nd a

motive for LSym f s in the cohomology of a Shimura surface A formal argument pro

duces over C an ab elian variety B of dimension T Q whose H contains a submotive

isomorphic to S y m f The Ho dge conjecture is invoked to descend B to a number eld L

and relate its l adic representations to those attached to f by Taylor From this p oint the

conclusion follows easily that the sought A is a factor of R B

It may b e hop ed that this result may b e proved by direct construction using for example

Shimura varieties attached to unitary groups However the construction ab ove leads to the

following general question Is the MumfordTate group of the category of motives dened by

Shimura varieties ie o ccuring in their cohomology simplyconnected

Christophe Breuil CNRS Universite dOrsay

Classication of group schemes and applications

padic Riemann conditions ShimuraTaniyamaWeil conjecture

Let p b e a prime p k a p erfect eld of char p W W k the Witt vectors

K Frac W K a nite totally ramied extension of K O the ring of integers and

a xed uniformizer in O

Let S b e the padic completion of W u Fil S the kernel of the map S O u

iN K

and S S the lifting of Frobenius induced by u u Let j one has

Fil S

Fil S S

Dene Mo dS to b e the category whose ob jects are

an S mo dule M

a submo dule Fil M Fil S M

(E (u)x)

a semilinear application Fil M M st sx s

(E (u))

s Fil S x M

where E u in the minimal p olynomial of and whose morphisms are the obvious com

patible ones

Finally let Mo dF I S b e the full sub category of Mo dS such that

M Sp S jI j

Fil M spans M over S

and dene a strongly divisible mo dule to b e any M in Mo dS st

M S

Fil M spans M over S

M Fil M has no ptorsion

Theorem There is an equivalence of categories b etween M odF I S and the category of

nite at pgroups over O such that k er p is still at over O for any n This equivalence

K K

preserves short exact sequences

Theorem There is an equivalence of categories b etween strongly divisible mo dules and

pdivisible groups over O

The pro of uses syntomic techniques introduced by Mazur Fontane Messing in the case

where K K Using this on can get

Theorem padic Riemann conditions Assume k F Then every crystalline repre

Q for K K with Ho dgeTate weights f g is isomorphic to T G sentation V of Gal

p p Z

some pdivisible group GO

This is already known for K K p Fontaine Messing Laaille

Corollary Let A b e an ab elian variety over K and k F

i A has go o d reduction over O T A Q is crystalline

K p p

ii A has semistable reduction over O T A Q is semistable

K p p

Corollary The p erio d map of Rap op ortZink for pdivisible groups is surjective

Corollary i was known Fontaine Coleman Iovita the implication of ii was

known b efore to o Fontaine The ob jects killed by p of Mo dF I S classify al l nite at

group schemes over O killed by p By developing a decent data formalism on such ob jects

and using it to compute the dimension of tangent spaces of sp ecic deformation problems

of p otential BarsottiTate representations joint work with B Conrad F Diamond and R

Taylor one can settle the last cases of the ShimuraTaniyamaWeil conjecture following of

course Wiles Taylors strategy Thus

Theorem Wiles Every ellipic curve over Q is mo dular

Kevin Buzzard Imp College London

Galois representations and weight mo dular forms

My talk was a rep ort on the current status of Taylors program for attacking Artins con

jecture for representations Gal Q Q GL C and also contained some remarks on some of

the technicalities b ehind one of the results used in the program The talk centered around a

theorem of myself and Taylor which broadly sp eaking was a weight analogue of Wiles

lifting theorem if is mo dular and lots of technical conditions hold then is mo dular

G GL Z etc

Q p

I explained how this lifting theorem already gave new examples of Artins conjecture and

also how mo dulo some technical details would give many more new examples The key

problem is that to get these many new examples one has to prove many standard results

ab out mo d p mo dular forms in the case when p The main standard result which is still

unproved is a companion forms result proved by Gross mo dulo unchecked compatibilities

for all primes and by Coleman and Voloch not assuming these compatibilities for p It

now seems that if this result were to b e unconditionally established for p then one would

Q Q GL C which b e able to prove that many icosahedral Galois representations Gal

were o dd would come from weight mo dular forms and would hence have the prop erty that

their Lfunctions had analytic continuation and functional equation

Henri Darmon McGill University Montreal

The padic Birch and SwinnertonDyer Conjecture

This talk summarizes the results of dierent works all joint with Massimo Bertolini and

Adrian Iovita and Michael Spie

Let E Q b e an elliptic curve and LE s its HasseWeil zetafunction Thanks to Breuils

lecture we know E is mo dular so there exists

H N E C

s s

and sLE s y f iy dy where if z dz so that f is a cusp form

of weight on n The Birch and SwinnertonDyer conjecture predicts

ord LE s rank E Q

But b oth conjectured inequalities are very mysterious

If E Q has multiplicative reduction there is a rigid analytic uniformization H

p p

E C with a padic arithmetic subgroup of SL Q Following an idea of Schneider

p p

and IovitaSpie we dene L E s by taking a kind of padic Mellin transform of the rigid

analytic mo dular form of weight on attached to E We are then able to

give formulae for L E relating these values to padic p erio ds and Heegner p oints

on E

extend the denition of L f to forms f of higher even weight k and express

L f in terms of an Linvariant of Teitelbaum and in terms of certain Heegner

cycles in the Chow groups of KugaSato varieties joint work with Iovita and Spie

prove ord L E s rank E Q in fact we supply a more precise lower b ound for

s p

ord L E s which is conjectually sharp

s p

Jonathan Dee Cambridge UK

Chow Groups for selfpro ducts of CM Elliptic Curves

I discuss certain prop erties of the image of the cycle class map for cycles of arbitrary co di

mension on the dfold selfpro duct of an elliptic curve over Q with complex multiplication

Theorem Let E jQ b e an elliptic curve with complex multiplication by O K quadratic

imaginary Put X E and assume E has go o d reduction at p p maxfi dg

d i

Then

Im cl Fil H X Q i

X p

i i

X Q i is the cycle class map where cl CH X H

p X Q

Furthermore if p do es not divide c then

CH X cl

tor s X Z

where c is a certain explicit pro duct of Lvalues

To prove this theorem we use a lo calglobal principle due to Nekovar in this setting which

reduces us to b ounding a certain Selmer group This is achieved by using Rubins pro of of

the Main Conjecture and an explicit recipro city law

I Fesenko Nottingham

Bizarre propgroups and CoatesGreenbergs problem

iq r

Let T f a t a a F g q p r With resp ect to substitution T is a

i i p

torsion free nitely generated propgroup in which every nontrivial closed subgroup of an

op en subgroup is op en

Theorem Let r p and F Q b e an unramied extension of degree q Then

there is a Galois extension LF with Gal LF T every such LF is an arithmetically

pronite extension and LF do esnt have innite padic Lie sub extensions

This answers armatively CoatesGreenbergs problem set in Ob erwolfach in

JeanMarc Fontaine Universite dOrsay

Semistable padic Galois representations

joint work with Pierre Colmez

Let K b e a complete eld of characteristic with p erfect residue eld of char p Let

K b e an alg closure of K G Gal K K The category of semistable representations

of G is a subtannakien category of the category of all padic representations and we have a

functor

V semistable representations ltered N mo dules

which is a fully faithfuld functor We prove that the essential image is as conjectured the

category of socalled weakly admissible ltered N mo dules

Alexander Goncharov Brown UniversityMPI

Galois groups and geometry of mo dular varieties

We study the map on Lie algebras induced by the map

Gal Q Q Aut L

where L is the pronilp otent Lie algebra over Q corresp onding to

P n f g V

let us denote it by g We formulate a precise conjecture on the structure of the bigraded

Lie algebra

D W

Gr g

where D and W are the socalled depth and weight ltration resp ectively

Guy Henniart Universite dOrsay

The Langlands conjecture for GL over padic elds

F b e an algebraic closure of F and Let p b e a prime number and F a nite extension of Q Let

W the Weil group of F over F The Langlands conjecture predicts the existence of a system

of canonical bijections for any integer n b etween the set G n of irreducible degree

n complex representations of W and the set A n of sup ercuspidal smo oth irreducible

complex representations of the lo cally compact totally disconnected group GL F Such

maps should b e compatible with lo cal class eld theory twisting by characters of

F taking contragredients and ab ove all preserve L and factors for pairs for G n

G n we should have

L s L s

s s

where is a nontrivial additive character of F and the factors on the right have b een dened

by Jacquet PiatetskiiShapito and Shalika

Last summer M Harris and R Taylor proved the existence of the required bijections provid

ing in fact a geometric lo cal mo del for the corresp ondence Their pro of was global and relied

on a detailed analysis of the bad reduction of some Shimura varieties attached to unitary

groups over number elds

I rep orted on a shorter pro of found in december which is still global but more straight

forward and uses Shimura varieties only at primes of go o d reduction where the situation was

analyzed by Kottwitz in the early s This shorter pro of however yields no geometric infor

mation Both global pro ofs back explicit information an explicitation of the corresp ondence

is under investigation work with C Bushnell

Uwe Jannsen Regensburg

Class eld theory for varieties over lo cal eld

Let X b e a smo oth pro jective variety over a nite extension K of Q and consider the

recipro city map

d ab

H X K X

d dim X into the ab elianized fundamental group Here

tame

k x K k y H X K cok er

xX y X

0 1

is the cokernel of the tame symbol where X is the set of p oints of dimension i on X and

the map is induced by the classical maps k x Gal k x jk x for the lo cal elds k x

ab ab

and the canonical maps Gal k xjk x X The following results were obtained in

joint work with Shuji Saito

Theorem Assume that the MilnorKato conjecture holds for K F l for all function

elds F of smo oth surfaces on X this holds for the prime l

a If X has go o d reduction then

w d w ab w

Zl H X K l X l for all w

b If X has semistable reduction with sp ecial bre Y there is an exact sequence

ab w w w d w

X l H Zl H Zl H X K l

Y Y d

where is the simplical complex describing the conguration of the irreducible comp onents

Y Y of Y provided

i l p or ii l p and Y is ordinary or iii l p and d p

Theorem Under the ab ove assumptions the groups k er Zl are nite of order

b ounded indep endent of w The same holds for the groups k er Zw if the MilnorKato

conjecture is true mo dulo all primes l

Guido Kings M unster

Geometry of elliptic p olylog and computation of l adic Eisenstein classes

In this talk we contruct a new geometric approach to the elliptic p olylogarithm sheaf

which allows to compute its l adic realizations This gives a description of the l adic Eisen

stein classes which is needed for the Tamagawa number conjecture for mo dular forms CM

elliptic curves or Dirichlet characters The idea is as follows Let E K b e an elliptic curve

t t

and l E E the multiplication by l Over E we construct a ramied covering exactly

ramied at E l in a norm compatible way To do this we use a covering of the general

t t

ized Jacobian for E with mo dulus E l which we pull back to E E l The covering is

s s

a T l torsor where T is the torus of the generalized Jacobian The resulting class

E l E l

t r t

s s

in H E E l T l comes via the b oundary map from a function E E l T

E l E l

Taking the lim lim and lim we get the class of the elliptic p olylogarithm

s r

Mark Kisin Sydney

Unit LFunctions and a Conjecture of Katz

Let X b e a nite type F scheme a complete lo cal Z algebra with nite residue eld

p p

For a adic lisse sheaf L on X one denes

dx dx

LX L det T jL

xjX j

where dx k x F is the geometric Frobenius and x is a geometric p oint over x

L is dened by The cohomological Lfunction LF H

i+1

F L LF H L det T jH X

p p

c c

We show a conjecture of Katz which says that the quotient LX LLF H L is a padic

analytic invertible function on the closed padic unit disc jT j

Laurent Laorgue CNRS Universite dOrsay

La corresp ondance de Langlands sur les corps de fonctions

Soit X une courb e pro jective lisse geometriquement connexe sur un corps ni F et F le corps

Z son group e de Weil des fonctions de X Soit A lanneau des adeles de F et W G

F F

On demontre quil existe une corresp ondance bijective preservant les fonctions L entre

lensemble des representations automorphes cuspidales de GL A r et lensemble des

representations l adiques continues de W irreductibles de dimension r comme conjecture

par Langlands

La methode generalise celle inauguree par Drinfeld dans sa preuve de la conjecture p our

r Elle consiste a etudier la cohomologie l adique des champs de Chtoucas de Drinfeld

de rang r audessus du p oint generique de X X avec laction des corresp ondances de Hecke

J Nekovar Cambridge UK

Duality theorems in Iwasawa theory

Duality theorems for Selmer complexes imply duality theorems for various complexes in

Iwasawa theory For example let p b e a prime K a number eld K K an extension

with Gal K K Z S any nite set of primes Write K K K K nite let

S b e the set of primes of K ab ove S

A P icO

p S K S

Put Z

Theorem There exists a canonical epimorphism in the category

mo dulespseudonull mo dules

l im A E xt l im A

S S

The same formalism allows one to describ e a variant of a padic height for Selmer complexes

in terms of a Bo ckstein map

Alexander Schmidt Heidelb erg

Singular Homology and Class Field Theory of Varieties over Finite Fields

The talk rep orts on joint work with M Spie We give a generalization of Katos and Saitos

unramied classeld theory for smo oth pro jective varieties over nite elds to the case of

smo oth quasipro jective varieties and tame coverings For this we use the singular homology

groups h X dened by Suslin

Let k b e a nite eld let X b e a smo oth pro jective and geometrically connected variety

t ab

X b e op en Let X b e the ab elianized tame fundamental group of X over k Let X

t ab

Let h X and X b e the kernel of the surjections h X h Sp eck Z and

t ab ab

X G Z resp ectively

Theorem There exists a natural homomorphism recipro city map

ab t

X r ec h X

t ab

which induces an isomorphism r ec h X X of nite ab elian groups which

ts into the commutative diagram

h X h X Z

r ec r ec

t ab t ab

X X Z

In particular r ec induces an isomorphism b etween the pronite completions of h X and

t ab

Leila Schneps

Galois Actions on Fundamental Groups

Whenever V is a variety dened over Q there is a canonical outer action of Gal Q jQ G

V Following ideas of Grothendieck we would like to on its geometric denoted

identify the image of G inside Out V We rst consider V P f g so that

V F the pronite free group on generators Because the Galois action preserves

inertia we obtain an injection

G Z F

via acting on F x y by x x y f y f In order to study the image of

G in Z F we introduce the automorphism group of a tower of fundamental groups

if we have a collection of Q varieties linked by Q morphisms f the geometric fundamental

group functor transforms it to a collection of pronite fundamental groups linked by induced

homomorphisms f and G maps to the automorphism group of the whole collection called

a tower

Two dierent towers give dierent theorems

Theorem F Pop If T is the tower corresp onding to the collection of P fn ptsQ g n

and if O ut T denotes the automorphism group preserving inertia then

G Out T

Theorem Drinfeld Schneps Lo chak Nakamura Hatcher If T is the tower of s

of mo duli spaces of Riemann surfaces then G O ut T and O ut T can b e explicitly

computed as a subset of Z F of elements satisfying a few simple prop erties

Future goal describ e G itself inside Z F

Ehud de Shalit Jerusalem

Residues on the BruhatTits building of GL Q and

d p

de Rham cohomology of padic symmetric spaces

Let K Q V a d dimensional vector space over K and

p K

X PV H

aPV

H the hyperplane a Drinfelds padic symmetric space of dimension d G

P GLV acts on X The de Rham cohomology H X closed rigid analytic forms mo dulo

exact k d was computed by Schneider and Stuhler Inv Math

We prop ose a somewhat dierent approach which also solves some questions left op en by

Schneider and Stuhler

Let T b e the BruhatTits building of G and T the oriented k simplices We dene the notion

of a harmonic k co chain as a certain map T K satisfying harmonicity conditions If

is a closed k form we dene its residue res along using a p owerseries expansion

in terms of appropriate co ordinates on X the op en set in X mapping onto j j under the

canonical reduction map X jT j

The co chain c res is harmonic it vanishes i is exact and the resulting map

c is an isomorphism

k k

H X C

dR har

onto the space of all harmonic k co chains

The pro of relies on a detailed combinatorial study of the subalgebra of logarithmic classes

a b V fg using ideas b orrowed from the theory of those obtained from dlog

hyperplane arrangements

Atsushi Shiho Universite dOrsay

Crystalline Fundamental Groups and padic Ho dge theory

In cohomology theory there is a comparison theorem which connects padic etale cohomology

and de Rham cohomology or crystalline cohomology and it is called padic Ho dge theory In

view of Delignes philosophy that the theory of prounip otent quotient of rational should

b e also motivic there should b e a similar comparison theorem also for rational and ratio

crys

nal We gave a denition of crystalline fundamental groups by the Tannaka dual

of the category of unip otent iso crystals and proved a rational version of BerthelotOgus

theorem comparison b etween de Rham and crystalline and crystalline conjecture a part

of padic Ho dge theory We can also prove the rational version of the ab ove theorems

and it seems that our metho d can b e used for the pro of of rational version of stable

conjecture

M Strauch M unster

On the JacquetLanglands corresp ondence

in the cohomology of the LubinTate deformation tower

Let F jQ b e a nite extension X a formal onedimensional o mo dule of F height h over F

p F p

a uniformizer of F K M o K GL o Dene the deformation functor

n h F h F

of deformations of X with Drinfeld levelnstructure over complete lo cal o algebras M

R fX g X b eing a formal o mo dule over F by M R with residue eld

F p K

n h n

R X X a quasiisogeny and o o X a Drinfeld basis Drinfeld has

F F

is representable by a nite at o u u algebra which itself rep shown that M

h K

b e the rigid analytic space asso c to resents M this is due to Lubin and Tate Let M

K K

where B is the group of quasiisogenies There is a natural action of K B on M M

K K

n n

of X Then dene for any op en compact K G GL F the space M as a quotient On

h K

i i

Q there is a smo oth action of G B W The construction of H lim H M C

F K p

c c

such representations is due to H Carayol generalizing work of Deligne F Q h Put

i i

H H Let b e a sup ercuspidal representation of G Then the following theorem

c c

holds by work of M Harris and R Taylor

Theorem Hom H IL hdiml W representation where IL is

G F

the representation of B corresp onding to via the JacquetLanglands corresp ondence

Using a Lefschetz trace formula for adic curves proved by R Hub er we show how to obtain

this theorem by purely lo cal means if h

For arbitrary h we show how to reduce this theorem to the existence of a suitable trace

formula for higherdimensional adic spaces work in progress

Takeshi Tsuji RIMS Kyoto Univ

Geometric Galois cohomology of B

crys

For a smo oth analytic space X the constant sheaf C has a canonical resolution which

allows us to compare the singular cohomology and the de Rham cohomology of X We want

an analogue of this argument in the padic case Let V b e a complete discrete valuation ring

of mixed characteristic p with a p erfect residue eld and assume that V is absolutely

unramied Let K b e the eld of fractions of V We consider a suciently small padically

complete and separated ring A formally smo oth over V Then the rings A B A and

A with their action of Gal AA A play the role of C in the padic case B

K X cr y s

where A denotes the maximal extension of A unramied in characteristic Around ten

A of the form acyclic resolution of years ago O Hyodo constructed a canonical A

and established a theory of variation of Ho dgeTate structures So on after that

N Tsuzuki established a B version In the case of B it is known that it also has a

dR cr y s

resolution B AB A

cr y s cr y s A

Theorem The resolution is A acyclic

As a corollary one can describ e the Geometric Galois cohomology of a crystalline represen

tation of A in terms of the asso ciated A mo dule with and Fil These results are

K K

based on the theory of almost etale extensions by G Faltings

Annette Werner M unster

Arakelov intersection numbers on P via buildings and symmetric spaces

Manins analogies continued

Manin raised the question if one could enrich the somewhat formal picture of Arakelov the

ory at the innite places by a suitable geometry at In fact he suggested certain

threedimensional hyperb olic manifolds as canditates for mo dels at innity for curves and

interpreted some intersection numbers on them via hyberb olic geometry He nds these man

ifolds by analogy with the nonarchimedean places In this talk we generalize some of the

results he showed for P to higherdimensional pro jective spaces

We calculate several nonarchimedean lo cal intersection indices of linear cycles in P via the

combinatorial geometry of the BruhatTits building for P GL Besides we show a formula

for Nerons lo cal height pairing applied to hyperplanes and zero cycles in the same spirit

This formula has an archimedean analogue using the dierential geometry of the symmetric

space Z SLn C S U n This might suggest to regard Z as some kind of mo del at

innity for P

Author of rep ort Matthias Strauch M unster

EMAIL ADDRESSES

G Banaszak banaszakmathamuedupl

M Bertolini massimodimatunipvit

A Besser b esseramathbguacil

D Blasius blasiusmathuclaedu

C Breuil breuilmathupsudfr

D Burns davidburnskclacuk

K Buzzard buzzardicacuk

T Chinburg tedmathup ennedu

P Colmez colmezensfr

H Darmon darmonmathmcgillca

J Dee jatddpmmscamacuk

C Deninger deningemathunimuensterde

I Fesenko ibfmathsnottacuk

JM Fontaine fontainemathupsudfr

EU Gekeler gekelermathunisbde

S Goncharov sashamathbrownedu

G Harder harderdiophantmathunibonnde

G Henniart guyhenniartmathupsudfr

A Hub er hubermathunimuensterde

U Jannsen uwejannsenmathematikuniregensburgde

G Kings kingsmathunimuensterde

M Kisin markkmathsusydeduau

L Laorgue laurentlaorguemathupsudfr

A Langer langeramathunimuensterde

J Nekovar nekovardpmmscamacuk

W Niziol niziolmathutahedu

F Pop p opmathunib onnde

W Raskind raskindmathuscedu

J Ritter rittermathuniaugsburgde

N Schappa schappamathustrasbgfr

L Schneps schnepsdmiensfr

A Schmidt schmidtmathiuniheidelbergde

CG Schmidt csmap ccsmathematikunikarlsruhede

A Scholl ajscholldurhamacuk

E de Shalit deshalitmathhujiacil

S Shiho atsushishihomathupsudfr

M Spie mksmathsnottacuk

M Strauch straucmmathunimuensterde

M Taylor martintaylorumistacuk

T Tsuji tsujikurimskyotouacjp

N Wach wachmathustrasbgfr

A Werner wernermathunimuensterde

K Wingb erg wingb ergmathiuniheidelb ergde

J P Wintenberger wintenbmathustrasbgfr

EW Zink zinkmathematikhuberlinde

Tagungsteilnehmer

Dr David J Burns ProfDr Grzegorz Banaszak

Department of Mathematics Fac of Math and Comp Sci

Kings College London Adam Miciewics University

University of London Matejki

Strand

Poznan

GBLondon WCR LS POLAND

ProfDr Kevin Buzzard

ProfDr Massimo Bertolini

Department of Mathematics

Imp erial College

University of Pavia

Huxley Building

Via Abbiategrasso

Queens Gate

I Pavia

GBLondon SW BZ

ProfDr Amnon Besser

ProfDr Ted C Chinburg

Department of Mathematics and

Department of Mathematics

Computer Science

University of Pennsylvania

BenGurion University of the Negev

South rd Street

POB

Philadelphia PA

BeerSheva

USA

ISRAEL

ProfDr Pierre Colmez

ProfDr Don Blasius

Lab oratoire de Mathematiques

Dept of Mathematics

de lEcole Normale Sup erieure

University of California

URA

Hilgard Avenue

rue dUlm

Los Angeles CA

F Paris

USA

ProfDr Henri Rene Darmon

ProfDr Christophe Breuil

Dept of Mathematics and Statistics

Mathematiques

McGill University

Universite de Paris Sud Paris XI

Sherbro oke Street West

Centre dOrsay Batiment

Montreal PQ HA K

F Orsay Cedex

CANADA

ProfDr G unter Harder Jonathan Dee

Mathematisches Institut Dept of Pure Mathematics and

Universitat Bonn Mathematical Statistics

Beringstr University of Cambridge

Mill Lane

Bonn

GBCambridge CB SB

ProfDr Christopher Deninger

ProfDr Guy Henniart

Mathematisches Institut

Mathematiques

Universitat M unster

Universite de Paris Sud Paris XI

Einsteinstr

Centre dOrsay Batiment

M unster

F Orsay Cedex

ProfDr Ivan B Fesenko

School of Math Sciences

University of Nottingham

Dr Annette Hub erKlawitter

GBNottingham NG RD

Mathematisches Institut

Universitat M unster

Einsteinstr

ProfDr JeanMarc Fontaine

M unster

Mathematiques

Universite de Paris Sud Paris XI

Centre dOrsay Batiment

F Orsay Cedex

ProfDr Uwe Jannsen

Fakultat f ur Mathematik

Universitat Regensburg

ProfDr ErnstUlrich Gekeler

Universitatsstr

Fachbereich Mathematik

Universitat des Saarlandes

Regensburg

Postfach

Saarbr ucken

Guido Kings ProfDr Alexander Goncharov

Mathematisches Institut Department of Mathematics

Universitat M unster Brown University

Einsteinstr

Providence RI

M unster USA

Dr Florian Pop Mark Kisin

Mathematisches Institut co ProfDr Deninger

Universitat Bonn Mathematisches Institut

Beringstr Universitat M unster

Einsteinstr

Bonn

M unster

ProfDr Wayne Raskind

Dept of Mathematics DRB

University of Southern California

ProfDr Laurent Laorgue

W Place

Mathematiques

Universite de Paris Sud Paris XI

Los Angeles CA

Centre dOrsay Batiment

USA

F Orsay Cedex

ProfDr J urgen Ritter

Institut f ur Mathematik

Universitat Augsburg

Andreas Langer

Augsburg

Mathematisches Institut

Universitat M unster

Einsteinstr

ProfDr Norb ert Schappacher

UFR de Mathematique et

M unster

dInformatique

Universite Louis Pasteur

rue Rene Descartes

Dr Jan Nekovar

F Strasb ourg Cedex

Dept of Pure Mathematics and

Mathematical Statistics

Cambridge University

Dr Alexander Schmidt

Mill Lane

Mathematisches Institut

Universitat Heidelb erg

GBCambridge CB SB

Im Neuenheimer Feld

Heidelb erg

ProfDr Wieslawa Niziol

ProfDr ClausG unther Schmidt Dept of Mathematics

Mathematisches Institut II University of Utah

Universitat Karlsruhe

Salt Lake City UT

Karlsruhe USA

Matthias Strauch Dr Leila Schneps

Mathematisches Institut Lab oratoire de Mathematiques

Universitat M unster Universite de FrancheComte

Einsteinstr Route de Gray

M unster F Besancon Cedex

ProfDr Martin J Taylor

Dr Anthony J Scholl

Dept of Mathematics

Dept of Mathematical Sciences

UMIST University of Manchester

The University of Durham

Institute of Science a Technology

Science Lab oratories

P O Box

South Road

GBManchester M QD

GBDurham DH LE

ProfDr Takeshi Tsuji

Research Institute for

ProfDr Ehud de Shalit

Mathematical Sciences

Institute of Mathematics

Kyoto University

The Hebrew University

Kitashirakawa Sakyoku

GivatRam

Kyoto

Jerusalem

JAPAN

ISRAEL

ProfDr Nathalie Wach

UFR de Mathematique et

ProfDr Atsushi Shiho

dInformatique

MathematiquesBat

Universite Louis Pasteur

Universite ParisSud

rue Rene Descartes

F Orsay Cedex

F Strasb ourg Cedex

Dr Michael Spie

Dr Annette Werner

Mathematisches Institut

Universitat Heidelb erg

Universitat M unster

Im Neuenheimer Feld

Einsteinstr

Heidelb erg

M unster

ProfDr ErnstWilhelm Zink ProfDr Kay Wingb erg

Institut f ur Reine Mathematik Mathematisches Institut

Fachbereich Mathematik Universitat Heidelb erg

HumboldtUniversitat Berlin Im Neuenheimer Feld

Berlin Heidelb erg

ProfDr JeanPierre Wintenberger

Institut de Mathematiques

Universite Louis Pasteur

rue Rene Descartes

F Strasb ourg Cedex