Documenta Mathematica
Journal der
Deutschen Mathematiker-Vereinigung Gegr¨ undet 1996
Extra Volume
A Collection of Manuscripts Written in Honour of
Andrei A. Suslin
on the Occasion of His Sixtieth Birthday
Editors:
I. Fesenko, E. Friedlander,
A. Merkurjev, U. Rehmann
ver¨offentlicht Forschungsarbeiten aus allen mathematischen Gebieten und wird in traditioneller Weise referiert. Es wird indiziert durch Mathematical Reviews, Science Citation Index Expanded, Zentralblatt f¨ur Mathematik.
Artikel k¨onnen als TEX-Dateien per E-Mail bei einem der Herausgeber eingereicht werden. Hinweise f¨ur die Vorbereitung der Artikel k¨onnen unter der unten angegebe- nen WWW-Adresse gefunden werden.
Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, publishes research manuscripts out of all mathematical fields and is refereed in the traditional manner. It is indexed in Mathematical Reviews, Science Citation Index Expanded, Zentralblatt f¨ur Mathematik.
Manuscripts should be submitted as TEX -files by e-mail to one of the editors. Hints for manuscript preparation can be found under the following web address.
http://www.math.uni-bielefeld.de/documenta Gesch¨aftsf¨uhrende Herausgeber / Managing Editors:
Alfred K. Louis, Saarbr¨ucken louis@num.uni-sb.de
Ulf Rehmann (techn.), Bielefeld rehmann@math.uni-bielefeld.de Peter Schneider, M¨unster pschnei@math.uni-muenster.de Herausgeber / Editors:
Christian B¨ar, Potsdam baer@math.uni-potsdam.de Don Blasius, Los Angeles blasius@math.ucla.edu Joachim Cuntz, M¨unster cuntz@math.uni-muenster.de Patrick Delorme, Marseille delorme@iml.univ-mrs.fr Gavril Farkas, Berlin (HU) farkas@math.hu-berlin.de Edward Frenkel, Berkeley frenkel@math.berkeley.edu Friedrich G¨otze, Bielefeld goetze@math.uni-bielefeld.de Ursula Hamenst¨adt, Bonn ursula@math.uni-bonn.de Lars Hesselholt, Cambridge, MA (MIT) larsh@math.mit.edu Max Karoubi, Paris karoubi@math.jussieu.fr
Stephen Lichtenbaum Stephen Lichtenbaum@brown.edu Eckhard Meinrenken, Toronto mein@math.toronto.edu
Alexander S. Merkurjev, Los Angeles merkurev@math.ucla.edu Anil Nerode, Ithaca anil@math.cornell.edu
Thomas Peternell, Bayreuth Thomas.Peternell@uni-bayreuth.de Eric Todd Quinto, Medford Todd.Quinto@tufts.edu
Takeshi Saito, Tokyo t-saito@ms.u-tokyo.ac.jp Stefan Schwede, Bonn schwede@math.uni-bonn.de Heinz Siedentop, M¨unchen (LMU) h.s@lmu.de
Wolfgang Soergel, Freiburg soergel@mathematik.uni-freiburg.de ISBN 978-3-936609-48-6 ISSN 1431-0635 (Print) ISSN 1431-0643 (Internet)
SPARC
Leading Edge
Documenta Mathematicais a Leading Edge Partner of SPARC, the Scholarly Publishing and Academic Resource Coalition of the As- sociation of Research Libraries (ARL), Washington DC, USA.
Address of Technical Managing Editor: Ulf Rehmann, Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, Postfach 100131, D-33501 Bielefeld, Copyright c 2010 for Layout: Ulf Rehmann.
Typesetting in TEX.
Extra Volume: Andrei A. Suslin’s Sixtieth Birthday, 2010
Preface 1
Denis Benois
Infinitesimal Deformations and thel-Invariant 5–31 Mikhail V. Bondarko
Motivically Functorial Coniveau Spectral Sequences;
Direct Summands of Cohomology of Function Fields 33–117 Manuel Breuning and David Burns
On Equivariant Dedekind Zeta-Functions ats= 1 119–146 V. Chernousov
Variations on a Theme of Groups Splitting by a Quadratic Extension and Grothendieck-Serre Conjecture for
Group SchemesF4 with Trivial g3 Invariant 147–169 Vincent Franjou and Wilberd van der Kallen
Power Reductivity over an Arbitrary Base 171–195 Eric M. Friedlander and Julia Pevtsova
Generalized Support Varieties
for Finite Group Schemes 197–222
Thomas Geisser
On Suslin’s Singular Homology and Cohomology 223–249 Detlev W. Hoffmann
Dimensions of Anisotropic
Indefinite Quadratic Forms II 251–265
Kevin Hutchinson, Liqun Tao Homology Stability
for the Special Linear Group of a Field
and Milnor-Witt K-theory 267–315
Bruno Kahn
Cohomological Approaches toSK1
andSK2 of Central Simple Algebras 317–369 Nikita A. Karpenko
with an Appendix by Jean-Pierre Tignol
Hyperbolicity of Orthogonal Involutions 371–392 Marc Levine
Slices and Transfers 393–443
Aurel Meyer, Zinovy Reichstein Some Consequences of
the Karpenko-Merkurjev Theorem 445–457
K-Theory and the Enriched Tits Building 459–513 Ivan Panin and Konstantin Pimenov
Rationally Isotropic Quadratic Spaces
Are Locally Isotropic: II 515–523
S. Saito and K. Sato
A p-adic Regulator Map and Finiteness Results
for Arithmetic Schemes 525–594
Michael Spieß
Twists of Drinfeld–Stuhler Modular Varieties 595–654 V. Valtman and S. Vostokov
Artin-Hasse Functions
and their Invertions in Local Fields 655–660 Alexander Vishik
Rationality of Integral Cycles 661–670
Vladimir Voevodsky
Cancellation Theorem 671–685
Go Yamashita
Bounds for the Dimensions
ofp-Adic Multiple L-Value Spaces 687–723
Preface
Over four decades, Andrei Suslin has conducted inspirational research at St. Pe- tersburg University (LOMI) and Northwestern University. Andrei’s impact on algebraic K-theory, motivic cohomology, central simple algebras, cohomology of groups, and representation theory have fundamentally changed these subjects.
Many of the best results in these areas are due to Andrei, many more were achieved using his ideas and guidance. Andrei’s influence extends beyond his published achievements, for he has been most generous in sharing his ideas and insights. With great admiration, this volume of Documenta Mathematica is dedicated to him.
St. Petersburg memories, Sasha Merkurjev
The Boarding School # 45 was a unique special place. It collected talented pupils in the North-West region of the Soviet Union. It was the only way into mathematics for many people living outside of big cities. Suslin taught at this school during 3 years when he was an undergraduate student. His style made a tremendous impact on me that I have never experienced later. Not only on me – for example, I just recently met my class-mate Sasha Koldobskiy (he is professor at the University of Missouri) and he shares the same feelings.
Needless to say that already at that time I decided to study algebra. Such early decisions were not exceptional: Nikita Karpenko asked me to be his advisor when he was a 9th year student at the School # 45.
Andrei’s passion for mathematics and his systematic approach were a model for us. We saw him reading algebra books like Bourbaki commutative algebra in a bus or metro. During short breaks between lessons he draw complicated diagrams in the notebook (standard thin 2 kopeks notebooks where Andrei used to record all his math) – that time Andrei was working on a problem in finite geometry and combinatorics. I guess that work was not successful and at the beginning of the senior year Andrei realized that he has nothing yet done for the diploma work to be completed in 9 months. That is how he turned to Serre’s conjecture concerning modules over polynomial rings.
During boring meetings we had to sit at, Andrei would ask me to give him problems to solve from recent mathematical olympiads, and often my list ended before the meeting was over. Andrei was a winner of the International Mathe- matical Olympiad in 1967.
The “olympiad spirit” has an interesting consequence: Andrei considers every mathematical problem as a personal challenge. That is why there are not so many Suslin’s conjectures: by making a conjecture Andrei admits that he failed to prove it himself.
Andrei’s impact of mathematicians has been tremendous, not only his own graduate students but on many others fortunate to be around him. I remember spontaneous seminars (for many hours) Andrei started when people randomly get together in his room at LOMI. I remember his lectures on the foundations of motivic cohomology in the late 80’s, when it was rather an improvisation at the board than lectures. Two of Andrei’s graduate students, Vanya Panin and Serge Yagunov, are organizers of this birthday celebration; other people who can call Andrei an informal advisor include Sasha Smirnov, Sasha Nenashev, myself, . . . During these seminars Andrei generously shared his ideas. (Markus Rost is another personality of this type.)
Immediately after his graduation, Andrei was hired as an assistant professor at the University (so he has never been a graduate student). He worked on Serre’s conjecture and tried to hide from the rest of the university world – at least he did not propose themes for students’ work, and I was not able to get him as thesis advisor.
Andrei liked to work at night – this habit comes from the time when he lived in an apartment shared by several families (with one bathroom and kitchen), so he could only work in the kitchen after midnight.
The most funny story about Andrei (unfortunately not for publishing) is that once he was a member of the Congress of the Young Communist League (he was the only doctor of sciences in the country younger 28) and he was given a speech to read about Brezhnev helping him to prove Serre’s Conjecture. As an exchange he was promised a separate apartment but it did not work out.
Perspective of a friend and colleague, Eric Friedlander Andrei has been my close friend for many years. We first met in Oberwolfach in the late 1970’s. Andrei’s English was perfect; not only did he speak and understand the language, but he understood subtle nuances which astonished me. We talked mathematics, but also about many other matters. This was the time his mathematical legend was already being established.
Perhaps few remember that Andrei was an “all Leningrad” gymnast. This showed when he lectured, for he seemed more poised at the blackboard. Some of us have never learned, despite much trying, to imitate his style of speaking slowly, writing very large symbols on the blackboard, all the while conveying elegantly and efficiently the essence of his mathematics.
A few years later, Andrei and I both visited University of Paris 7. An early memory of that year followed Andrei’s talk and gold medal at the College de France. We wandered around Paris at 7:30pm looking for dinner. All restau- rants were empty, but all were reserved for the night, just as had been the case of restaurants in the USSR. One morning Andrei called me to say that during the night he proved the Quillen-Lichtenbaum Conjecture for algebraically closed fields of positive characteristic and asked if I would photocopy his manuscript at IHES. Andrei stood at the entrance of the peripherique on the fringe of
Paris, handing through my car window his coffee-stained manuscript as the car briefly paused before quickly merging into traffic. What did this Russian to American exchange look like to an observer? When he first talked about this result in a Paris seminar, the audience broke tradition to give him an ovation.
The 1986 ICM in Berkeley was the “Mathematical Congress of Absent Rus- sians”. The world mathematical community eagerly anticipated the remark- able, almost mythical creators of so much new mathematics. Sadly, Andrei was among those not allowed to attend, but I was given a manuscript of his plenary address. This manuscript consisted of page after page of new results on algebraic K-theory. After spending time with Andrei in Paris, I had the privilege of visiting the Suslin family in their St. Petersburg apartment; my achievement was explaining the colloquial English in a popular cartoon series, not quite equal to Andrei’s explanations of mathematical lectures given in Rus- sian which we attended in Novosibirsk. Food memories include the delicious
”Russian salad” and the rich soup of cepes (from the woods near the Suslin dacha) prepared by Olga Suslina. A measure of time passing has been watching Andrei’s daughters Olga and Maria grow from young girls to successful adults with children of their own.
Andrei visited M.I.T. and the University of Chicago in the early 1990’s. To my overwhelming delight and benefit, Andrei decided to join the Northwestern faculty in 1995. A frequent image which comes to mine is of Andrei pacing outside my office ignoring whatever weather Chicago was throwing us, while I stayed warm and dry by scribbling on a blackboard. The best of those times was our extended effort to prove finite generation of certain cohomology rings;
this was a question that I had thought about for years, and the most important step I took towards its solution was to consult Andrei. Vladimir Voevodsky was briefly our colleague at Northwestern. Indeed, a few years earlier, I had arranged for Andrei to meet Vladimir, recognizing that their different styles and powerful mathematical talents could be blended together in a very fruitful manner.
So many mathematicians over the years have benefited from Andrei’s insights and confidence. If someone mentioned a result, then typically Andrei would say he is sure it is right. On the other hand, should he need the result he would produce his own proof – typically improving the statement as well as the proof – or find a counter-example. With me, perhaps Andrei was a bit more relaxed for he would occasionally tell me something was nonsense and even occasionally admit after extended discussion that he was wrong. Those interactions are among my best memories of our days together at Northwestern.
Andrei’s generosity extended to looking after me on the ski slopes, willingness to drive to the airport at an awful hour, and other matters of daily life. Our friendship has been the most remarkable aspect of my mathematical career.
I. Fesenko, E. Friedlander, A. Merkurjev, U. Rehmann
Infinitesimal Deformations and the
ℓ-Invariant
To Andrei Alexandrovich Suslin, for his 60th birthday
Denis Benois
Received: November 11, 2009 Revised: January 22, 2010
Abstract. We give a formula for the generalized Greenberg’sℓ-invariant which was constructed in [Ben2] in terms of derivatives of eigenvalues of Frobenius.
2000 Mathematics Subject Classification: 11R23, 11F80, 11S25, 11G40, 14F30
Keywords and Phrases: p-adic representation, (ϕ,Γ)-module,L-function Introduction
0.1. Let M be a pure motive over Q with coefficients in a number field E.
Assume that the L-function L(M, s) is well defined. Fixinig an embedding ι : E ֒→Cwe can consider it as a complex-valued Dirichlet seriesL(M, s) =
P∞ n=0
ann−s which converges fors≫0 and is expected to admit a meromorphic continuation toCwith a functional equation of the form
Γ(M, s)L(M, s) = ε(M, s) Γ(M∗(1),−s)L(M∗(1),−s)
where Γ(M, s) is the product of some Γ-factors and the ε-factor has the form ε(M, s) =abs.
Assume that M is critical and that L(M,0) 6= 0.Fix a finite place λ|p of E and assume that the λ-adic realization Mλ of M is semistable in the sense of Fontaine [Fo3]. The (ϕ, N)-module Dst(Mλ) associated to Mλ is a finite dimensionalEλ-vector space equipped with an exhaustive decreasing filtration FiliDst(Mλ), a Eλ-linear bijective frobenius ϕ : Dst(Mλ) −→ Dst(Mλ) and a nilpotent monodromy operator N such that N ϕ = p ϕ N. We say that a (ϕ, N)-submoduleD ofDst(Mλ) is regular if
Dst(Mλ) = D⊕Fil0Dst(Mλ)
as Eλ-vector spaces. The theory of Perrin-Riou [PR] suggests that to any reg- ularDone can associate ap-adicL-functionLp(M, D, s) interpolating rational parts of special values of L(M, s). In particular, the interpolation formula at s= 0 should have the form
Lp(M, D,0) = E(M, D)L(M,0) Ω∞(M)
where Ω∞(M) is the Deligne period ofM andE(M, D) is a certain product of Euler-like factors. Therefore one can expect that Lp(M, D,0) = 0 if and only ifE(M, D) = 0 and in this case one says thatLp(M, D, s) has a trivial zero at s= 0.
0.2. According to the conjectures of Bloch and Kato [BK], the Eλ-adic representationMλshould have the following properties:
C1)The Selmer groupsHf1(Mλ) andHf1(Mλ∗(1)) are zero.
C2) H0(Mλ) = H0(Mλ∗(1)) = 0 where we write H∗ for the global Galois cohomology.
Moreover one expects that
C3)ϕ : Dst(Mλ)−→Dst(Mλ) is semisimple (semisimplicity conjecture).
We also make the following assumption which is a direct generalization of the hypothesisU)from [G].
C4) The (ϕ,Γ)-module D†rig(Mλ) has no saturated subquotients of the form Um,n whereUm,n is the unique crystalline (ϕ,Γ)-module sitting in a non split exact sequence
0−→ RL(|x|xm)−→Um,n−→ RL(x−n)−→0, L=Eλ
(see§1 for unexplained notations).
In [Ben2], we extended the theory of Greenberg [G] toL-adic pseudo geometric representations which are semistable at p and satisfyC1-4). Namely to any regular D ⊂Dst(V) of a reasonably behaved representation V we associated an integer e > 0 and an element L(V, D) ∈ L which can be seen as a vast generalization of the L-invariants constructed in [Mr] and [G]. If V =Mλ we set L(M, D) =L(Mλ, D). A natural formulation of the trivial zero conjecture states as follows:
Conjecture. Lp(M, D, s) has a zero of ordereats= 0 and
(0.1) lim
s→0
Lp(M, D, s)
se = E+(M, D)L(M∗(1), D∗) L(M,0) Ω∞(M),
where E+(M, D) is the subproduct of E(M, D) obtained by ”excluding zero factors” and D∗ = Hom(Dst(V)/D,Dst(L(1))) is the dual regular module
(see [Ben2] for more details). We refer to this statement as Greenberg’s conjecture because if Mλ is ordinary at p it coincides with the conjecture formulated in [G], p.166. Remark that if Mλ is crystalline at p, Greenberg’s conjecture is compatible with Perrin-Riou’s theory ofp-adicL-functions [Ben3].
0.3. Consider the motiveMf attached to a normalized newformf = P∞
n=1
anqn of weight 2k on Γ0(N p) with (N, p) = 1. The complex L-function of Mf is L(f, s) = P∞
n=1
ann−s.The twisted motiveMf(k) is critical. The eigenvalues ofϕ acting onDst(Mf,λ(k)) areα=p−kapandβ=p1−kapwithvp(ap) =k−1. The unique regular submodule ofDst(Mf,k(k)) is D=Eλdwhere ϕ(d) =α dand Lp(Mf(k), D, s) =Lp(f, s+k) whereLp(f, s) is the classicalp-adicL-function associated to ap via the theory of modular symbols [Mn], [AV]. If ap =pk−1, the functionLp(f, s) vanishes ats=k. In this case several constructions of the L-invariant based on different ideas were proposed (see [Co1], [Tm], [Mr], [O], [Br]). Thanks to the work of many people it is known that they are all equal and we refer to [Cz3] and [BDI] for further information. AsMf(k) is self-dual (i.e.
Mf(k)≃Mf∗(1−k)) one hasL(Mf∗(1−k), D∗) =L(Mf(k), D) (see also section 0.4 below). Moreover it is not difficult to prove thatL(Mf(k), D) coincides with theL-invariant of Fontaine-MazurLFM(f) [Mr] ([Ben2], Proposition 2.3.7) and (0.1) takes the form of the Mazur-Tate-Teitelbaum conjecture
L′p(f, k) =L(f)L(f, k) Ω∞(f)
where we writeL(f) for an unspecifiedL-invariant and Ω∞(f) for the Shimura period off [MTT]. This conjecture was first proved by Greenberg and Stevens in the weight two case [GS1] [GS2]. In the unpublished note [St], Stevens generalized this approach to the higher weights. Other proofs were found by Kato, Kurihara and Tsuji (unpublished but see [Cz2]), Orton [O], Emerton [E] and by Bertolini, Darmon and Iovita [BDI]. The approach of Greenberg and Stevens is based on the study of families of modular forms and their p- adicL-functions. Namely, Hida (in the ordinary case) and Coleman [Co1] (in general) constructed an analytic family fx = P∞
n=1
an(x)qn of p-adic modular forms for x ∈ Cp passing through f with f = f2k. Next, Panchishkin [Pa]
and independently Stevens (unpublished) constructed a two-variablep-adicL- function L-functionLp(x, s) satisfying the following properties:
• Lp(2k, s) =Lp(f, s).
• Lp(x, x−s) =− Ns−x
Lp(x, s).
•Lp(x, k) = (1−pk−1ap(x)−1)L∗(x) whereL∗p(x) is ap-adic analytic function such thatL∗p(2k) =L(f, k)/Ω∞(f).
From these properties it follows easily that
L′p(f, k) =−2dlogap(2k)L(f, k) Ω∞(f),
where dlogap(x) =ap(x)−1dap(x)
dx . Thus the Mazur-Tate-Teitelbaum conjec- ture is equivalent to the assertion that
(0.2) L(f) =−2dlogap(2k).
This formula was first proved for weight two by Greenberg and Stevens. In the higher weight case several proofs of (0.2) have been proposed:
1. By Stevens [St], working with Coleman’sL-invariantLC(f) defined in [Co1].
2. By Colmez [Cz5], working with the Fontaine-Mazur’s L-invariantLFM(f) defined in [Mr].
3. By Colmez [Cz6], working with Breuil’sL-invariantLBr(f) defined in [Br].
4. By Bertolini, Darmon and Iovita [BDI], working with Teitelbaum’s L- invariantLT(f) [Tm] and Orton’sL-invariantLO(f) [O].
0.4. In this paper, working with theL-invariant defined in [Ben2] we generalize (0.2) to some infinitesimal deformations of pseudo geometric representations.
Our result is purely algebraic and is a direct generalization of Theorem 2.3.4 of [GS2] using the cohomology of (ϕ,Γ)-modules instead Galois cohomology.
Let V be a pseudo-geometric representation with coefficients in L/Qp which satisfies C1-4). Fix a regular submoduleD. In view of (0.1) it is convenient to set
ℓ(V, D) =L(V∗(1), D∗).
Suppose thate= 1. Conjecturally this means that the p-adic L-function has a simple trivial zero. Then eitherDϕ=p−1 or (D∗)ϕ=p−1 has dimension 1 over L. To fix ideas, assume that dimLDϕ=p−1 = 1. Otherwise, as one expects a functional equation relating Lp(M, D, s) and Lp(M∗(1), D∗,−s) one can consider V∗(1) and D∗ instead V andD. We distinguish two cases. In each case one can expressℓ(V, D) directly in terms ofV andD.
• The crystalline case: Dϕ=p−1∩N Dst(V)ϕ=1
= {0}. Let D†rig(V) be the (ϕ,Γ)-module over the Robba ring RL associated to V [Ber1], [Cz1]. Set D−1 = (1−p−1ϕ−1)D and D0 = D. The two step filtration D−1 ⊂ D0 ⊂ Dst(V) induces a filtration
F−1D†rig(V)⊂F0D†rig(V)⊂D†rig(V)
such that gr0D†rig(V)≃ RL(δ) is the (ϕ,Γ)-module of rank 1 associated to a characterδ : Q∗p−→L∗ of the formδ(x) =|x|xmwithm>1.The cohomology of (ϕ,Γ)-modules of rank 1 is studied in details in [Cz4]. Let η : Q∗p−→L∗ be a continuous character. Colmez proved that H1(RL(η)) is a one dimensional L-vector space except for η(x) = |x|xm with m > 1 and η(x) = x−n with n60. In the exceptional casesH1(RL(η)) has dimension 2 and can be canon- ically decomposed into direct sum of one dimensional subspaces
(0.3) H1(RL(η))≃Hf1(RL(η))⊕Hc1(RL(η)), η(x) =|x|xm orη(x) =x−n
([Ben2], Theorem 1.5.7). The conditionC1)implies that
(0.4) H1(V)≃M
l∈S
H1(Ql, V) Hf1(Ql, V)
for a finite set of primes S. This isomorphism defines a one dimensional subspace H1(D, V) of H1(V) together with an injective localisation map κD : H1(D, V) −→ H1(RL(δ)). Then ℓ(V, D) is the slope of Im(κD) with respect to the decomposition ofH1(RL(δ)) into direct sum (0.3). Let
0−→V −→Vx−→L−→0
be an extension in the category of global Galois representations such that cl(x)∈H1(D, V) is non zero. We equipD†rig(Vx) with a canonical filtration
{0} ⊂F−1D†rig(Vx)⊂F0D†rig(Vx)⊂F1D†rig(Vx)⊂D†rig(Vx)
such that FiD†rig(Vx) = FiD†rig(V) for i = −1,0 and gr1D†rig(Vx) ≃ RL. Let VA,x be an infinitesimal deformation of Vx over A=L[T]/(T2) endowed with a filtrationFiD†rig(VA,x) such thatFiD†rig(V) =FiD†rig(VA,x)⊗AL.Write
gr0D†rig(VA,x)≃ RA(δA,x), gr1D†rig(VA,x)≃ RA(ψA,x) withδA,x, ψA,x: Q∗p−→A∗.
Theorem 1. Assume that d(δA,xψA,x−1)(u) dT
T=06= 0foru≡1 (modp2).Then
ℓ(V, D) =−log(u)dlog(δA,xψA,x−1)(p) dlog(δA,xψ−1A,x)(u)
T=0
(note that the right hand side does not depend on the choice ofu).
•The semistable case: Dϕ=p−1⊂N Dst(V)ϕ=1
.SetD−1= (1−p−1ϕ−1)D, D0=D andD1=N−1(Dϕ=p−1)∩Dst(V)ϕ=1.The filtration
D−1⊂D0⊂D1⊂Dst(V) induces a filtration
F−1D†rig(V)⊂F0D†rig(V)⊂F1D†rig(V)⊂D†rig(V)
Then gr0D†rig(V) ≃ RL(δ) and gr1D†rig(V) ≃ RL(ψ) where the characters δ and ψ are such that δ(x) = |x|xm and ψ(x) = x−n for some m > 1 and n>0.SetM =F1D†rig(V)/F−1D†rig(V) and consider the mapκD : H1(M)−→ H1(RL(ψ)) induced by the projectionM −→ RL(ψ).The image ofκD is a one dimensionalL-subspace ofH1(RL(ψ)) andℓ(V, D) is the slope of Im(κD) with respect to (0.3).
Assume thatVA is an infinitesimal deformation ofV equipped with a filtration FiD†rig(VA) such that FiD†rig(V) = FiD†rig(VA)⊗A L. Write gr0D†rig(VA) ≃ RA(δA) and gr1D†rig(VA)≃ RA(ψA).
Theorem 2. Assume that (0.5) d(δAψA−1)(u)
dT
T=0
6
= 0 foru≡1 (modp2).
Then
ℓ(V, D) =−log(u)dlog(δAψ−1A ) (p) dlog(δAψA−1)(u) T=0. Remark that in the semistable caseℓ(V, D) =L(V, D).
For classical modular forms the existence of deformations having the above properties follows from the theory of Coleman-Mazur [CM] together with deep results of Saito and Kisin [Sa], [Ki]. Applying Theorem 2 to the representation Mf,λ(k) we obtain a new proof of (0.2) with the Fontaine-MazurL-invariant.
Remark that the local parameter T corresponds to the weight of a p-adic modular form and (0.5) holds automatically. In the general case the existence of deformations satisfying the above conditions should follow from properties of eigenvarieties of reductive groups [BC].
The formulations of Theorems 1 and 2 look very similar and the proof is essentially the same in the both cases. The main difference is that in the crystalline case the ℓ-invariant is global and contains information about the localisation mapH1(V)−→H1(Qp, V).In the proof of Theorem 1 we consider Vx as a representation of the local Galois group but the construction of Vx
depends on the isomorphism (0.4). In the semistable case the definition of ℓ(V, D) is purely local and the hypothesis C1-2) can be omitted. However C1-2) are essential for the formulation of Greenberg conjecture because (0.1) is meaningless ifL(M,0) = 0. One can compare our results with Hida’s paper [Hi] where the case of ordinary representations over totally real ground field is studued.
Here goes the organization of this paper. The §1 contains some background material. In section 1.1 we review the theory of (ϕ,Γ)-modules and in section 1.2 recall the definition of theℓ-invariant following [Ben2]. The crystalline and semistable cases of trivial zeros are treated in §2 and §3 respectively. I would like to thank Pierre Parent for several very valuable discussions which helped me with the formulation of Theorem 1 and the referee for pointing out several inaccuracies in the first version of this paper.
It is a great pleasure to dedicate this paper to Andrei Alexandrovich Suslin on the occasion of his 60th birthday.
§1. The ℓ-invariant 1.1. (ϕ,Γ)-modules. ([Fo1], [Ber1], [Cz1])
1.1.1. Let pbe a prime number. Fix an algebraic closureQp of Qp and set GQp = Gal(Qp/Qp).We denote by Cp thep-adic completion of Qp and write
| · |for the absolute value onCp normalized by|p|= 1/p.For any 06r <1 set B(r,1) ={z∈Cp|p−1/r6|z|<1}.
Let χ : GQp −→Z∗p denote the cyclotomic character. Set HQp = ker(χ) and Γ = GQp/HQp. The characterχ will be often considered as an isomorphism χ : Γ→∼ Z∗p. LetL be a finite extension ofQp. For any 06r <1 we denote by B†rig,L,r the ring of p-adic functions f(π) = P
k∈Z
akπk (ak ∈ L) which are holomorphic on the annulusB(r,1).The Robba ring overLis defined asRL= S
r
B†rig,L,r . Recall that RL is equipped with commuting, L-linear, continuous actions of Γ and a frobeniusϕwhich are defined by
γ(f(π)) =f((1 +π)χ(γ)−1), γ∈Γ, ϕ(f(π)) =f((1 +π)p−1).
Sett= log(1 +π) = X∞ n=1
(−1)n−1πn
n .Remark thatγ(t) =χ(γ)tandϕ(t) =p t.
A finitely generated free RL-module D is said to be a (ϕ,Γ)-module if it is equipped with commuting semilinear actions of Γ and ϕ and such that RLϕ(D) = D. The last condition means simply that ϕ(e1), . . . , ϕ(ed) is a basis ofDife1, . . . , ed is.
Let δ : Q∗p −→ L∗ be a continuous character. We will write RL(δ) for the (ϕ,Γ)-moduleRLeδ of rank 1 defined by
ϕ(eδ) =δ(p)eδ, γ(eδ) =δ(χ(γ))eδ, γ∈Γ.
For any Dwe letD(χ) denote the ϕ-module Dendowed with the action of Γ twisted by the cyclotomic characterχ.
Fix a topological generator γ ∈ Γ. For any (ϕ,Γ)-module D we denote by Cϕ,γ(D) the complex
0−→D−→f D⊕D−→g D−→0
withf(x) = ((ϕ−1)x,(γ−1)x) andg(y, z) = (γ−1)y−(ϕ−1)z([H1], [Cz4]).
We shall writeH∗(D) for the cohomology ofCϕ,γ(D).The main properties of these groups are the following
1) Long cohomology sequence. A short exact sequence of (ϕ,Γ)-modules 0−→D′−→D−→D′′−→0
gives rise to an exact sequence
0−→H0(D′)−→H0(D)−→H0(D) ∆
0
−−→H1(D′)−→ · · · −→H2(D′′)−→0.
2) Euler-Poincar´e characteristic. Hi(D) are finite dimensionalL-vector spaces and
χ(D) = X2 i=0
(−1)idimLHi(D) = −rg(D).
(see [H1] and [Li]).
3) Computation of the Brauer group. The map cl(x)7→ −
1−1
p −1
(logχ(γ))−1res(xdt)
is well defined and induces an isomorphism inv : H2(RL(χ)) →∼ L (see [H2]
[Ben1] and [Li]).
4) The cup-products. LetD and M be two (ϕ,Γ)-modules. For alli and j such thati+j62 define a bilinear map
∪ : Hi(D)×Hj(M)−→Hi+j(D⊗M) by
cl(x)∪cl(y) = cl(x⊗y) ifi=j= 0,
cl(x)∪cl(y1, y2) = cl(x⊗y1, x⊗y2) ifi= 0, j= 1,
cl(x1, x2)∪cl(y1, y2) = cl(x2⊗γ(y1)−x1⊗ϕ(y2)) ifi= 1,j= 1, cl(x)∪cl(y) = cl(x⊗y) ifi= 0,j= 2.
These maps commute with connecting homomorphisms in the usual sense.
5) Duality. LetD∗= HomRL(D,RL).Fori= 0,1,2 the cup product (1.1) Hi(D)×H2−i(D∗(χ))−→∪ H2(RL(χ))≃L
is a perfect pairing ([H2], [Li]).
1.1.2. Recall that a filtered (ϕ, N)-module with coefficients in L is a finite dimensional L-vector spaceM equipped with an exhausitive decreasing filtra- tion FiliM, a linear bijective map ϕ : M −→ M and a nilpotent operator N : M −→ M such that ϕN = p ϕN. Filtered (ϕ, N)-modules form a ⊗- category which we denote by MFϕ,N.A filtered (ϕ, N)-module M is said to
be a Dieudonn´e module if N = 0 on M. Filtered Dieudonn´e modules form a full subcategory MFϕ of MFϕ,N. It is not difficult to see that the series log(ϕ(π)/πp) and log(γ(π)/π) (γ∈Γ) converge inRL.Let logπbe a transcen- dental element over the field of fractions ofRLequipped with actions ofϕand Γ given by
ϕ(logπ) =plogπ+ log ϕ(π)
πp
, γ(logπ) = logπ+ log γ(π)
π
.
Thus the ring RL,log = RL[logπ] is equipped with natural actions of ϕ and Γ and the monodromy operator N = −
1−1
p −1
d
dlogπ. For any (ϕ,Γ)- moduleDset
Dst(D) = (D⊗RLRL,log[1/t])Γ
witht= log(1+π).ThenDst(D) is a finite dimensionalL-vector space equipped with natural actions ofϕandNsuch thatN ϕ=p ϕN.Moreover, it is equipped with a canonical exhaustive decreasing filtration FiliDst(D) which is induced by the embeddings ιn : B†,rrig,L ֒→ L∞[[t]], n ≫ 0 constructed in [Ber1] (see [Ber2] for more details). Set
Dcris(D) =Dst(D)N=0= (D[1/t])Γ. Then
dimLDcris(D)6dimLDst(D)6rg(D)
and one says thatDis semistable (resp. crystalline) if dimLDcris(D) = rg(D) (resp. if dimLDst(D) = rg(D)). IfD is semistable, the jumps of the filtration FiliDst(D) are called the Hodge-Tate weights ofDand the tangent space ofD is defined astD(L) =Dst(D)/Fil0Dst(D).
We let denote by Mϕ,Γpst andMϕ,Γcris the categories of semistable and crystalline representations respectively. In [Ber2] Berger proved that the functors
( 1.2) Dst : Mϕ,Γpst −→MFϕ,N, Dcris : Mϕ,Γcris−→MFϕ are equivalences of⊗-categories.
1.1.3. As usually, H1(D) can be interpreted in terms of extensions. Namely, to any cocycleα= (a, b)∈Z1(Cϕ,γ(D)) one associates the extension
0−→D−→Dα−→ RL−→0
such thatDα =D⊕ RLewith ϕ(e) =e+aand γ(e) =e+b.This defines a canonical isomorphism
H1(D)≃Ext1(RL,D).
We say that cl(α)∈H1(D) is crystalline if dimLDcris(Dα) = dimLDcris(D)+1 and define
Hf1(D) ={cl(α)∈H1(D) | cl(α) is crystalline}.
It is easy to see that Hf1(D) is a subspace ofH1(D). IfD is semistable (even potentially semistable), one has
H0(D) = Fil0Dst(D)ϕ=1,N=0,
dimLHf1(D) = dimLtD(L) + dimLH0(D) (1.3)
(see [Ben2], Proposition 1.4.4 and Corollary 1.4.5). Moreover, Hf1(D) and Hf1(D∗(χ)) are orthogonal complements to each other under duality (1.1) ([Ben2], Corollary 1.4.10).
1.1.4. LetDbe semistable (ϕ,Γ)-module of rankd. Assume thatDst(D)ϕ=1= Dst(D) and that the all Hodge-Tate weights ofDare>0.SinceN ϕ=pϕNthis implies thatN = 0 onDst(D) andD is crystalline. The results of this section are proved in [Ben2] (see Proposition 1.5.9 and section 1.5.10). The canonical mapDΓ−→ Dcris(D) is an isomorphism and thereforeH0(D)≃ Dcris(D) =DΓ has dimension doverL. The Euler-Poincar´e characteristic formula gives
dimLH1(D) =d+ dimLH0(D) + dimLH0(D∗(χ)) = 2d.
On the other hand dimLHf1(D) = d by (1.3). The group H1(D) has the following explicit description. The map
iD : Dcris(D)⊕ Dcris(D)−→H1(D), iD(x, y) = cl(−x,logχ(γ)y)
is an isomorphism. (Remark that the sign −1 and logχ(γ) are normalizing factors.) We let denote iD,f and iD,c the restrictions of iD on the first and second summand respectively. Then Im(iD,f) =Hf1(D) and we setHc1(D) = Im(iD,c).Thus we have a canonical decomposition
H1(D)≃Hf1(D)⊕Hc1(D) ([Ben2], Proposition 1.5.9).
Now consider the dual module D∗(χ). It is crystalline, Dcris(D∗(χ))ϕ=p−1 = Dcris(D∗(χ)) and the all Hodge-Tate weights ofD∗(χ) are60.Let
[, ]D : Dcris(D∗(χ))× Dcris(D)−→L denote the canonical pairing. Define
iD∗(χ) : Dcris(D∗(χ))⊕ Dcris(D∗(χ))−→H1(D∗(χ)) by
iD∗(χ)(α, β)∪iD(x, y) = [β, x]D−[α, y]D.
As before, let iD∗(χ), f and iD∗(χ), c denote the restrictions of iD on the first and second summand respectively. FromHf1(D∗(χ)) =Hf1(D)⊥it follows that Im(iD∗(χ), f) =Hf1(D∗(χ)) and we setHc1(D∗(χ)) = Im(iD∗(χ), c).
Write∂for the differential operator (1 +π) d dπ.
Proposition 1.1.5. LetRL(|x|xm)be the(ϕ,Γ)-moduleRLeδ associated to the characterδ(x) =|x|xm(m>1). Then
i) Dcris(RL(|x|xm)) is the one-dimensional L-vector space generated by t−meδ. Moreover Dcris(RL(|x|xm)) = Dcris(RL(|x|xm))ϕ=p−1 and the unique Hodge-Tate weight ofRL(|x|xm)is−m.
ii) H0(RL(|x|xm)) = 0 and H1(RL(|x|xm)) is the two-dimensional L-vector space generated by α∗m = −
1−1
p
cl(αm) and βm∗ =
1−1 p
logχ(γ) cl(βm)where
αm=(−1)m−1 (m−1)! ∂m−1
1 π+1
2, a
eδ
witha∈ R+L =RL∩L[[π]]such that(1−ϕ)a= (1−χ(γ)γ) 1
π+1 2
and
βm=(−1)m−1 (m−1)! ∂m−1
b,1
π
eδ
withb∈ RLsuch that(1−ϕ) 1
π
= (1−χ(γ)γ)b.Moreoverim,f(1) =α∗mand im,c(1) =βm∗ where im denotes the map i defined in 1.1.4 for RL(|x|xm). In particular,Hf1(RL(|x|xm))is generated byα∗mandHc1(RL(|x|xm))is generated byβm∗.
iii) Letx= cl(u, v)∈H1(RL(|x|xm)).Then x=acl(αm) +bcl(βm) witha= res(utm−1dt)andb= res(vtm−1dt).
iv) The map
Resm : RL(|x|xm)−→L, Resm(α) =−
1−1
p −1
(logχ(γ))−1res αtm−1dt induces an isomorphisminvm : H2(RL(|x|xm))≃L.Moreover
invm(ωm) = 1 where ωm= (−1)m
1−1 p
logχ(γ)
(m−1)!cl ∂m−1(1/π) Proof. The assertions i) and ii) are proved in [Cz4], sections 2.3-2.5 and [Ben2], Theorem 1.5.7 and (16). The assertions iii) and iv) are proved in [Ben2], Proposition 1.5.4 iii) Corollary 1.5.5.
1.1.6. In [Fo1], Fontaine worked out a general approach to the classification of p-adic representations in terms of (ϕ,Γ)-modules. Thanks to the work of Cherbonnier-Colmez [CC] and Kedlaya [Ke] this approach allows to construct an equivalence
D†rig : RepL(GQp)−→M´ϕ,Γet
between the category ofL-adic representations ofGQp and the categoryMϕ,Γ´et of ´etale (ϕ,Γ)-modules in the sense of [Ke]. IfV is aL-adic representation of GQp, define
Dst(V) =Dst(D†rig(V)), Dcris(V) =Dcris(D†rig(V)).
Then Dst and Dcris are canonically isomorphic to classical Fontaine’s func- tors [Fo2], [Fo3] defined using the rings Bst andBcris ([Ber1], Theorem 0.2).
The continuous Galois cohomologyH∗(Qp, V) =Hcont∗ (GQp, V) is functorially isomorphic toH∗(D†rig(V)) ([H1], [Li]). and under this isomorphism
Hf1(D†rig(V))≃Hf1(Qp, V)
where Hf1(Qp, V) = ker(H1(Qp, V)−→H1(Qp, V ⊗Bcris)) is Hf1of Bloch and Kato [BK].
1.2. The ℓ-invariant.
1.2.1. The results of this section are proved in [Ben2], 2.1-2.2. Denote by Q(S)/Qthe maximal Galois extension of Q unramified outside S∪ {∞}and set GS = Gal(Q(S)/Q).IfV is aL-adic representation of GS we writeH∗(V) for the continuous cohomology ofGS with coefficients inV. IfV is potentially semistable atp, set
Hf1(Ql, V) =
(ker(H1(Ql, V)−→H1(Qnrl , V) ifl6=p, Hf1(D†rig(V)) ifl=p.
The Selmer group of Bloch and Kato is defined by
Hf1(V) = ker H1(V)−→M
l∈S
H1(Ql, V) Hf1(Ql, V)
! .
Assume thatV satisfies the conditionC1-4)of0.2.
The Poitou-Tate exact sequence together with C1)gives an isomorphism
(1.4) H1(V)≃ M
l∈S
H1(Ql, V) Hf1(Ql, V).
Recall that a (ϕ, N)-submoduleD ofDst(V) is said to be regular if the canon- ical projectionD−→tV(L) is an isomorphism. To any regularD we associate a filtration onDst(V)
{0} ⊂D−1⊂D0⊂D1⊂Dst(V) setting
Di =
(1−p−1ϕ−1)D+N(Dϕ=1) ifi=−1,
D ifi= 0,
D+Dst(V)ϕ=1∩N−1(Dϕ=p−1) if i= 1.
By (1.2) this filtration induces a filtration on D†rig(V) by saturated (ϕ,Γ)- submodules
{0} ⊂F−1D†rig(V)⊂F0D†rig(V)⊂F1D†rig(V)⊂D†rig(V).
SetW =F1D†rig(V)/F−1D†rig(V).In [Ben2], Proposition 2.1.7 we proved that
(1.5) W ≃W0⊕W1⊕M,
whereW0andW1are direct summands of gr0
D†rig(V)
and gr1
D†rig(V) of ranks dimLH0(W∗(χ)) and dimLH0(W) respectively. MoreoverM seats in a non split exact sequence
0−→M0
−→f M −→g M1−→0 with rg(M0) = rg(M1),gr0
D†rig(V)
=M0⊕W0 and gr1
D†rig(V)
=M1⊕ W1. Set
e= rg(W0) + rg(W1) + rg(M0).
Generalizing [G] we expect that thep-adicL-functionLp(V, D, s) has a zero of ordereats= 0.
If W0 = 0, the main construction of [Ben2] associates to V and D an ele- ment L(V, D) ∈ L which can be viewed as a generalization of Greenberg’s L-invariant to semistable representations. Now assume that W1 = 0. Let D∗ = Hom(Dst(V)/D,Dst(Qp(1))) be the dual regular space. As the decom- positions (1.5) for the pairs (V, D) and (V∗(1), D∗) are dual to each other, one can define
ℓ(V, D) =L(V∗(1), D∗).
In this paper we do not review the construction of theL-invariant but give a direct description of ℓ(V, D) in terms ofV and D in two important particular cases.
1.2.2. The crystalline case: W =W0 (see [Ben2], 2.2.6-2.2.7 and 2.3.3).
In this caseW is crystalline,W1=M = 0 andF0D†rig(V) =F1D†rig(V).From the decomposition (1.5) it is not difficult to obtain the following description of Hf1(Qp, V) in the spirit of Greenberg’s local conditions:
(1.6) Hf1(Qp, V) = ker H1(F0D†rig(V))−→ H1(W) Hf1(W)
! .
Let H1(D, V) denote the inverse image of H1(F0D†rig(V))/Hf1(Qp, V) under the isomorphism (1.4). Thus one has a commutative diagram
(1.7) H1(D, VN)NNNNNNN//NHNN1N(F'' 0D†rig(V))
H1(D†rig(V))
where the vertical map is injective ([Ben2], section 2.2.1). From (1.6) it follows that the composition map
κD : H1(D, V)−→H1(F0D†rig(V))−→H1(W)
is injective. By construction,Dcris(W) =D/D−1=Dϕ=p−1. AsD is regular, the Hodge-Tate weights ofW are60.Thus one has a decomposition
iW : Dcris(W)⊕ Dcris(W)≃Hf1(W)⊕Hc1(W)≃H1(W).
Denote bypD,f andpD,c the projection ofH1(W) on the first and the second direct summand respectively. We have a diagram
Dcris(W)
H1(D, V)
ρqD,fqqqqqqq88 qq κD
//
ρD,c
&&
MM MM MM MM
MM H1(W)
pD,f
OO
pD,c
Dcris(W)
whereρD,c is an isomorphism. Then ℓ(V, D) = detL
ρD,f◦ρ−D,c1 | Dcris(W) .
1.2.3. The semistable case: W =M (see [Ben2], 2.2.3-2.2.4 and 2.3.3). In this caseW is semistable ,W0=W1= 0 and
(1.8) Hf1(Qp, V) = ker
H1(F1D†rig(V))−→H1(M1) .
Let H1(D, V) be the inverse image of H1(F1D†rig(V))/Hf1(Qp, V) under the isomorphism (1.4). Consider the exact sequence
H1(M0) h1(f) //H1(M) h1(g) //H1(M1) ∆
1
//H2(M0) //0.
H1(D, V)
κD
OO ¯κD
88r
rr rr rr rr r
By (1.8), the map ¯κD is injective and it is not difficult to prove that the image ofH1(D, V) inH1(M1) coincides with Im(h1(g)) ([Ben2], section 2.2.3).
Thus in the semistable case the position ofH1(D, V) inH1(M1) is completely determined by the the restriction of V on the decomposition group atp. By construction, Dst(M1) = D1/D where (D1/D)ϕ=1 = D1/D and the Hodge- Tate weights ofM1are>0.Again, one has an isomorphism
iM1 : Dcris(M1)⊕ Dcris(M1)≃Hf1(M1)⊕Hc1(M1)≃H1(M1) which allows to construct a diagram
Dst(M1)
Im(h1(g))
ρqD,fqqqqqqq88 qq κD
//
ρD,c
&&
MM MM MM MM
MM H1(M1)
pD,f
OO
pD,c
Dst(M1).
Then
(1.9) ℓ(V, D) =L(V, D) = detL
ρD,f◦ρ−D,c1 | Dst(M1) .
From (1.5) it is clear that if e = 1 then eitherW = W0 with rg(W0) = 1 or W =M with rg(M0) = rg(M1) = 1.We consider these cases separately in the rest of the paper.
§2. The crystalline case
2.1. Let A = L[T]/(T2) and let VA be a free finitely generated A-module equipped with a A-linear action of GS. One says that VA is an infinitesimal deformation of ap-adic representationV ifV ≃VA⊗AL.WriteRA=A⊗LRL and extend the actions of ϕ and Γ to RA by linearity. A (ϕ,Γ)-module over RA is a free finitely generated RA-module DA equipped with commuting semilinear actions ofϕand Γ and such thatRAϕ(DA) =DA.We say thatDA
is an infinitesimal deformation of a (ϕ,Γ)-moduleDoverRL ifD=DA⊗AL.
2.2. LetV be ap-adic representation ofGS which satisfies the conditionsC1- 4) and such that W =W0.Moreover we assume that rg(W) = 1.Thus W is a crystalline (ϕ,Γ)-module of rank 1 withDcris(W) =Dcris(W)ϕ=p−1 and such that Fil0Dcris(W) = 0.This implies that
(2.1) W ≃ RL(δ) with δ(x) =|x|xm, m>1.
(see for example [Ben2], Proposition 1.5.8). Note that the Hodge-Tate weight of W is −m. The L-vector space H1(D, V) is one dimensional. Fix a basis cl(x)∈H1(D, V).We can associate to cl(x) a non trivial extension
0−→V −→Vx−→L−→0.
This gives an exact sequence of (ϕ,Γ)-modules
0−→D†rig(V)−→D†rig(Vx)−→ RL−→0.
From (1.7) it follows that there exists an extension in the category of (ϕ,Γ)- modules
0−→F0D†rig(V)−→Dx−→ RL−→0 which is inserted in a commutative diagram
0 //F0D†rig(V) //
Dx //
RL //
=
0
0 //D†rig(V) //D†rig(Vx) //RL //0.
Define a filtration
{0} ⊂F−1D†rig(Vx)⊂F0D†rig(Vx)⊂F1D†rig(Vx)⊂D†rig(Vx) byFiD†rig(Vx) =FiD†rig(V) fori=−1,0 and F1D†rig(Vx) =Dx. Set
Wx=F1D†rig(Vx)/F−1D†rig(Vx).
