Galois deformation theory for norm fields and its arithmetic applications

385 Pages·2009·1.79 MB·English

Galois deformation theory for norm fields and its
arithmetic applications
by
Wansu Kim
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
(Mathematics)
in The University of Michigan
2009
Doctoral Committee:
Professor Brian D. Conrad, Co-chair
Associate Professor Stephen M. DeBacker, Co-chair
Professor Jeffrey C. Lagarias
Professor Dan Boneh, Stanford University
To my mentor in life, Mr. Daisaku Ikeda,
to a great teacher in mathematics, Brian Conrad,
to my mom and my sister,
and to whoever lent me support.
ii
ACKNOWLEDGEMENTS
The author deeply thanks Brian Conrad for his guidance. The author especially
appreciates his careful listening of my results and numerous helpful comments. The
author would also like to thank Gebhard B¨ockle and Urs Hartl for various helpful
comments. The author thanks Tong Liu for providing his idea to prove Corollary
12.2.6 when p = 2.
iii
TABLE OF CONTENTS
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
CHAPTER
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation and overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Structure of the Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Notations/Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
II. Frobenius modules and Hodge-Pink theory . . . . . . . . . . . . . . . . . . . 19
2.1 Rigid-analytic objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 ϕ-modules of finite P-height . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Hodge-Pink structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 Filtered isocrystals, crystalline GK-representations, and resum´e of [52] . . . 41
III. Hodge-Pink theory and rigid analytic ϕ-vector bundles . . . . . . . . . . . . 55
3.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Equvalence of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
IV. Weakly admissible Hodge-Pink structure . . . . . . . . . . . . . . . . . . . . . 69
4.1 Review of slopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2 “Dwork’s trick” for ϕ-modules . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 ϕ-vector bundle pure of slope 0 and weak admissibility . . . . . . . . . . . . 81
V. π -adic G -representation of finite P-height . . . . . . . . . . . . . . . . . . . 87
0 K
5.1 E´tale ϕ-modules and π -adic representations of G . . . . . . . . . . . . . . 87
0 K
5.2 Main theorem and G -representations of finite P-height . . . . . . . . . . . 96
K
VI. Some non-archimedean functional analysis . . . . . . . . . . . . . . . . . . . . 111
6.1 Rigid-analytic disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.2 Newton polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.3 Proof of Proposition 4.3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
VII. Effective local shtukas and π -divisible groups . . . . . . . . . . . . . . . . . . 144
0
7.1 Local shtukas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
iv
7.2 Classification of finite locally free group schemes with trivial Verschiebung . 153
7.3 Effective local shtukas and π -divisible groups . . . . . . . . . . . . . . . . . 162
0
7.4 Some commutative algebra over an adic ring . . . . . . . . . . . . . . . . . . 173
VIII. Torsion G -representations of P-height 6h . . . . . . . . . . . . . . . . . . . 177
K
8.1 Torsion ϕ-modules and torsion G -representation of P-height 6h . . . . . . 177
K
8.2 ϕ-modules with coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
8.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
IX. “Raynaud’s theory” for torsion ϕ-modules . . . . . . . . . . . . . . . . . . . . 205
9.1 Classification of rank-1 objects in (ModFI/S)6h . . . . . . . . . . . . . . . . 205
F
9.2 S-lattices of P-height 6h . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
9.3 The case of small h and small ramification . . . . . . . . . . . . . . . . . . . 215
9.4 Torsion Galois representations . . . . . . . . . . . . . . . . . . . . . . . . . . 222
X. Categories co-fibered in groupoids . . . . . . . . . . . . . . . . . . . . . . . . . 229
10.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
10.2 The 2-Yoneda lemma and representibility . . . . . . . . . . . . . . . . . . . . 238
10.3 Deformation and framed deformation groupoids . . . . . . . . . . . . . . . . 244
10.4 2-categorical limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
XI. Deformations for G -representations of P-height 6h . . . . . . . . . . . . . 260
K
11.1 Deformations and S -lattice of P-height 6h. . . . . . . . . . . . . . . . . . 261
A
11.2 Generic fibers of deformation rings. . . . . . . . . . . . . . . . . . . . . . . . 275
11.3 Local structure of the generic fiber of deformation ring . . . . . . . . . . . . 286
11.4 “Ordinary” and “formal” components . . . . . . . . . . . . . . . . . . . . . . 304
11.5 Connected components: h=1 Case . . . . . . . . . . . . . . . . . . . . . . . 328
11.6 Application to flat deformation rings . . . . . . . . . . . . . . . . . . . . . . 334
11.7 Representability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
XII. Integral p-adic Hodge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
12.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
12.2 Galois representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
12.3 Proof of Lemma 12.2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
v
CHAPTER I
Introduction
1.1 Motivation and overview
1.1.1 p-adic local Galois representations
Since the pioneering work of Wiles on the modularity of semi-stable elliptic curves
overQ, manyclassesof2-dimensional(modporp-adic)globalGaloisrepresentations
are known to “come from” modular forms. One of the main difficulties of proving
modularity lies in the study of local deformation problems with various p-adic Hodge
theory conditions, for which one needs to understand Galois-stable Z -lattices in
p
(potentially) semi-stable p-adic representations and their reductions mod pn. On the
other hand, “integral p-adic Hodge theory” is much more delicate than “classical”
p-adic Hodge theory, which makes it hard to study deformations satisfying various
p-adic Hodge theory conditions.
This paper introduces a new technique of using the norm fields to study defor-
mations and mod p reductions in p-adic Hodge theory, which is explained below.
Let K /Q be a finite extension. Choose a uniformizer π ∈ o , and consider an
p K
√
infinite Kummer-type extension K ∞ := K ( p∞ π). We put GK := Gal(K /K )
and G := Gal(K /K ). Kisin [52] showed that the restriction to G of a
K∞ ∞ K∞
semi-stable G -representation with Hodge-Tate weights in [0,h] is so-called a G -
K K
∞
1
2
representation “of height 6 h.”1 The precise definition will be given later in Defini-
tion 5.2.8. The point is that integral theory for G -representations “of height 6 h”
K
∞
is much simpler than integral p-adic Hodge theory, and that we lose no information
by restricting crystalline G -representations to G .2 See §2.4 for a summary of
K K
∞
Kisin [52].
In order to study (or even, to define) deformations “of height 6 h” one needs
to define and study torsion representations “of height 6 h,” which is carried out in
§8–§9 of this paper. One of the main results of this paper is the existence of universal
G -deformation rings “of height 6 h” for any positive integer h:
K
∞
Theorem(11.1.2). LetFbeafiniteextensionofF andletρ¯ beanF-representation
p ∞
of G of finite dimension. Then there exists a complete local noetherian W(F)-
K
∞
algebra R(cid:3),6h with residue field F and a framed deformation of ρ¯ over R(cid:3),6h
ρ¯∞ ∞ ρ¯∞
which is universal among all the framed deformations of ρ¯ with “height 6 h.”
∞
If End (ρ¯ ) ∼= F then there exists a complete local noetherian W(F)-algebra R6h
GK∞ ∞ ρ¯∞
with residue field F and a deformation of ρ¯ over R6h which is universal among all
∞ ρ¯∞
the deformations of ρ¯ with “height 6 h.”
∞
The existence of such G -deformation rings is surprising because the usual
K
∞
‘unrestricted’ G -deformation functor has a infinite-dimensional tangent space (so
K
∞
‘unrestricted’ G -deformation rings do not exist in the category of complete local
K
∞
noetherian rings); see §11.7.1 for the proof of this claim. Note that G does not
K
∞
satisfy the cohomological finiteness condition that is usually used to prove the finite-
dimensionality of the tangent space of interesting Galois deformation functors.
1Laterinthispaper, weusetheterminologyP-height insteadofheight whereP(u)isanEisensteinpolynomial
over the maximal unramified subextension K0 of K such that P(π) = 0. This is to avoid confusion with the
analogousnotionofheightwhichusesthep-adiccyclotomicextensioninsteadofaninfiniteKummer-typeextension.
2Thereisasemi-stableanalogueofthisstatement. Roughlyspeaking,itsaysthatbyrestrictingtheGK-action
ofasemi-stablerepresentationtoGK∞,weonlylosethemonodromyoperatorofthecorrespondingfiltered(ϕ,N)-
module.
3
Letρ¯beafinite-dimensionalF-representationofGK suchthatρ¯|GK∞ ∼= ρ¯∞. Then
“restricting the G -action to G ” defines natural maps from R(cid:3),6h constructed
K K∞ ρ¯∞
in the above theorem into crystalline/semi-stable framed deformation rings3 of ρ¯
with Hodge-Tate weights in [0,h]. (If End (ρ¯ ) ∼= F then we obtain the same
GK∞ ∞
result for deformation rings without framing.) By using these maps and analyzing
the structure of G -deformation rings constructed above, we obtain the following
K
∞
results on crystalline/semi-stable deformation rings.
• The “ordinary” condition cuts out a union of connected components in (the Q -
p
fiber of) a crystalline or semi-stable (framed) deformation ring with Hodge-Tate
weights in [0,h] (where the crystalline and semi-stable deformation rings are as
defined by Kisin [55] and Tong Liu [59]). This is done in Proposition 11.4.18.
• Assumedim ρ¯= 2. LetR(cid:3),v bethequotientoftheflatframeddeformationring
F
fl
with the property that the determinant of the action of the inertia group I is
K
equal to the p-adic cyclotomic character4. Kisin gave a complete description of
the connected components of SpecR(cid:3),v[1], which is used as the main technical
fl p
ingredient for the proof of his modularity lifting theorem [51, 53]. Assuming
p > 2, the author gives a new proof of Kisin’s description of the connected
components of SpecR(cid:3),v[1], which was crucially used in Kisin’s modularity
fl p
liftingtheorem[51,53]. Theideaisto“resolve”SpecR(cid:3),v usingtheBreuil-Kisin
fl
classification of finite flat group schemes. This paper presents another method
to resolve SpecR(cid:3),v using G -deformation rings, so we eliminate the Bruil-
K
fl ∞
Kisin classification from the proof of Kisin’s modularity theorem. The virtue
of this new method is that it works more uniformly in the case p = 2 (after
3Acrystalline/semi-stable(framed)deformationring“overQp”wasdefinedbyKisin[55],andlaterTongLiu[59]
defineditwithoutinvertingpWewilluseTongLiu’sdefinition,whichrecoversKisin’sringafterinvertingp.
4Thisconditioncanbethoughtofasfixingap-adicHodgetype.
4
minor modifications), while the Breuil-Kisin classification of finite flat group
schemes is quite problematic when p = 2. Kisin needs a separate paper [53] to
prove the classification of connected finite flat group schemes over a 2-adic base,
which uses Zink’s theory of windows and displays, and the full proof of Serre’s
conjecture by Khare-Wintenberger uses the modularity of 2-adic Barsotti-Tate
liftings. See §11.6 for more details.
We digress to record the following result of separate interest, which is obtained as
a byproduct of the study of torsion representations “of height 6 h.” Observe that a
semi-simple mod p representation of G can be uniquely recovered from its restriction
K
to G . Indeed, since any semi-simple mod p representation of G is tame, this
K K
∞
assertion follows from the fact that the extension K /K does not have any non-
∞
trivial tame subextension. By studying restrictions to G , we thereby obtain an
K
∞
explicit description of mod p crystalline characters with Hodge-Tate weights in [0,h]
for any positive h. (See Proposition 9.4.8 for the case when the residue field of K
is big enough. The author plans to generalize this results to accommodate “descent
data for a tame extension” in a subsequent work.) Even the case h = 1 (i.e., finite
flat mod p characters) is interesting. Savitt [70] obtained the same result for the case
p > 2 and h = 1 via elaborate computations with Breuil modules, but the author’s
argument is much simpler and works in the case p = 2 as well (in addition to allowing
any h > 1).
This result is a first step towards understanding the reduction mod p of crystalline
G -representations up to semisimplification, since any absolutely irreducible mod p
K
representation of G arises as an “unramified induction” of a character.
K
5
1.1.2 Equi-characteristic analogue
There exists an equi-characteristic “analogue” of Kisin’s theory [52], which his-
torically came first as initiated by Genestier-Lafforgue [35] and Hartl [39, 41] in an
attempt to find an equi-characteristic analogue of Fontaine’s theory of crystalline
representations. To explain this we first introduce some notations. We fix a formal
power series ring F [[π ]], which will play the role of Z (and π will play the role
q 0 p 0
∼
of p). We also fix a finite field k, a complete discrete valuation ring o = k[[u]]
K
with the fraction field K ∼= k((u)) and a local map F [[π ]] → o which makes o
q 0 K K
a finite F [[π ]]-module. In particular, this specifies an embedding F ,→ k. Let
q 0 q
G denote the absolute Galois group for K. Genestier-Lafforgue and Hartl studied
K
F [[π ]]-representations of G which can be viewed as analogues of crystalline repre-
q 0 K
sentations, and their theory bears an incredible resemblance with the class of p-adic
G -representations “of finite height.”
K
∞
Before we discuss the work of Genestier-Lafforgue [35] and Hartl [39, 41], let
us explain why their theory can be regarded as an equi-characteristic analogue of
Fontaine’s theory of crystalline representations. (The idea presented below is also
found in Hartl’s work [39, 41].) If one wants to find a class of F [[π ]]-representations
q 0
of G which can be viewed as an “analogue” of crystalline representations (or
K
Barsotti-Tate representations), then the natural candidate is the π -adic Tate mod-
0
ule of a “π -divisible group” G over o . But it turns out that in order to get a
0 K
nice theory we need more assumptions on the π -divisible groups. We say that a
0
π -divisible group G is of “finite height”5 if the Verschiebung of G vanishes6 and
0
5Hartl calls it a divisible Anderson module in [41, §3.1]. A π0-divisible formal Lie group of height 6 1 is also
knownasaDrinfeldformalFq[[π0]]-module,andthesehavebeenwidelystudiedsincebeingintroducedbyDrinfeld
in[25].
6The π0-divisible group associated to a Drinfeld module or to any π0-divisible formal Lie group has vanishing
Verschiebung,sothisisnotarestrictiveassumption. See[34,Ch.I,Prop2.1.1]forthecaseofπ0-divisibleformalLie
groups.