Motivic Classification of Regular Equivalued Orbits in the Exceptional Group G(2)

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2015-05-20 Motivic Classification of Regular Equivalued Orbits in the Exceptional Group G(2)

Nicholson, Jason

Nicholson, J. (2015). Motivic Classification of Regular Equivalued Orbits in the Exceptional Group G(2) (Unpublished doctoral thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/27014 http://hdl.handle.net/11023/2259 doctoral thesis

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Motivic Classiﬁcation of Regular Equivalued Orbits in the Exceptional Group G(2)

Jason Scott Nicholson

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

GRADUATE PROGRAM IN MATHEMATICS AND STATISTICS

CALGARY, ALBERTA

May, 2015

c Jason Scott Nicholson 2015

Abstract

We exhibit a motivic parameterization of conjugacy classes of equivalued regular

semisimple elements in the Lie algebra of the exceptional group G(2) over local ﬁelds with residual characteristic at least 5.

ii Acknowledgements

Many lessons are learned and much personal growth occurs during a PhD program, but two insights seem appropriate to share here. First, completing a PhD is not only a result of intellectual prowess. What carries a student through to the end are traits common to those who successfully complete any challenging endeavour: perseverance, self-discipline, courage, fortitude, stamina, faith – an ‘unbending intent’ to succeed.

Second, and most importantly, is the realization that a PhD is not an individual effort, but a group venture merely spearheaded by the recipient of the degree. It is from this perspective that I would like to specifically acknowledge a few of the most important members of my “group”, knowing there are many more whose contributions, big or small, have been essential. To these I offer my heartfelt gratitude.

Above all, I want to thank my supervisor Clifton Cunningham for his patience and understanding of my situation which was a great support to me; for his high standards, tremendous knowledge, and masterful teaching which trained and drove me; and for his passion for and great love of mathematics which always inspired and energized me in our many excellent conversations. It has been both a privilege and a pleasure to work with a world class mind.

Thanks also go especially to my parents Keith and Kathleen Nicholson. In many

iii ways this PhD is as much theirs as mine. I literally could not have done it without

their support on all levels: physically, by providing me with a beautiful place to live,

food to eat and all manner of money and incidentals during much of the duration of my

studies; intellectually, by providing sage guidance both academically and otherwise;

emotionally, by being there for me during all the ups and downs of life and studies;

and spiritually, by being a rock of unconditional love and support throughout.

Additionally, I want to give thanks to Puneet Kapur for being there with wise

council and a dose of reality during several crises, and to Claude Laﬂamme who

quietly and deftly rescued me on several occasions and uplifted me constantly with

his silent unconditional support.

Thanks also to everyone in the Mathematics and Statistics department: to the

other graduate and undergraduate students for their friendship and warmth, to the

faculty for being like an extended family to me, and to the support staﬀ, especially

Yanmei Fei, for their help and kindness.

Finally, I want to thank all the health care providers of various ilks in my life, especially Michael Kricken, who kept me balanced and able to continue throughout, the people in my building(s), especially Armin Heyrati, for their selﬂess help and support, and my extended family, especially Laura Coggles, for their constant, palpable good wishes.

iv Dedication

To my parents

v Table of Contents

Acknowledgements iii

Dedication v

Table of Contents vi

List of Tables ix

List of Figures x

List of Symbols and Nomenclature xi

Epigraph xvii

Introduction xviii

1 Statement of the main result 1 1.1 Root datum of type G2 ...... 1 1.2 Weylgroup ...... 4 1.3 Chevalleygroupscheme ...... 6 1.4 Chevalley bases and Structure Coeﬃcients ...... 6 1.5 Equivalued/Goodelements...... 9 1.6 Thickenedorbits ...... 10 1.7 DeﬁnableSubassignments ...... 10 1.8 Statementofthemainresult...... 12

2 Motivic classiﬁcation of thickened stable good orbits 13 2.1 Polynomials from R ...... 14 2.2 Steinbergquotient ...... 17 2.3 Parameterizationofstablegoodorbits ...... 17 2.4 Steinbergbydepth ...... 17

vi 2.5 Maximaltori ...... 20 2.6 Algebras attached to regular equivalued elements ...... 21 2.7 Factorizations ...... 23 2.8 Galoisrepresentations ...... 26 2.9 ProofofProposition2.1 ...... 28

3 Factorizations of coverings and deﬁnable subsets 30 3.1 Fractionaldepth0 ...... 30 3.1.1 Case: w =1...... 32 3.1.2 Case: w = w1 ...... 32 3.1.3 Case: w = w2 ...... 35 3 3.1.4 Case: w =(w2w1) ...... 37 2 3.1.5 Case: w =(w2w1) ...... 38 3.1.6 Case: w = w2w1 ...... 38 1 5 3.2 Fractional depth 6 or 6 ...... 39 1 2 3.3 Fractional depth 3 or 3 ...... 39 3.3.1 Case: w =1...... 40 3 3.3.2 Case: w =(w2w1) ...... 40 1 3.4 Fractional depth 2 ...... 41 3.4.1 Case: w =1...... 42 3.4.2 Case: w = w2 ...... 42 2 3.4.3 Case: w =(w2w1) ...... 44

4 Galois cohomology: H1(K,W ) 46 4.1 Fractionaldepth0 ...... 46 4.1.1 Case: w =1 W ...... 47 ∈ 0 4.1.2 Case: w (w ) W ...... 47 ∈ 1 ⊂ 0 4.1.3 Case: w (w ) W ...... 48 ∈ 2 ⊂ 0 4.1.4 Case: w =(w w )3 W ...... 49 2 1 ∈ 0 4.1.5 Case: w =(w w )2 W ...... 50 2 1 ∈ 0 4.1.6 Case: w = w2w1 W0 ...... 51 1 5 ∈ 4.2 Fractional depth 6 or 6 ...... 53 1 2 4.3 Fractional depth 3 or 3 ...... 54 4.3.1 Case: w =1...... 54 3 4.3.2 Case: w =(w2w1) ...... 56 1 4.4 Fractional depth 2 ...... 57 4.4.1 Case: w =1 W ...... 58 ∈ 1/2 4.4.2 Case: w (w ) W ...... 58 ∈ 2 ⊂ 1/2 4.4.3 Case: w =(w w )2 W ...... 59 2 1 ∈ 1/2

vii 5 Galois cohomology of maximal tori 61 5.1 Fractionaldepth0 ...... 62 5.1.1 Case: w =1...... 62 5.1.2 Case: w = w1 ...... 62 5.1.3 Case: w = w2 ...... 64 3 5.1.4 Case: w =(w2w1) ...... 66 2 5.1.5 Case: w =(w2w1) ...... 66 5.1.6 Case: w = w2w1 ...... 67 1 5 5.2 Fractional depth 6 or 6 ...... 68 1 2 5.3 Fractional depth 3 or 3 ...... 70 5.3.1 Case: w =1...... 70 3 5.3.2 Case: w =(w2w1) ...... 71 1 5.4 Fractional depth 2 ...... 72 5.4.1 Case: w =1...... 72 5.4.2 Case: w = w2 ...... 72 2 5.4.3 Case: w =(w2w1) ...... 75

6 Proof of the main result 76 6.1 Kostantsection ...... 76 6.2 Tate-Nakayama ...... 77 6.3 Proofofthemainresult ...... 79 6.4 Application to stable orbit representatives ...... 82 6.5 Futurework...... 84

Bibliography 85

A Representation of G(2) in SO(8) 87

viii List of Tables

1.1 Action of Weyl group W on Xˇ Λ...... ˇ 5 1.2 Structure Coeﬃcients for g ...... ⊂ 8

2.1 The process of r-reduction produces Pr(λ) and Pr′(λ) from P (λ) and P ′(λ) over Z[t]...... 18 2.2 The coordinate ring Z[tr] = Z[Φr] and the sets Φr, the groups Wr indicating a section of 1 W r W W 1, and the discriminant → → → r → dr Z[tr]usingnotationfromTable2.1...... 20 2.3 Factorizations∈ of P (λ) over Z[S ] for r Z...... 22 r r ∈ 2.4 Factorizations of Pr(λ) and Pr′(λ) over Z[Sr] for r Z...... 24 w6∈ 2.5 The splitting extension of lifts of Rs(λ) for s Sr (k) and s = µr,w(x) 27 2.6 Representatives ρ Z1 (K,W ) for H1 (K,W∈), for all s Sw(k), s = s ∈ tr tr ∈ r µr,w(x)...... 29 6.1 H1(K,T ) for tori T determined by s Sw(k)...... 78 s s ∈ r

ix List of Figures

1.1 Root system and fundamental weights of type G2, from [Bou68, p. 276]. 3 1.2 SubgrouplatticeoftheWeylgroup ...... 4

x List of Symbols and Nomenclature

Symbol Deﬁnition Page

K localnon-Archimedeanfield xix G, G(2) Chevalley group scheme of type G2 xix G2 typeofexceptionalrootsystem xix g, g(2) Liealgebraschemeoftype G2 associated to G xix S smoothlocusoftheSteinbergquotient xix S(K) K-rational points on S xix T tamely ramified algebraic torus associated to s Sw xix s ∈ r s stable orbit variety associated to s S xix O 1 ∈ H (K,Ts) first cohomology set of Gal K/K¯ over Ts equals xix Hom(Gal(K/K¯ ),Ts)/Ts-conj SO(8) (so(8)) specialorthogonalgroup(Liealgebra)ofrank8 xx definable sub- subassignment h with a formula φ such that F xx m ∀ n ∈ assignment Fieldf , h(F ) is the set of all points in F ((t)) F Zr satisfying φ × × g(r) definable subassignment of equivalued regular xx semisimple elements of g of depth r Br definable subassignment parameterizing thickened xx good regular semisimple adjoint orbits in g(K) ν family of maps of definable subassignments g(r) xx r → Br F k residue field of K, k ∼= q xx νr/K specialization of νr determined by K xxi π a uniformizer of K 1 ordK integral valuation on K 1 K× invertible elements in K 1 Λ lattice 1

xi Symbol Deﬁnition Page

Xˇ cocharacter lattice 1 <,> pairing Λˇ Λ Z given by = δ 1 × → i j i,j R root system of type G2 2 ∆ basis α ,α of the root system R 2 { 1 2} e1,e2,e3 basis of X, image of ǫ1,ǫ2,ǫ3 Λ 2 {R } short roots α ,α ,α{ in R}∈with α := e , 2 short { 1 3 5} 1 − 1 α3 := α1 + α2 = e2, α5 :=2α1 + α2 = e3 R long roots α ,α− ,α in R with α := e e , 2 long { 2 4 6} 2 1 − 2 α4 :=3α1 + α2 = e3 e1, α6 :=3α1 +2α2 = e3 e2 α ‘longest’ root with− respect to the basis ∆− = 2 α1,α2 A type{ of} family of root systems for n Z 2 e n ∈ Rˇ dual root system of R 2 αˇ , αˇ , αˇ short roots in Rˇ withα ˇ := 2f + f + f , 3 { 1 3 5} 1 − 1 2 3 αˇ3 :=α ˇ1 +3ˇα2 = f1 2f2 + f3,α ˇ5 :=2ˇα1 +3ˇα2 = f f +2f − − 1 − 2 3 αˇ2, αˇ4, αˇ6 long roots in Rˇ withα ˇ2 := f1 f2,α ˇ4 :=α ˇ1 +ˇα2 = 3 { } f f ,α ˇ :=α ˇ +2ˇα = f −f 3 − 1 5 1 2 3 − 2 ∆ˇ thebasis αˇ , αˇ of the dual root system Rˇ 3 { 1 2} (X,R, X,ˇ Rˇ) semisimple root datum of type G2 3 Q(R) root lattice of(latticegenerated by) R 4 Q(Rˇ) rootlatticeof Rˇ 4 w1,w2 generated by w1 and w2 4 hW i Weyl group w ,w for R 4 h 1 2i sα reﬂection across the hyperplane of root α 4 Cn cyclic group of order n 4 Dn dihedral group of order 2n 4 Sn symmetric group on n elements 4 V4 Klein 4-group 4 greg regular semisimple elements of g 4 w1 equal to sαˇ1 5 w2 equal to sαˇ2 5 Xα Chevalley basis element corresponding to root α 6 Nα,β structure coeﬃcient corresponding to the ordered 6 pair of roots (α, β) α, β ordered pair of roots in R R 7 R{ + } positive roots in R × 7 α β if β α R+ for roots α, β R 7 ≺ − ∈ ∈

xii Symbol Deﬁnition Page

Z[g] coordinateringof g 9 reg tX (K) theCartansubalgebracontaining X g (K) 9 good of slope r X greg(K) is if ord (α(X)) = r for∈ each root 9 ∈ K of g(K) relative to tX (K) depth of X is r if X is good of slope r; see [CCGS11, 9 Def 2.1] g(r, K) thesetofgoodelementsin greg(K) of depth r 9 (X) orbitof X greg(K) 10 O ∈ reg r(X) thickened orbit of X g (K) where the 10 O depth of X is r ∈ tX (K)r+ the elements of depth strictly greater than r in 10 tX (K) K the ring of integers of K 11 OKint elements x K with ord (x) Z 11 ∈ K ∈ Deff the category of definable subassignments over 12 fields containing a field f µr familyofmapsofdefinablesubassignments 13 µr/K specialization of µr determined by K 13 Sr classifies r-reductions of characteristic polyno- 13 mials of regular equivalued elements X g(K) 1 ∈ of depth r, for each r 6 Z st ∈ reg r (X) thickenedstableorbitof X g (K) 13 SpecO spectrum(setofprimeideals)ofaring∈ 14 t := Spec(Z[Xˇ]) Cartan subalgebra Spec(Z[Xˇ]) 14 W Z[t] W elements of Z[t] invariant under the action of 14 | | W and localized at W t/W another notation for| the| Steinberg (adjoint) 14 quotient S

Q(λ) rootpolynomial α R =(λ α) 14 ∈ − P (λ) shortrootpolynomial α R (λ α) 15 Q ∈ short − s coeﬃcient of P (λ) with s = e2e2e3 15 1 Q 1 1 2 3 s2 coeﬃcient of P (λ) with s2 = e1e2 + e2e3 + e3e1 15

P ′(λ) longrootpolynomial α R (λ α) 15 ∈ long − PX (λ):= Ps(λ) if(s1,s2)= s = µ(X) 16 Q 2 s1′ coeﬃcient of P ′(λ) with s1′ = (e1 e2) (e2 16 e )2(e e )2 = (27s +4s3) − − 3 3 − 1 − 1 2 s2′ coeﬃcient of P ′(λ) with s2′ = 3(e1e2 + e2e3 + 16 e3e1)=3s2

xiii Symbol Definition Page d discriminant of P (λ) and t 16 g/G another notation for the Steinberg (adjoint) quo- 17 tient S ∼= t/W D any pre-image of d Z[t] under Z[g] Z[t] 17 µ : greg S restriction of g t/W∈ to greg → 17 K¯ → separable closure→ of the field K 17 s(K) stableorbitin g(K) related to s S 17 Ostable orbit adjoint conjugacy class over K¯ ∈ 17 r-reduction process introduced in [CH04, 3.1] taking 17 P (λ) K[λ] to R(λ) k[λ§] such that K[λ]/(P∈(λ)) depends only on∈ R r integer part of r Q 18 ⌊r⌋ fractional part of∈r Q 18 {fractional} another name for the∈ fractional part of r Q 18 depth ∈ R (λ) r-reduction of P (λ) for each X g(r, K) 18 r X ∈ Φr the set of ‘roots’ of Qr(λ) 18 tr defined by Z[tr] W = Z[Φr] over Z W 19 reg reg| | | | tr defined by Z[tr ]= Z[Φr]d 19 W Wr Wr quotient of W for which Z[Φr] = Z[Φr] 19 r W w W w(f)= f, f Φr 19 {reg ∈ | ∀ ∈ } Sr tr /Wr 19 µr/K µr/K (X) Sr(k) is the r-reduction of PX (λ) for 19 X g(r, K∈ ) ∈ indexed root sextuple for tori (X, , X,ˇ , , ρ) where ρ 20 data Z1(K,W ); see [Spr09, ∅16.2]∅ ∅ ∈ Z1(K,W ) setof1-cocyclesof K/K¯ § over W 20 Ktr a tamely ramified closure of K 20 1 tr Htr(K,W ) 1-cohomologysetof K /K over W 20 Qs(λ) equals α R (λ α(X′)) K[λ] where X′ 20 reg ∈ − ∈ ∈ t (K¯ ) and s = µ(X′) K splittingQ extension of Q (λ) for each s S (k) 21 s s ∈ r ρs tame Galois representation from Gal(K/K¯ ) to 21 W for each s Sw(k) ∈ r Rs(λ) r-reduction of Ps(λ) 21 g deg(Rs) 23 Rs,i(λ) irreduciblefactorof Rs(λ) in k[λ] 23 Is index set of irreducible factors in Rs(λ) 23

xiv Symbol Deﬁnition Page

R˙ s,i(λ) anyliftof Rs,i(λ) under r-reduction 23 g equals deg R for each i I 23 i s,i ∈ s K(n) unique unramiﬁed extension of K of de- 23 gree n Z ∈ ζi root of Rs,i(λ) 23 (gi) ζ˙i lift in K of ζi 23 I index set of partitions into w -orbits 24 w h i Ri w -orbit in R for i Iw 24 h i reg ∈ Sw factorization of t S 24 reg → Sr,w factorization of tr Sr 25 µ : S S scheme morphism with→ w W 25 r,w r,w → r ∈ r w w′ partial order on W that implies the ex- 25 ≤ r istence of a canonical map S ′ S r,w → r,w over Sr w Sr deﬁnable subset given by 25 Sw := µ (S ) ′ µ ′ (S ′ ) r r,w r,w \∪w

X equals X α1 + X α2 77 − − − 1 t2 t after base change to Z[2− ] 77

xv Symbol Deﬁnition Page

1 1 #H (K,Ts) cardinalityof H (K,Ts) 78 An the group corresponding to each 78 1 H (K,Ts), interpreted as a definable set 1 hr : Wr N hr(w)=#H (K,Ts) for Ts correspond- 78 → 1 ing to each w Wr and r 6 Z ∈w ∈ g(r, w) thefibreof Sr ֒ Sr under the map 79 of definable subassignments→ µ : g(r) r → Sr w w µr : g(r, w) Sr maps of definable subassignments for 79 → 1 every r 6 Z and w Wr w ∈ w ∈ δr/K function δr/K : g(r, w, K) Ahr(w) for 79 1 → every r 6 Z and w Wr w ∈ w ∈w Br definable set Br := Sr Ahr(w) 80 w w w w× w νr/K : g(r, K) Br (k) defined as νr/K := µr/K δr/K : 80 → w × g(r, K) Br (k) → w (x, a) equals r(X) if µr/K (X) = (x, a) 81 O w O w ∈ Br (k)= Sr (k) Ahr(w) × w Br definable set w Wr Sr Ahr(w) 81 w ∈ × νr : g(r) Br map of definable subassignments 81 → ` νw g g r (r, K) w Wr (r, w, K) w→ ∈ → w Wr Br Br ∈ → Sr definable subassignment` given by 82 `r S (K)= (s1,s2) S(K) ordK (s1)= 6r and{ ord (s ∈) 2r | K 2 ≥⌈ ⌉} resr restriction map of definable subassign- 82 ments

xvi Epigraph

Anything is possible if one wants it with unbending intent...

Carlos Castaneda, The Second Ring of Power [Cas77, p. 123]

xvii Introduction

The major step in the theory of conjugacy classes of semisimple algebraic groups was taken by Steinberg in 1965 [Ste65, 6] when he introduced a parameterization of the § conjugacy classes by a variety commonly called the adjoint quotient but which we will refer to as the Steinberg quotient. This was done over an algebraically closed ﬁeld, though, and this is a limitation for our purposes.

Langlands [Lan79] then parameterized the rational conjugacy classes within a stable conjugacy class and, over local and non-Archimedean ﬁelds, showed how to calculate the parameterizing object. His motivation was the desire to calculate the

Arthur-Selberg trace formula, in particular “...an analysis of local orbital integrals to which the sum over a global stable conjugacy class is not directly amenable.”([Lan79, p. 701]) As the name suggests, orbital integrals are integrals over the (semisimple) conjugacy classes – orbits – of elements in a connected semisimple group.

His method of resolving this problem was to look at calculating the error terms left over when the orbital inegrals are calculated over stable conjugacy classes. How- ever, another solution appeared when T.C. Hales showed that p-adic rational orbital integrals are motivic [Hal04] and may be computed using the technique of motivic integration introduced by M. Kontsevich in 1995.

xviii Orbital integrals were clariﬁed in classical groups in 2004 by C. Cunningham

and T.C. Hales [CH04], and in 2011 by Cluckers, Cunningham, Gordon and Spice

[CCGS11]. One of their last contributions was to outline some steps towards writing

a computer program to produce their results, one of the beneﬁts of the motivic ap-

proach. Building on results from [CH04] on good (read equivalued) orbital integrals,

Step 1 in this plan is a motivic parameterization of thickened good adjoint orbits in

the Lie algebra of the p-adic group. However, that paper was limited to symplec-

tic and special orthogonal groups because it relied on the classiﬁcation of regular

semisimple adjoint orbits given in [Wal01], which, while eminently motivic in nature,

only treats classical groups.

Here we have given a recipe which could also be automated, but may be used for

any linear algebraic group – we have only used the information given in the Dynkin

diagram. As a demonstration we perform the calculations for the Chevalley group

scheme G of type G2: a motivic parameterization of thickened good adjoint orbits in the Lie algebra of G. This result should be viewed as a basic part of the infrastructure

needed to compute regular semisimple orbital integrals on this Lie algebra over local

ﬁelds K.

In broad strokes, our approach to this problem is familiar: we use the Steinberg

quotient S over K to parameterize stable orbit varieties , with s S(K), of regular Os ∈ semisimple elements; we ﬁnd a stable conjugacy class of maximal tori T G attached s ⊂ to s S(K); we compute H1(K,T ) to detect how many adjoint orbits appear in the ∈ s stable orbit (K); and we use the Kostant section for g = Lie G over K to put a Os group structure on the torsor (K)/G(K) of adjoint orbits in stable orbits. Os A novelty of the approach in this thesis, however, is that all this is done in a way

xix which is independent of the local ﬁeld K, except that its residual characteristic must be at least 5, and without making use of any representation of the group G, relying instead only on the root datum for G. In particular, in this thesis we make no use of arcane knowledge of the exceptional group G(2), no use of the representation of G(2)

in SO(8), and no use of Bruhat-Tits theory; everything in this thesis is derived directly

from the root datum of type G2. This is all made possible by the use of r-reduction, as

1 developed in [CH04], which in turn rests on Krasner’s lemma, to show that H (K,Ts) does not change under p-adically small perturbations of s S(K) and show further ∈ that what ‘p-adically small’ means here can be expressed in the language of Pas.

Making this precise leads to thickened orbits, a notion appearing ﬁrst in [CH04] and then clariﬁed in [CCGS11]. Putting all these pieces together proves the main result of the thesis, Theorem 1.1, giving the motivic parameterization of thickened good regular semisimple adjoint orbits in g(K).

The result is a simple motivic gadget – a map of deﬁnable subassignments – which is built directly from the Chevalley group scheme G as determined by its root datum and a Chevalley basis, which is independent of any representation of G and any local

ﬁeld but which, after the choice of a local ﬁeld K with residual characteristic of at least

5, parameterizes all thickened good regular semisimple adjoint orbits in g(K). The promised map of definable subassignments is exhibited in Theorem 1.1 and described informally here, where g is a Chevalley Lie algebra scheme of type G2: We find a family of maps of definable subassignments

r Q, ν : g(r) B ∀ ∈ r → r

xx such that if K is a local field and 6 is invertible in its residue field k then each νr specializes to a surjective function ν : g(r, K) B (k) for which the fibres are r/K → r thickened orbits of good elements in g(K) and all such orbits arise in this way. The point of this thesis is not just to promise the existence of ν : g(r) B but to r → r actually exhibit it.

While this thesis only considers the Chevalley group scheme G of type G2, the strategy used here adapts to any Chevalley group scheme. It is hoped that this strategy will be implemented in the near future.

xxi 1 Chapter 1

Statement of the main result

We begin with a brief review of basic facts about the Chevalley group scheme G of type G2 and its Lie algebra. We then state the main result of the thesis, the proof of which will occupy Chapters 2 and 6.

Throughout the thesis we write K for a non-Archimedean local ﬁeld, k for its

residue ﬁeld, and π for a uniformizer of K, when we need to introduce one. Let ordK

be a valuation on K so that ordK (K×)= Z.

1.1 Root datum of type G2

Consider the lattices

Λ= x ǫ + x ǫ + x ǫ x ,x ,x Z { 1 1 2 2 3 3 | 1 2 3 ∈ } and

Λ=ˇ y f + y f + y f y ,y ,y Z { 1 1 2 2 3 3 | 1 2 3 ∈ } with pairing Λˇ Λ Z given by = δ . Now, set ǫ = ǫ + ǫ + ǫ Λ × → i j i,j 1 2 3 ∈ and consider the quotient lattice X =Λ/ xǫ x Z and the sub-lattice Xˇ = y { | ∈ } { ∈ 2

Λˇ =0 with pairing Xˇ X Z given by | } × →

y x + y x + y x . 1 1 2 2 3 3 1 1 2 2 3 3 7→ 1 1 2 2 3 3

Consider the root system R X given by ⊂

R = α , α , (α + α ), (2α + α ), (3α + α ), (3α +2α ) {± 1 ± 2 ± 1 2 ± 1 2 ± 1 2 ± 1 2 }

where, writing ei for the image of ǫi in Λ,

α := e α := e e 1 − 1 2 1 − 2 α := α + α = e α :=3α + α = e + e 3 1 2 − 2 4 1 2 − 1 3 α :=2α + α = e α˜ = α :=3α +2α = e + e . 5 1 2 3 6 1 2 − 2 3

This is a root system of type G2. Note that the short roots satisfy the identities

1 α = ( 2e + e + e ) 1 3 − 1 2 3 1 α + α = (e 2e + e ) 1 2 3 1 − 2 3 1 2α + α = ( e e +2e ) 1 2 3 − 1 − 2 3

1 in X Z[3− ]. ⊗ The ‘longest’ root with respect to the basis ∆ = α ,α is α = 3α +2α and { 1 2} 1 2 the fundamental weights are ̟ = 2α + α and ̟ = α. The short roots and long 1 1 2 2 e roots in R, e

R = α , α , α R = α , α , α , short {± 1 ± 3 ± 5} long {± 2 ± 4 ± 6} 3

Figure 1.1: Root system and fundamental weights of type G2, from [Bou68, p. 276].

each form root systems of type A2.

The dual root system Rˇ Xˇ is given by ⊂

Rˇ = αˇ , αˇ , (ˇα +α ˇ ), (ˇα +2ˇα ), (ˇα +3ˇα ), (2ˇα +3ˇα ) {± 1 ± 2 ± 1 2 ± 1 2 ± 1 2 ± 1 2 }

with αˇ := 2f + f + f αˇ := f f 1 − 1 2 3 2 1 − 2 αˇ +3ˇα = f 2f + f αˇ +α ˇ = f + f 1 2 1 − 2 3 1 2 − 1 3 2ˇα +3ˇα = f f +2f αˇ +2ˇα = f + f . 1 2 − 1 − 2 3 1 2 − 2 3 The Cartan matrix for (R, Rˇ) with reference to the pair of bases ∆ = α ,α and { 1 2} ∆=ˇ αˇ , αˇ is the matrix { 1 2}

<αˇ ,α > <αˇ ,α > 2 3 1 1 1 2 −   =   . <αˇ2,α1 > <αˇ2,α2 > 1 2   −     

The quartuple (X,R, X,ˇ Rˇ) is a semisimple root datum of type G2; compare with 4

[Bou68, p. 220].

The character lattice X coincides with the root lattice Q(R) (the lattice generated by R) and the cocharacter lattice Xˇ coincides with the coroot lattice Q(Rˇ) (the

lattice generated by Rˇ). In this way we see that the root datum (X,R, X,ˇ Rˇ) is

simultaneously of adjoint type and simply connected.

1.2 Weyl group

The Weyl group W for R may be apprehended through the action of W = s α R h α | ∈ i on X given by s (x) = x <α,x>αˇ . Using the Cartan matrix above, we see that α − s (α )= α and s (α )=3α + α while s (α )= α + α and s (α )= α . α1 1 − 1 α1 2 1 2 α2 1 1 2 α2 2 − 2

Henceforth we will adopt the notation w1 := sα1 and w2 := sα2 .

Figure 1.2: Subgroup lattice of the Weyl group

We remark that W = w ,w is the dihedral group D of order 12, generated h 1 2i 6

by w2 and w2w1, for example. The element w2w1 is Coxeter, and only all non-trivial 5

powers of w2w1 are regular and regular elliptic elements of W . Thus, the regular and regular elliptic numbers for W are 2, 3 and 6, the latter being the Coxeter number of

W .

With reference to Figure 1.2, The normal proper subgroups of W are:

1. w w = C , w , (w w )2) = S , w , (w w )2) = S ; h 1 2i ∼ 6 h 1 1 2 i ∼ 3 h 2 1 2 i ∼ 3 2. (w w )2 = C ; and h 1 2 i ∼ 3 3. (w w )3 = C . h 1 2 i ∼ 2

Of the three normal subgroups of index 2, no two are conjugate.

Table 1.1: Action of Weyl group W on Xˇ Λˇ ⊂ w W w(y1,y2,y3) ∈

w2w1 ( y3, y1, y2) − − − 5 (w2w1) = w1w2 ( y2, y3, y1) − − −

2 (w2w1) (y2,y3,y1)

4 2 (w2w1) =(w1w2) (y3,y1,y2)

3 3 (w2w1) =(w1w2) ( y1, y2, y3) − − −

w2 = sα2 (y2,y1,y3)

w1w2w1 = sα4 (y3,y2,y1)

w2w1w2w1w2 = sα6 (y1,y3,y2)

w1 = sα1 ( y1, y3, y2) − − −

w2w1w2 = sα3 ( y3, y2, y1) − − −

w1w2w1w2w1 = sα5 ( y2, y1, y3) − − −

1 (y1,y2,y3) 6

The Weyl group may also be apprehended through the action of s αˇ ∆ˇ on Xˇ h αˇ | ∈ i given by s (y)= y αˇ. Using the description above of Xˇ as a sub-lattice of Λ,ˇ αˇ − we have s : y f + y f + y f y f y f y f and s : y f + y f + y f αˇ1 1 1 2 2 3 3 7→ − 1 1 − 3 2 − 2 3 αˇ2 1 1 2 2 3 3 7→

y2f1 + y1f2 + y3f3. For use below, we record the action of W on Xˇ in Table 1.1, in which elements of W are separated by conjugacy classes and where we use the

notation w1 := sαˇ1 and w2 := sαˇ2 ; context makes this notation unambiguous.

1.3 Chevalley group scheme

Let G be a Chevalley group scheme over Z determined by the root datum (X,R, X,ˇ Rˇ);

see [Che61] and [Gro96]. We remark that G Z Spec(Q ) is a split, connected ×Spec( ) p reductive algebraic group with root datum (X,R, X,ˇ Rˇ) for every prime p. Let g be the Lie algebra scheme of G [CR10].

1.4 Chevalley bases and Structure Coeﬃcients

A Chevalley basis [Che55] for g is a function R g, α X , with the following → 7→ α

properties: for every α R, the triple (Xα, [Xα,X α],X α) is an sl2-triple over Z; ∈ − − the union Xα α R [Xα,X α] α ∆ is a basis for g; [Xα,Xβ] = 0 unless { | ∈ } ∪ { − | ∈ } α + β = 0 or α + β R; if α + β R then [X ,X ] = N X for integers N ∈ ∈ α β α,β α+β α,β called the structure coeﬃcients of the Chevalley basis.

The structure coeﬃcients satisfy the following relations:

(i) N = N α, β R. α,β − β,α ∈ Nα,β Nβ,γ Nγ,α (ii) (γ,γ) = (α,α) = (β,β) if α,β,γ R satisfy α + β + γ = 0. ∈ 7

2 (iii) Nα,βN α, β = (p + 1) , α,β R − − − ∈ Nα.β Nγ,δ Nβ,γ Nα,δ Nγ,αNβ,δ (iv) (α+β,α+β) + (β+γ,β+γ) + (γ+α,γ+α) =0 if α,β,γ,δ R satisfy α + β + γ + δ and if no pair are opposite. ∈ Here p is the greatest integer such that β pα R, and ( , ) is the standard inner − ∈ product on R. Moreover N = (p+1) so ﬁnding the structure coeﬃcients amounts α,β ± to determining the sign.

To calculate a Chevalley basis for g, we follow [Car72, 4.1-4.2], from which we §§ recall the following notions:

(s) a special (s) ordered pair of roots α, β R R is one in which α + β R { } ∈ × ∈ and 0 α β, where α β β α R+ = xα + yα R x> 0 or x = ≺ ≺ ≺ ⇔ − ∈ { 1 2 ∈ | 0 y > 0 ;1 and ⇒ } (es) an extraspecial (es) pair of roots α, β R R is a special pair such that, for { }∈ × all special pairs γ,δ with α + β = γ + δ, α γ. { } Choosing the sign of the structure coeﬃcients for the extraspecial pairs of roots α, β { }

uniquely determines the structure coeﬃcients Nα,β for all pairs [Car72, Prop. 4.2.2]; we set sign(N ) = +1 for all extraspecial pairs α, β . α,β { } Then the calculation of the Chevalley basis is algorithmic:

(1) Calculate the special ordered pairs of roots; in our case the pairs α ,α + α , { 1 1 2} α , 2α + α , α + α , 2α + α . { 1 1 2} { 1 2 1 2} (2) Determine which special pairs are extraspecial; in our case all of the pairs

α ,α + α , α , 2α + α , α + α , 2α + α . { 1 1 2} { 1 1 2} { 1 2 1 2} (3) Set sign(N ) = +1 for all extraspecial pairs α, β . α,β { }

1In fact, any total ordering on R will give an order relation that will work. The relation we have chosen is convenient as we already have the basis ∆ of R. ≺ 8

Table 1.2: Structure Coeﬃcients for g 3α1 2α2 3α1 α2 2α1 α2 α1 α2 α2 α1 − − − − − − − − − − 3α1 2α2 − − 3α1 α2 +1 − − 2α1 α2 +3 +3 − − α1 α2 -3 +2 − − α2 -1 -1 − α1 -3 -2 +1 − α1 -1 -2 +3 α2 -1 -1 α1 + α2 -1 +2 -1 +3 2α1 + α2 +1 +1 +2 -2 3α1 + α2 +1 +1 -1 3α1 +2α2 +1 +1 -1 -1

α1 α2 α1 + α2 2α1 + α2 3α1 + α2 3α1 +2α2 3α1 2α2 +1 +1 -1 -1 − − 3α1 α2 +1 -1 -1 − − 2α1 α2 +2 -2 -1 -1 − − α1 α2 -3 +1 -2 +1 − − α2 +1 +1 − α1 -3 +2 +1 − α1 -1 +2 +3 α2 +1 +1 α1 + α2 -2 +3 2α1 + α2 -3 -3 3α1 + α2 -1 3α1 +2α2 9

(4) Apply condition (i) to the extraspecial pairs.

(5) Apply condition (iii) to the extraspecial pairs.

(6) Apply condition (iii) to the pairs produced in (4).

(7) Apply condition (ii) to the pairs α, β produced in (4), (5) and (6) with γ = { } α β. − − (8) Apply condition (iv) to calculate the sign of a pair α, β outside of those { } produced in (4), (5), (6) and (7).

(9) Apply conditions (i), (ii) and (iii) to any new pairs produced by (8).

(10) Repeat steps (8) and (9) until signs are calculated for all pairs.

Using this algorithm, and making the choice indicated above, the complete list of structure coeﬃcients in our case is given in Table 1.2.

1.5 Equivalued/Good elements

Let greg ֒ g be the open subscheme of regular semisimple elements obtained by → localizing g at the discriminant D Z[g] (the coordinate ring of g) which will be ∈ computed in Sections 2.1 and 2.2.

Let K be a local ﬁeld. An element X greg(K) is called good of slope r if it is ∈

equivalued in the following sense: ordK (α(X)) = r for each root of g(K) relative to

tX (K), the Cartan subalgebra containing X. In this case, the depth of X is r; see [CCGS11, Def 2.1] for more detail. As they amount to the same thing, we will only use the term ‘depth’ henceforth. Because we are only interested in good elements which are also regular semisimple, we henceforth shorten ‘good and regular semisimple’ to

‘good’. We write g(r, K) for the set of good elements in greg(K) of depth r. 10

1.6 Thickened orbits

Suppose X greg(K) and let r = depth(X). The thickened orbit of X greg(K) is ∈ ∈ the set

(X):= (X + Y ), Or O Y t (K) + ∈ X[ r where (X + Y ) is the G(K)-adjoint-orbit of X + Y in g(K) [CCGS11, Def. 2.5]. O

1.7 Deﬁnable Subassignments

There is one more deﬁnition we must make – that of a deﬁnable subassignment; we

refer to [GY09, 5.2.1-4]. Given the categories Field of ﬁelds containing a ﬁeld f §§ f

and Set of sets, deﬁne a functor h[m,n,r]= hA ((t))m An Zr : F ieldf Set by f × f × →

n r h[m,n,r](F )= hA ((t))m An Zr (F ) := F ((t)) F Z f × f × × ×

n for some field F containing f, and where Af is affine n-space over f. In general, a subassignment h of the functor : C Set between any category C F → and Set is a collection of subsets h(C) (C) for each C C. To define a definable ⊂ F ∈ subassignment, we need the following.

A formal language is a set of strings made up of certain symbols. The formal L languages we are interested in here are the Language of Rings, Presburger’s Language, and the Language of Denef-Pas.

The Language of Rings is made up from the following symbols: countably many symbols for variables x ,x ,...,x ,..., ‘0’, ‘1’, ‘ ’, ‘+’, ‘=’, and parentheses ‘(’ and 1 2 n × ‘)’, the existential quantiﬁer ‘ ’, and the logical operations ‘ ’, ‘=’, and ‘ ’. ∃ ∧ 6 ∨ 11

Presburger’s Language is made up from the following symbols: countably many symbols for variables over Z x ,x ,...,x ,..., ‘0’, ‘1’, ‘+’, ‘ ’, and for each d = 1 2 n ≤ 2, 3,... a symbol ‘ ’ denoting x y mod d, and the same symbols for quantiﬁers, ≡d ≡ logical operations and parentheses as in the Language of Rings.

The Language of Denef-Pas is an extension of the ﬁrst two languages for valued

ﬁelds. It has three sorts of variables: variables over the residue ﬁeld whose accom-

panying symbols are those of the Language of Rings with symbols for every rational

number (so formulas can have coefficients in Q), variables over the value group whose accompanying symbols are those of Presburger’s language along with the symbol ‘ ’, ∞ and finally variables over the valued field itself whose accompanying symbols are those of the Language of Rings plus the symbols ord and ac, defined below.

Denote the ring of integers of K by and a ﬁxed uniformizer by π. Let res : OK k be the residue map, and let Kint = x K ord (x) Z . The angular OK → { ∈ | K ∈ } int ordx component is a function ac : K k¯× given by ac(0) = 0 and ac(x) = res(x/π ). →

Finally, since we are concerned here only with elements of Fieldf , we also add a symbol for each element of f((t)), a case which we note with the phrase “formulas with coeﬃcients on f((t))”.

A subassignment h of h[m,n,r]isa definable subassignment if there is a formula φ in the Language of Denef-Pas with coefficients in f((t)) where m, n, r are the numbers of free variables of the valued field, the residue field, and the value sort, respectively, such that for every F Field , h(F ) is the set of all points in F ((t))m F n Zr ∈ f × ×

satisfying φ. Then a morphism of deﬁnable subassignments from h1 to h2 to be a

deﬁnable subassignment d such that d(C) is the graph of a function from h1(C) to

h2(C) for each object C in C. The category of deﬁnable subassignments is denoted 12

by Deff . Following [CH04, Lemma 5.1] we see that, for every r Q, there is a formula φ ∈ r in the language of Denef-Pas such that for every F Field , g(r, F ) is the set of all ∈ f points in F ((t))m F n Zr satisfying φ . Let g(r) be the deﬁnable subassignment × × r of equivalued regular semisimple elements of g of depth r.

1.8 Statement of the main result

Theorem 1.1. Let G be a Chevalley group scheme of type G2 and let g be its Lie algebra. Every Chevalley basis for g determines a family of maps of deﬁnable subassignments

r Q, ν : g(r) B ∀ ∈ r → r

such that if K is a local ﬁeld and 6 is invertible in the residue ﬁeld k of K then

1 νr/K− (νr/K (X)) is a thickened orbit in g(r, K), where νr/K is the specialization determined by K, and every thickened orbit of regular equivalued elements in g(r, K) arises in this way. 13 Chapter 2

Motivic classiﬁcation of thickened

stable good orbits

As a step toward proving Theorem 1.1, in this chapter we classify thickened stable

good orbits in g(K). Suppose X greg(K) and let r = depth(X). We are now able ∈ to state the main result of this chapter.

Proposition 2.1. Let G be a Chevalley group scheme of type G2 and let g be its Lie algebra. For every r Q there is a map of deﬁnable subassignments ∈

µ : g(r) S r → r

with the following property: if K is a local ﬁeld and 6 is invertible in its residue ﬁeld k, then the specialization µ : g(r, K) S (k) is surjective; and r/K → r

st 1 X g(r, K), (X)= µ− (µ (X)); ∀ ∈ Or r/K r/K

moreover, every thickened stable orbit in g(r, K) arises in this way.

Note that Proposition 2.1 does not require the choice of a Chevalley basis for g,

in contrast to Theorem 1.1. 14

2.1 Polynomials from R

Consider t := Spec(Z[Xˇ]). Since we have introduced a pair of lattices (X, Xˇ) through

the pair of lattices (Λ, Λ),ˇ it is natural to use the basis for Λˇ introduced above to

determine a set of generators for the coordinate ring of t:

Z Z [t] ∼= [y1,y2,y2]/(y1 + y2 + y3).

With reference to the action of the Weyl group W on Xˇ in Section 1.2, invariant theory gives

W Z[t] W = Z[s1,s2]6, | | ∼

2 2 3 where s1 = y1y2y3 and s2 = y1y2 + y2y2 + y3y1. However, because Xˇ = Q(Rˇ), (since G(2) is adjoint), it is more natural to use the basis αˇ , αˇ for Rˇ, as introduced above, to determine generators for the coordinate { 1 2} Z Z Z Z ring: [t] ∼= [z1,z2], where [y1,y2,y2]/(y1 + y2 + y3) ∼= [z1,z2] is determined by

z1αˇ1 + z2αˇ2 = y1f1 + y2f2 + y3f3.

Consider the polynomial Q(λ) over Z[t] deﬁned by

Q(λ):= (λ α) . − α R Y∈ Here we view each α R as an element of Z[t]= Z[z ,z ] according to the identiﬁca- ∈ 1 2

tion α = α(z1αˇ1 + z2αˇ2) = z1<αˇ1,α> + z2<αˇ2,α>. Note that, with this notation,

Z[t]2 = Z[R]2.

Since W stabilizes R, we see that the coeﬃcients of Q(λ) lie in Z[t]W so, in fact,

W Q(λ) lies in Z[t] [λ]. We will ﬁnd the coeﬃcients of Q(λ). Since W stabilizes Rshort, 15

it follows that the polynomial over Z[t] deﬁned by

P (λ):= (λ α) − α R ∈Yshort also lies in Z[t]W [λ]. A simple calculation shows

P (λ)= λ6 +2s λ4 + s2λ2 s 2 2 − 1

where

s := αβ = α α α α α α = e e + e e + e e 2 1 3 − 3 5 − 5 1 1 2 2 3 3 1 α=β α1, α3,α5 R 6 ∈{− X− }⊂ short

and

2 2 2 3 s1 := α = α = e1e2e3.

α α1, α3,α5 R α Rshort ∈{− −Y }⊂ short ∈Y We will sometimes use the notation P (λ):= λ6 +2s λ4 + s2λ2 s . s1,s2 2 2 − 1

Likewise, since W stabilizes Rlong, it follows that the polynomial over Z[t] deﬁned by

P ′(λ):= (λ α) − α R ∈Ylong also lies in Z[t]W [λ]. A simple calculation shows that

6 4 2 2 P ′(λ):= λ +2s′ λ +(s′ ) λ s′ , 2 2 − 1 16

P λ P ′ ′ λ so s′1,s2 ( )= s1,s2 ( ), where

s′ = αβ = α α α α α α = 3(e e + e e + e e )=3s 2 2 4 − 4 6 − 6 2 1 2 2 3 3 1 2 α=β α2,α4, α6 R 6 ∈{ X− }⊂ long

and

2 2 2 2 3 s′ := α = α =(e e ) (e e ) (e e ) = (27s +4s ). 1 1 − 2 2 − 3 3 − 1 − 1 2 α α2,α4, α6 R α Rlong ∈{ Y− }⊂ long ∈Y

Returning to Q(λ) Z[t]W [λ], note that the constant term of Q(λ) is ∈

2 2 2 2 2 2 2 2 2 s s′ = α α = α = e e e (e e ) (e e ) (e e ) , 1 1 1 2 3 1 − 2 2 − 3 3 − 1 α R α R α R ∈Yshort ∈Ylong Y∈

2 which is precisely the discriminant of P (λ) and thus of t; we set d := s s′ = 27s 1 1 − 1 − 4s s3 Z[t]W . We now see that Z[treg]W Z[treg] is given by 1 2 ∈ →

W W W W Z[t] Z[t] [λ]/(Q(λ)) = Z[t] [λ]/(P (λ)) Z[t] [λ]/(P ′(λ)). d → d ∼ d ⊕ d

This deﬁnes the map treg treg/W , denoted by µ : treg S henceforth. → →

If we pick a K-rational point X on t and replace (s1,s2) with s = µ(X) then we

may write PX (λ):= Ps(λ) and PX′ (λ):= Ps′(λ). We remark that, in this context,

2 λ P (λ) = det(λ adg(X)), X −

is the characteristic polynomial of X g(K). ∈ 17

2.2 Steinberg quotient

Let g g/G = t/W be the Steinberg quotient for g [CR10]. Let D Z[g] be any pre- → ∼ ∈ image of d Z[t] under the quotient Z[g] Z[t]. Then greg := Spec(Z[g] ), which is ∈ → D independent of the choice for D, is the open subscheme of regular semisimple elements in g. We write µ : greg S for the restriction of g t/W to greg. → →

2.3 Parameterization of stable good orbits

One begins by working over a separable closure K¯ and recalling the classical result

[Ste65] that adjoint orbits in greg over K¯ are classiﬁed by the regular part S of the

Steinberg quotient over K¯ . The fibres of the Steinberg map µ : greg S define → subvarieties g, for s S. Then one observes that S is in fact defined over Os ⊂ ∈ K and if s S(K) then is also defined over K. The K-variety may be ∈ Os Os apprehended as the quotient of G by the maximal torus T G containing X, for X ⊆ any X greg(K) with µ(X)= s. The set (K) is commonly called a stable orbit in ∈ Os g(K).

2.4 Steinberg by depth

One of the key tools in this thesis is r-reduction, as introduced in [CH04, 3.1]. § N N 1 Originally, r-reduction took a polynomial P = λ +α1λ − +...+αng over K, whose

g g 1 roots λ K¯ all satisﬁed ord (λ ) = r, to a polynomial R = λ + a λ − + ... + a i ∈ K i 1 g over k in a combinatorial manner: r Q, g,ℓ,n,L,N Z; N 1; g 1; r ∈ ∈ ≥ ≥ ≥ 0; r = L/N; g = gcd(L,N); ℓ = L/g; n = N/g. It was then shown that the splitting

ﬁeld of P depended only R. Here we use it schematically: r-reducing the polynomial 18

Q(λ)= P (λ)P ′(λ) introduced in Section 2.1 ﬁeld independently, and then specializing to a ﬁeld as required.

From the form of the polynomial QX (λ) it follows that g(r, K) is empty unless r 1 Z. Henceforth, we suppose r 1 Z and write r to be the integer part of r ∈ 6 ∈ 6 ⌊ ⌋ and r to be the fractional part, or fractional depth, of r, so r = r + r and { } { } ⌊ ⌋ 1 1 1 2 5 r 0, , , , , . Since Q (λ) = P (λ)P ′ (λ) in K[λ], we may calculate the { } ∈ { 6 3 2 3 6 } X X X r-reduction of PX (λ) and PX′ (λ) separately.

The process of r-reduction produces, for each r 1 Z, a quotient treg S of ∈ 6 r → r aﬃne schemes which recovers the quotient treg S when r Z, as we now explain. → ∈

Table 2.1: The process of r-reduction produces Pr(λ) and Pr′(λ) from P (λ) and P ′(λ) over Z[t].

r P (λ), P ′(λ) { } r r

6 4 2 2 6 4 2 2 0 λ +2s2λ + s2λ s1 = (λ α) λ +2s2′ λ +(s2′ ) λ s1′ = (λ α) − α R − − α R − ∈ short ∈ long Q Q

1 5 2 2 2 2 2 2 , λ s = λ α α α λ s′ = λ α α α 6 6 − 1 − 1 3 5 − 1 − 2 4 6 1 2 2 2 , λ s =(λ α α α )(λ + α α α ) λ s′ =(λ α α α )(λ + α α α ) 3 3 − 1 − 1 3 5 1 3 5 − 1 − 2 4 6 2 4 6 1 3 2 2 2 2 2 3 2 2 2 2 2 λ +2s λ + s λ s =(λ α )(λ α )(λ α ) λ +2s′ λ +(s′ ) λ s′ =(λ α )(λ α )(λ α ) 2 2 2 − 1 − 1 − 3 − 5 2 2 − 1 − 2 − 4 − 6

From Table 2.1 we see how r-reduction produces from Q(λ)= P (λ)P ′(λ) a poly-

1 nomial Q (λ) = P (λ)P ′(λ) over Z[t], for each r Z. Let Φ Z[t] be the set of r r r ∈ 6 r ⊂ ‘roots’ of Q (λ) for each r 1 Z; see Table 2.2 for a list of the sets Φ for each r 1 Z. r ∈ 6 r ∈ 6 19

Deﬁne t tr over Z W by → | |

Z[tr] W = Z[Φr] W Z[R] W = Z[t] W | | | | ⊆ | | | |

reg reg reg reg and t tr by Z[tr ] W = Z[Φr]dr Z[R] W d = Z[t ] W where dr is the → | | ⊆ | | | | restriction of d from Z[t] to Z[t ] with the factor W for convenience. See Chapter r | | 3 for more detail and explicit examples. Note that the action of W on Z[t] descends

to Z[Φr].

The covering group of treg treg is a quotient W of W for which Z[Φ ]W = → r r r Z[Φ ]Wr ; the kernel of W W is W r := w W w(f) = f, f Φ . In fact, r → r { ∈ | ∀ ∈ r} in each case there is a natural section of 1 W r W W 1, as indicated in → → → r → Table 2.2. Deﬁne the aﬃne scheme

reg Sr = tr /Wr

Z Z treg Wr Z t Wr by [Sr]:= [ r ] = [ r]dr .

Let K be a local ﬁeld and suppose 6 is invertible in its residue ﬁeld k. By construc-

tion, Sr(k) classiﬁes r-reductions of characteristic polynomials of regular equivalued elements X g(K) of depth r, for each r 1 Z. Deﬁne ∈ ∈ 6

µ : g(r, K) S (k) r/K → r as follows: for X g(r, K), let µ (X) be the element of S (k) corresponding to the ∈ r/K r r-reduction of PX (λ). 20

Table 2.2: The coordinate ring Z[tr]= Z[Φr] and the sets Φr, the groups Wr indicating r a section of 1 W W Wr 1, and the discriminant dr Z[tr] using notation from Table 2.1.→ → → → ∈

r ′ r W Wr Z[Φr] dr = W s1s { } | | 1

′ 2 3 0 1 W Z[α1,α3,α5] Z[α2,α4,α6] W s1s = 324s 48s1s ⊗ | | 1 − 1 − 2 (Φr = R)

1 5 Z 2 2 2 Z 2 2 2 ′ 2 6 , 6 W 1 [α1α3α5] [α2α4α6] W s1s1 = 324s1 2 2⊗ 2 2 2 2 | | − (Φr = α α α ,α α α ) { 1 3 5 2 4 6} 1 2 2 3 ′ 2 , w2, (w2w1) = S3 (w2w1) = C2 Z[α1α3α5] Z[α2α4α6] W s1s = 324s 3 3 h i ∼ h i ∼ ⊗ | | 1 − 1 (Φr = α1α3α5,α2α4α6 ) { } 1 3 2 Z 2 2 2 Z 2 2 2 ′ 2 3 2 (w2w1) = C2 w2, (w2w1) = S3 [α1,α3,α5] [α2,α4,α6] W s1s1 = 324s1 48s1s2 h i ∼ h i ∼ ⊗2 2 2 | | − − (Φr = α1,α3,α5, α2,α{ 2,α2 ) 2 4 6}

2.5 Maximal tori

Recall that isomorphism classes of tori over K that embed into G over K as a maximal torus are classiﬁed by H1(K,W ) and thus determined, up to isomorphism, by indexed root data of the form (X, , X,ˇ , , ρ) where ρ Z1(K,W ) = Hom(Gal(K/K¯ ),W ); ∅ ∅ ∅ ∈ see [Spr09, 16.2]. § In this thesis we are concerned only with tamely ramiﬁed maximal tori, so we

1 tr will restrict our attention to Htr(K,W ) = Hom(Gal(K /K),W )/W -conj. Here we see how all such data arise from elements of greg(K) under the hypothesis that 6 is

invertible in the residue ﬁeld of K.

Suppose X greg(K). Since all Cartans are conjugate over K¯ to t, and since ∈ reg conjugation preserves depth, X′ t (K¯ ) for some conjugate X′. Let s = µ(X′) and ∈ consider

Q (λ)= (λ α(X′)) K[λ]; s − ∈ α R Y∈ 21

this is a specialization of Q(λ) Z[t][λ], introduced in Section 2.1. Then ∈

K := K(α(X′) α R) s | ∈

is the splitting extension of Q (λ)= P (λ)P ′(λ). Since α(X′) K¯ , there is a natural s s s ∈ action of Gal(K/K¯ ) on the root values α(X′) α R and since the symmetry { | ∈ } group of Q (λ) is W , there is a homomorphism ρ : Gal(K/K¯ ) W , unique up s s → to W -conjugacy, so that σ(α(X′)) = ρ (σ)(α)(X′) for each α R. Note that, up s ∈ to W -conjugation, the roots α(X′) are determined by s through the splitting of the polynomial Q (λ). In this way we see that every element X greg(K) determines s ∈ s S(K) and thence [ρ ] H1(K,W ) by way of Q (λ) K[λ]. In this way we deﬁne ∈ s ∈ s ∈ a function S(K) H1(K,W ), from stable conjugacy classes of elements in greg(K) → to stable conjugacy class of Cartans in g, by s [ρ ]. However, in order to compute 7→ s the splitting extension K , for each s S (k), we must determine the irreducible s ∈ r factors of Rs(λ), the r-reduction of Ps(λ). The next few sections explain how to do that.

2.6 Algebras attached to regular equivalued elements

In this section we prepare for a study of the function S(K) H1(K,W ) . → The coordinate ring of the ﬁbre of µ : treg S above a K-rational point s → ∈ S(K) is the K-algebra K[λ]/(Q (λ)). Now suppose X g(r, K) and s = µ(X) s ∈ ∈ S(K). By [CH04, 3.2], K[λ]/(Q (λ)) is completely determined by µ (X) S (k). § s r/K ∈ r Now [CH04, 3.2] also shows that the irreducible factors of Q (λ) correspond to the § s irreducible factors of its r-reduction in the following way. Let Rs(λ) be the r-reduction 22

Table 2.3: Factorizations of P (λ) over Z[S ] for r Z. r r ∈ w W Pr(λ) Z[Sr][λ] ∈ ∈

6 4 2 2 w2w1 λ +2s2λ + s λ s1 2 − 5 6 4 2 2 (w2w1) λ +2s2λ + s λ s1 2 −

2 3 3 (w2w1) λ + s2λ + α1α3α5 λ + s2λ α1α3α5 − 4 3 3 (w2w1) λ + s2λ + α1α3α5 λ + s2λ α1α3α5 −

3 2 2 2 2 2 2 (w2w1) λ α λ α λ α − 1 − 3 − 5

2 2 sα2 = w2 λ (α1 + α3)λ + α1α3 λ +(α1 + α3)λ + α1α3 (λ α5)(λ + α5) − − 2 2 sα4 = w1w2w1 λ (α1 α5)λ α1α5 (λ α3)(λ + α3) λ +(α1 α5)λ α1α5 − − − − − − 2 2 sα6 = w2w1w2w1w2 (λ α1)(λ + α1) λ (α3 α5)λ α3α5 λ +(α3 α5)λ α3α5 − − − − − −

2 2 2 2 sα1 = w1 λ α λ (α3 + α5)λ + α3α5 λ +(α3 + α5)λ + α3α5 − 1 − 2 2 2 2 sα3 = w2w1w2 λ (α1 + α5)λ α1α5 λ α λ +(α1 + α5)λ α1α5 − − − 3 − 2 2 2 2 sα5 = w1w2w1w2w1 λ (α1 + α3)λ α1α3 λ +(α1 + α3)λ α1α3 λ α − − − − 5

1 (λ α1)(λ + α1)(λ α3)(λ + α3)(λ α5)(λ + α5) − − − 23

of Ps(λ) and let Rs′ (λ) be the r-reduction of Ps′(λ). Set g = deg(Rs) = deg(Rs′ ). In the numerology r (g,ℓ,n) of [CH04, 3.1], we have ng = deg(P ) and 2ng = deg(Q)= 7→ § W R λ R λ R λ . Now let s( ) = i Is s,i( ) be the decomposition of s( ) into irreducible | | ∈

factors in k[λ]; likewise,QRs′ (λ) = i I′ Rs,i′ (λ). We will study the index sets Is and ∈ s ˙ ˙ Is′ , below. Let Rs,i(λ) (resp. Rs,i′ Q(λ)) be any lift of Rs,i(λ) (resp. Rs,i′ (λ)). Then [CH04, 3.2] gives §

K[λ] K[λ] K[λ]/(Q (λ)) = . s ′ i⊕Is ˙ i⊕I ˙ ∈ (Rs,i(λ)) s (Rs,i′ (λ)) M ∈

For each i I , let g = deg R and let K(gi) be the unramiﬁed extension of K of ∈ s i s,i ′ ¯ (gi) degree gi in K; likewise deﬁne gi′ = deg Rs,i′ and K . Then

(g ) n (g′) n K[λ]/(Q (λ)) = K i ( πℓζ˙ ) K i ( πℓζ˙ ), s i ′ i′ i⊕Is i⊕I ∈ s q M ∈ q

where π is any uniformizer for K, independent of the choice of root ζi of Rs,i(λ) (resp.

(g ) (g′) ζ′ of R′ (λ)) and of the lift ζ˙ K i (resp. ζ˙′ K i ). i s,i i ∈ i ∈

In order to pin down K[λ]/(Qs(λ)) more precisely, we must get information about

R λ R λ R λ ′ R λ the decompositions s( ) = i Is s,i( ) and s′ ( ) = i I s,i′ ( ) into irreducible ∈ ∈ s polynomials and their dependenceQ on s S (k). That isQ the topic of the next section. ∈ r

2.7 Factorizations

As a sort of warm-up to the problem of ﬁnding all decompositions of the r-reduction

of Q (λ) K[λ], thus determining the index set I appearing above, in this section s ∈ s we ﬁnd all decompositions of Q(λ) Z[S][λ]. It is enough to ﬁnd all decompositions ∈ 24

of P (λ) Z[S][λ]. ∈

w W R R w Each element determines a partition of = i Iw i into -orbits. ∈ ∈ h i The factorizations of P are listed in Table 2.3, taking r `= 0. The composition { } treg S S is a factorization of µ : treg S and all factorizations of µ arise in → w → → this manner. Each w W thus determines a factorization treg S S of treg S ∈ → w → → corresponding to factorizations of P . We note that Sw ∼= Sw′ over S if and only if if

w′ is W -conjugate to w.

Table 2.4: Factorizations of P (λ) and P ′(λ) over Z[S ] for r Z. r r r 6∈ ′ r w Wr Pr(λ) Z[Sr][λ] P (λ) Z[Sr][λ] { } ∈ ∈ r ∈

1 2 3 2 2 3 ′ 2 ′ 2 ′ (w2w1) λ +2s2λ + s λ s1 λ +2s λ +(s ) λ s 2 2 − 2 2 − 1 1 4 3 2 2 3 ′ 2 ′ 2 ′ (w2w1) λ +2s2λ + s λ s1 λ +2s λ +(s ) λ s 2 2 − 2 2 − 1

1 2 2 2 2 2 2 2 2 2 2 2 2 sα2 = w2 (λ (α + α )λ + α α )(λ α ) (λ α )(λ (α + α )λ + α α ) 2 − 1 3 1 3 − 5 − 2 − 4 6 4 6 1 2 2 2 2 2 2 2 2 2 2 2 2 sα4 = w1w2w1 (λ (α + α )λ + α α )(λ α ) (λ α )(λ (α + α )λ + α α ) 2 − 1 5 1 5 − 3 − 4 − 2 6 2 6 1 2 2 2 2 2 2 2 2 2 2 2 2 sα6 = w2w1w2w1w2 (λ (α + α )λ + α α )(λ α ) (λ α )(λ (α + α )λ + α α ) 2 − 5 3 3 5 − 1 − 6 − 2 4 2 4

1 1 (λ α2)(λ α2)(λ α2) (λ α2)(λ α2)(λ α2) 2 − 1 − 3 − 5 − 2 − 4 − 6

1 2 3 2 2 2 2 2 2 2 2 , (w2w1) λ α α α λ α α α 3 3 − 1 3 5 − 2 4 6

1 2 , 1 (λ α1α3α5)(λ + α1α3α5) (λ α2α4α6)(λ + α2α4α6) 3 3 − −

1 , 5 1 λ α2α2α2 λ α2α2α2 6 6 − 1 3 5 − 2 4 6

The method used above to determine all factorizations of the polynomial P (λ) over

Z[t] may be applied to the polynomials P (λ) over Z[t ], for each r 1 Z. Chapter 3 r r ∈ 6 25

lists the morphisms

µ : treg S r r → r

and the factors

µ : S S , r,w r,w → r

for every r 1 Z and every w W . The results are summarized in Table 2.4 ∈ 6 ∈ r where the case r = 0 is omitted because that case corresponds to Table 2.3. Again { } arguing as above, we see that the factorizations in Tables 2.3 and 2.4 correspond to factorizations treg S S of treg S , for w W . r → r,w → r r → r ∈ r

reg reg t tr

µ W Sw µr Wr Sr,w

µw µr,w

S Sr

Now ﬁx r 1 Z and consider the family of scheme morphisms µ : S ∈ 6 { r,w r,w → S w W . The partition of Φ into w -orbits determines a partial order (<) r | ∈ r} r h i on W , corresponding to ‘ﬁner’ factorizations of P : w w′ the factorization r r ≤ ⇔ corresponding to w divides into the factorization corresponding to w′. This is not the

same as the Bruhat order. Thus 1 is minimal and the Coxeter elements w1w2 and w w are maximal. For instance, if r Z then 1

w S := µ (S ) ′ µ ′ (S ′ ). r r,w r,w \∪w

The deﬁnable subsets Sw S are also recorded in Chapter 3. r ⊆ r w The deﬁnable subsets S S determine the index sets I and I′ appearing in r ⊆ r s s K[λ]/(Q (λ)), as follows. Suppose s S (k). Then s Sw(k) for a unique w W . s ∈ r ∈ r ∈ r

This w determines the factorization of Rs(λ) and Rs′ (λ) into irreducible polynomials

over k and thus the index sets Is and Is′ appearing in K[λ]/(Qs(λ)) .

2.8 Galois representations

Having found all irreducible factors of R (λ), for every s S (k), we may now ﬁnd s ∈ r the splitting extensions Ks; Table 2.5 records the results. Following the strategy of Section 2.5, Table 2.6 records a tame Galois represen-

tation ρ : Gal(K/K¯ ) W for each s Sw(k) and thus defines a tamely ramified s → ∈ r algebraic torus T for each s Sw(k). Here we say a few words about the calculation s ∈ r 1 of Htr(K,W ) above, using the inflation-restriction sequence

1 H1(k,W ) H1 (K,W ) H1 (Knr,W )Fr. → → tr → tr

First, we observe that

1 H (k,W ) ∼= W/W -conj,

since Gal(k/k¯ ) = Zˆ, and H1(k,W ) H1 (K,W ) is injective [Ser02]. Thus, the part ∼ → tr of H1 (K,W ) corresponding to the case r = 0 is exactly the image of H1(k,W ) tr { } 1 1 nr in Htr(K,W ), which is H (Gal(K /K),W ); clearly, this is the unramiﬁed part of

1 Htr(K,W ). We ﬁx a lift σ of Frobenius. Next, we observe that

H1 (Knr,W )Fr = W [q 1]/W -conj tr ∼ − 27

Table 2.5: The splitting extension of lifts of R (λ) for s Sw(k) and s = µ (x) s ∈ r r,w r w Wr lift of Ks { } ∈ Rs(λ)

6 2r 4 4r 2 2 6r (6) ˙ 0 w2w1 λ +2π x˙ 2λ + π x˙ 2λ π x˙ 1 K = K(ζ) 6 4 2 2 − 6 4 2 2 λ +2x2λ + x λ x1 ζ +2x2ζ + x ζ x1 =0 2 − 2 − 2 3 2r 3r 3 2r 3r (3) 0 (w2w1) (λ + π x˙ 2λ + π x˙ 1)(λ + π x˙ 2λ π x˙ 1) K = K(ζ˙) 3 3 − 3 (λ + x2λ + x1)(λ + x2λ x1) ζ + x2ζ + x1 =0 − 3 2 2r 2 2r 2 2r (2) 0 (w2w1) (λ π x˙ 1)(λ π x˙ 2)(λ π x˙ 3) K = K(ζ˙) − 2 −2 2 − 2 (λ x1)(λ x2)(λ x3) ζ x1 =0 − − − − 2 2r 2 r 2r 2 r 2r (2) 0 w1 (λ π x˙ 1)(λ + π x˙ 3λ + π x˙ 2)(λ π x˙ 3λ + π x˙ 2) K = K(ζ˙) − 2 2 2 − 2 (λ x1)(λ + x3λ + x2)(λ x3λ + x2) ζ x1 =0 − − − 2 r 2r 2 r 2r 2 2r 2 (2) ˙ 0 w2 (λ + π x˙ 2λ + π x˙ 1)(λ π x˙ 2λ + π x˙ 1)(λ π x˙ 2) K = K(ζ) 2 −2 2 2− 2 (λ + x2λ + x1)(λ x2λ + x1)(λ x ) ζ + x2ζ + x1 =0 − − 2 2 2r 2 2 2r 2 2 2r 2 0 1 (λ π x˙ 1)(λ π x˙ 2)(λ π x˙ 3) K −(λ2 x2)(λ−2 x2)(λ2 −x2) − 1 − 2 − 3

1 2 6 2r 4 4r 2 2 6r (3) ˙ 2 (w2w1) λ +2π x˙ 2λ + π x˙ 2λ π x˙ 1 K (qπζ) 3 2 2 − 3 2 2 λ +2x2λ + x λ x1 ζ +2x2ζ + x ζ x1 =0 2 − 2 −

1 2 2r 4 2r 2 4r (2) ˙ √ 2 w2 (λ π x˙ 3)(λ π x˙ 2λ + π x˙ 1) K (qπζ, πx˙ 3) − −2 2 (λ x3)(λ x2λ + x1) ζ x2ζ + x1 =0 − − − 1 2 2r 2 2r 2 2r 1 (λ π x˙ 1)(λ π x˙ 2)(λ π x˙ 3) K(√πx˙ 1, √πx˙ 2, √πx˙ 3) 2 − − − (λ x1)(λ x2)(λ x3) − − −

1 2 3 6 6r (2) 3 ˙ (2) 3 2 ˙ 3 , 3 (w2w1) λ π x˙ 1 K ( qπζ), K ( qπ ζ) −2 2 λ x1 ζ x1 =0 − − 1 2 6 6r 2 √3 √3 3 , 3 1 λ π x˙ 1 K( πx˙ 1, πx˙ 1), −2 2 3 2 3 − 2 λ x K( π x˙ 1, π x˙ 1) − 1 p p−

1 5 6 6r √6 6 , 6 1 λ π x˙ 1 K(ζ3, πx˙ 1), − 6 5 λ x1 K(ζ3, π x˙ 1) − p 28

tr nr Zˆ Z ¯ as pointed sets, since Gal(K /K ) ∼= ( / p)(1) as a Gal(k/k)-module. We ﬁx a topological generator τ for Gal(Ktr/Knr). Then, for every ρ Z1 (K,W ), ∈ tr

1 q ρ(στσ− )= ρ(τ) .

This makes it easy to build ρ from ρ(σ) and ρ(τ). Case-by-case calculations are given in Chapter 4; Table 2.6 records the results.

2.9 Proof of Proposition 2.1

To see that µ : g(r, K) S (k) is surjective and that its ﬁbres are thickened r/K → r stable orbits, we argue as in [CH04, Thm 4.4]. Suppose s S (k). Then s Sw(k) ∈ r ∈ r for a unique w W . Then t :=Lie T admits an embedding into g as a Cartan ∈ s s subalgebra. Let P (λ) K[λ] be any lift of P (λ) k[λ]. Then P determines a ∈ s ∈ stable conjugacy class (K) g(K) that intersects t (K). Any X t (K) (K) Os ⊂ s ∈ s ∩Os maps to s under µ . This shows that µ : g(r, K) S (k) is surjective. It is r/K r/K → r st 1 clear that Z (X) implies µ (Z) = µ (X). To see that µ− (µ (X)) is a ∈ Or r/K r/K r/K r/K

thickened stable orbit we suppose µr/K (X)= µr/K (Y ). Then, up to stable conjugacy,

X,Y t (K) and P (λ) and P (λ) have the same r-reduction, so X Y t (K) + , ∈ s X Y − ∈ s r by [CH04, Cor 3.11], so Y st(X). ∈O To see that the collection of functions µ : g(r, K) S (k), for K and k as r/K → r above, define a map of definable subassignments µ : g(r) S , it is sufficient to r → r st 1 observe that (s):= µ− (s) is definable and depends on s S in a definable way. Or r ∈ r Both statements are clear. 29

Table 2.6: Representatives ρ Z1 (K,W ) for H1 (K,W ), for all s Sw(k), s ∈ tr tr ∈ r s = µr,w(x).

r w Ks/K ρs(τ) W ρs(σ) W Gal(Ks/K) { }1 w ∈tr nr ∈ r Z w Wr s S (k) τ Gal(K /K ) σ Fr iso type ∈ 6 ∈ ∈ r ∈ 7→

(6) 0 w2w1 K 1 w2w1 C6

2 (3) 2 0 (w2w1) K 1 (w2w1) C3

3 (2) 3 0 (w2w1) K 1 (w2w1) C2

(2) 0 w1 K 1 w1 C2

(2) 0 w2 K 1 w2 C2

0 1 K 1 1 1

1 2 (3) ˙ 3 2 2 (w2w1) K (qπζ) (w2w1) (w2w1) C6 3 2 2 ζ +2x2ζ + x ζ x1 =0 2 −

1 (2) ˙ √ 3 2 w2 K (qπζ, πx˙ 1) (w2w1) w2 V4 2 ζ x2ζ + x3 =0 − 1 √ √ √ 3 2 1 K( πx˙ 1, πx˙ 2, πx˙ 3) (w2w1) 1 C2

1 2 3 (2) 3 ˙ (2) 3 2 ˙ 2 3 , 3 (w2w1) K ( qπζ), K ( qπ ζ) (w2w1) 1,q 1(3) C3,q 1(3) 2 ≡ ≡ ζ x1 =0 w2,q 2(3) S3,q 2(3) − ≡ ≡

1 2 3 3 2 2 , 1 K(ζ3, √πx˙1), K(ζ3, π x˙1) (w2w1) 1,q 1(3) C3,q 1(3) 3 3 p ≡ ≡ w2,q 2(3) S3,q 2(3) ≡ ≡

1 5 6 6 5 , 1 K(ζ3, √πx˙ 1), K(ζ3, π x˙ 1) w2w1 1,q 1(3) C6,q 1(3) 6 6 p ≡ ≡ w2,q 2(3) D6,q 2(3) ≡ ≡ 30 Chapter 3

Factorizations of coverings and deﬁnable subsets

1 In this chapter we calculate the coverings S S of schemes over Z[6− ] that r,w → r appeared in Section 2.7, then use these morphisms to give explicit descriptions of the

. deﬁnable subsets Sw ֒ S , for every r 1 Z and every w W r → r ∈ 6 ∈ r

reg tr

µr Wr Sr,w

µr,w

3.1 Fractional depth 0

If the fractional depth of r is 0 (so r is an integer) then

6 4 2 2 6 4 2 2 P (λ)= λ +2s λ + s λ s and P ′(λ)= λ +2s′ λ +(s′ ) λ s′ . r 2 2 − 1 r 2 2 − 1

1 Thus, Φr = R and Wr = W . Thus, Z[6− ][tr]= Z[Φr] Wr = Z[R]6. | | Observe that Z[R] = Z[α ,α ,α ] = Z[y ,y ,y ] /(y + y + y ) under y = α , 6 ∼ 1 3 5 6 ∼ 1 2 3 6 1 2 3 1 − 1 31

y = α and y = α . Consequently, 2 − 3 3 5

reg tr = Spec(Z[y1,y2,y3]Dr /(y1 + y2 + y3)) and Sr = Spec(Z[s1,s2]dr ) where D =6y2y2y2(y y )2(y y )2(y y )2 and d = 12s (27s +4s3). Using r 1 2 3 1 − 2 2 − 3 3 − 1 r − 1 1 2 this notation, the morphism µ : treg S is given by r r → r

µ : treg S r r → r (y ,y ,y ) (y2y2y2,y y + y y + y y ). 1 2 3 7→ 1 2 3 1 2 2 3 3 1

Z Z Z Of course, it is also true that [R]6 ∼= [α2,α4,α6]6 ∼= [y1′ ,y2′ ,y3′ ]6/(y1′ + y2′ + y3′ )

under y′ = α , y′ = α and y′ = α . Moreover, s s′ and s s′ deﬁnes an 1 2 2 4 3 − 6 1 7→ 1 2 7→ 2 isomorphism S′ := Spec(Z[s′ ,s′ ] ′ ) Spec(Z[s ,s ] ) = S, with d′ deﬁned in the 1 2 dr → 1 2 dr r 6 obvious way; indeed, the inverse to s s′ and s s′ is given by s′′ = 3 s and 1 7→ 1 2 7→ 2 1 1 3 s2′′ =3 s2. Set

reg Z ′ Z ′ tr′ := Spec( [y1′ ,y2′ ,y3′ ]Dr /(y1′ + y2′ + y3′ )) and Sr′ := Spec( [s1′ ,s2′ ]dr )

reg reg ′Then the inclusion Z[R ] ֒ Z[R] induces isomorphisms t t′ and S S short → → → compatible with the map µ : treg S. We choose to work with the short roots → exclusively, for the remainder of this section, dealing with the case r = 0. { } 32

3.1.1 Case: w =1

Since all orbits in R under the action of w = 1 are singletons, the element w = 1 h i determines the factorization

P (λ)= λ6 +2s λ4 + s2λ2 s =(λ2 α2)(λ2 α2)(λ2 α2). r 2 2 − 1 − 1 − 3 − 5

Thus, S = treg = treg and S S is µ = µ = µ : treg S which, with r,w r r,w → r r,w r → reference to the notation above, is given by

Z[s ,s ] Z[y ,y ,y ] /(y + y + y ) 1 2 dr → 1 2 3 Dr 1 2 3

with s y2y2y2 and s y y + y y + y y . 1 7→ 1 2 3 2 7→ 1 2 2 3 3 1

Aside: In this case, the cover treg S is the identity on treg. r → r,w r

The deﬁnable subset Sw S attached to the morphism of aﬃne schemes µ : r ⊂ r r treg S is µ (treg) S , which is to say, r → r r r ⊂ r

S1 = (s ,s ) S (y ,y ,y ), y +y +y =0, (s ,s )=(y2y2y2,y y +y y +y y ) . r { 1 2 ∈ |∃ 1 2 3 1 2 3 1 2 1 2 3 1 2 2 3 3 1 }

3.1.2 Case: w = w1

The action of w1 on R determines the factorization

P (λ)= λ6 +2s λ4 + s2λ2 s =(λ2 α2)(λ2 α λ + α α )(λ2 + α λ + α α ). r 2 2 − 1 − 1 − 6 3 5 6 3 5 33

Set x = α2 and x = α α and x = α + α = α ; then x2 4x = x . Then 1 1 2 3 5 3 3 5 6 3 − 2 1 µ : S S is given by r,w r,w → r

Z[s ,s ] Z[x ,x ,x ] /(x2 4x x ) 1 2 dr → 1 2 3 Dr,w 3 − 2 − 1

with s x x2 and 2s 2x x2 x where D = µ# (d ). 1 7→ 1 2 2 7→ 2 − 3 − 1 r,w r,w r Aside: The cover treg S is given by x y2 and x y y and x y y . r → r,w 1 7→ 1 2 7→ − 2 3 3 7→ 3 − 2 Since 1

Sw1 = µ (S ) µ (S ). r r,w1 r,w1 \ r,1 r,1

Case: w = w2w1w2

Since w2w1w2 is conjugate to w1, this case is nothing more than a re-labelling of the

case w = w1, above. The action of w2w1w2 on R determines the factorization

P (λ)= λ6 +2s λ4 + s2λ2 s =(λ2 α λ + α α )(λ2 α2)(λ2 + α λ + α α ). r 2 2 − 1 − 4 1 5 − 3 4 1 5

2 Consequently, if we set x1 = α3 and x2 = α1α5 and x3 = α1 + α5 = α4 then x2 4x = x , as above. Thus, µ : S S is given, again in this case, by 3 − 2 1 r,w r,w → r

Z[s ,s ] Z[x ,x ,x ] /(x2 4x x ) 1 2 dr → 1 2 3 Dr,w 3 − 2 − 1

with s x x2 and 2s 2x x2 x and D = µ# (d ), as in the case w , 1 7→ 1 2 2 7→ 2 − 3 − 1 r,w r,w r 1 above. 34

Aside: However, in this case, the cover treg S is given by x y2 and x y y r → r,w 1 7→ 3 2 7→ 1 3 and x y y . 3 7→ 3 − 1

As above, since 1

set Sw2w1w2 w S attached to the morphism of aﬃne schemes µ : S r ⊂ r r,w2w1w2 r,w2w1w2 →

Sr in this case is

Sw2w1w2 = µ (S ) µ (S ). r r,w2w1w2 r,w2w1w2 \ r,1 r,1

Case: w = w1w2w1w2w1

Since w1w2w1w2w1 is conjugate to w1, this case is, again, nothing more than a re-

labelling of the case w = w1, above. The action of w1w2w1w2w1 on R determines the factorization

P (λ)= λ6 +2s λ4 + s2λ2 s =(λ2 α λ + α α )(λ2 + α λ + α α )(λ2 α2). r 2 2 − 1 − 5 1 3 5 1 3 − 5

2 Consequently, if we set x1 = α5 and x2 = α1α3 and x3 = α1 + α3 = α5 then x2 4x = x , as above. Thus, µ : S S is given in this case by 3 − 2 1 r,w r,w → r

Z[s ,s ] Z[x ,x ,x ] /(x2 4x x ) 1 2 dr → 1 2 3 Dr,w 3 − 2 − 1

with s x x2 and 2s 2x x2 x and D = µ# (d ), as in the case w , 1 7→ 1 2 2 7→ 2 − 3 − 1 r,w r,w r 1 above.

Aside: However, in this case, the cover treg S is given by x y2 and x y y r → r,w 1 7→ 5 2 7→ 1 3 and x y y . 3 7→ 3 − 1

As above, since 1 < w1w2w1w2w1, is the only chain to w1w2w1w2w1 in Wr,

the deﬁnable subset Sw1w2w1w2w1 S attached to the morphism of aﬃne schemes r ⊂ r 35

µ : S S in this case is r,w1w2w1w2w1 r,w1w2w1w2w1 → r

Sw1w2w1w2w1 = µ (S ) µ (S ). r r,w1w2w1w2w1 r,w1w2w1w2w1 \ r,1 r,1

3.1.3 Case: w = w2

The action of w2 on R determines the factorization

P (λ)= λ6 +2s λ4 + s2λ2 s =(λ2 α λ + α α )(λ2 + α λ + α α )(λ α )(λ + α ). r 2 2 − 1 − 5 1 3 5 1 3 − 5 5

Set x = α α and x = α . Then µ : S S is given by 1 1 3 2 − 5 r,w r,w → r

Z[s ,s ] Z[x ,x ] 1 2 dr → 1 2 Dr,w

with s x2x2 and 2s 2x x2 and D = µ# (d ). 1 7→ 1 2 2 7→ 1 − 2 r,w r,w r Aside: The cover treg S is given by x y y and x y . r → r,w 1 7→ 1 2 2 7→ − 3 Since 1

Sw2 = µ (S ) µ (S ). r r,w2 r,w2 \ r,1 r,1

Case: w = w1w2w1

Since w1w2w1 is conjugate to w2, this case is a mere re-labelling of the case w = w2,

above. The action of w1w2w1 on R determines the factorization

P (λ)= λ2 α λ α α (λ α )(λ + α ) λ2 + α λ α α . r − 3 − 1 5 − 3 3 3 − 1 5 36

Set x = α α and x = α . Then µ : S S is as above: 1 − 1 5 2 − 3 r,w r,w → r

Z[s ,s ] Z[x ,x ] 1 2 dr → 1 2 Dr,w

with s x2x2 and 2s 2x x2 and D = µ# (d ). 1 7→ 1 2 2 7→ 1 − 2 r,w r,w r Aside: Unlike the case above, here the cover treg S is given by x y y and r → r,w 1 7→ 1 3 x y . 2 7→ 2 The only chain to w w w in W is 1

Sw1w2w1 = µ (S ) µ (S ). r r,w1w2w1 r,w1w2w1 \ r,1 r,1

Case: w = w2w1w2w1w2

Since w2w1w2w1w2 is conjugate to w2, this case is again a mere re-labelling of the

case w = w2, above. The action of w2w1w2w1w2 on R determines the factorization

P (λ)=(λ α )(λ + α ) λ2 α λ α α λ2 + α λ α α . r − 1 1 − 1 − 3 5 1 − 3 5 Set x = α α and x = α . Then µ : S S is as above: 1 − 3 5 2 − 1 r,w r,w → r

Z[s ,s ] Z[x ,x ] 1 2 dr → 1 2 Dr,w

with s x2x2 and 2s 2x x2 and D = µ# (d ). 1 7→ 1 2 2 7→ 1 − 2 r,w r,w r Aside: Unlike the case above, here the cover treg S is given by x y y and r → r,w 1 7→ 2 3 x y . 2 7→ 1 37

The only chain to w2w1w2w1w2 in Wr is 1 < w2w1w2w1w2, so the deﬁnable sub-

set Sw2w1w2w1w2 S attached to the morphism of aﬃne schemes µ : r ⊂ r r,w2w1w2w1w2 S S is r,w2w1w2w1w2 → r

Sw2w1w2w1w2 = µ (S ) µ (S ). r r,w2w1w2w1w2 r,w2w1w2w1w2 \ r,1 r,1

3 3.1.4 Case: w =(w2w1)

3 The action of (w2w1) on R determines the factorization

P (λ)= λ6 +2s λ4 + s2λ2 s =(λ2 α2)(λ2 α2)(λ2 α2). r 2 2 − 1 − 1 − 3 − 5

2 2 2 Set x1 = α1 and x2 = α3 and x3 = α5. Let Ir,w be the ideal in Z[x1,x2,x3] generated by the relation x2 + x2 + x2 = 2(x x + x x + x x ). Then µ : S S is given 1 2 3 1 2 2 3 3 1 r,w r,w → r by

Z[s ,s ] Z[x ,x ,x ] /I 1 2 dr → 1 2 3 Dr,w r,w

with s x x x and 2s x + x + x and D = µ# (d ). 1 7→ 1 2 3 − 2 7→ 1 2 3 r,w r,w r

Aside: The cover treg S is given by x y2 and x y2 and x y2. r → r,w 1 7→ 1 2 7→ 2 3 7→ 3

3 3 The only chain to (w2w1) in Wr is 1 < (w2w1) . So, the deﬁnable subset

3 (w2w1) Sr S attached to the morphism of aﬃne schemes µ 3 : S 3 S ⊂ r r,(w2w1) r,(w2w1) → r is

3 (w2w1) S = µ 3 (S 3 ) µ (S ). r r,(w2w1) r,(w2w1) \ r,1 r,1 38

2 3.1.5 Case: w =(w2w1)

2 The action of (w2w1) on R determines the factorization

P (λ)= λ6 +2s λ4 + s2λ2 s =(λ3 + s λ + α α α )(λ3 + s λ α α α ). r 2 2 − 1 2 1 3 5 2 − 1 3 5

Set x = α α α and x = α α α α α α . Then µ : S S is given by 1 1 3 5 2 1 3 − 3 5 − 5 1 r,w r,w → r

Z[s ,s ] Z[x ,x ] 1 2 d → 1 2 D

with s x2 and s x . 1 7→ 1 2 7→ 2 Aside: The cover treg S is given by x y y y and x y y + y y + y y . r → r,w 1 7→ 1 2 3 2 7→ 1 2 2 3 3 1 2 The complete list of elements less than (w2w1) in Wr is: 1, w2, w1w2w1 and

2 (w2w1) w w w w w . So, the deﬁnable subset Sr S attached to the morphism of 2 1 2 1 2 ⊂ r

aﬃne schemes µ 2 : S 2 S is r,(w2w1) r,(w2w1) → r

2 (w2w1) S = µ 2 S 2 µ (S ), r r,(w2w1) r,(w2w1) \ r,w r,w w (w2) ∈[ where (w2) denotes the conjugacy class of w2 in Wr.

3.1.6 Case: w = w2w1

Since w2w1 acts transitively on R, this element of Wr determines no factorization of P (λ). Thus, S = S in this case and µ : S S is the identity on S . r,w r r,w r,w → r r Aside: In this case, the cover treg S is exactly µ . r → r,w r 39

All elements of order less than 6 are less than w2w1 in Wr are, so

w2w1 S = S µ 2 (S 2 ) µ 3 (S 3 ) µ (S ) r r \  r,(w2w1) r,(w2w1) ∪ r,(w2w1) r,(w2w1) r,w r,w  w (w1) ∈[  

1 5 3.2 Fractional depth 6 or 6

1 5 If the fractional depth of r is 6 or 6 then

2 2 2 2 2 2 Q (λ)= P (λ)P ′(λ)=(λ s )(λ s′ )=(λ α α α )(λ α α α ), r r r − 1 − 1 − 1 3 5 − 2 4 6 so Φ = α2α2α2,α2α2α2 and W = 1. Recall the notation s = α2α2α2 and r { 1 3 5 2 4 6} r 1 1 3 5 2 2 2 s1′ = α2α4α6 from Section 2.1; set Dr =6s1s1′ . Then

reg tr = Spec(Z[s1,s1′ ]Dr )= Sr and µ : treg S is the identity map, as are µ : S S and treg S and r r → r r,1 r,1 → r r → r,1 1 Sr = Sr.

1 2 3.3 Fractional depth 3 or 3

1 2 If the fractional depth of r is 3 or 3 then

2 2 2 2 2 2 2 2 Q (λ)= P (λ)P ′(λ)=(λ α α α )(λ α α α ). r r r − 1 3 5 − 2 4 6 40

Thus, in this case, Φ = α α α ,α α α so W = (w w )3 = C . Set y = α α α r { 1 3 5 2 4 6} r h 2 1 i ∼ 2 1 3 5 2 2 2 2 and y′ = α2α4α6 so y = s1 and y′ = s1′ . Set Dr =6y y′ and dr =6s1s1′ . Then

reg tr = Spec(Z[y,y′]Dr ) and Sr = Spec(Z[s1,s1′ ]dr ),

reg # 2 # 2 and µ : t S is given by µ : s y and µ : s′ y′ . r r → r r 1 7→ r 1 7→

3.3.1 Case: w =1

The element 1 W determines the factorizations ∈ r

P (λ)=(λ α α α )(λ + α α α ) and P ′(λ)=(λ α α α )(λ + α α α ). r − 1 3 5 1 3 5 r − 2 4 6 2 4 6

Thus, S = treg and S S is µ = µ : treg S . Recall that µ#(s )= y2 and r,w r r,w → r r,w r r → r r 1 # 2 µr (s1′ )= y′ .

Aside: The cover treg S is the identity on treg. r → r,w r The deﬁnable subset S1 S attached to the morphism of aﬃne schemes µ : r ⊂ r r treg S is µ (treg) S , which is to say, r → r r r ⊂ r

1 2 2 S = (s ,s′ ) S y,y′, y = s ,y′ = s′ . r { 1 1 ∈ r | ∃ 1 1}

3 3.3.2 Case: w =(w2w1)

3 The element (w w ) W determines the trivial factorization of P (λ) and P ′(λ). 2 1 ∈ r r r Thus, S S and µ : S S is the identity on S . r,w ⊂ r r,w r,w → r r Aside: In this case, the cover treg S is exactly µ . r → r,w r 41

The deﬁnable subset attached to µr,1 is

3 (w2w1) S = µ 3 S 3 µ (S ), r r,(w2w1) r,(w2w1) \ r,1 r,1 which is to say,

3 (w2w1) 2 2 S = (s ,s′ ) S y,y′, y = s or y′ = s′ . r { 1 1 ∈ r | ∀ 6 1 6 1}

1 3.4 Fractional depth 2

1 If the fractional depth of r is 2 then

3 2 2 3 2 2 Q (λ)= P (λ)P ′(λ)=(λ +2s λ + s λ s )(λ +2s′ λ +(s′ ) λ s′ ). r r r 2 2 − 1 2 2 − 1

Thus, in this case, Φ = α2,α2,α2,α2,α2,α2 and W = w , (w w )2 = S . Set r { 1 3 5 2 4 6} r h 2 2 1 i ∼ 3 2 2 2 2 2 2 y1 = α1, y2 = α3, y3 = α5; also set y1′ = α2, y2′ = α4, y3′ = α6. Then

2 2 2 y1 + y2 + y3 = 2(y1y2 + y2y3 + y3y1);

let Ir be the ideal in Z[y1,y2,y3] generated by this relation. Set Dr = 6y1y2y3 and d = 12s (27s +4s3). Then r − 1 1 2

reg tr = Spec(Z[y1,y2,y3]Dr /Ir) and Sr = Spec(Z[s1,s2]dr )

The morphism µ : treg S is given by s y y y and 2s y + y + y . If r r → r 1 7→ 1 2 3 − 2 7→ 1 2 3

tr′ and Sr′ are the schemes deﬁned using y1′ , y2′ and y3′ in place of y1, y2 and y3 then 42

t t t r′ ∼= r and Sr′ ∼= Sr. Accordingly, we work only with Pr(λ), r and Sr, below.

3.4.1 Case: w =1

The element 1 W determines the factorization ∈ r

P (λ)= λ3 +2s λ2 + s2 s =(λ α2)(λ α2)(λ α2). r 2 2 − 1 − 1 − 3 − 5

Thus, S = treg and S S is µ = µ : treg S . r,w r r,w → r r,w r r → r Aside: The cover treg S is the identity on treg. r → r,w r The deﬁnable subset S1 S attached to the morphism of aﬃne schemes µ : r ⊂ r r treg S is µ (treg) S , which is to say, r → r r r ⊂ r

S1 = (s ,s ) S (y ,y ,y ), s = y y y , 2s = y + y + y . r { 1 2 ∈ r | ∃ 1 2 3 1 1 2 3 − 2 1 2 3}

3.4.2 Case: w = w2

The element w W determines the factorization 2 ∈ r

P (λ)= λ3 +2s λ2 + s2λ s =(λ2 (α2 + α2)λ + α2α2)(λ α2) r 2 2 − 1 − 1 3 1 3 − 5

2 2 2 2 2 2 Set x1 = α1α3 and x2 = α1 + α3 and x3 = α5. Then (x2 + x3) = 4(x1 + x2x3); let I be the ideal in Z[x ,x ,x ] generated by this relation. Then µ : S S is r,w 1 2 3 r,w r,w → r given by

Z[s ,s ] Z[x ,x ,x ] /I 1 2 dr → 1 2 3 Dr r,w

with s x x and 2s x + x and D =6x x . 1 7→ 1 3 − 2 7→ 2 3 r 1 3 43

Aside: In this case, the cover treg S is x y y , x y + y and x y . r → r,w 1 7→ 1 2 2 7→ 1 2 3 7→ 3

The deﬁnable subset attached to µr,w2 is

Sw2 = µ (S ) µ (S ). r r,w2 r,w2 \ r,1 r,1

Case: w = w1w2w1

Since w is conjugate to w this case is like the case w , above. The element w w w 2 2 1 2 1 ∈

Wr determines the factorization

P (λ)= λ3 +2s λ2 + s2 s =(λ2 (α2 + α2)λ + α2α2)(λ α2) r 2 2 − 1 − 1 5 1 5 − 3

2 2 2 2 2 Set x1 = α1α5 and x2 = α1 +α5 and x3 = α3. Then the relation determining the ideal I in Z[x ,x ,x ] is the same as that in the case w , above; likewise, µ : S S r,w 1 2 3 2 r,w r,w → r is again given by

Z[s ,s ] Z[x ,x ,x ] /I 1 2 dr → 1 2 3 Dr r,w

with s x x and 2s x + x and D =6x x . 1 7→ 1 3 − 2 7→ 2 3 r 1 3

Aside: In this case, the cover treg S is x y y , x y + y and x y . r → r,w 1 7→ 1 3 2 7→ 1 3 3 7→ 2

The deﬁnable subset attached to µr,w1w2w1 is

Sw1w2w1 = µ (S ) µ (S ). r r,w1w2w1 r,w1w2w1 \ r,1 r,1 44

Case: w = w2w1w2w1w2

Since w is conjugate to w2 this case is also like the case w2, above. The element w = w w w w w W determines the factorization 2 1 2 1 2 ∈ r

P (λ)= λ3 +2s λ2 + s2 s =(λ2 (α2 + α2)λ + α2α2)(λ α2) r 2 2 − 1 − 3 5 3 5 − 1

2 2 2 2 2 Set x1 = α3α5 and x2 = α3 +α5 and x3 = α1. Then the relation determining the ideal I in Z[x ,x ,x ] is the same as that in the case w , above; likewise, µ : S S r,w 1 2 3 2 r,w r,w → r is again given by

Z[s ,s ] Z[x ,x ,x ] /I 1 2 dr → 1 2 3 Dr r,w with s x x and 2s x + x and D =6x x . 1 7→ 1 3 − 2 7→ 2 3 r 1 3 Aside: In this case, the cover treg S is x y y , x y + y and x y . r → r,w 1 7→ 2 3 2 7→ 2 3 3 7→ 1

The deﬁnable subset attached to µr,w2w1w2w1w2 is

Sw2w1w2w1w2 = µ (S ) µ (S ). r r,w2w1w2w1w2 r,w2w1w2w1w2 \ r,1 r,1

2 3.4.3 Case: w =(w2w1)

The element (w w )2 W determines the trivial factorization of P (λ)= λ3 +2s λ2 + 2 1 ∈ r r 2 s2λ s . Thus, S = S and µ : S S is the identity on S . 2 − 1 r,w r r,w r,w → r r Aside: In this case, the cover treg S is exactly µ . r → r,w r

2 The deﬁnable subset attached to µr,(w2w1) is

2 (w2w1) S = µ 2 S 2 µ (S ), r r,(w2w1) r,(w2w1) \ r,w r,w w (w2) ∈[ 45

where (w2) denotes the conjugacy class of w2 in Wr. 46 Chapter 4

Galois cohomology: H1(K,W )

In this chapter we review the calculations summarized in Tables 2.5 and 2.6. Recall

that ‘fractional depth’ refers to the fractional part r of the depth r 1 Z. To define { } ∈ 6 ρ we will exploit the action of Gal(K /K) on the fibre in Xˇ K through the second s s ⊗ s component, which determines an action of W on Xˇ K through the first component. ⊗ s

4.1 Fractional depth 0

If r = 0 and s =(s ,s ) S (K) then { } 1 2 ∈ r

6 4 2 2 6 4 2 2 Q (λ)= P (λ)P ′(λ)=(λ +2s λ + s λ s )(λ +2s′ λ +(s′ ) λ s′ ). s s s 2 2 − 1 2 2 − 1

Recall the partition

w Sr(k)= Sr (k) w Wr a∈ introduced in Section 2.7 with supporting calculations presented in Chapter 3. Then

each s S (k) lies in Sw(k) for a unique w W . We will ﬁnd the splitting extension ∈ r r ∈ r K of every polynomial with r-reduction Q (λ), for every s Sw(k). In the cases s s ∈ r below, we consider only Ps(λ) since it contains all the needed information. 47

4.1.1 Case: w =1 W ∈ 0 Suppose s S1(k). Then s = µ (x) for some x = (x ,x ,x ) S (k); see Sec- ∈ r r,1 1 2 3 ∈ r,1 tion 3.1.1. Then

P (λ)= λ6 +2s λ4 + s2λ2 s =(λ2 x2)(λ2 x2)(λ2 x2) s 2 2 − 1 − 1 − 2 − 3

so Ps(λ) splits in k[λ], and any lift

P (λ)= λ6 +2π2rs˙ λ4 + π4rs˙2λ2 π6rs˙ =(λ2 π2rx˙ 2)(λ2 π2rx˙ 2)(λ2 π2rx˙ 2). 2 2 − 1 − 1 − 2 − 3 splits in K[λ].

In this case, ρ : Gal(K/K¯ ) W is trivial. s →

4.1.2 Case: w (w ) W ∈ 1 ⊂ 0 Suppose w lies in the conjugacy class of w in W and s Sw(k); without loss of 1 0 ∈ r generality, suppose w = w . Then s = µ (x) for some x = (x ,x ,x ) S (k). 1 r,w1 1 2 3 ∈ r,w1 Then

P (λ)= λ6 +2s λ4 + s2λ2 s =(λ2 x )(λ2 x λ + x )(λ2 + x λ + x ) s 2 2 − 1 − 1 − 3 2 3 2

is the decomposition of Ps(λ) into irreducible monic factors in k[λ]. Let ζ be a root of the irreducible factor λ2 x of P (λ). Then [k(ζ): k]=2. Let K(2) be the unique − 1 s unramiﬁed extension of K of degree 2. Then the factors of any lift

P (λ)=(λ2 π2rx˙ )(λ2 + πrx˙ λ + π2rx˙ )(λ2 πrx˙ λ + π2rx˙ ) − 1 3 2 − 3 2 48

are also irreducible and the splitting extension of this polynomial is K(2) = K(ζ˙)

where ζ˙ is any lift of ζ k(ζ) to K(2). ∈ With reference to Section 5.1.2, pick y =(y ,y ,y ) Xˇ K(2) = t(K(2)), regular, 1 2 3 ∈ ⊗ reg (2) (2) such that its image under µ (2) : t (K ) S (K ) is a lift of s = µ (x) r/K → r r,w ∈ S (k). Then, without loss of generality, y = ζ˙ with ζ = √x . Let σ Gal(K(2)/K) r 1 1 ∈ be the element deﬁned by σ(ζ˙)= ζ˙. Then, comparing the form of P (λ) with P (λ) − s r from Section 5.1.2, we have

σ(y ,y ,y )=( y , y , y )= w (y ,y ,y ). 1 2 3 − 1 − 3 − 2 1 1 2 3

In this way we determine a homomorphism Gal(K/K¯ ) W with ρ (σ)= w . Since → s 1 Gal(K(2)/K) = σ , this determines ρ : Gal(K/K¯ ) W with kernel Gal(K/K¯ (2)) h i s → and image w W . h 1i⊂

4.1.3 Case: w (w ) W ∈ 2 ⊂ 0

Suppose w lies in the conjugacy class of w in W and s Sw(k); without loss of 2 r ∈ r generality, suppose w = w . Then s = µ (x) for some x = (x ,x ) S (k). 2 r,w2 1 2 ∈ r,w2 Then

P (λ)= λ6 +2s λ4 + s2λ2 s =(λ2 + x λ + x )(λ2 x λ + x )(λ + x )(λ x ) s 2 2 − 1 2 1 − 2 1 2 − 2

is the decomposition of Ps(λ) into irreducible monic factors in k[λ]. Let ζ be a root of 2 x2+√x 4x1 2 − 2− the irreducible quadratic factor λ + x2λ + x1 of Ps(λ); write ζ = 2 . Then [k(ζ) : k] = 2. Let K(2) be the unique unramiﬁed extension of K of degree 2. Then 49

the factors of any lift

P (λ)=(λ2 + πrx˙ λ + π2rx˙ )(λ2 πrx˙ λ + π2rx˙ )(λ2 π2rx˙ 2) 2 1 − 2 1 − 1

are also irreducible and the splitting extension of this polynomial is K(2) = K(ζ˙)

where ζ˙ is any lift of ζ k(ζ) to K(2). ∈ With reference to Section 3.1.3, pick y =(y ,y ,y ) Xˇ K(2) = t(K(2)), regular, 1 2 3 ∈ ⊗ reg (2) (2) such that its image under µ (2) : t (K ) S (K ) is a lift of s = µ (x) r/K → r r,w ∈ S (k). Let σ Gal(K(2)/K) be non-trivial. Then σ( x2 4x ) = x2 4x . r ∈ 2 − 1 − 2 − 1

Then, comparing the form of Ps(λ) with Pr(λ) from Sectionp 3.1.3, we havep

σ(y1,y2,y3)=(y2,y1,y3)= w2(y1,y2,y3).

In this way we determine a homomorphism Gal(K/K¯ ) W with ρ (σ)= w . Since → s 2 Gal(K(2)/K) = σ , this determines ρ : Gal(K/K¯ ) W with kernel Gal(K/K¯ (2)) h i s → and image w W . h 2i⊂

4.1.4 Case: w =(w w )3 W 2 1 ∈ 0

3 3 (w2w1) (w2w1) Suppose s Sr (k), so s = µ 3 (x) for some x = (x ,x ,x ) Sr (k). ∈ r,(w2w1) 1 2 3 ∈ Then

P (λ)= λ6 +2s λ4 + s2λ2 s =(λ2 x )(λ2 x )(λ2 x ) s 2 2 − 1 − 1 − 2 − 3

is the decomposition of Ps(λ) into irreducible monic factors in k[λ]. Let ζ be any root of the irreducible quadratic factor λ2 x of P (λ); write ζ = √x . Then − 1 s 1 [k(ζ) : k] = 2. Let K(2) be the unique unramiﬁed extension of K of degree 2. Then 50

the factors of any lift

P (λ)=(λ2 π2rx˙ )(λ2 π2rx˙ )(λ2 π2rx˙ ) − 1 − 2 − 3

are also irreducible and the splitting extension of this polynomial is K(2) = K(ζ˙)

where ζ˙ is any lift of ζ k(ζ) to K(2). ∈ With reference to Section 3.1.4, pick y =(y ,y ,y ) Xˇ K(2) = t(K(2)), regular, 1 2 3 ∈ ⊗ reg (2) (2) such that its image under µ (2) : t (K ) S (K ) is a lift of s = µ (x) r/K → r r,w ∈ S (k). Let σ Gal(K(2)/K) be non-trivial; then σ(ζ) = ζ. Then, comparing the r ∈ − form of Ps(λ) with Pr(λ) from Section 3.1.4, we have

σ(y ,y ,y )=( y , y , y )=(w w )3(y ,y ,y ). 1 2 3 − 1 − 2 − 3 2 1 1 2 3

In this way we determine a homomorphism Gal(K/K¯ ) W with ρ (σ)=(w w )3. → s 2 1 Since Gal(K(2)/K)= σ , we have now determined ρ : Gal(K/K¯ ) W with kernel h i s → Gal(K/K¯ (2)) and image (w w )3 W . h 2 1 i⊂

4.1.5 Case: w =(w w )2 W 2 1 ∈ 0

2 2 (w2w1) (w2w1) Suppose s Sr (k). Then s = µ 2 (x) for some x = (x ,x ) Sr (k). ∈ r,(w2w1) 1 2 ∈ Then

P (λ)= λ6 +2s λ4 + s2λ2 s =(λ3 + x λ + x )(λ3 + x λ x ). s 2 2 − 1 2 1 2 − 1

is the decomposition of Ps(λ) into irreducible monic factors in k[λ]. Let ζ be any root

3 (3) of the irreducible cubic factor λ + x2λ + x1 of Ps(λ). Then [k(ζ): k]=3. Let K 51

be the unique unramiﬁed extension of K of degree 3. Then the factors of any lift

P (λ)=(λ3 + π2rx˙ λ + π3rx˙ )(λ3 + π2rx˙ λ π3rx˙ ) 2 1 2 − 1

are also irreducible and the splitting extension of this polynomial is K(3) = K(ζ˙)

where ζ˙ is any lift of ζ k(ζ) to K(3). ∈

With reference to Section 3.1.5, pick y =(y ,y ,y ) Xˇ K(3) = t(K(3)), regular, 1 2 3 ∈ ⊗ reg (3) (3) such that its image under µ (3) : t (K ) S (K ) is a lift of s = µ (x) r/K → r r,w ∈ S (k). Let σ Gal(K(3)/K) be non-trivial, hence a generator. Then, comparing the r ∈

form of Ps(λ) with Pr(λ) from Section 3.1.5, we have

2 σ(y1,y2,y3)=(y2,y3,y1)=(w2w1) (y1,y2,y3).

In this way we determine a homomorphism Gal(K/K¯ ) W with ρ (σ)=(w w )2. → s 2 1 Since Gal(K(3)/K)= σ , we have now determined ρ : Gal(K/K¯ ) W with kernel h i s → Gal(K/K¯ (3)) and image (w w )2 W . h 2 1 i⊂

4.1.6 Case: w = w w W 2 1 ∈ 0

If s =(s ,s ) Sw2w1 (k) then 1 2 ∈ r

P (λ)= λ6 +2s λ4 + s2λ2 s = λ6 +2x λ4 + x2λ2 x s 2 2 − 1 2 2 − 1 52

(6) is irreducible in k[λ]. Let ζ be any root of Ps(λ). Then [k(ζ) : k] = 6. Let K be the unique unramiﬁed extension of K of degree 6. Then any lift

P (λ)= λ6 +2π2rx˙ λ4 + π4rx˙ 2λ2 π6rx˙ K[λ] 2 2 − 1 ∈

is also irreducible, wherex ˙ , x˙ are any lifts of x ,x k. The splitting 1 2 ∈ OK 1 2 ∈ extension of this polynomial in K[λ] is K(6) = K(ζ˙) where ζ˙ is any lift of ζ k(ζ) to ∈ K(6).

The lift P (λ) K[λ] above determines a W -conjugacy class of homomorphisms ∈ ρ : Gal(K /K) W as follows. Split P (λ) K[λ] in K(6): s s → ∈

P (λ)=(λ2 λ2)(λ2 λ2)(λ2 λ2). − 1 − 2 − 3

With reference to Section 3.1.6, pick y = (y ,y ,y ) Xˇ K(6) = t(K(6)), regular, 1 2 3 ∈ ⊗ reg (6) (6) such that its image under µ (6) : t (K ) S (K ) is a lift of s = µ (x) r/K → r r,w ∈ S (k). Let σ Gal(K(6)/K) be a generator. Then, comparing the form of P (λ) with r ∈ s

Pr(λ) from Section 3.1.6, we have

σ(y ,y ,y )=( y , y , y )= w w (y ,y ,y ). 1 2 3 − 3 − 1 − 2 2 1 1 2 3

In this way we determine a homomorphism Gal(K/K¯ ) W with ρ (σ) = w w . → s 2 1 Since Gal(K(6)/K)= σ , we have now determined ρ : Gal(K/K¯ ) W with kernel h i s → Gal(K/K¯ (6)) and image w w W . h 2 1i⊂ 53

1 5 4.2 Fractional depth 6 or 6

1 5 If r = or and s =(s ,s′ ) S (K) then { } 6 6 1 1 ∈ r

Q (λ)= P (λ)P ′(λ)=(λ x )(λ x′ ) s s s − 1 − 1

1 for x = (x ,x′ ) S (k). Since W = 1, there is only one case to consider: S = S 1 1 ∈ r r r r

and Ps(λ) and Ps′(λ) are evidently irreducible; see Section 2.7 and 3.2. Then, for any

liftsx ˙ , x˙ ′ , the sextic factors of 1 1 ∈OK

6 6r 6 6r Q(λ)=(λ π x˙ )(λ π x˙ ′ ) − 1 1

6 6 are irreducible. The splitting extension Ks of this lift is K(ζ3, √πx˙ 1)= K(ζ3, πx˙ 1′ )

1 6 5 6 5 5 if r = and is K(ζ , √π x˙ )= K(ζ , π x˙ ′ ) if r = . p { } 6 3 1 3 1 { } 6 p Next, we see how s determines a representation ρ : Gal(K /K) W , unique up s s → to W -conjugation. Split λ6 π6rx˙ in K : − 1 s

λ6 π6rx˙ =(λ θ)(λ ζ θ)(λ ζ2θ)(λ + θ)(λ + ζ θ)(λ + ζ2θ). − 1 − − 3 − 3 3 3

where θ = πr √6 x˙ if r = 1 and θ = πr √6 π5x˙ if r = 5 . Set y = θ, y = ζ θ 1 { } 6 1 { } 6 1 2 3 and y = ζ2θ. With reference to the notation of Table 2.6, deﬁne σ Gal(K /K) by 3 3 ∈ s σ(ζ )= ζ2 if q 2 mod 3 and σ(ζ )= ζ if q 1 mod 3; then 3 3 ≡ 3 3 ≡

(y1,y3,y2)= w2w1w2w1w2(y1,y2,y3), q 2 mod3; σ(y1,y2,y3)=  ≡  (y1,y2,y3) q 1 mod3. ≡   54

Deﬁne τ Gal(K /K) by τ(θ)= ζ θ where ζ := ζ2, a primitive sixth root-of-unity ∈ s 6 6 − 3

in Ks; then, τ(y ,y ,y )=( y , y , y )= w w (y ,y ,y ). 1 2 3 − 3 − 1 − 2 2 1 1 2 3

Since

1 q στσ− = τ ,

this completely deﬁnes a homomorphism Gal(K /K) W . We conjugate this homo- s → morphism by w w to deﬁne ρ : Gal(K/K¯ ) W appearing in Table 2.5; note that 1 2 s → the image W of ρ is w w if q 1mod 3 while the image of ρ is w ,w w = W s s h 2 1i ≡ s h 2 2 1i if q 2 mod 3. ≡

1 2 4.3 Fractional depth 3 or 3

1 2 If r = or and s =(s ,s′ ) S (K) then { } 3 3 1 1 ∈ r

2 2 Q (λ)= P (λ)P ′(λ)=(λ s )(λ s′ ). s s s − 1 − 1

In this case, W = (w w )3 . r h 2 1 i

4.3.1 Case: w =1

1 Suppose s S (k). Then, using Section 3.3.1, s = µ (x) is given by µ (x ,x′ ) = ∈ r r,1 r,1 1 1 2 2 (x1,x1′ ). Thus,

P (λ)=(λ x )(λ + x ) and P ′(λ)=(λ x′ )(λ + x′ ). s − 1 1 s − 1 1 55

so Ps(λ) splits in k[λ]. Consider a lift to K[λ]:

3 3r 3 3r 3 3r 3 3r P (λ)=(λ π x˙ )(λ + π x˙ ) and P ′(λ)=(λ π x˙ ′ )(λ + π x˙ ′ ). − 1 1 − 1 1

3 1 3 2 The splitting extension K of P (λ)P ′(λ) is K(ζ , √πx˙ ) if r = and K(ζ , √π x˙ ) s 3 1 { } 3 3 1 if r = 2 . { } 3

We now deﬁne the representation ρ : Gal(K /K) W appearing in Table 2.5. s s →

Split P (λ) in Ks:

(λ3 π3rx˙ )(λ3 + π3rx˙ )=(λ θ)(λ ζ θ)(λ ζ2θ)(λ + θ)(λ + ζ θ)(λ + ζ2θ), − 1 1 − − 3 − 3 3 3

where θ = πr √3 πx˙ if r = 1 and θ = πr √3 π2x˙ if r = 2 and where ζ is a primitive 1 { } 3 1 { } 3 3 2 third root-of-unity in Ks. As above, set y1 = θ, y2 = ζ3θ and y3 = ζ3 θ, and deﬁne σ Gal(K /K) by σ(ζ )= ζ2 if q 2 mod 3 and σ(ζ )= ζ if q 1 mod 3; then ∈ s 3 3 ≡ 3 3 ≡

(y1,y3,y2)= w2w1w2w1w2(y1,y2,y3), q 2 mod3; σ(y1,y2,y3)=  ≡  (y1,y2,y3) q 1 mod3. ≡   Deﬁne τ Gal(K /K) by τ(θ)= ζ θ; then, ∈ s 3

2 τ(y1,y2,y3)=(y2,y3,y1)=(w2w1) (y1,y2,y3).

Since

1 q στσ− = τ ,

this completely deﬁnes a homomorphism Gal(K /K) W . We conjugate this homo- s → 56

morphism by w w to deﬁne ρ : Gal(K/K¯ ) W appearing in Table 2.5; note that 1 2 s → the image W of ρ is (w w )2 if q 1 mod 3 while the image of ρ is w , (w w )2 s s h 2 1 i ≡ s h 2 2 1 i if q 2 mod 3. ≡

3 4.3.2 Case: w =(w2w1)

3 (w2w1) Suppose s Sr (k). Recall from Section 3.3.2 that µ 3 : S S is the ∈ r,(w2w1) r,w → r identity. Here,

2 2 Q (λ)= P (λ)P ′(λ)=(λ s )(λ s′ ), s s s − 1 − 1

2 and P (λ) and P ′(λ) are irreducible in k[λ]. Let ζ be a root of P (λ)= λ s ; thus, s s s − 1 2 ζ = √s ; let ζ′ be a root of P ′(λ)= λ s′ ; thus, ζ′ = s′ . Consider a lift of P (λ) 1 s − 1 1 s to K[λ]: p

P (λ)= λ6 π6rs˙ =(λ3 π3rζ˙)(λ3 + π3rζ˙). − 1 −

Then the splitting extension K of P (λ) is K(2)( 3 πζ˙) if r = 1 and K(2)( 3 π2ζ˙) if s { } 3 q q r = 2 . Let ζ be a root of the irreducible factor λ2 x of P (λ). Again [k(ζ): k]=2, { } 3 − 1 s and K(2) = K(ζ˙) where ζ˙ is any lift of ζ k(ζ) to K(2). ∈ We now deﬁne the representation ρ : Gal(K /K) W appearing in Table 2.5. s s →

Split P (λ) in Ks:

(λ3 π3rζ˙)(λ3 + π3rζ˙)=(λ θ)(λ ζ θ)(λ ζ2θ)(λ + θ)(λ + ζ θ)(λ + ζ2θ), − − − 3 − 3 3 3

where θ = πr 3 πζ˙ if r = 1 and θ = πr 3 π2ζ˙ if r = 2 and where ζ is a primitive { } 3 { } 3 3 q q third root-of-unity in K . As above, deﬁne σ Gal(K /K) by σ(ζ ) = ζ2 if q 2 s ∈ s 3 3 ≡ 57

mod 3 and σ(ζ )= ζ if q 1 mod 3; then 3 3 ≡

(y1,y3,y2)= w2w1w2w1w2(y1,y2,y3), q 2 mod3; σ(y1,y2,y3)=  ≡  (y1,y2,y3) q 1 mod3. ≡   Deﬁne τ Gal(K /K) by τ(θ)= ζ θ; then, ∈ s 3

2 τ(y1,y2,y3)=(y2,y3,y1)=(w2w1) (y1,y2,y3).

Since

1 q στσ− = τ ,

this completely deﬁnes a homomorphism Gal(K /K) W . We conjugate this homo- s → morphism by w w to deﬁne ρ : Gal(K/K¯ ) W appearing in Table 2.5; note that 1 2 s → the image W of ρ is (w w )2 if q 1 mod 3 while the image of ρ is w , (w w )2 s s h 2 1 i ≡ s h 2 2 1 i if q 2 mod 3. ≡

1 4.4 Fractional depth 2

If r = 1 and s =(s ,s ) S (K) then { } 2 1 2 ∈ r

P (λ)= λ3 +2s λ2 + s2λ s s 2 2 − 1

and W = w , (w w )2 . Any lift of P (λ) to K[λ] takes the form r h 2 2 1 i s

P (λ)= λ6 +2π2rs˙ λ4 + π4rs˙2λ2 π6rs˙ . 2 2 − 1 58

4.4.1 Case: w =1 W ∈ 1/2 Suppose s = (s ,s ) S1(k). Then, with reference to Section 3.4.1, s = µ (x) for 1 2 ∈ r r,1 x =(x ,x ,x ) S (k) and 1 2 3 ∈ r,1

P (λ)= λ3 +2s λ2 + s2λ s =(λ x )(λ x )(λ x ), s 2 2 − 1 − 1 − 2 − 3

so Ps(λ) splits in k[λ]. Consider a lift:

P (λ)=(λ2 π2rx˙ )(λ2 π2rx˙ )(λ2 π2rx˙ ) − 1 − 2 − 3

Then the splitting extension of P (λ) is Ks = K(√πx˙ 1, √πx˙ 2, √πx˙ 3). Set y1 = √πx˙ 1 and y = √πx˙ and y = √πx˙ . From the structure of S1(k) we ﬁnd that if σ 2 2 3 3 r ∈ Gal(K /K) is non-trivial, then σ(y ) = y and σ(y ) = y and σ(y ) = y , so s 1 − 1 2 − 2 3 − 3 Gal(K /K)= σ and, moreover, s h i

σ(y ,y ,y )=( y , y , y )=(w w )3(y ,y ,y ). 1 2 3 − 1 − 2 − 3 2 1 1 2 3

This deﬁnes Gal(K /K) W by σ (w w )3 and thus deﬁnes ρ : Gal(K/K¯ ) W s → 7→ 2 1 s → for s S1 (k) in Table 2.6. ∈ 1/2

4.4.2 Case: w (w ) W ∈ 2 ⊂ 1/2 Suppose s = (s ,s ) Sw(k). Then, with reference to Section 3.4.2, s = µ (x) for 1 2 ∈ r r,w x =(x ,x ,x ) S (k) where (x + x )2 = 6(x + x x ). Then 1 2 3 ∈ r,w 2 3 1 2 3

P (λ)=(λ2 x λ + x )(λ x ). s − 2 1 − 3 59

2 x2+√x 4x1 Let ζ be a root of the irreducible polynomial λ2 x λ + x , so ζ = 2− ; set − 2 1 2 2 x2 √x 4x1 − 2− ζ′ = 2 . Then k(ζ)/k is a splitting extension for Ps(λ).

Consider a lift of Ps(λ) to K[λ]:

P (λ)=(λ4 π2rx˙ λ2 + π4rx˙ )(λ2 π2rx˙ ). − 2 3 − 1

(2) r The splitting extension for P (λ) over K is Ks = K ( πζ,˙ √πx˙ 1). Set y1 = π πζ˙, r ˙ r q q y2 = π πζ′, y3 = π √πx˙ 3. Then q

σ(y1,y2,y3)=(y2,y1,y3)= w2(y1,y2,y3)

and

τ(y ,y ,y )=( y , y , y )=(w w )3(y ,y ,y ) 1 2 3 − 1 − 2 − 3 2 1 1 2 3

1 q generate Gal(K /K). Since στσ− = τ , this deﬁnes Gal(K /K) W with σ w s s → 7→ 2 and τ (w w )3 with image W = V . This deﬁnes ρ : Gal(K/K¯ ) W in this case, 7→ 2 1 s ∼ 4 s → as appearing in Table 2.6.

4.4.3 Case: w =(w w )2 W 2 1 ∈ 1/2

2 (w2w1) Suppose s =(s ,s ) Sr (k). Then, with reference to Section 3.4.3, 1 2 ∈

P (λ)= λ3 +2s λ2 + s2λ s s 2 2 − 1

(3) is irreducible. Let ζ1, ζ2, ζ3 be roots of this polynomial; let ζ˙1, ζ˙2, ζ˙3 be lifts to K .

r r r Then y1 = π πζ˙1, y2 = π πζ˙2, y3 = π πζ˙3 are roots of a lift of Ps(λ) to P (λ). q q q 60

(3) r The splitting extension Ks of this lift is K (π πζ˙), where ζ is any root of Ps(λ).

q 1 q The Galois group Gal(Ks/K) is generated by σ and τ with στσ− = τ where

2 σ(y1,y2,y3)=(y2,y3,y1)=(w2w1) (y1,y2,y3)

and

τ(y ,y ,y )=( y , y , y )=(w w )3(y ,y ,y ). 1 2 3 − 1 − 2 − 3 2 1 1 2 3

This determines ρ : Gal(K/K¯ ) W in this case, with kernel Gal(K /K) and image s → s W = (w w )2, (w w )3 = w w = C . s h 2 1 2 1 i h 2 1i ∼ 6 61 Chapter 5

Galois cohomology of maximal tori

w In Chapter 3 we found the sets Sr (k) and in Chapter 4 we found the Galois representation ρ : Gal(K/K¯ ) W for every s Sw(k), and therefore a torus T over K s → ∈ r s 1 which embeds into G over K as a maximal torus. In this chapter we ﬁnd H (K,Ts) and therefore ﬁnd the cardinality of G(K)-conjugacy classes of embeddings of Ts into G over K; the results of this chapter are summarized in Table 6.1 where they are used to prove Theorem 1.1.

1 To determine H (K,Ts) we use Tate-Nakayama ([Lan79, p. 3] or [Ser02] more generally):

1 ˇ trWs =0 ˇ H (K,Ts)= X /XWs ,

¯ where Ws = ρs(Gal(K/K)) (so Ws ∼= Gal(Ks/K), since Ks = ker ρs) and

Xˇ trWs =0 = y Xˇ w(y)=0 and Xˇ = w(y) y y X,wˇ W . { ∈ | } Ws h − | ∈ ∈ si w Ws X∈

If W > 2, we use (the logical ‘or’) to separate the non-trivial cases in the | s| ∨ calculations. 62

5.1 Fractional depth 0

5.1.1 Case: w =1

If s S1(k) then W = 1; see Table 2.6, Section 4.1.1 and Section 3.1.1. In this case, ∈ r s

y Xˇ w(y)=0 trW =0 w Ws Xˇ s /XˇW = { ∈ | ∈ } s w(y) y y X,wˇ W h − P| ∈ ∈ si (y1,y2,y3) w 1 w(y1,y2,y3)=(0, 0, 0) = | ∈{ } ˇ nw(y1,y2,y3) P(y1,y2,y3) y X,w 1 o h − | ∈ ∈ { }i (y ,y ,y ) (y ,y ,y )=(0, 0, 0) = { 1 2 3 | 1 2 3 } (y ,y ,y ) (y ,y ,y ) h 1 2 3 − 1 2 3 i (0, 0, 0) = { } = 0 (0, 0, 0) ∼ h i

5.1.2 Case: w = w1

If s Sw1 (k) then W = w = 1,w ; see Table 2.6, Section 4.1.2 and Section 5.1.2. ∈ r s h 1i { 1} In this case,

y Xˇ w(y)=0 trW =0 w Ws Xˇ s /XˇW = { ∈ | ∈ } s w(y) y y X,wˇ W h − P| ∈ ∈ si (y ,y ,y ) w(y ,y ,y )=(0, 0, 0) 1 2 3 w 1,w1 1 2 3 = | ∈{ } ˇ nw(y1,y2,y3) P(y1,y2,y3) y X,w 1,w1 o h − | ∈ ∈ { }i (y ,y ,y ) (y ,y ,y )+( y , y , y )=(0, 0, 0) = { 1 2 3 | 1 2 3 − 1 − 3 − 2 } (y ,y ,y ) (y ,y ,y ), ( y , y , y ) (y ,y ,y ) h 1 2 3 − 1 2 3 − 1 − 3 − 2 − 1 2 3 i (y ,y ,y ) (0,y y ,y y )=(0, 0, 0) = { 1 2 3 | 2 − 3 3 − 2 } (0, 0, 0), ( 2y , y y , y y ) h − 1 − 3 − 2 − 2 − 3 i (y ,y ,y ) y = y , y = y y = 2y = { 1 2 3 | 2 3 1 − 2 − 3 − 2} ( 2y ,y ,y ) h − 1 1 1 i ( 2y ,y ,y ) Z = { − 2 2 2 } = = 0 ( 2y ,y ,y ) ∼ Z ∼ h − 1 1 1 i 63

Case: w = w2w1w2

Since w2w1w2 is conjugate to w1, this case is nothing more than a re-labelling of the

case w = w , as in Section . If s Sw2w1w2 (k) then W = w w w = 1,w w w ; 1 ∈ r s h 2 1 2i { 2 1 2} see Section 4.1.2 and Section 5.1.2. In this case,

y Xˇ w(y)=0 trW =0 w Ws Xˇ s /XˇW = { ∈ | ∈ } s w(y) y y X,wˇ W h − P| ∈ ∈ si (y ,y ,y ) w(y ,y ,y )=(0, 0, 0) 1 2 3 w 1,w2w1w2 1 2 3 = | ∈{ } ˇ nw(y1,y2,y3) P(y1,y2,y3) y X,w 1,w2w1w2o h − | ∈ ∈ { }i (y ,y ,y ) (y ,y ,y )+( y , y , y )=(0, 0, 0) = { 1 2 3 | 1 2 3 − 3 − 2 − 1 } (y ,y ,y ) (y ,y ,y ), ( y , y , y ) (y ,y ,y ) h 1 2 3 − 1 2 3 − 3 − 2 − 1 − 1 2 3 i (y ,y ,y ) (y y , 0,y y )=(0, 0, 0) = { 1 2 3 | 1 − 3 3 − 1 } (0, 0, 0), ( y y , 2y , y y ) h − 3 − 1 − 2 − 1 − 3 i (y ,y ,y ) y = y , y = y y = 2y = { 1 2 3 | 1 3 2 − 1 − 3 − 1} (y , 2y ,y ) h 2 − 2 2 i (y , 2y ,y ) Z = { 1 − 1 1 } = = 0 (y , 2y ,y ) ∼ Z ∼ h 2 − 2 2 i

Case: w = w1w2w1w2w1

Since w2w1w2w1w2 is conjugate to w1, this case is nothing more than a re-labelling

of the case w = w , above. If s Sw1w2w1w2w1 (k) then W = w w w w w = 1 ∈ r s h 1 2 1 2 1i 1,w w w w w ; see Section 4.1.2 and Section 5.1.2. In this case, { 1 2 1 2 1}

y Xˇ w(y)=0 trW =0 w Ws Xˇ s /XˇW = { ∈ | ∈ } s w(y) y y X,wˇ W h − P| ∈ ∈ si (y ,y ,y ) w(y ,y ,y )=(0, 0, 0) 1 2 3 w 1,w1w2w1w2w1 1 2 3 = | ∈{ } ˇ wn(y1,y2,y3) P(y1,y2,y3) y X,w 1,w1w2w1w2w1o h − | ∈ ∈ { }i (y ,y ,y ) (y ,y ,y )+( y , y , y )=(0, 0, 0) = { 1 2 3 | 1 2 3 − 2 − 1 − 3 } (y ,y ,y ) (y ,y ,y ), ( y , y , y ) (y ,y ,y ) h 1 2 3 − 1 2 3 − 2 − 1 − 3 − 1 2 3 i 64

(y ,y ,y ) (y y ,y y , 0) = (0, 0, 0) = { 1 2 3 | 1 − 2 2 − 1 } (0, 0, 0), ( y y , y y , 2y ) h − 2 − 1 − 1 − 2 − 3 i (y ,y ,y ) y = y , y = y y = 2y = { 1 2 3 | 1 2 3 − 1 − 2 − 1} (y ,y , 2y ) h 3 3 − 3 i (y ,y , 2y ) Z = { 1 1 − 1 } = = 0 (y ,y , 2y ) ∼ Z ∼ h 3 3 − 3 i

5.1.3 Case: w = w2

If s Sw2 (k) then W = w = 1,w ; see Table 2.6, Section 4.1.3 and Section 3.1.3. ∈ r s h 2i { 2} In this case,

y Xˇ w(y)=0 trW =0 w Ws Xˇ s /XˇW = { ∈ | ∈ } s w(y) y y X,wˇ W h − P| ∈ ∈ si (y ,y ,y ) w(y ,y ,y )=(0, 0, 0) 1 2 3 w 1,w2 1 2 3 = | ∈{ } ˇ nw(y1,y2,y3) P(y1,y2,y3) y X,w 1,w2 o h − | ∈ ∈ { }i (y ,y ,y ) (y ,y ,y )+(y ,y ,y )=(0, 0, 0) = { 1 2 3 | 1 2 3 2 1 3 } (y ,y ,y ) (y ,y ,y ), (y ,y ,y ) (y ,y ,y ) h 1 2 3 − 1 2 3 2 1 3 − 1 2 3 i (y ,y ,y ) (y + y ,y + y , 2y )=(0, 0, 0) = { 1 2 3 | 1 2 2 1 3 } (0, 0, 0), (y y ,y y , 0) h 2 − 1 1 − 2 i (y ,y ,y ) y = y , y =0 = { 1 2 3 | 2 − 1 3 } (y y , (y y ), 0) h 2 − 1 − 2 − 1 i (y , y , 0) Z = { 1 − 1 } = = 0 (y y , (y y ), 0) ∼ Z ∼ h 2 − 1 − 2 − 1 i

Case: w = w1w2w1

Since w1w2w1 is conjugate to w2, this case is nothing more than a re-labelling of the

case w = w , above. If s Sw1w2w1 (k) then W = w w w = 1,w w w ; see 2 ∈ r s h 1 2 1i { 1 2 1} 65

Table 2.6, Section 4.1.3 and Section 3.1.3. In this case,

y Xˇ w(y)=0 trW =0 w Ws Xˇ s /XˇW = { ∈ | ∈ } s w(y) y y X,wˇ W h − P| ∈ ∈ si (y ,y ,y ) w(y ,y ,y )=(0, 0, 0) 1 2 3 w 1,w1w2w1 1 2 3 = | ∈{ } ˇ nw(y1,y2,y3) P(y1,y2,y3) y X,w 1,w1w2w1o h − | ∈ ∈ { }i (y ,y ,y ) (y ,y ,y )+(y ,y ,y )=(0, 0, 0) = { 1 2 3 | 1 2 3 3 2 1 } (y ,y ,y ) (y ,y ,y ), (y ,y ,y ) (y ,y ,y ) h 1 2 3 − 1 2 3 3 2 1 − 1 2 3 i (y ,y ,y ) (y + y , 2y ,y + y )=(0, 0, 0) = { 1 2 3 | 1 3 2 3 1 } (0, 0, 0), (y y , 0,y y ) h 3 − 1 1 − 3 i (y ,y ,y ) y = y , y =0 = { 1 2 3 | 3 − 1 2 } (y y , 0, (y y )) h 3 − 1 − 3 − 1 i (y , 0, y ) Z = { 1 − 1 } = = 0 (y y , 0, (y y )) ∼ Z ∼ h 3 − 1 − 3 − 1 i

Case: w = w2w1w2w1w2

Since w2w1w2w1w2 is conjugate to w2, this case is nothing more than a re-labelling

of the case w = w , above. If s Sw2w1w2w1w2 (k) then W = w w w w w = 2 ∈ r s h 2 1 2 1 2i 1,w w w w w ; see Table 2.6, Section 4.1.3 and Section 3.1.3. In this case, { 2 1 2 1 2}

y Xˇ w(y)=0 trW =0 w Ws Xˇ s /XˇW = { ∈ | ∈ } s w(y) y y X,wˇ W h − P| ∈ ∈ si (y ,y ,y ) w(y ,y ,y )=(0, 0, 0) 1 2 3 w 1,w2w1w2w1w2 1 2 3 = | ∈{ } ˇ wn(y1,y2,y3) P(y1,y2,y3) y X,w 1,w2w1w2w1w2o h − | ∈ ∈ { }i (y ,y ,y ) (y ,y ,y )+(y ,y ,y )=(0, 0, 0) = { 1 2 3 | 1 2 3 1 3 2 } (y ,y ,y ) (y ,y ,y ), (y ,y ,y ) (y ,y ,y ) h 1 2 3 − 1 2 3 1 3 2 − 1 2 3 i (y ,y ,y ) (2y ,y + y ,y + y )=(0, 0, 0) = { 1 2 3 | 1 2 3 3 2 } (0, 0, 0), (0,y y ,y y ) h 3 − 2 2 − 3 i (y ,y ,y ) y = y , y =0 = { 1 2 3 | 3 − 2 1 } (0,y y , (y y )) h 3 − 2 − 3 − 2 i (0,y , y ) Z = { 2 − 2 } = = 0 (0,y y , (y y )) ∼ Z ∼ h 3 − 2 − 3 − 2 i 66

3 5.1.4 Case: w =(w2w1)

3 (w2w1) 3 3 If s Sr (k) then W = (w w ) = 1, (w w ) ; see Table 2.6, Section 4.1.4 ∈ s h 2 1 i { 2 1 } and Section 3.1.4. In this case,

y Xˇ w(y)=0 trW =0 w Ws Xˇ s /XˇW = { ∈ | ∈ } s w(y) y y X,wˇ W h − P| ∈ ∈ si (y ,y ,y ) 3 w(y ,y ,y )=(0, 0, 0) 1 2 3 w 1,(w2w1) 1 2 3 = | ∈{ } ˇ 3 nw(y1,y2,y3) P(y1,y2,y3) y X,w 1, (w2w1) o h − | ∈ ∈ { }i (y ,y ,y ) (y ,y ,y )+( y , y , y )=(0, 0, 0) = { 1 2 3 | 1 2 3 − 1 − 2 − 3 } (y ,y ,y ) (y ,y ,y ), ( y , y , y ) (y ,y ,y ) h 1 2 3 − 1 2 3 − 1 − 2 − 3 − 1 2 3 i (y ,y ,y ) (0, 0, 0) = (0, 0, 0) = { 1 2 3 | } (0, 0, 0), ( 2y , 2y , 2y ) h − 1 − 2 − 3 i (y ,y ,y ) y ,y arbitrary, y = y y = { 1 2 3 | 1 2 3 − 1 − 2} ( 2y , 2y , 2y ) h − 1 − 2 − 3 i (y ,y , y y ) = { 1 2 − 1 − 2 } ( 2y1, 2y2, 2y1 +2y2) hZ− Z − Z Z i = × = ∼ 2Z 2Z ∼ 2Z × 2Z ×

2 5.1.5 Case: w =(w2w1)

2 (w2w1) 2 2 4 If s Sr (k) then W = (w w ) = 1, (w w ) , (w w ) ; see Table 2.6, Sec- ∈ s h 2 1 i { 2 1 2 1 } tion 4.1.5 and Section 3.1.5. In this case,

y Xˇ w(y)=0 trW =0 w Ws Xˇ s /XˇW = { ∈ | ∈ } s w(y) y y X,wˇ W h − P| ∈ ∈ si (y ,y ,y ) 2 4 w(y ,y ,y )=(0, 0, 0) 1 2 3 w 1,(w2w1) ,(w2w1) 1 2 3 = | ∈{ } ˇ 2 4 wn(y1,y2,y3) P(y1,y2,y3) y X,w 1, (w2w1) .(w2w1)o h − | ∈ ∈ { }i (y ,y ,y ) (y ,y ,y )+(y ,y ,y )+(y ,y ,y )=(0, 0, 0) = { 1 2 3 | 1 2 3 2 3 1 3 1 2 } (0, 0, 0), (y y ,y y ,y y ), (y y ,y y ,y y ) h 2 − 1 3 − 2 1 − 3 3 − 1 1 − 2 2 − 3 i 67

(y ,y ,y ) y + y + y =0 = { 1 2 3 | 1 2 3 } (y y , y 2y , 2y + y ), ( y 2y ,y y , 2y + y ) h 2 − 1 − 1 − 2 1 2 − 2 − 1 1 − 2 2 1 i (y ,y + v , 2y v ) = { 1 1 1 − 1 − 1 } (v, 3y1 2v, 3y1 + v) ( v 3y1, v, 2v +3y1) h −Z −Z Z ∨ − − − i = × = Z 3Z 3Z Z ∼ 3Z × ∨ ×

5.1.6 Case: w = w2w1

If s Sw2w1 (k) then W = w w = 1,w w , (w w )2, (w w )3, (w w )4, (w w )5 ; ∈ r s h 2 1i { 2 1 2 1 2 1 2 1 2 1 } see Table 2.6, Section 4.1.6 and Section 3.1.6. In this case,

y Xˇ w(y)=0 trW =0 w Ws Xˇ s /XˇW = { ∈ | ∈ } s w(y) y y X,wˇ W h − P| ∈ ∈ si (y ,y ,y ) w(y ,y ,y )=(0, 0, 0) 1 2 3 w w2w1 1 2 3 = | ∈h i ˇ nw(y1,y2,y3) P(y1,y2,y3) y X,w w2w1 o h − | ∈ ∈h ii (y ,y ,y )+( y , y , y )+(y ,y ,y ) 1 2 3 − 3 − 1 − 2 2 3 1    +( y1, y2, y3)+(y3,y1,y2)+( y2, y3, y1)=(0, 0, 0)  = − − − − − −  (y1,y2,y3) (y1,y2,y3), ( y3, y1, y2) (y1,y2,y3),  − − − − −    (y2,y3,y1) (y1,y2,y3), ( y1, y2, y3) (y1,y2,y3),  * − − − − − +   (y3,y1,y2) (y1,y2,y3), ( y2, y3, y1) (y1,y2,y3)  − − − − −   (y1,y2,y3) (0, 0, 0) = (0, 0, 0)  =  { | }  (0, 0, 0), (y ,y ,y ), (y y ,y y ,y y ), 2 3 1 2 − 1 3 − 2 1 − 3   * ( 2y1, 2y2, 2y3), (y3 y1,y1 y2,y2 y3), (y3,y1,y2) +  − − − − − −    68

(y ,y ,y ) y = y y = { 1 2 3 | 3 − 1 − 2} (y ,y ,y ) (y y ,y y ,y y ) ( 2y , 2y , 2y ) 2 3 1 ∨ 2 − 1 3 − 2 1 − 3 ∨ − 1 − 2 − 3   * (y3 y1,y1 y2,y2 y3) (y3,y1,y2) +  ∨ − − − ∨  (y1,y2, y1 y2) =  { − − }  (y , y y ,y ) (v, 3y 2v, 3y v) 2 − 1 − 2 1 ∨ − 1 − 1 −    ( 2y1, 2y2, 2y1 +2y2) ( v 3y1, v, 2v +3y1)  * ∨ − − ∨ − − − +   ( y1 y2,y1,y2)  ∨ − −   Z Z Z Z  =  × = × =0 Z Z Z 3Z 2Z 2Z 3Z Z Z Z ∼ Z Z ∼ × ∨ × ∨ × ∨ × ∨ × ×

1 5 5.2 Fractional depth 6 or 6

Suppose s S (k). Refer to Table 2.6, Section 4.2 and Section 3.2. ∈ r

If q 1(3) then W = 1,w w = 1,w w , (w w )2, (w w )3, (w w )4(w w )5 ≡ s h 2 1i { 2 1 2 1 2 1 2 1 2 1 } ˇ trWs =0 ˇ In this case, X /XWs =0 as in the w = w2w1 case of fractional depth 0.

If q 2(3) then W = w ,w w = w ,w = 1,w ,w ,w w ,w w ,w w w , ≡ s h 2 2 1i h 2 1i { 2 1 2 1 1 2 2 1 2 w w w ,w w w w ,w w w w ,w w w w w ,w w w w w ,w w w w w w = D . 1 2 1 2 1 2 1 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1} ∼ 6 Thus, in this case,

y Xˇ w(y)=0 trW =0 w Ws Xˇ s /XˇW = { ∈ | ∈ } s w(y) y y X,wˇ W h − P| ∈ ∈ si (y ,y ,y ) w(y ,y ,y )=(0, 0, 0) 1 2 3 w w2,w1 1 2 3 = | ∈{h i} ˇ nw(y1,y2,y3)P(y1,y2,y3) y X,w w2,w1 o h − | ∈ ∈h ii 69

(y ,y ,y )+(y ,y ,y )+( y , y , y ) 1 2 3 2 1 3 − 1 − 3 − 2   +( y , y , y )+( y , y , y )+( y , y , y )  − 3 − 1 − 2 − 2 − 3 − 1 − 3 − 2 − 1       +(y3,y2,y1)+(y2,y3,y1)+(y3,y1,y2) 

 +(y1,y3,y2)+( y2, y1, y3)+( y1, y2, y3)=(0, 0, 0)  =  − − − − − −     (y ,y ,y ) (y ,y ,y ), (y ,y ,y ) (y ,y ,y ),  1 2 3 − 1 2 3 2 1 3 − 1 2 3   ( y , y , y ) (y ,y ,y ), ( y , y , y ) (y ,y ,y ),  1 3 2 1 2 3 3 1 2 1 2 3   − − − − − − − −     ( y2, y3, y1) (y1,y2,y3), ( y3, y2, y1) (y1,y2,y3),   − − − − − − − −    * (y3,y2,y1) (y1,y2,y3), (y2,y3,y1) (y1,y2,y3), +  − −   (y3,y1,y2) (y1,y2,y3), (y1,y3,y2) (y1,y2,y3)   − −     ( y , y , y ) (y ,y ,y ), ( y , y , y ) (y ,y ,y )   2 1 3 1 2 3 1 2 3 1 2 3   − − − − − − − −   (y1,y2,y3) (0, 0, 0) = (0, 0, 0)  =  { | }  (0, 0, 0), (y y ,y y , 0), 2 − 1 1 − 2   ( 2y , y y , y y ), ( y y , y y , y y ),  1 3 2 2 3 3 1 1 2 2 3   − − − − − − − − − − −     ( y2 y1, y3 y2, y1 y3), ( y3 y1, 2y2, y1 y3),   − − − − − − − − − − −    * (y3 y1, 0,y1 y3), (y2 y1,y3 y2,y1 y3), +  − − − − −   (y3 y1,y1 y2,y2 y3), (0,y3 y2,y2 y3),   − − − − −     ( y y , y y , 2y ), ( 2y , 2y , 2y )   2 1 1 2 3 1 2 3   − − − − − − − −   (y1,y2,y3) y1,y2 arbitrary, y3 = y1 y2  =  { | − − }  (0, 0, 0), (v , v , 0), ( 2y ,y ,y ), (y ,y ,y ), (y ,y ,y ), 3 − 3 − 1 1 1 2 3 1 3 1 2    (y2, 2y2,y2), (v2, 0, v2), (y2 y1, y1 2y2, 2y1 + y2)   − − − − −    * ( y2 2y1,y1 y2,y1 +2y2), (0,v1, v1), (y3,y3, 2v3), +  − − − − −   ( 2y1, 2y2, 2y3)   − − −      70

(y ,y , y y = { 1 2 − 1 − 2} (v , v , 0) ( 2y ,y ,y ) (y , y y ,y ) 3 − 3 ∨ − 1 1 1 ∨ 2 − 1 − 2 1    ( y1 y2,y1,y2) (y2, 2y2,y2) (v2, 0, v2)   ∨ − − ∨ − ∨ −    * (v3, 3y1 2v3, 3y1 + v3) ( v3 3y1, v3, 2v3 +3y1) +  ∨ − − ∨ − − −   (0,v1, v1) (y3,y3, 2v3) ( 2y1, 2y2, 2y1 +2y2)   ∨ − ∨ − ∨ − −   Z Z Z Z =  × = × = 0 ∼ Z Z ∼ Z 0 Z 0 Z Z Z Z Z 0 Z 0 × × ∨ × ∨ × ∨ × ∨ × ∨ ×    Z 3Z 3Z Z Z 0 Z 0 2Z 2Z   ∨ × ∨ × ∨ × ∨ × ∨ ×    1 2 5.3 Fractional depth 3 or 3

5.3.1 Case: w =1

Suppose s S1(k). The cases below refer to Table 2.6, Section 3.3.1 and Section 4.3.1. ∈ r

If q 1(3) then W = 1, (w w )2 = (w w )2 = 1, (w w )2, (w w )4 . In this ≡ s h 2 1 i h 2 1 i { 2 1 2 1 } Z case, Xˇ trWs =0/Xˇ = as in the w =(w w )2 case of fractional depth 0. Ws 3Z 2 1

If q 2(3) then W = w , (w w )2 = S . In this case, ≡ s h 2 2 1 i ∼ 3

y Xˇ w(y)=0 trW =0 w Ws Xˇ s /XˇW = { ∈ | ∈ } s w(y) y y X,wˇ W h − P| ∈ ∈ si (y ,y ,y ) 2 w(y ,y ,y )=(0, 0, 0) 1 2 3 w w2,(w2w1) 1 2 3 = | ∈{h i} ˇ 2 nw(y1,y2,y3) P(y1,y2,y3) y X,w w2, (w2w1) o h − | ∈ ∈h ii 71

(y1,y2,y3)+(y2,y1,y3)+(y3,y2,y1)    +(y1,y3,y2)+(y2,y3,y1)+(y3,y1,y2)=(0, 0, 0)  =  (y1,y2,y3) (y1,y2,y3), (y2,y1,y3) (y1,y2,y3),  − −    (y3,y2,y1) (y1,y2,y3), (y1,y3,y2) (y1,y2,y3)  * − − +   (y2,y3,y1) (y1,y2,y3), (y3,y1,y2) (y1,y2,y3)  − −   (y1,y2,y3) y1 + y2 + y3 =0  =  { | }  (0, 0, 0), (y y ,y y , 0), 2 − 1 1 − 2    (y3 y1, 0,y1 y3), (0,y3 y2,y2 y3)  * − − − − +   (y2 y1,y3 y2,y1 y3), (y3 y1,y1 y2,y2 y3)  − − − − − −   (y1,y2,y3) y1,y2 arbitrary, y3 = y1 y2  =  { | − − } (0, 0, 0), (v , v , 0), (v , 0, v ), (0,v , v ), 3 − 3 2 − 2 1 − 1   * (y2 y1, y1 2y2, 2y1 + y2), ( y2 2y1,y1 y2,y1 +2y2) +  − − − − − −  (y1,y2, y1 y2 =  { − − }  (v , v , 0) (v , 0, v ) (0,v , v ) 3 − 3 ∨ 2 − 2 ∨ 1 − 1   * (v3, 3y1 2v3, 3y1 + v3) ( v3 3y1, v3, 2v3 +3y1) +  ∨ − − ∨ − − −  Z Z =  ×  Z  0 Z 0 Z 0 Z 3Z 3Z Z  Z× {Z}∨ Z× { }∨ × { }∨ × ∨ × = × = ∼ Z 3Z ∼ 3Z ×

3 5.3.2 Case: w =(w2w1)

3 (w2w1) Suppose s Sr (k). The calculations in this case are identical to those in the ∈ 1 2 w = 1 case for fractional depth 3 or 3 above. 72

1 5.4 Fractional depth 2

5.4.1 Case: w =1

Suppose s S1(k). Then W = 1, (w w )3 = 1, (w w )3 ; see Table 2.6, Sec- ∈ r s h 2 1 i { 2 1 } Z Z tion 3.4.1 and Section 4.4.1. In this case, Xˇ trWs =0/Xˇ = as in the Ws 2Z × 2Z 3 w =(w2w1) case of fractional depth 0.

5.4.2 Case: w = w2

Suppose s Sw2 (k). Then W = w , (w w )3 = 1,w ,w w w w w , ∈ r s h 2 2 1 i { 2 1 2 1 2 1 w w w w w w = V ; see Table 2.6, Section 3.4.2 and Section 4.4.2. In this case, 2 1 2 1 2 1} ∼ 4

y Xˇ w(y)=0 trW =0 w Ws Xˇ s /XˇW = { ∈ | ∈ } s w(y) y y X,wˇ W h − P| ∈ ∈ si (y ,y ,y ) 3 w(y ,y ,y )=(0, 0, 0) 1 2 3 w w2,(w2w1) 1 2 3 = | ∈{ } ˇ 3 nw(y1,y2,y3) P(y1,y2,y3) y X,w w2, (w2w1) o h − | ∈ ∈ { }i

(y1,y2,y3) (y1,y2,y3)+(y2,y1,y3)  

 +( y2, y1, y3)+( y1, y2, y3)=(0, 0, 0)  = − − − − − −

 (y ,y ,y ) (y ,y ,y ), (y ,y ,y ) (y ,y ,y ),  1 2 3 − 1 2 3 2 1 3 − 1 2 3   * ( y2, y1, y3) (y1,y2,y3), ( y1, y2, y3) (y1,y2,y3) +  − − − − − − − −  (y1,y2,y3) (0, 0, 0) = (0, 0, 0) =  { | }  (0, 0, 0), (y y ,y y , 0), 2 − 1 1 − 2   * ( y2 y1, y1 y2, 2y3), ( 2y1, 2y2, 2y3) +  − − − − − − − −    73

(y ,y ,y ) y ,y arbitrary, y = y y = { 1 2 3 | 1 2 3 − 1 − 2} (0, 0, 0), (v , v , 0), 3 − 3   * (y3,y3, 2y3), ( 2y1, 2y2, 2y3) +  − − − −  (y1,y2, y1 y2 =  { − − }  (v3, v3, 0), (y3,y3, 2y3), ( 2y1, 2y2, 2y1 +2y2) h − Z Z − − Z− Z i = × = Z 0 Z 0 2Z 2Z ∼ 2Z × 2Z × { }∨ × { }∨ ×

Case: w = w1w2w1

Since w1w2w1 is conjugate to w2, this case is nothing more than a re-labelling of the

case w = w , above. Suppose s Sw1w2w1 (k). Then W = w w w , (w w )3 = 2 ∈ r s h 1 2 1 2 1 i 1,w w w ,w w w ,w w w w w w = V ; see Table 2.6, Section 3.4.2 and Sec- { 1 2 1 2 1 2 2 1 2 1 2 1} ∼ 4 tion 4.4.2. In this case,

y Xˇ w(y)=0 trW =0 w Ws Xˇ s /XˇW = { ∈ | ∈ } s w(y) y y X,wˇ W h − P| ∈ ∈ si (y ,y ,y ) 3 w(y ,y ,y )=(0, 0, 0) 1 2 3 w w1w2w1,(w2w1) 1 2 3 = | ∈{ } ˇ 3 wn(y1,y2,y3) P(y1,y2,y3) y X,w w1w2w1, (w2w1)o h − | ∈ ∈ { }i

(y1,y2,y3) (y1,y2,y3)+(y3,y2,y1)  

 +( y3, y2, y1)+( y1, y2, y3)=(0, 0, 0)  = − − − − − −

 (y ,y ,y ) (y ,y ,y ), (y ,y ,y ) (y ,y ,y ),  1 2 3 − 1 2 3 3 2 1 − 1 2 3   * ( y3, y2, y1) (y1,y2,y3), ( y1, y2, y3) (y1,y2,y3) +  − − − − − − − −  (y1,y2,y3) (0, 0, 0) = (0, 0, 0) =  { | }  (0, 0, 0), (y y , 0,y y ), 3 − 1 1 − 3   * ( y3 y1, 2y2, y1 y3), ( 2y1, 2y2, 2y3) +  − − − − − − − −    74

(y ,y ,y ) y ,y arbitrary, y = y y = { 1 2 3 | 1 2 3 − 1 − 2} (0, 0, 0), (v , 0, v ), 2 − 2   * (y2, 2y2,y2), ( 2y1, 2y2, 2y3) +  − − − −  (y1,y2, y1 y2 =  { − − }  (v2, 0, v2), (y2, 2y2,y2), ( 2y1, 2y2, 2y1 +2y2) h − Z Z− − Z− Z i = × = Z 0 Z 0 2Z 2Z ∼ 2Z × 2Z × { }∨ × { }∨ ×

Case: w = w2w1w2w1w2

Since w2w1w2w1w2 is conjugate to w2, this case is nothing more than a re-labelling

of the case w = w , above. Suppose s Sw2w1w2w1w2 (k). Then W = w w w w w , 2 ∈ r s h 2 1 2 1 2 (w w )3 = 1,w w w w w ,w ,w w w w w w = V ; see Table 2.6, Section 3.4.2 2 1 i { 2 1 2 1 2 1 2 1 2 1 2 1} ∼ 4 and Section 4.4.2. In this case,

y Xˇ w(y)=0 trW =0 w Ws Xˇ s /XˇW = { ∈ | ∈ } s w(y) y y X,wˇ W h − P| ∈ ∈ si (y ,y ,y ) 3 w(y ,y ,y )=(0, 0, 0) 1 2 3 w w2w1w2w1w2,(w2w1) 1 2 3 = | ∈{ } ˇ 3 wn(y1,y2,y3) P(y1,y2,y3) y X,w w2w1w2w1w2, (w2w1)o h − | ∈ ∈ { }i

(y1,y2,y3) (y1,y2,y3)+(y1,y3,y2)  

 +( y1, y3, y2)+( y1, y2, y3)=(0, 0, 0)  = − − − − − −

 (y ,y ,y ) (y ,y ,y ), (y ,y ,y ) (y ,y ,y ),  1 2 3 − 1 2 3 1 3 2 − 1 2 3   * ( y1, y3, y2) (y1,y2,y3), ( y1, y2, y3) (y1,y2,y3) +  − − − − − − − −  (y1,y2,y3) (0, 0, 0) = (0, 0, 0) =  { | }  (0, 0, 0), (0,y y ,y y ), 3 − 2 2 − 3   * ( 2y1, y3 y2, y2 y3), ( 2y1, 2y2, 2y3) +  − − − − − − − −    75

(y ,y ,y ) y ,y arbitrary, y = y y = { 1 2 3 | 1 2 3 − 1 − 2} (0, 0, 0), (0,v , v ), 1 − 1   * ( 2y1,y1,y1), ( 2y1, 2y2, 2y3) +  − − − −  (y1,y2, y1 y2 =  { − − }  (0,v1, v1), ( 2y1,y1,y1), ( 2y1, 2y2, 2y1 +2y2) h − Z − Z − Z− Z i = × = Z 0 Z 0 2Z 2Z ∼ 2Z × 2Z × { }∨ × { }∨ ×

2 5.4.3 Case: w =(w2w1)

2 (w2w1) 2 3 Suppose s Sr (k). Then W = (w w ) , (w w ) = w w ; see Table 2.6, ∈ s h 2 1 2 1 i h 2 1i ˇ trWs =0 ˇ Section 3.4.3 and Section 4.4.3. In this case, X /XWs = 0 as in the w = w2w1 case of fractional depth 0. 76 Chapter 6

Proof of the main result

In this chapter we prove the main result in this thesis, stated again here for convenience.

Theorem 1.1. Let G be a Chevalley group scheme of type G2 and let g be its Lie algebra. Every Chevalley basis for g determines a family of maps of deﬁnable subassignments

r Q, ν : g(r) B ∀ ∈ r → r such that if K is a local ﬁeld and 6 is invertible in the residue ﬁeld k of K then the specialization νr/K determined by K is surjective and

1 (X)= ν− (ν (X)). Or r/K r/K

6.1 Kostant section

In this section we recall the Kostant section κ : S greg of the Steinberg map → µ : greg S. → Following [Kos63] (and a nice pr´ecis in [Kot99, 2.4]), set X = X + X and § + α1 α2 77

X = X α1 + X α2 . Using the structure coeﬃcients of Table 1.2, the centralizer − − − ker ad(X ) of X in g is found to be the linear span of − −

X α1 X α2 ,X α . { − − − − } e

1 Passing from Z to Z[2− ] and using [Kos63, Prop 19], we ﬁnd that the restriction of

1 g t/W to X+ + kerad(X ) is an isomorphism of Z[2− ]-schemes. In this way we → −

ﬁnd that the Kostant section t2/W g2, with image X+ + kerad(X ), is given by → −

s1 s2 (s1,s2) Xα1 + Xα2 + X α (X α1 X α2 ) . 7→ 4 − − 2 − − − e

The restriction of µ to X+ + kerad(X ) is not an isomorphism of schemes over Z, − 1 but is after base change to Z[2− ]. This recipe for the section t /W g of the base 2 → 2 1 change of g t/W to Z[2− ] depends only on the basis ∆ = α ,α for R and → { 1 2} the Chevalley basis X α R . We write κ : S greg for the restriction of the { α | ∈ } 2 → 2

Kostant section to S2.

6.2 Tate-Nakayama

In Chapter 2 we attached a torus T to every s S (k) using the partition S = s ∈ r r Sw T w Wr r of deﬁnable sets of Section 2.7. The torus s was determined by the ∈ cocycle` (just a homomorphism, actually) ρ : Gal(K/K¯ ) W deﬁned in Section 2.8. s → 1 This made it easy to calculate H (K,Ts) using Tate-Nakayama, and the calculation

1 of Xˇ trWs =0/Xˇ , for every r Z and w W and s Sw(k), was carried out in Ws ∈ 6 ∈ r ∈ r 1 Chapter 5. The groups H (K,Ts) are listed in Table 6.1, from which we see that 78

H1(K,T ) depends only on the conjugacy class of w W for which s Sw(k). s ∈ r ∈ r 1 Recall that H (K,Ts) classiﬁes G(K)-conjugacy classes of subtori of G of type Ts.

Table 6.1: H1(K,T ) for tori T determined by s Sw(k). s s ∈ r 1 r w H (K,Ts) hr(w) { }1 1 r Z w Wr =#H (K,Ts) ∈ 6 ∈

0 w2w1 0 1 2 0 (w2w1) Z/3Z 3 3 0 (w2w1) Z/2Z Z/2Z 4 × 0 w1 0 1 0 w2 0 1 0 1 0 1

1 2 2 (w2w1) 0 1 1 w2 Z/2Z Z/2Z 4 2 × 1 1 Z/2Z Z/2Z 4 2 ×

1 2 3 Z Z 3 , 3 (w2w1) /3 3 1 2 Z Z 3 , 3 1 /3 3

1 5 6 , 6 1 0 1

1 From Table 6.1 we note that the group H (K,Ts) is determined by its car-

1 1 1 dinality: if #H (K,Ts) = 1 then H (K,Ts) is trivial; if #H (K,Ts) = 3 then H1(K,T ) = Z/3Z; and if #H1(K,T ) = 4 then H1(K,T ) = Z/2Z Z/2Z. For each s ∼ s s ∼ × n 1, 3, 4 , let A be the corresponding group, interpreted as a definable set. Using ∈ { } n these facts we define h : W N, for each r 1 Z, by the data appearing in the r r → ∈ 6 final column of Table 6.1. 79

6.3 Proof of the main result

For each w W , let g(r, w) ֒ g(r) be the fibre of Sw ֒ S under the map of ∈ r → r → r definable subassignments µ : g(r) S from Proposition 2.1. By pull-back, the r → r S w partition r = w Wr Sr defines a partition ∈ ` g(r)= g(r, w)

w Wr a∈ and maps of deﬁnable subassignments

µw : g(r, w) Sw. r → r

We now deﬁne a function

δw : g(r, w, K) A r/K → hr(w)

for every r 1 Z and w W . Suppose X g(r, w, K); set s = µ(X). Using ∈ 6 ∈ r ∈ the Kostant section, set X := κ(µ(X)) and note that X g(r, K) and X is stably 0 0 ∈ conjugate to X . The relationship between the stable orbit (K) and the G(K)- 0 Os orbit (X ) of X is found by computing the connecting homomorphism of the long O 0 0 exact sequence in Galois cohomology

δX 1 T (K) G(K) (K) 0 H1(K,T ) H1(K,G) X0 Os X0 80

derived from the short exact sequence of K-varieties

1 TX0 G G/TX0 1.

1 Since H (K,G) = 0, the Galois cohomology of TX0 measures how many G(K)-orbits lie in (K): with the choice of X (K) as a base point, the torsor (K)/G(K) Os 0 ∈Os Os 1 1 becomes a group isomorphic to H (K,TX0 ). By Tate-Nakayama, H (K,TX0 ) may be ¯ calculated directly from the action of Gal(K/K) on the cocharacter lattice X (TX0 ). ∗ 1 Indeed, since TX0 = Ts, we have already determined the group H (K,TX0 ), above. In particular, from Table 6.1 we see

H1(K,T ) A . X0 ∼= hr(w)

Since X is stably conjugate to X , we have X (X ), so the connecting homo- 0 ∈ Os 0 morphism δ : (X ) H1(K,T ) sends X to an element of H1(K,T ). In this X0 Os 0 → X0 X0 way we have deﬁned the function

δw : g(r, w, K) A . r/K → hr(w)

w Note that δr/K is clearly surjective.

Set

Bw := Sw A ; r r × hr(w)

note that this is a deﬁnable set. The argument above shows that

νw := µw δw : g(r, K) Bw(k) r/K r/K × r/K → r 81

is surjective.

w Moreover, arguing as in the proof of Proposition 2.1, we see that the ﬁbre of νr/K above νw (X) Bw(k), for X g(r, w, K), is precisely the thickened orbit of X in r/K ∈ r ∈ g(K):

w 1 w (X)=(ν )− (ν (X)). Or r/K r/K

This justifies the notation (x, a) for (X) if µw (X)=(x, a) Bw(k)= Sw(k) O Or r/K ∈ r r × A . Since thickened orbits are definable and since the dependence of (x, a) in hr(w) O (x, a) Sw(k) A is definable, the functions νw : g(r, K) Bw(k) define a ∈ r × hr(w) r/K → r map of definable subassignments,

νw : g(r) Bw. r → r

Set

B := Sw A ; r r × hr(w) w Wr a∈ note that this too is a deﬁnable set. Let

ν : g(r) B r → r

be the map of definable subassignments defined by composing the isomorphism of g g definable subassignments (r, K) w Wr (r, w, K) with the coproduct of the maps → ∈ νw g w ` w r : (r, w, K) Br and the isomorphism of definable subassignments w Wr Br → ∈ → B . Then ν : g(r) B is a map of definable subassignments and if 6` is invertible r r → r in the residue field of K then the specialization ν : g(r, K) B (k) is surjective, r/K → r 82

and

1 (X)= ν− (ν (X)) Or r/K r/K

for every X g(r, K). This completes the proof of Theorem 1.1. ∈

6.4 Application to stable orbit representatives

We conclude by explaining how to use this thesis to enumerate representatives for

stable conjugacy classes of good (equivalued) elements in Lie G(2) over K, assuming only that the residual characteristic of K is at least 5.

For each r 1 Z, consider the deﬁnable subassignment Sr S given by the ∈ 6 ⊂ specializations

Sr(K)= (s ,s ) S(K) ord (s )=6r and ord (s ) 2r . { 1 2 ∈ | K 1 K 2 ≥⌈ ⌉}

Recall the deﬁnition of Sr from Section 2.4 and the map of deﬁnable subassignments µ : g(r) S appearing in Proposition 2.1. Let r → r

res : Sr S r → r

be the map of deﬁnable subassignments given by the surjective specializations

res : Sr(K) S (k) r/K → r 83 where 1 (res6r(s1), res 2r (s2)) r =0, ⌈ ⌉ { } 2 resr/K (s1,s2):=   3 1 1 2 5 (res6r(s1), res6r( 3 s1)) r = , , , . − { } 6 3 3 6  reg Then the image of g(r, K) under the Steinberg quotient µK : g (K) S(K) is → precisely Sr(K) and µ : g(r) S factors through res : r → r r

µ greg(K) K S(K)

def’ble def’ble µ res K |g(r,K) r r/K g(r, K) S (K) Sr(k)

µr

Now, suppose s S (k). Then s Sw(k) S (k) for a unique w W . This ∈ r ∈ r ⊆ r ∈ r parameterizes the components of s by s = µ (x) for x S (k). Lets ˙ be any lift r,w ∈ r,w of s S (k) to Sr(K); thus, res (˙s)= s. Using Section 6.1, we see that ∈ r r/K

s˙1 s˙2 κ(˙s)= Xα1 + Xα2 + X α (X α1 X α2 ) 4 − − 2 − − − e

lies in g(r, K). Letting s range over Sr(k), the set

1 κ(˙s) g(r, K) s˙ res− (s), s S (k) { ∈ | ∈ r/K ∈ r } is a set of representatives for the stable orbits in g(r, K). 84

6.5 Future work

The techniques presented in this thesis may also be used to produce a complete list of Cartan subalgebras of g(K). We leave that for another day. 85

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Representation of G(2) in SO(8)

Traditionally, getting a handle on G(2) has been diﬃcult. Elie´ Cartan noted in

1914 [Car14] that G(2) was the automorphism group of the octonions. Springer

[Spr09, 17.4] realizes this using a direct sum of 2 2-matrices over a ﬁeld. § × A standard, but even more involved, way of saying this has been through the use

of Cayley algebras: G(2) is the automorphism group of a Cayley algebra where the

Cayley algebra is built, for example, from 2 2-matrices this time over a 3-dimensional × cross product algebra over Q.

Using this Cayley algebra, Bump and Joyner [BJ87, 1] deﬁne a group we call G(2) §

of automorphisms of type G2 and its Lie algebra g(2) = Lie G(2). They show G(2) embeds in SO(8) the special orthogonal group, so g(2) embeds in so(8) = Lie SO(8)

the special orthogonal Lie algebra.

Then they oﬀer a Chevalley basis for g(2) in so(8) which we used for doing explicit

calculations. It is, in our notation:

01000 0 0 0 000000 0 0 00000 0 0 0 001000 0 0  00011 0 0 0   000000 0 0  0 0 0 0 0 −1 0 0 000000 0 0 Xα = , Xα = , 1  0 0 0 0 0 −1 0 0  2  000000 0 0       00000 0 0 0   000000 −1 0   −     00000 0 0 1   000000 0 0   00000 0 0 0   000000 0 0          88

0 0 −1000 0 0 000110 0 0 00 0 110 0 0 000001 0 0  00 0 000 0 0   000000 −1 0  00 0 000 −1 0 000000 0 −1 Xα +α = , X2α +α = , 1 2  00 0 000 −1 0  1 2  000000 0 −1       00 0 000 0 1   000000 0 0       00 0 000 0 0   000000 0 0   00 0 000 0 0   000000 0 0          0000010 0 0000001 0 0000000 0 0000000 −1  0000000 −1   0000000 0  0000000 0 0000000 0 X3α +α = , and X3α +2α = 1 2  0000000 0  1 2  0000000 0       0000000 0   0000000 0       0000000 0   0000000 0   0000000 0   0000000 0          with the Chevalley basis elements for the negative roots being simply the transpose

10000000 0 −1000 0 0 0  00200000  00000000 of those for the positive roots. So [Xα ,X α ] = and 1 1  00000000  −    0 0 000 −2 0 0     00000010   0 0 000 0 0 −1      00 0 000 0 0 01 0 000 0 0  0 0 −1000 0 0  00 0 000 0 0 [Xα ,X α ] = . 2 2  00 0 000 0 0  −    00 0 001 0 0   −   00 0 000 1 0   00 0 000 0 0     