Galois representations of orthogonal rigid local systems
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Galois representations of orthogonal rigid lo al systems Von der Universität Bayreuth zur Erlangung des Grades eines Doktors der Naturwissenshaften (Dr. rer. nat) genehmigte Abhandlung von Mihael Shulte (geb. Maier) aus Karlsruhe 1. Gutahter: Prof. Dr. Mihael Dettweiler (Universität Bayreuth) 2. Gutahter: Prof. Dr. Stefan Wewers (Universität Ulm) Tag der Einreihung: 17. April 2012 Tag des Kolloquiums: 10. July 2012
Contents 1 Intro dution 5 1.1 Einleitung ......................................... 10 1.2 Aknowledgement ..................................... 15 1.3 Eidesstattlihe Erklärungen ............................... 16 1.4 Notation .......................................... 17 2 Preliminary Results 19 2.1 Galois Representations .................................. 19 2.2 Étale Fundamental Group Funtor π´et 1 ......................... 23 2.3 Weil Conjeture ...................................... 31 2.4 Crystalline Representations ............................... 33 2.5 Semi-Simpliation .................................... 39 3 Intro dution to MCχ 41 3.1 The Middle Convolution ................................. 41 3.2 The Numerology of MCχ ................................ 44 3.3 Constrution of Hm,ℓ ................................... 47 4 Ho dge Strutures and Middle Convolution 51 4.1 Ho dge Strutures ..................................... 51 4.2 Variations of Hodge Struture .............................. 52 4.3 Extensions of Variations of Ho dge Struture . . . . . . . . . . . . . . . . . . . . . . 54 5 Motivi Desription of Hm,ℓ 57 5.1 Setting ........................................... 57 5.2 The Motivi Interpretation of MCχ ........................... 59 5.3 Appliation to Hm,ℓ ................................... 60 5.4 Analytiation of Hm,ℓ .................................. 65 6 Irreduibility of ρm 67 6.1 Lifting Irreduibility ................................... 67 6.2 Serre's Results on Charaters of GQ .......................... 68 6.3 Groups of Lie Type .................................... 69 6.4 Irreduibility of the modℓ Representation ....................... 73 7 Potential Automorphy of Sp eializations 79 7.1 Langlands Corresp ondene ................................ 79 7.2 Mo dular Lifting ...................................... 80 Bibliography 85 3
1 Intro dution For over 5000 years the rational numb ers have b een known and have been a ornerstone of mathematis sine then. The onsideration of p olynomials and the symmetries of their roots by the group of automorphisms was initiated by Galois and marks the starting p oint of mo dern numb er theory. The Galois group of a p olynomial eno des muh far-reahing information. But what ab out the absolute Galois group GQ , whih is the group of automorphisms of the algebrai losure of Q ? Its struture is still a mystery and far from b eing well-understo od. In order to get partial answers, there are various dierent approahes. First we have a lo ok at the fator groups of the absolute Galois group. Esp eially there is the unsolved question, if every nite group is of this type, known as the inverse Galois problem (see [MM99℄). A seond promising approah are Galois representations, i.e. ontinuous homomorphisms of GQ to matrix groups GLn(Qℓ) . Often it is p ossible to onstrut these in a similar way for all ℓ , whih yields under some irumstanes weakly ompatible systems. These are families of ℓ -adi Galois representations for eah prime number ℓ . They have the prop erty that a Frob enius morphism is mapp ed in suh a way that its harateristi p olynomial is rational and indep endent of ℓ for almost all representations. It is p ossible to desribe many arithmeti ob jets by weakly ompatible systems, for example Galois representations on the Tate mo dules of ellipti urves, whih are ruial to the pro of of Fermat's onjeture, f. Theorem of Wiles [Wil95℄. The ℓ -adi Galois representations have interesting onnetions in two diretions. First we an assign to an irreduible, weakly ompatible system of ℓ -adi Galois representations an analyti funtion, known as L -funtion. If the system is automorphi its L -funtion is equal to an analyti L -funtion. Therefore prop erties like meromorphi ontinuation to the whole omplex plane and the funtional equation an b e transferred. The other way around the Langlands orresp ondene is onjetured (f. [BBG + 03℄), whih says that eah L -funtion asso iated to a weakly ompatible system is obtained by suh an automorphi representation. This is still mostly unproven. In [BLGGT10℄ Barnet-Lamb, Gee, Geraghty and Taylor reently provided some interesting to ols for pro ofs in this diretion. The other onnetion is to geometry. In order to ontrol the Galois group, one often ho oses families of Galois representations given by lisse étale sheaves (see Corollary 2.2.12). In suh families it is p ossible to give far reahing information of the absolute Galois group on the stalks by top ologial means, the so alled mono dromy. A very important ase are the motivi families of Galois representations, whih desrib e variations of ℓ -adi ohomology groups for variable ℓ . 5
1. Introdution An example is the variation of H1 ´et of the Legendre family of ellipti urves Eλ given by Y2=X(X−1)(X−λ) on A1 Q\{0,1} . More general for ellipti urves E the orresp onding Galois representation on the rst ohomology ρℓ:GQ−→ GL(H1 ´et(E,Zℓ)) ∼ =GL2(Zℓ)//GL2(Qℓ) is dual to the Galois representation on the Tate mo dule Tℓ(E) . Serre's op en image Theorem gives information on the size of the image of the representation. Théorème ( [Ser72℄, (7) ): If E is an el lipti urve without omplex multipliation dened over a number eld K and ρℓ:GK−→ GL2(Qℓ) the orresponding representation, then for almost al l prime numbers ℓ we have im (ρℓ) = GL2(Zℓ). Let λ b e a geometri p oint whih is dened over Q suh that Eλ has no omplex multipliation. Then the sp eialization H1 ´et(Eλ,Qℓ) is a GQ -mo dule on whih the group GQ ats maximally for almost all ℓ . The seond imp ortant prop erty is that we get a weakly ompatible system of representations (ρℓ)ℓ prime whih is automorphi in the sense of the Langlands program. This was shown in 2001 by Breuil, Conrad, Diamond, Taylor [BCDT01℄ and Wiles [Wil95℄ in the pro of of the Taniyama-Shimura-Weil onjeture, whih is now known as mo dularity theorem, and is true for all ellipti urves. The ambition of this work is to prove similar results for higher dimensional generalizations of the Legendre family. The key observation is that the mono dromy of the Legendre family is given by a rigid loal system whih is the solution of a Piard-Fuhs equation, a sp eial hypergeometri dierential equation: λ(1 −λ)f′′ + (1 −2λ)f′−1 4f= 0. In this ontext, rigid means that the lo al system admits no deformations, whih is equivalent to rig(F) = 2 if the system is irreduible (see Theorem 3.1.5). In general it is p ossible to desrib e rigid lo al systems by Katz' theory of middle onvolution MCχ (Chapters 5,6 of [Kat96℄). Let χ:π´et 1(Gm,K)−→ Qℓ × b e a geometrially non-trivial one dimensional representation and F a lisse étale sheaf on A1 K\S , where S⊆A1 K is nite. Then one obtains a lisse étale sheaf MCχ(F) on A1 K\S . If F is irreduible and rigid, then MCχ(F) is irreduible and rigid as well but usually rk(MCχ(F)) 6= rk(F) . 6
In summary we get the following onstrutive statement, whih is known as Katz algorithm: Theorem ([Kat96℄): Let F be an irreduible rigid loal system of Qℓ -modules on A1 K\S . There exist n∈N0 , loal systems L0,...,Ln of Qℓ -modules of rank one on A1 K\S and representations χ1,...,χn:π´et 1(Gm,K)−→ Qℓ × , suh that F=Ln⊗MCχn(... L2⊗MCχ2(L1⊗MCχ1(L0) ) ... ). For S={0,1} if L0 has monodromy tuple (−1,−1,1) (f. Chapter 2.2) and χ=−1 , the unique quadrati harater, we get the Legendre family MC−1(L0) . This is the starting point of an innite family (Hm,ℓ)m∈N0 of lo al systems of Qℓ -mo duls on A1 K with i∗Hm,ℓ =Ln⊗MC−1(... L2⊗MC−1(L1⊗MC−1(L0) ) ... ), where we have i:A1 K\ {0,1}//A1 K and Lj with mono dromy tuple (1,−1,−1) for j o dd resp etively Lj with (−1,1,−1) for j6= 0 even. This onstrution yields the following result: Theorem (f. 3.3.1) Let ℓ be a prime number and K an algebraial ly losed eld with har (K)∤2ℓ . Then, for any m∈N0 there exists a ohomologial ly rigid Hm,ℓ ∈Tℓ(K) of generi rank m+1 , a Qℓ -sheaf on A1 K whih is lisse on i:A1 K\{0,1}//A1 K . If m is even, then Hm,ℓ has orthogonal monodromy, and if m is odd, Hm,ℓ has sympleti monodromy, i.e. there is an orthogonal respetively sympleti pairing Hm,ℓ ×Hm,ℓ −→ Qℓ. The monodromy tuple of i∗Hm,ℓ has the fol lowing Jordan normal form: at 0 : J1(1)m 2⊕J1(−1)m 2+1 for 2 |m, J2(1)m+1 2for 2 ∤m, at 1 : J2(1)m 2⊕J1(−1) for m≡0 mod 4, J1(1)m−1 2⊕J2(−1) ⊕J1(−1)m−1 2for m≡1 mod 4, J3(1) ⊕J2(1)m 2−1for m≡2 mod 4, J2(1) ⊕J1(1)m−3 2⊕J1(−1)m+1 2for m≡3 mod 4, at ∞ : Jm+1(1). Here Jn(λ) denotes the upp er triangular Jordan blo k of length n and eigenvalue λ . 7
1. Introdution We want to note that the ase rk(H6,ℓ) = 7 is of sp eial interest. Its examination answered a question of Serre on the existene of motivi Galois groups of type G2 (f. [Ser94℄). By this it was possible to onstrut suh motives (f. Dettweiler, Katz and Reiter [DR10℄). The motivi desription of rigid lo al systems by Katz yields lisse étale sheaves Hm,ℓ , whose analytiations ome from variations of Ho dge struture. Theorem (f. 5.4.3) Let Hm,ℓ be as in Theorem 3.3.1 . Then there exists a loal system of Z -modules Gm on A1 C\{0,1} underlying a polarized variation of Z -Hodge struture (Gm,F•,∇) on C\{0,1} pure of weight m suh that (i∗Hm,ℓ)an ∼ =Gm⊗Qℓ. The indued isomorphism on the stalks (i∗Hm,ℓ)an x∼ =Gm⊗Qℓx for x∈C\{0,1} is given by the omparison isomorphism between étale ohomology and singular ohomology: 1 2(1 −σ)ker Hm ´et (Xx,Qℓ)→Hm ´et (Dx,Qℓ)∼ =1 2(1 −σ)ker (Hm B(X(C)x,Z)→Hm B(D(C)x,Z)) ⊗Qℓ. Moreover, the Hodge ltration of Gm has maximal length. In this way we get, as in the Legendre ase, families of weakly ompatible systems of Galois representations ρm of Q by three transformations, namely by tensoring it with its determinant, sp eializing like in Theorem 6.4.1 and nally semi-simpliation: ρm= (ρm,ℓ)ℓ prime := ((ρi∗Hm,ℓ ⊗det(ρi∗Hm,ℓ )) ◦ιx)ss ℓ prime . Here ιx:GK//π´et 1(A1 K\{0,1}) denotes the sp eialization map for a xed x∈A1 K\{0,1} , whih omes from the morphism {x}//A1 K\{0,1} (f. Setion 2.2). These representations ρm have the property that they fator over Zℓ and over a speial orthogonal group for even m or a sympleti group for o dd m . If m is even, by tensoring the initial system of representations with the ylotomi harater χℓ to the p ower m 2 we get systems of weight 0 representations. The redution mo d ℓ is dened as dρm,ℓ := ρi∗Hm,ℓ ⊗det(ρi∗Hm,ℓ )⊗χm 2 ℓ:π1 ´et(A1 Q\{0,1})−→ SOm+1(Fℓ). Theorem (f. 6.4.1) Let x∈A1 Q\ {0,1} , suh that there exist odd prime numbers p, q 6=ℓ satisfying νp(x)<0 but ℓ∤νp(x) and νq(x−1) >0 but ℓ∤νq(x−1) . Then the fol lowing holds: If m∈N0 even and m≥12 then Ωm+1(Fℓ)⊆im ( dρm,ℓ ◦ιx) for almost al l prime numbers ℓ , where dρm,ℓ ◦ιx:GQ−→ SOm+1(Fℓ) is the speialization at x . If m= 6 then for almost al l ℓ , we have im ( dρm,ℓ ◦ιx) = G2(Fℓ) 8
Therefore it is possible to sp eialize in suh a way, that we obtain an irreduible representation. Finally Theorem 7.2.2 and the motivi desription in Chapter 5give the automorphy over a numb er eld. Theorem ([BLGGT10℄, Thm.5.3.1): Suppose that K is a CM (or total ly real) eld and that (ρℓ)ℓ prime is an irreduible, total ly odd, essential ly onjugate self-dual, regular, weakly ompatible system of ℓ -adi representations of K . Then there is a nite, CM (or total ly real), Galois extension L|K suh that the restrition of (ρℓ)ℓ prime to GL is automorphi. The motivi desription of rigid lo al systems by Katz shows that these systems of representations are rystalline for almost all prime numb ers ℓ . Under the assumptions of Theorem 6.4.1 using the work of Barnet-Lamb, Gee, Geraghty and Taylor, we obtain the following result. A weaker statement was proved in [GMHK10℄. Theorem (f. 7.2.4) For m= 6 or m∈N0 even, m≥12 and K=Q the irreduible, weakly ompatible system ρm= (ρm,ℓ)ℓ prime of Galois representations is potential ly automorphi. 9
1. Introdution 1.3 Eidesstattlihe Erklärungen Hiermit versihere ih, dass ih die hier vorliegende zur Promotion eingereihte Arb eit mit dem Titel Galois representations of orthogonal rigid loal systems selbstständig verfasst, nur die angegeb enen Quellen und Hilfsmittel b enutzt und wörtlih o der inhaltlih üb ernommene Stellen als solhe gekennzeihnet hab e. Ih versihere an Eides statt, dass diese Angaben wahr sind und dass ih nihts vershwiegen hab e. Mir ist b ekannt, dass die falshe Abgabe einer Versiherung an Eides statt mit Freiheitsstrafe bis zu drei Jahren o der mit Geldstrafe b estraft wird. Bayreuth, den 16. April 2012 Hiermit erkläre ih, dass ih bisher keine Promotionsversuhe mit dieser o der einer anderen Dissertation unternommen hab e. Die Arb eit wurde bisher weder im Inno h Ausland in gleiher o der ähnliher Form einer anderen Prüfungsb ehörde vorgelegt. Bayreuth, den 16. April 2012 Hiermit b estätige ih, dass ih keinerlei Hilfe von gewerblihen Promotionsb eratern bzw. -vermittlern oder ähnlihen Dienstleistern in Anspruh genommen hab e, no h künftig in Anspruh nehmen werde. Bayreuth, den 16. April 2012 16
1.4. Notation 1.4 Notation N={1,2,...} natural numb ers N0={0,1,2, . . .} natural numb ers with zero K, L elds K algebrai losure of K Ksep separable losure of K in K GK= Gal(Ksep/K) absolute Galois group of K ΣK set of nite plaes of K Kv ompletion of K at v kv residue eld of K at v Cv=ˆ Kv the ompletion of the algebrai losure of Kv Iw inertia group at w∈ΣL\{0} for a eld Extension L/K Itame K:= πtame 1(Gm,K) tame inertia group as in Denition 2.2.4 Knr v maximal unramied extension AK,IK adele ring of K and idele group of K ℓ prime numb er χ one dimensional ℓ -adi Galois representation χℓ ylotomi harater 1,−1 trivial and quadrati rank one representation Lχ Kummer sheaf asso iated to χ L middle extension sheaf on A1 K MCχ middle onvolution funtor as in Denition 3.1.2 MTL middle tensor pro dut as in Denition 3.1.6 Tℓ(K) ategory of sp eial Qℓ -sheaves as in Denition 3.1.3 i:U//A1 K inlusion of an op en dense subset of the ane line R ommutative ring with 1 M(m×n, R)m times n matries over the ring R GL(V) group of invertible endomorphisms of the vetor spae V An K,Ga,K,Gm,K,GLn(K),SLn(K) algebrai groups O n(K) orthogonal group SOn(K) sp eial orthogonal group Ωn(K) derived group of SOn(K) Spn(K) sympleti group G2(K) a sp oradi group Jn(λ) upp er triangular Jordan blo k of length n and eigenvalue λ ιx sp eialization map to x (f. Setion 2.2) Hm,ℓ sp eial Qℓ -sheaf onstruted in Setion 3.3 ρm,ℓ :GK−→ GLm+1(Qℓ)ℓ -adi Galois representation onstruted in Setion 7.2 dρm,ℓ :π´et 1(A1 Q\{0,1})−→ SOm+1(Fℓ) weight 0 representation of π´et 1(A1 Q\{0,1}) (see Setion 6.4) 17
2 Preliminary Results In this hapter, we want to x the notation used in this work. In addition we give an overview of onepts and theorems losely related to Galois representations. 2.1 Galois Representations Let K b e a eld and denote an algebrai losure by K . If L/K is a Galois extension (not neessarily nite), we get the Galois group Gal(L/K) := AutK(L) . The group is equipp ed with a natural top ology, the Krull top ology. This is the ase b eause Gal(L/K) is a topologial group as pro jetive limit of the disrete nite Galois groups of the nite Galois sub extensions. Therefore the absolute Galois group GK:= Gal(Ksep/K) is a pronite group, where Ksep denotes the separable losure of K in K . We will regard all o urring algebrai extensions of K as subelds of K . If K is a p erfet eld, the algebrai and the separable losure oinide. For a xed prime numb er ℓ , we have the ℓ -adi integers Zℓ:= lim ←− Z/ℓnZ and the eld of ℓ -adi rational numb ers Qℓ:= Quot(Zℓ) = Zℓ[1 ℓ] , whih is the ompletion of Q with resp et to the ℓ -adi disrete absolute value. This valuation extends uniquely to the algebrai losure Qℓ . The ℓ -adi distane given by this valuation indues for n∈N a top ology on M(n×n, Qℓ) . Beside the Zariski top ology on GLn(Qℓ) we get thereby another struture as top ologial group, whih we will use in the following denition. This yields a natural ontinuous ation of this top ologial group on Qℓ n equipp ed with any norm, esp eially the ℓ -adi one. This onstrution of the ℓ -adi top ology is suitable for any nite dimensional Qℓ -vetor spae V . For further details on the following denitions see [Ser68℄. Denition 2.1.1 For a eld K an ℓ -adi Galois representation is a homomorphism ρ:GK−→ GL(V) of topologial groups from the absolute Galois group of K to the general linear group of a nite dimensional Qℓ -vetor spae V equipped with the ℓ -adi topology. The dimension of V is al led the rank of ρ . This is the same as a Qℓ -vetor spae V equipp ed with the ℓ -adi top ology and a ontinuous GK -op eration. Two representations ρ, ρ′:GK−→ GL(V) are equivalent , if there exists a linear map φ∈GL(V) suh that φ−1◦ρ(g)◦φ=ρ′(g) for all g∈GK . 19
2. Preliminary Results An imp ortant example of an ℓ -adi Galois representation of GQ of rank one is the ylotomi harater χℓ:GQ−→ GL1(Qℓ) = Qℓ × . More preisely, it maps to Z× ℓ in the following way: For eah n∈N , we have a lo ok at the ylotomi extension Q(ζℓn) for a primitive ℓn -th ro ot of unity ζℓn . Then Gal(Q(ζℓn)/Q)∼ =(Z/ℓnZ)× , whih an b e hosen in suh a way that it ts together with the isomorphism for smaller n . Indep endent of the hoies, we get a ompatible system of ontinuous group homomorphisms whih gives rise to the harater. This onstrution an b e generalized to a eld K with harateristi unequal to ℓ . One of the main prop erties of the Galois representations ρHm,ℓ onstruted in Setion 3.3 and Setion 7.2 is the existene of a long unip otent element in its image. Denition 2.1.2 We say that a representation ρ:G−→ GL(V) for a group G and an n -dimensional vetor spae V over a eld K has a long unip otent element , if there exists an element g∈G suh that the Jordan normal form of ρ(g) is Jn(1) over K , where Jn(1) denotes a Jordan blok of length n to the eigenvalue 1 . For a go o d introdution to the onepts of algebrai numb er theory, have a lo ok at [Neu99℄. If K is a numb er eld, i.e. a nite extension of Q , then ΣK denotes the set of nite plaes , whih is the set of normalized non-arhimedean valuations of K . We identify ΣK\{0} with the set of non-trivial prime ideals of OK . For v∈ΣK\{0} we have two elds: the nite eld kv:= OK/v of harateristi pv and the ompletion via the indued metri Kv:= Quot(lim ←−(OK/vn)) , as eah plae orresp onds to a normalized disrete valuation. The adele ring AK of K is dened as AK:= Y v|∞ Kv |{z } =:AK,∞ ×Y′ v∈ΣK\{0} Kv, where AK,∞ is the produt of the ompletions of K aording to the valuation given by the Arhimedean plaes and Q′ is the restrited pro dut, i.e. almost all entries are in the rings of integers OKv . The idele group IK is the group of units A× K of the adele ring. For a nite Galois extension L/K and w∈ΣL\ {0} suh that w|v , i.e. w⊇vOL , we obtain two anonial subgroups of the Galois group Gal(L/K) , the deomposition group Dw:= {σ∈Gal(L/K)|σw =w} and a normal subgroup of Dw the inertia group Iw:= {σ∈Dw|σ(x)−x∈w∀x∈ OL} . Fixing an embedding of K in Kv , we get a natural emb edding of GKv in GK , whih orresponds to ho osing a nite plae w in K extending v and therefore xing GKv as a sp ei deomp osition group Dw . Setting lw:= OL/w , we have a short exat sequene of nite groups 1−→ Iw−→ Dw−→ Gal(lw/kv)−→ 1. 20
2.1. Galois Representations For a nite plae w6= 0 of L there is a unique nite plae v of K , suh that w|v . The extension L/K is alled unramied at w if [L:K] = [lw:kv] . In this ase we have Iw= 1 . For a nite plae v6= 0 of K multiple nite plaes w of L may exist, suh that w|v . The eld extension L/K is alled unramied at v if [L:K] = [lw:kv] for eah of them (equivalently one of them, as we have a Galois extension). Otherwise the nite plaes are alled ramied and for eah suh extension there is only a nite numb er of them. For a general eld K and L an algebrai extension, L/K is unramied at a non-arhimedean valuation v of K , if for eah nite eld extension L′/K inside L/K and eah valuation w′ of L′ extending v , l′ w′|kv is separable and [L′:K] = [l′ w′:kv] , otherwise L/K is alled ramied at v . In the numb er eld ase, Gal(lw/kv) is a nite yli group generated by the Frob enius. If w∈ΣL\ {0} is unramied, we have Dw∼ =Gal(lw/kv) and we an talk of a Frob enius element in the deomp osition group as well. For v∈ΣK\ {0} unramied and w, w′∈ΣL\ {0} suh that w, w′|v , there is an element σ∈Gal(L/K) mapping one to the other, i.e. σw =w′ . Therefore the orresponding deomposition groups are onjugated , i.e. σDwσ−1=Dw′ , as well as the Frob enius elements. The other way around, for onjugates of Frob enius elements we have orresp onding plaes of L . If we generalize to an arbitrary algebrai Galois extension L/K , the set of nite plaes ΣL is the pro jetive limit of the system of nite plaes of the nite sub extensions of L/K . This is dened via the following onnetion morphisms: whenever we have a sub extension L/L1/L2/K , we map w1∈ΣL1 to w2∈ΣL2 , where w2 is the unique plae suh that w1|w2 . The inertia and deomp osition group an b e dened as pro jetive limits in the same way. Denition 2.1.3 For an ℓ -adi Galois representation ρ of a number eld K , we say that ρ is unramied at v∈ΣK\ {0} , if ρ(Iw) = 1 for any valuation w of Ksep extending v . Let ρ be unramied at v∈ΣK\ {0} , then the Frob enius element Frobv,ρ in the representation ρ at v is the onjugay lass in GL(V) of the images of the Frobenius element in Dw for any w∈ΣKsep \{0} extending v : 1//Iw// ρ|Iw Dw _ ////Gal(lw/kv)// 1 GK ρ 1//GL(V) . As the Frob enius element Frobv,ρ is a onjugay lass, its harateristi p olynomial fv,ρ(x) := det(1 ·x−Frobv,ρ)∈Qℓ[x] 21
2. Preliminary Results is well-dened. An ℓ -adi Galois representation is rational (resp etively integral ) if at almost all nite plaes v it is unramied, i.e. fv,ρ(x) exists, and the harateristi p olynomial has rational (resp etively integral) o eients. We will keep to the language of Rihard Taylor (f. [BLGGT10℄), for systems of ℓ -adi Galois representations. Denition 2.1.4 a) Let ℓ, ℓ′ be prime numbers. A rational ℓ -adi Galois representation ρ and a rational ℓ′ -adi Galois representation ρ′ of the same number eld K are ompatible at v∈ΣK\ {0} if they are both unramied at v and the harateristi polynomials fv,ρ(x) = fv,ρ′(x)∈Q[x] oinide. b) A weakly ompatible system (ρℓ)ℓ prime of Galois representations of a number eld K onsists of a family of rational, semi-simple ℓ -adi Galois representations ρℓ of K for eah prime number ℓ and a nite set S⊂ΣK , suh that the fol lowing holds: 1. For v∈ΣK\S and prime numbers ℓ, ℓ′ unequal to the harateristi of kv , the representations ρℓ, ρℓ′ are ompatible at v . 2. For v∈ΣK and ℓ equal to the harateristi pv of kv , the representation ρℓ is deRham in v and rystal line in v if v6∈ S (f. Denition 2.4.8 ). 3. For eah embedding τ:K//Q the τ -Hodge-Tate numbers of ρℓ are independent of ℓ (f. Denition 2.4.9 ). ) A weakly ompatible system (ρℓ)ℓ prime is al led irreduible if there is a set P of prime numbers of Dirihlet density 1 , i.e. lim s→1+ |log(s−1)|−1X ℓ∈P ℓ−s= 1, suh that for al l ℓ∈P the representation ρℓ is irreduible. It is also p ossible to extend this denition by ho osing a numb er eld M instead of Q . In this ase the family is indexed by the set of nite plaes of M and harateristi p olynomials in the ring M[x] are allowed. As this is not neessary for this work, we omit this and refer to the more general [BLGGT10℄, Denition 1.1. If ρ= (ρℓ)ℓ prime is a weakly ompatible system and S⊂ΣK the nite exeptional set. For a nite plaes v∈ΣK\S the harateristi p olynomials fv,ρℓ(x) of the Frob enius elements oinide in Q[x] for almost all ℓ , whih will b e alled fv,ρ(x) . This will b e the key ingredient in Setion 7.1 to dene an L -funtion for a sp eial kind of weakly ompatible systems of ℓ -adi Galois representations. 22
2.2. Étale Fundamental Group Funtor π´et 1 2.2 Étale Fundamental Group Funtor π´et 1 This introdution to the étale fundamental funtor π´et 1 from the ategory of No etherian separated onneted shemes to the ategory of groups is as in the rst hapter of [FK88℄. Denition 2.2.1 a) A ring homomorphism f:A−→ B of loal ommutative rings with unit is unramied , if f(mA)·B=mB and the indued eld extension A/mA−→ B/mB is nite and separable. b) Let X,Y be Noetherian separated shemes. The morphism f:Y −→ X is étale , if the fol lowing onditions are satised: 1. f is loal ly of nite type. 2. for every point x∈ X the morphism f♯ x:OY,f(x)−→ OX,x is at, unramied and makes OX,x a nitely generated OY,f(x) -algebra. For a No etherian separated sheme X , we all a No etherian separated sheme Y with an étale morphism X −→ Y an étale extension of X . We denote the full sub ategory of étale extensions of X in the ategory Sch(X) of shemes over X by Ét (X) (then every morphism in Ét (X) is étale, f. [FK88℄, Remark 2.2.). A morphism of Noetherian separated shemes is a overing if it is nite and étale. Again the full sub ategory Cov(X) of overings over X in Ét (X) has only morphisms whih are overings. This is b eause an étale morphism is nite, if and only if it is prop er (see page 282 of [FK88℄ and [Har06℄, Corollary 4.8 (e) ). If we x a geometri p oint s: spec(Ω) −→ X ( Ω separably losed), we get the asso iated funtor of geometri p oints over s Cov(X)−→ Sets,Y 7→ Y(s) := HomX(spec(Ω),Y). A pointed overing of (X, s) is a pair (Y, α) onsisting of Y ∈ Ob (Cov(X)) and an α∈ Y(s) . These form the ategory Cov(X, s) together with the mapping of p ointed overing spaes f: (Y1, α1)−→ (Y2, α2) whih is an X -morphism f:Y1−→ Y2 satisfying f◦α1=α2 . For a onneted Y ∈ Ob (Cov(X)) , we have |AutX(Y)| ≤ |Y(s)|, as there is at most one morphism from a p ointed overing sheme to a onneted p ointed sheme (see [FK88℄, (1)). Now we will have a lo ok at the ase when there exists exatly one morphism. 23
2. Preliminary Results Denition 2.2.2 For a Noetherian separated sheme X and a geometri point s of X a Galois overing is a onneted overing sheme Y over X if |AutX(Y)|=|Y(s)|. This leads to the full sub ategory Gal(X, s) of Galois overings in Cov(X, s) . Sine b etween two ob jets there is at most one morphism, we obtain a total ordering on the isomorphism lasses. Furthermore we get that the ob jets form an inverse system. For an X -morphism f:Z −→ Y b etween two Galois overings and σ∈AutX(Z) , there is exatly one σ′∈AutX(Y) suh that f◦σ=σ′◦f . This mapping denes a surjetive group homomorphism (see [FK88℄, (4)) and hene an inverse system of groups. Denition 2.2.3 For a Noetherian separated sheme X and a geometri point s of X , we dene the étale fundamental group (a pronite group) as the fol lowing inverse limit of nite groups with the disrete topology: π´et 1(X, s) := lim ←− (Y,α)∈ Ob (Gal(X,s)) AutX(Y). Then π´et 1 b eomes a ovariant funtor from the ategory of p ointed shemes to the ategory of pronite groups by onstruting suitable morphisms b etween the inverse systems out of a morphism of shemes (see [FK88℄, A1.3). The tame fundamental group is a fator group of the étale fundamental group. This group will b e of imp ortane b eause ontinuous representations of πtame 1(X, s) give nie ontinuous representations of π´et 1(X, s) . Denition 2.2.4 The tame fundamental group πtame 1(X, s) of a Noetherian separated sheme X and a geometri point s of X is the projetive limit of al l pointed Galois overings whih are tamely ramied i.e. for eah geometri point α: spec(Ω) −→ Y the ardinality |{σ∈AutX(Y)|σ◦α=α}| is invertible in OY,α . If X is onneted and s′ is another geometri p oint of X , the étale fundamental groups are isomorphi: π´et 1(X, s)∼ =π´et 1(X, s′) (see [FK88℄, A1.2). In this ase we write π´et 1(X) := π´et 1(X, s) and view it as a funtor from the ategory of onneted shemes to the ategory of isomorphism lasses of pronite groups. This is valid as well for the tame fundamental group. We dene the tame inertia group Itame K:= πtame 1(Gm,K) . 24
2.2. Étale Fundamental Group Funtor π´et 1 If we x a Noetherian separated onneted sheme and a geometri p oint s of X together with a overing Y , there is a natural ontinuous π´et 1(X, s) -ation on Y(s) . We will now explain this ation in more detail. By [FK88℄, (2) and (3) there is a Galois overing Z of X dominating Y , i.e. suh that there is an X -morphism Y −→ Z . Cho osing an α∈ Z(s) , we get a p ointed Galois overing (Z, α) of (X, s) and a natural bijetion HomX(Z,Y)−→ Y(s), f 7→ f◦α. Therefore the anonial right ation of AutX(Z) on HomX(Z,Y) yields a right ation on Y(s) . As we have a disrete group, this ation is ontinuous and an b e extended to a ontinuous right ation of π´et 1(X) on Y(s) via the anonial pro jetion π´et 1(X)////π´et 1(X)/π´et 1(Z, α) . For a dierent hoie of α we obtain a dierent ation, but this transformation is the same as a onjugation in π´et 1(X, s) . Furthermore it is indep endent of the hoie of Z , as for two hoies there is a third dominating them. Prop osition 2.2.5 Let X be a Noetherian separated onneted sheme. The assignment Y 7→ Y(s) establishes an equivalene between the ategory of overing spaes of X and the ategory of nite ontinuous π´et 1(X) -sets ([FK88℄, A I.5 ). In order to generalize the onept of sheaves the following denition was given by Artin in [Art62℄, Denition 1.1.1. Denition 2.2.6 A Grothendiek top ology onsists of a ategory T and a set Cov T of families {Ui φi −→U}i∈I of maps in T al led overings (where in eah overing the range U of the maps φi is xed) satisfying a) if φ is an isomorphism then {φ} ∈ Cov T ; b) if {Ui→U}i∈I∈Cov T and {Vij →Ui}j∈Ji∈Cov T for eah i then the family {Vij →U}i∈I,j∈J obtained by omposition is in Cov T ; ) if {Ui→U}i∈I∈Cov T and V→U∈ Mor (T) is arbitrary then Ui×UV exists and {Ui×UV→V}i∈I∈Cov T . As Ét (X) fullls all these prop erties for a Noetherian separated sheme X this yields an example of a Grothendiek top ology. 25
2. Preliminary Results Theorem 2.3.2 Let X be a smooth and projetive sheme over the nite eld Fq and X:= X ×spec Fqspec Fq . a) The polynomials Pj(t) = det(1 −t·Frob⋆|Hj(X,Qℓ))∈Qℓ[t] have rational integer oeients. These are independent of ℓ . b) The eigenvalues λ of Frob⋆|Hj(X,Qℓ) , and thus the reiproal roots of Pj(t) , al l have the omplex absolute value |λ|=qj 2. ) There is a funtional equation for ZX(t) = 2 dim X Q j=0 Pj(t)(−1)j+1 , namely ZX1 qjt=ǫ·qj 2χ(X)·tχ(X)·ZX(t). Here χ(X) = 2 dim X P j=0 (−1)jdim Hj(X,Qℓ) is the Euler harateristi of X and ǫ=(1 2 6 | j, (−1)N2|j, where N is the multipliity of the eigenvalue qj 2 of Frob⋆|Hj(X,Qℓ) . Let X b e a nitely generated No etherian separated sheme over the eld Fq and G a onstrutible sheaf of Qℓ -vetor spaes on X . For a geometri p oint α: spec(Fq)−→ X the residue eld κ(α) is nite and beause of that it denes an element in Gal(Fq/κ(α)) , the Frob enius fα:x7→ x|κ(x)| . This element ats on the stalk Gα of G and on the stalk Gα of the sheaf G=G ⊗Fq . Denition 2.3.3 a) We al l the sheaf G puntually pure of weight j if for al l suh geometri points α of X the eigenvalues of f−1 α:Gα−→ Gα are algebrai numbers whose omplex onjugates λ have omplex absolute value |λ|=qj 2d(α), d(α) = [κ(α) : Fq] . b) The sheaf G is al led mixed of weight less or equal to j if G has a ltration 0 = F(0) ⊂ F(1) ⊂...⊂ F(r)=G for whih al l fator sheaves F(ν)/F(ν−1) are puntual ly pure of weight less or equal to j . The following statement is [Del80℄, Théorème 3.3.1. Theorem 2.3.4 Let f:X −→ Y be a morphism of nitely generated shemes over Fq , and let G be a mixed sheaf of weight less or equal j on X . Then the diret image sheaves with ompat support Rnf!G are mixed of weight less or equal j+n . 32
2.4. Crystalline Representations 2.4 Crystalline Representations In order to present the onept of rystalline representations, it is neessary to dene the graded rings BdR, Bcris, Bst and BHT intro dued by Fontaine [Fon02℄ with their natural GKv -ation for a lo al eld Kv . The rst step is to study Witt vetors. 2.4.1 Witt vetors In the artile [Wit37℄, whih was published in 1937, Witt generalized the onstrution of Zℓ out of Fℓ for a given prime numb er ℓ to general ommutative rings. For a xed prime numb er ℓ and n∈N0 we dene the n -th Witt polynomial wn:= n X j=0 ℓjXℓn−j j∈Z[X0,...,Xn]. Denition and remark 2.4.1 Let ℓ be a prime number and A a ommutative ring. Then the fol lowing holds: a) For eah n∈N0 , there exist polynomials sn, mn∈Z[Y0,...,Yn, Z0,...,Zn] , suh that wn(s0,...,sn) = wn(Y0,...,Yn) + wn(Z0,...,Zn) and wn(m0,...,mn) = wn(Y0,...,Yn)·wn(Z0,...,Zn). b) The fol lowing onvention denes a ring struture on AN0 , whih is al led the ring of Witt vetors W(A) (an)n∈N0+ (bn)n∈N0:= ( sn(a0,...,an, b0,...,bn) )n∈N0, (an)n∈N0·(bn)n∈N0:= ( mn(a0,...,an, b0,...,bn) )n∈N0. ) If A is a perfet eld of harateristi ℓ , then W(A) is a omplete disrete valuation ring and its residue eld is A . For n∈N this struture an b e restrited to An by pro jetion on the rst n terms, the Witt vetors of length n . The ring W should b e viewed as the unique ovariant funtor from the ategory of rings to itself, for whih the following map is a homomorphism: W(A)−→ AN0 (an)n∈N07→ (wn(a0,...,an) )n∈N0 . 33
2. Preliminary Results For a more detailed version of the following approah, we refer to [FO08℄. Let Kv b e a lo al eld, that is a omplete disrete valuation eld, whose residue eld kv is p erfet of harateristi ℓ > 0 . The most imp ortant ase is as in Setion 2.1, where K is a numb er eld, v∈ΣK\{0} a plae of K and Kv the ompletion at v . Then the valuation v on Kv an b e uniquely extended to Kv as onsequene of Chevalley's Extension Theorem (f. [EP05℄). Kv might not b e omplete but by Krasner's lemma its ompletion Cv:= ˆ Kv is algebraially losed. For the rings of integers OKv:= x∈Kv|v(x)≥0 and OCv:= {x∈Cv|v(x)≥0} we get ompat rings with the following isomorphi fator rings of harateristi ℓ OKv/ℓOKv∼ =OCv/ℓOCv. Now we sp eialize A as the pro jetive limit of OCv/ℓOCv Frobℓ ←− OCv/ℓOCv Frobℓ ←− OCv/ℓOCv Frobℓ ←− ... i.e. A=n(a(n))n∈N0∈(OCv/ℓOCv)N0(a(n))ℓ=a(n−1) ∀n∈No and get a p erfet ring of harateristi ℓ . The ring A arries a anonial valuation indued by v , whih will not b e disussed in detail here, and is a omplete valuation ring with resp et to it. The ation of GKv on Kv an b e extended ontinuously to Cv and restrits to a lo al ation on OCv . Therefore A is endowed with a natural struture as GKv -mo dule, whih ommutes with the Frob enius on A and gives nally an ation of GKv on W(A) . For a= (a(n))n∈N0∈A we dene ea:= lim n→∞(g a(n)ℓn )∈ OCv , where g a(n)∈ OCv is some lift of a(n)∈ OCv/ℓOCv . It is easy to hek that the limit exists and that it is independent of the hoies. This yields a map θ:W(A)−→ OCv (an)n∈N07→ ∞ P n=0 ℓnfan whih is an epimorphism of GKv -mo dules. This result is obtained by restriting θ to the rst n omp onents of W(A) and then using the pro jetive limit pro ess. For n∈N0 , we suessively hoose a(n)∈ OCv , suh that we get a ompatible system of ℓn+1 -ro ots of ℓ , i.e. a(n) is a zero of Xℓn+1 −ℓ and a(n)= (a(n+1))ℓ . By pro jetion this denes a series of non-zero elements ℓ(n)∈ OCv/ℓOCv and therefore an element ℓ= (ℓ(n))n∈N0∈A for whih eℓ=ℓ . Then the kernel of θ is a prinipal ideal generated by ξ:= (−ℓ, 1,0,...)∈W(A) , whih denes a ξ -adi top ology on W(A) and W(A)[(0,1,0,...)−1] . 34
2.4. Crystalline Representations 2.4.2 The GKv -mo dule BdR : The ompletion of W(A)[(0,1,0,...)−1] in the ξ -adi top ology is the disrete valuation ring B+ dR with maximal ideal (ξ) and residue eld B+ dR/(ξ)∼ =Cv . Denition 2.4.2 The eld B+ dR is dened as the eld of frations of B+ dR : BdR := Quot(B+ dR) = Quot lim ←− n W(A)[(0,1,0,...)−1] ξn!. It has a natural dereasing ltration FilmBdR =ξmB+ dR for m∈Z . 2.4.3 The GKv -mo dule Bcris : We dene Acris to b e the ℓ -adi ompletion of the divided p ower envelop e of W(A) with resp et to (ξ) , i.e.: Acris := (∞ X n=0 wn ξn n!wn∈W(A), wn−→ 0 for n−→ ∞)⊂B+ dR. This is the same as taking the pro jetive limit lim ←− n A0 cris/(0,1,0,...)nA0 cris ∼ =Acris where A0 cris := (N X n=0 wn ξn n!N∈N0, wn∈W(A))⊂W(A)[(0,1,0,...)−1]. In this way we obtain the following subring B+ cris := Acris[(0,1,0,...)−1]⊂B+ dR . Again we ho ose suessively a(n)∈ OCv , suh that we get a non-trivial ompatible system of ℓn -th ro ots of unity, i.e. a(n) is a zero of Xℓn−1 , a(1) 6= 1 and a(n)= (a(n+1))ℓ , whih yields an element ε∈A . As (ε−1,0,...)∈ Ker (θ) = (ξ) we have log((ε, 0,...)) := ∞ X n=1 (−1)n+1 (ε−1,0,...)n n∈B+ dR. Denition 2.4.3 By loalizing and taking the subspae grading by BdR , we dene the graded ring Bcris := B+ cris[log((ε, 0,...))−1] = Acris[log((ε, 0,...))−1]⊂BdR. 35
2. Preliminary Results 2.4.4 The GKv -mo dule Bst : We set log((−ℓ, 0,...)) := − ∞ P n=1 ξn n·(0,1,0,...)n∈B+ dR (for a omplete approah to the logarithm see [FO08℄, Setion 6.1.3). This element is transendental over Quot(Bcris) . Denition 2.4.4 The ring Bst is dened as the Bcris -subalgebra of BdR generated by t:= log((−ℓ, 0,...)) : Bst := Bcris[t] = Bcris[log((−ℓ, 0,...))] = Bcris "− ∞ X n=1 ξn n·(0,1,0,...)n#. Remark 2.4.5 We have Bcris ⊆Bst ⊆BdR , whih gives that Bcris and Bst are domains. Eah ring is stable under the ation of GKv on BdR , whih is obtained by onsidering the projetive limit of the ations on W(A)[(0,1,0,...)−1] ξn . As A is of harateristi ℓ we have Fℓ//A , and by the funtoriality of W we get that W(Fℓ) = Zℓ//W(A) , whih leads to Qℓ⊆Bcris ⊆Bst ⊆BdR. 2.4.5 The GKv -mo dule BHT : Now we hoose an emb edding τ:K//Q , whih is the same as a ontinuous inlusion Kv//Qℓ=Kv . This yields a Qℓ -vetor spae struture on Kv and therefore on Cv by ontinuous extension. Q_ oO //Qℓ _ nN ~~| | | | | | | | K o τ > > > > > > > > //Kvp A A A A Q//Qℓ The element t has been hosen in suh a way that for an element g∈GKv , we have g·t=χℓ(g)t . Here χℓ:GKv−→ Q× ℓ⊂Bcris is the ylotomi harater, whih is dened as in 2.1 by the op eration on the ℓn -th ro ots of unity. The element t generates the maximal ideal of B+ dR and therefore the grading on BdR . As a eld BdR is isomorphi to Cv((t)) (the isomorphism dep ends on the hoie of ε ). 36
2.4. Crystalline Representations For a j∈Z the j -th Tate Twist Cv(j) of Cv is Cv viewed as GKv -mo dule twisted by the j -th p ower of the ylotomi harater, so that g·c=χj ℓ(g)g(c) for g∈GKv and c∈Cv . This is imp ortant b eause it is isomorphi as GKv -mo dule to the i -th graded omp onent of BdR : grjBdR = FiljBdR/Filj+1BdR =ξjB+ dR/ξj+1B+ dR =tjB+ dR/tj+1B+ dR ∼ =Cv(j). Denition 2.4.6 The Ho dge-Tate ring BHT is dened as the diret sum of al l Tate twists of Cv , whih is BHT =Cv[t, t−1] = M j∈Z grjBdR and the GKv -ation of BdR restrits in the fol lowing way g·X j∈Z cjtj=X j∈Z χℓ(g)jg(cj)tj for g∈GKv and cj∈Cv unequal to 0 only for a nite number of j∈Z . The residue eld kv lies inside OKv/ℓOKv and is xed by the GKv -ation. This denes the eld K0:= Quot(W(kv)) , whih is the xed eld of Bcris and Bst with resp et to the GKv -ation K0=BGKv cris =BGKv st ⊆Kv=BGKv dR =BGKv HT . Then Kv is a totally ramied extension of K0 and b oth elds oinide if K is unramied in v . Denition 2.4.7 For an ℓ -adi Galois representation ρ:GKv−→ GL(V) of Kv , like in Denition 2.1.1 , we dene Qℓ -vetor spaes, the ltered Dieudonné mo dules Dcris(V) := (Bcris ⊗V)GKv, Dst(V) := (Bst ⊗V)GKv, DdR(V) := (BdR ⊗V)GKv and DHT(V) := (BHT ⊗V)GKv as invariants of the tensor produts under the ation of the absolute Galois group. The rst two are free K0⊗Qℓ -mo dules and the seond ones are free Kv⊗Qℓ -mo dules. 37
2. Preliminary Results Then we have the following natural inequalities rank K0⊗QℓDcris(V)≤ rank K0⊗QℓDst(V)≤ rank Kv⊗QℓDdR(V) ≤ rank Kv⊗QℓDHT(V)≤dimQℓV. Denition 2.4.8 Let ρ be an ℓ -adi Galois representation of K on V , v∈ΣK\{0} suh that ℓ=pv= har (kv) . Then we get an ℓ -adi Galois representation ρ|GKv:GKv−→ GL(V) by xing an embedding K//Kv . ρ is al led Ho dge-Tate at v , if rank Kv⊗QℓDHT(V) = dimQℓV , deRham at v , if rank Kv⊗QℓDdR(V) = dimQℓV , semi-stable at v , if rank K0⊗QℓDst(V) = dimQℓV , rystalline at v , if rank K0⊗QℓDcris(V) = dimQℓV , and we have: ρ rystal line at v⇒ρ semi-stable at v⇒ρ deRham at v⇒ρ Hodge-Tate at v. The grading on DHT(V) is given by the degree in t , so that eah graded piee gr−jDHT(V) = (Cv(−j)⊗V)GKv an b e haraterized by the ylotomi harater ating on it to the j -th p ower. The dimensions of these spaes play an imp ortant role in the lassiation of representations. Denition 2.4.9 Let ρ= (ρℓ)ℓ prime be a system of Galois representations of a number eld K , where ρℓ:GK−→ GL(Vℓ) where Vℓ is a Qℓ -vetor spae. For v∈ΣK\{0} , ℓ=pv= har (kv) and a xed embedding K//Kv a GKv -ation on Vℓ and as before by τ:K//Q a GKv -ation on Cv is given. The τ -Ho dge-Tate numb ers hv,j(ρ)∈N0 for an integer j are dened as hv,j(ρ) = dimQℓ(Cv(−j)⊗Vℓ)GKv. 38
2.5. Semi-Simpliation 2.5 Semi-Simpliation The semi-simpliation of a representation ρ:G−→ GL(V) , where V is a nite-dimensional L -vetor spae, is done by viewing V as an L[G] -mo dule. Then there is a nite, stritly dereasing hain of submo dules V=V0⊃V1⊃...⊃Vo={0} suh that Vj/Vj+1 is a simple L[G] -mo dule. The fators are alled Jordan-Hölder fators and are uniquely dened up to p ermutation. This denes a unique semi-simple L[G] -mo dule o−1 L j=0 Vj/Vj+1 and therefore a semi-simple representation ρss :G−→ o−1 M j=0 Vj/Vj+1. The following part is out of [Wor02℄, 2.1. Statement of the prop osition . Let K b e a numb er eld, v∈ΣK\ {0} a nite plae of K and p= har (kv) . For m∈N oprime to p we get a natural emb edding of the m -th ro ots of unity in the maximal unramied extension Knr v of Kv in kv∼ =Fp : µm={ζ∈Knr v|ζm= 1}// s && M M M M M M M M M M M M MOKv ⊂Kv kv∼ =Fp . Let π b e a uniformizer of Knr v , then Knr v(π1 m) is a totally ramied extension of Knr v of degree m and we get a pro jetive system of maps e Ψm: Gal(Knr v(π1 m)/Knr v)−→ µm with (e Ψm(σ))(π1 m) = σ(π1 m) for σ∈Gal(Knr v(π1 m)/Knr v) . Its pro jetive limit is Ψ = lim ←− p∤me Ψm:Itame Kv= lim ←− p∤m Gal(Knr v(π1 m)/Knr v)−→ lim ←− p∤m µm As in [Wor02℄ on page 4 we dene its natural pro jetions Ψq−1:Itame Kv−→ µq−1⊆Knr v for a p -p ower q , whih dene Ψ uniquely. 39
2. Preliminary Results Let {m1,...,mr} b e the set of indies where the ltration of Dcris(V) jumps, i.e. grmDcris(V)6= 0 , and with dj:= rank K0⊗QℓgrmjDcris(V)∈N the multipliity of mj , i.e. the rank of the asso iated quotient. Prop osition 2.5.1 ( [Wor02℄, Prop.3 ) Assume that for v∈ΣK\{0} , Kv is absolutely unramied, i.e. Kv/Q is unramied, and let w be the unique extension of v to Kv . If the ℓ -adi Galois representation ρ:GKv−→ GL(V) fators through Zℓ , is rystal line and if the length of the ltration on Dcris(V) is less than ℓ , then the fol lowing holds: a) The semi-simpliation of the modℓ redued Iw -module V , via ρ|Iw:Iw−→ GL(V) , is wel l-dened and the ation of Iw fators through the tame quotient Itame Kv . b) For a simple subquotient W of the Iw -module V of dimension d one has EndFℓ(W)∼ =Fℓd . Fixing an isomorphism gives W the struture of a one dimensional Fℓd -vetor spae on whih Itame Kv ats via multipliation with Ψi0+...+id−1ℓd−1 ℓd−1 , where the indies −ij run through {m1,...,ms} suh that eah omponent of (m1,...,ms) (ounted with multipliities) appears as some index −ij for some subquotient. Corollary 2.5.2 ( [Wor02℄, Cor.4 ) The automorphisms dened by g∈Itame Kv , where the tame fundamental group is viewed as a subgroup (not unique) of the inertia group Iw , satisfy detFℓρ(g) = Ψℓ−1(g)−S, where S:= s P k=1 dkmk . 40
3 Intro dution to MCχ In Setion 2.2, we saw that it is p ossible to onstrut Galois representations by onstruting lisse Qℓ -sheaves. In order to do so, we use the middle onvolution as a geometri op eration. We will follow Chapter 2 in [Kat96℄. 3.1 The Middle Convolution Let K b e a eld and ℓ a prime numb er unequal to the harateristi of K . We x an algebrai group A over K with multipliation map µ:A×A−→ A . We denote by Db c(A,Qℓ) the bounded derived ategory of onstrutible Qℓ -sheaves on A , whih is onstruted by taking the ategory of b ounded hain omplexes and lo alizing the quasi-isomorphisms (for more details see [Kat96℄, Setion 2.2). For two ob jets F,G ∈ Db c(A,Qℓ) , we have the exterior tensor pro dut F⊠G:= pr ∗ 1F ⊗ pr ∗ 2G ∈ Db c(A ×A,Qℓ), and the shift by m∈Z to the left F[m] . As usual the onstrutible Qℓ -sheaves are emb edded as degree 0 ob jets. Denition 3.1.1 The ! -onvolution of F,G is dened as the right derivative of the diret image with ompat support by the multipliation map µ F ∗!G:= Rµ!(F⊠G)∈Db c(A,Qℓ) and their ∗ -onvolution as the right derivative of the diret image by the multipliation map µ F ∗∗G:= Rµ∗(F⊠G)∈Db c(A,Qℓ) of the exterior tensor produt. For eah onstrutible sheaf X on A , the supp ort supp(X) is the losure of the set {a∈A| Xa6= 0} and therefore a variety of some dimension. An element F ∈ Db c(A,Qℓ) is alled a perverse sheaf , 41
3. Introdution to MCχ ohomologially rigid, and dene indutively Hm+1,ℓ := (MTi∗L(1,−1)(MC−1(Hm,ℓ)) 2 |m, MTi∗L(−1,1)(MC−1(Hm,ℓ)) 2 ∤m∈Tℓ(K). By Theorem 3.1.7, we have that Hm,ℓ is ohomologially rigid. That Hm,ℓ resp ets an orthogonal resp etively sympleti form is a onsequene of Poinaré duality (see [DR99℄, Corollary 5.10). The rest of the pro of is an indution on m with the help of Prop osition 3.2.1. For m= 0 the mono dromy tuple of i∗Hm,ℓ is (J1(−1),J1(−1),J1(1)) ∈GL1(Qℓ)3 , whih is of the given form. So we start with Hm,ℓ whose loal mono dromy is of the predited type. Therefore we an determine the rank of Hm+1,ℓ : rk(Hm+1,ℓ) = rk(MC−1(Hm,ℓ)) = 2(m+ 1) −e1(0,−1,Hm,ℓ)−e1(1,1,Hm,ℓ)−e1(∞,−1,Hm,ℓ) = 2(m+ 1) −m 2−m 2−0m≡0 mod 4 2(m+ 1) −m+1 2−m−1 2−0m≡1 mod 4 2(m+ 1) −m 2−m 2−0m≡2 mod 4 2(m+ 1) −m+1 2−m−1 2−0m≡3 mod 4 =m+ 2 For the alulation of the loal mono dromy at 0 , we have two ases. If m is even, the lo al mono dromy of Hm,ℓ is of the form J1(1)m 2⊕J1(−1)m 2+1 : J1(1)m 2⊕J1(−1)m 2+1 MC−1 0⊕J2(1)m 2+1 MTi∗L(1,−1) J2(1)(m+1)+1 2 If m is o dd, then J2(1)m+1 2MC−1 J1(−1)m+1 2⊕J1(1)m+3 2 MTi∗L(−1,1) J1(1)m+1 2⊕J1(−1)m+1 2+1. At 1 we start with the ase m≡0 mod 4 and therefore we get: J2(1)m 2⊕J1(−1) MC−1 J1(−1)m 2⊕J2(1)⊕J1(1)m 2 MTi∗L(1,−1) J1(1)(m+1)−1 2⊕J2(−1)⊕J1(−1)(m+1)−1 2 for m≡1 mod 4 : J1(1)m−1 2⊕J2(−1) ⊕J1(−1)m−1 2MC−1 0⊕J3(1) ⊕J2(1)m−1 2 MTi∗L(−1,1) J3(1) ⊕J2(1)m+1 2−1 for m≡2 mod 4 : J3(1)⊕J2(1)m 2−1MC−1 J2(−1)⊕J1(−1)m 2−1⊕J1(1)m 2+1 MTi∗L(1,−1) J2(1)⊕J1(1)(m+1)−3 2⊕J1(−1)(m+1)+1 2 for m≡3 mod 4 : J2(1) ⊕J1(1)m−3 2⊕J1(−1)m+1 2MC−1 J1(−1) ⊕0⊕J2(1)m+1 2 MTi∗L(−1,1) J1(−1) ⊕J2(1)m+1 2. 48
3.3. Constrution of Hm,ℓ The Jordan normal form of the lo al mono dromy of i∗Hm,ℓ at ∞ is of the form Jm+1(1) . Then we get by the previous remark that the lo al mono dromy of MC−1(Hm,ℓ) is Jm+2(1) . For L(−1,1) and L(1,−1) the lo al mono dromy at ∞ is −1 . Altogether we have: Jm+1(1) MC−1 Jm+2(−1) MTi∗L(1,−1) MTi∗L(−1,1) Jm+2(1). Corresp onding to the onstruted lisse Qℓ -sheaf i∗Hm,ℓ on A1 K\ {0,1} , we have a ontinuous representation ρi∗Hm,ℓ :π´et 1(A1 K\ {0,1})−→ GL((i∗Hm,ℓ)x) for x∈A1 K\ {0,1} (see Corollary 2.2.12). This map an b e tensored by the following ontinuous one dimensional representation det(ρi∗Hm,ℓ ) : π´et 1(A1 K\{0,1})−→ {±1} ⊂ Qℓ ×, γ 7→ det(ρi∗Hm,ℓ (γ)), whih is in our ase det(ρi∗Hm,ℓ (γ0)) = det(ρi∗Hm,ℓ (γ1)) = (1m6≡ 0 mod 4 −1m≡0 mod 4 . We get a representation ρi∗Hm,ℓ ⊗det(ρi∗Hm,ℓ ) : π´et 1(A1 K\{0,1})−→ SL((i∗Hm,ℓ)x), whih fators through SO((i∗Hm,ℓ)x) for even m and through Sp((i∗Hm,ℓ)x) for o dd m . Using the Corollary again, we get the Qℓ -sheaf ] Hm,ℓ := Vρi∗Hm,ℓ ⊗det(ρi∗Hm,ℓ ) on A1 K\{0,1} . This will lead us to the wanted Galois representation in Setion 7.2. Theorem 3.3.2 For m∈N0 and a prime number ℓ , the Zariski losure of the monodromy group of ] Hm,ℓ is as fol lows: a) Sp(W) for m odd, b) G2(W) for m= 6 , ) SO(W) for m even and m6= 6 . 49
3. Introdution to MCχ Proof: Sine Hm,ℓ ∈Tℓ(K) , the Qℓ -sheaf i∗] Hm,ℓ is irreduible on A1 K\{0,1} . Therefore we have that H=ρi∗Hm,ℓ ⊗det(ρi∗Hm,ℓ )(γ0), ρi∗Hm,ℓ ⊗det(ρi∗Hm,ℓ )(γ1) is irreduible and onneted by the presene of the long unip otent element. By the onsiderations ab ove the group H leaves a sympleti form invariant if m is o dd, i.e. H≤Sp(W) , and an orthogonal form if m is even, i.e. H≤SO(W) . Moreover the presene of the long unip otent element, given by the mono dromy at ∞ , implies that H is tensor indeomp osable and by [SS97℄, Theorem B, we have the following p ossibilities for maximal losed redutive subgroups of Sp(W) resp etively SO(W) ontaining H : (a) A1<Sp(W) resp etively SO(W) (for p= 0 or p > h ), (b) SO(W).2<Sp(W) (for p= 2 ), () G2<SO7 (resp etively Sp6 if p= 2 ), (d) A2.2<Sp8 (for p= 2 ), (e) B3<SO8 . Here p= har (Qℓ) = 0 , whih ontradits ase (b) and (d). The ase (e) is not p ossible b eause for m= 7 a sympleti form is resp eted. For m= 6 the laim was proved in [DR10℄, Theorem 1. For m6= 6 the presene of the unip otent mono dromy element at 1 rules out the ase H≤A1= PSL2(W) . This proves the laim. 50
4 Ho dge Strutures and Middle Convolution The following intro dution to Ho dge theory is taken out of [PS08℄. 4.1 Ho dge Strutures Let b e m∈Z and R⊆R a No etherian ring, suh that R⊗Q is a eld and VR a nitely generated R -mo dule. Denition 4.1.1 A (pure) R -Ho dge struture of weight m on VR is a diret sum deomposition VC:= VR⊗C=M p+q=m Vp,q with Vp,q =Vq,p omplex vetor spaes for eah p, q . The numbers hp,q(V) := dimCVp,q are al led the Ho dge numbers of the Hodge struture. A morphism of Ho dge strutures f:VR−→ WR is an R -linear map suh that its omplexiation fC=f⊗ id C preserves types, i.e. fC(Vp,q)⊆Wp,q . Dening a Ho dge struture of weight m on a nite dimensional omplex vetor spae V is the same as giving a Hodge ltration F• of V . That is a dereasing ltration of omplex vetor spaes, suh that Fp∩Fq={0} for p+q=m+ 1 . A Ho dge ltration is asso iated to a Ho dge struture by Fp:= M r≥p Vr,s and vie versa Vp,q := Fp∩Fq. The free rank one R -mo dules whih arry a Ho dge struture are all of even weight and up to isomorphism of the following form: Denition 4.1.2 A Ho dge struture of Tate , denoted by Z(n) , for n∈Z is the Z -module (2πi)nZ⊂C with sum deomposition Z(n)⊗C=V−n,−n , whih has therefore weight −2n . If we have an R -Hodge struture VR of weight m , the Tate twist VR(n) is an R -Hodge struture of weight m−2n . It has VR⊗(2πi)nZ as underlying R -module, while VR(n)p,q =Vp−n,q−n R . 51
4. Ho dge Strutures and Middle Convolution 4.2 Variations of Ho dge Struture Ho dge strutures o ur quite naturally on the lo al systems onstruted by middle onvolution. In this ase, one gets a whole sheaf of Ho dge strutures tting together, whih is alled a variation of Ho dge struture (VHS). Denition 4.2.1 Let X be a omplex manifold. A variation of R -Ho dge struture (VR,F•,∇) of weight m on X onsists of the fol lowing data: a loal system VR of nitely generated R -modules on X , a nite dereasing ltration F• of the holomorphi vetor bund le V:= VR⊗ OX by holomorphi subbund les (the Ho dge ltration ). These data should satisfy the fol lowing onditions: a) for eah x∈ X the ltration F• x of Vx∼ =VR,x ⊗C denes a Hodge struture of weight m on the nitely generated R -module VR,x , b) the onnetion ∇:V−→ V⊗Ω1 X , whose sheaf of horizontal setions is VC , satises the Griths' transversality ondition ∇(Fp)⊆ Fp−1⊗Ω1 X. A morphism of variations of Ho dge struture is a morphism of loal systems whih preserves types, i.e. agrees with the ltrations. We x a omplex manifold X , a base p oint x∈ X and a Ho dge struture V . For eah group homomorphism ρ:πtop 1(X, x)−→ Aut(V) , we get a lo ally onstant variation of Ho dge struture. This is the same metho d as in Corollary 2.2.10, gluing V as stalk in eah p oint and gluing the lo al Ho dge ltration. This prop erty haraterizes the lo al system obtained by representation preserving types. Therefore we get ∇(Fp)⊆ Fp⊗Ω1 X . By VX we denote the variation for the trivial representation. For a xed Ho dge struture V , the Weil operator C is the C -linear automorphism of V , suh that for all v∈Vp,q we have C(v) = ip−q·v . Denition 4.2.2 a) A p olarization of an R -Hodge struture VR of weight m is an R -valued bilinear form Q:VR⊗VR−→ R whih is (−1)m -symmetri and suh that 1. The orthogonal omplement of Fn is Fm−n+1 for al l n∈Z , 2. The hermitian form Q(C(·),·) : VC⊗VC−→ R on VC is positive-denite. 52
4.2. Variations of Ho dge Struture b) A p olarization of a variation of R -Hodge struture V of weight m on X is a morphism of variations Q:V⊗V−→ R(−m)X whih indues on eah bre a polarization of the orresponding R -Hodge struture of weight m . Theorem 4.2.3 Let X be a ompat Kähler manifold. Let Hp,q(X) be the spae of ohomology lasses whose harmoni representative is of type (p, q) . There is a diret sum deomposition Hm dR (X, C) := Hm dR (X)⊗C=M p+q=m Hp,q(X). Moreover Hp,q(X) = Hq,p(X) . If we denote the losed Kähler form of X with ω and the dimension of X with n , then the Ho dge-Riemann form on Hm dR (X, C) is the bilinear form Q(α, β) = (−1)m(m−1) 2ZX α∧β∧ωn−m, whih is a p olarization of the previously dened pure R -Ho dge struture on Hm dR (X, C) of weight m (f. [PS08℄, Theorem 1.33). The standard examples are geometri variations of Ho dge struture (see page 507-508 in [SZ85℄). Remark 4.2.4 a) Given a smooth, proper holomorphi mapping f:X−→ S with X as above a Kähler manifold. Then Rmf∗Q is the underlying system of a variation of Hodge struture of weight m , dened over Q , in whih F•(s) is the usual Hodge ltration of the ohomology of the bre Hm dR (Xs,C) . By adjusting the up-produt on ohomology by the use of the Kähler lass and its (at) primitive deomposition, one obtains a polarization over R for Rmf∗R in the geometri ase. If X is a family of algebrai varieties, then the polarization is in fat dened over Q . b) By [SZ85℄, Remark 3.3 , the polarized struture passes on to subvariations, kernel and images of funtorial morphism of ohomology. Given a variation of Hodge struture, it is in some ases possible to extend the struture to a punture of the underlying spae. This struture is not pure any more, but onsists of the sum of multiple Ho dge strutures of dierent weights, hene it is a mixed Ho dge struture (MHS). 53
4. Ho dge Strutures and Middle Convolution Denition 4.2.5 An R -mixed Ho dge struture on VR onsists of two ltrations: an inreasing ltration by rational vetor spaes on VR⊗Q , the weight ltration W• and a dereasing ltration F• by omplex vetor spaes on VC=VR⊗C , the Ho dge ltration . The Hodge ltration indues a pure K:= (R⊗Q) -Hodge struture of weight m on eah graded piee GrW m(VR⊗Q) = Wm/Wm−1 by Fp(GrW m(VR⊗Q)⊗C) = (Fp∩Wm⊗C+Wm−1⊗C)/(Wm−1⊗C). The Ho dge numb ers are the dimensions of the graded piees of this indued grading: hp,q(V) := dimCGrp F(GrW p+q(VR⊗Q)⊗C) = dimCFp(GrW p+q(VR⊗Q)⊗C)/Fp+1(GrW p+q(VR⊗Q)⊗C). For two nitely generated R -modules VR, VR ′ with R -mixed Hodge strutures a morphism f:VR−→ VR ′ (of weight 0 ) is an R -linear map, whih indues for m∈Z morphisms of Hodge strutures by GrW m(f) : GrW m(VR⊗Q)−→ GrW m(VR ′⊗Q). A graded p olarization on an R -mixed Hodge struture is a polarization of eah GrW m(VR⊗Q) , whih has a pure (R⊗Q) -Hodge struture. 4.3 Extensions of Variations of Ho dge Struture In order to get some information on the Ho dge struture onstruted by middle onvolution, whih will b e introdued in the next setion, it is helpful to have a lo ok at the limit struture at the singularities. This is p ossible as the struture an b e extended in some ases by the work of Shmid. We will summarize the results of [Sh73℄. Every omplex lo al system an b e seen as a holomorphi vetor bundle with a at (and therefore integrable) onnetion. In fat, if the ground spae is omplex analyti b oth ategories are equivalent. Let (V,∇) b e a holomorphi vetor bundle on the puntured disk D∗ equipp ed with an integrable onnetion. An extension (e V,e ∇) of the bundle to D is said to b e logarithmi at 0 if ∇ extends to a morphism e ∇:e V −→ e V ⊗Ω1 D(log z) whih satises Leibniz' rule, i.e. e ∇(fs) = fe ∇(s) + s⊗ d f for every lo al setion f of OD and s of e V . The Poinaré residue map an b e dened as R: Ω1 D(log z)−→ Oz∼ =C, ω =η∧ d z z+η′7→ η(0), suh that z= 0 is an equation for D and η, η′ not ontaining d z . This indues a C -linear endomorphism res 0(e ∇) of e V0 , the residue at 0 . 54
4.3. Extensions of Variations of Ho dge Struture If we x the anonial ontinuous setion τ:C/Z−→ [0,1) + iR⊂C , we get the following prop osition due to Manin (see [Del70℄). Prop osition 4.3.1 Let (V,∇) be a holomorphi vetor bund le on the puntured disk D∗ equipped with an integrable onnetion. There exists a unique extension e V of V , al led the anonial extension , to a vetor bund le on D suh that ∇ extends to a logarithmi onnetion e ∇ on e V whose residue at 0 has its eigenvalues in the image of τ , i.e. their real part is greater or equal to 0 and less than 1 . Let V b e a p olarized variation of C -Ho dge strutures of weight m on D∗ . Supp ose that the lo al mono dromy op erator is T∈GLn(C) , where T is unip otent and we have a dereasing ltration of holomorphi vetor bundles F• . Now we want to extend this Ho dge ltration F• to D so that we get something lose to a VHS. This extended ltration gives a mixed Ho dge struture in 0 , where the weight ltration an b e desrib ed very expliitly in the following way. As T is unip otent there is a nilpotent matrix N , suh that T= exp N . On a nite dimensional vetor spae every nilpotent endomorphism has a Jordan deomp osition and therefore an b e written as sum of Jordan blo ks to the eigenvalue 0 . There is an appropriate basis (v1,...,vj) of length j for eah Jordan blo k Jj(0) in the Jordan normal form of N . We an dene an inreasing ltration on the vetor spae by putting Wo:= {0}o≤ −j, hv1,...,v⌊o+j+1 2⌋i −j < o < j −1, hv1,...,vjij−1≤o. By adding these for the dierent blo ks and shifting it by an integer m , we get the following prop erties of the weight ltration, whih desrib e it uniquely. Denition and remark 4.3.2 Given a nilpotent endomorphism N of a nite dimensional vetor spae V , there exists a unique inreasing ltration W•=W•(N, m) of V , al led the weight ltration of N entred at m , with the properties a) N(Wo+2)⊆Wo, o ∈N0 b) the map No: GrW m+oV−→ GrW m−oV is an isomorphism for al l o∈N0 . Moreover, there is a Lefshetz-typ e deomp osition GrWV= m M o=0 o M r=0 NrPVm+o with PVm+o:= Ker (No+1 : GrW m+oV−→ GrW m−o−2V) and the endomorphism N has dimCPVm+o Jordan blo ks of size o+ 1, o = 0,...,m . 55
4. Ho dge Strutures and Middle Convolution As [Sh73℄, Theorem 6.16 we nd the following result: Theorem 4.3.3 Let V be a polarized variation of C -Hodge strutures of weight m on D∗ with loal monodromy operator T∈GLn(C) , suh that T is unipotent. Choose N∈GLn(C) nilpotent suh that exp N=T . The Hodge bund les F• of V extend to holomorphi subbund les e F• of the anonial extension e V , and the triple V Hdg 0:= (e V0, W•(N, m),e F• 0) is a mixed Hodge struture, al led the anonial bre . This is an important tool for the determination of the original variation of C -Ho dge strutures. For a unip otent T a nilp otent matrix N with exp N=T has the same Jordan blo k struture but eigenvalue 0 instead of 1 . Therefore the rational dimension of GrW m±o(e V0⊗Q) for o∈N0 is the numb er of Jordan blo ks of o dd length whose length is greater than o , if o is even, and the numb er of Jordan blo ks of even length whose length is greater than o , if o is o dd. Let us assume from now on that T is a long unip otent element, i.e. has Jordan normal form Jn(1) , where n is the rank of V . By the last theorem, we an extend a polarized variation V of C -Ho dge struture of weight m on P1 C\S ( S nite) to a xed s∈S by a mixed Ho dge struture V Hdg s= (e Vs, W•(1 2πi log T, m),e F• s). The weight ltration W•:= W•(1 2πi log T, m) on Vs indues therefore the following dimensions of the grading: dimQGrW j(Vs) = (1−n < j −m < n and 2|j−m−(n−1), 0 else . In general this do es not dene e F• s uniquely. But exatly in this stritly one and zero dimensional ase the dimensions of the Ho dge ltration are determined by the dimensions of the weight ltration. This is done by alulating the Ho dge numb ers of the mixed Ho dge strutures V Hdg s . As eah graduated piee is of dimension either 0 or 1 , there is exatly one Ho dge struture over Q of a given even weight m , namely Q(−m 2) with Ho dge type (m 2,m 2) . Sine N lowers the weight stritly by 2 , the Ho dge ltration has maximal length and the mixed Ho dge struture is uniquely dened by the weight ltration. Summarizing we obtain the following result: Corollary 4.3.4 Let V be a polarized variation of C -Hodge struture of weight m and rank n on P1 C\S ( S nite) and s∈S suh that the monodromy operator at s is a long unipotent element. Then the Hodge ltration of V has maximal length. 56
5 Motivi Desription of Hm,ℓ For a general intro dution to motives, have a look at [Jan94℄ or [And04℄. The following part is a lose adaption to hapter 8 of [Kat96℄. 5.1 Setting Let K b e an algebraially losed eld and ho ose o≥2 distint p oints s1,...,so∈A1 K . For a prime numb er ℓ and an integer N∈N , suh that har (K)∤Nℓ , we x a primitive N -th ro ot of unity in K and a primitive N -th ro ot of unity ζN in Q . We dene the rings RN,ℓ := Z[ζN,(Nℓ)−1]⊂Q and SN,o,ℓ := RN,ℓ[T1,...,To][∆−1]⊂Q(T1,...,To), where ∆ := Q i<j (Ti−Tj) . By the xation ab ove, we get a unique ring homomorphism ϕ:SN,o,ℓ −→ K with the prop erty, that ϕ(Ti) = si for i= 1,...,o and that ζN is mapp ed to the hosen N -th ro ot of unity in K . For m∈N0 we onsider the following ane spaes A(o, m + 1)RN,ℓ := spec RN,ℓ[T1,...,To, X1,...,Xm+1][∆−1 m+1] where ∆m+1 := Y i<j (Ti−Tj)· m+1 Y i=1 o Y j=1 (Xi−Tj)· m Y i=1 (Xi+1 −Xi) with natural pro jetions pr i:A(o, m + 1)RN,ℓ ////A1 SN,o,ℓ \ {T1,...,To} indued by the emb edding SN,o,ℓ[X]//RN,ℓ[T1,...,To, X1,...,Xm+1][∆−1 m+1] with X7→ Xi . On Gm,RN,ℓ with o ordinate Z , one has the Kummer overing of degree N of the equation YN=Z . Let µN(RN,ℓ) denote the N -th ro ots of unity, then ho osing an N -th primitive ro ot of unity in Qℓ is the same as ho osing an emb edding χ:µN(RN,ℓ)//Qℓ × . Then the overing denes a onneted µN(RN,ℓ) -torsor, whih gives a representation π´et 1(Gm,RN,ℓ )ε////µN(RN,ℓ)χ //Qℓ × and therefore the orresp onding Kummer sheaf Lχ . For any sheme G and any morphism f:G −→ Gm,RN,ℓ , we dene Lχ(f):= f∗Lχ . Let f:A(o, 2)RN,ℓ −→ Gm,RN,ℓ b e indued by the ring homomorphism X7→ X2−X1 . 57
5. Motivi Desription of Hm,ℓ The funtoriality of ( −1 )-omp onent of the higher diret image in the sense of Setion 5.2 (the notion extends in an obvious way to X and to D ) and again the exatness of the sequene yield the following hain of isomorphisms K=m= (Rm(φA)!Qℓ)−1 =m∼ =im Rm(φA)!Qℓ→Rm(φXA)∗Qℓ−1 ∼ =ker Rm(φXA)∗Qℓ→Rm(φDA)∗Qℓ−1. By Corollary 5.3.1 the sheaf K=m is lisse and the isomorphisms imply that ker Rm(φX)∗Qℓ→Rm(φD)∗Qℓ−1 is lisse to o. It follows from prop er base hange that ker Rm(φXA)∗Qℓ→Rm(φDA)∗Qℓ−1 is lisse, where DA:= ` i∈I Di,A and φDA=` i∈I φXA|Di,A . We laim that the natural map ψ: ker Rm(φXA)∗Qℓ→Rm(φDA)∗Qℓ−1−→ ker Rm(φXA)∗Qℓ→Rm(φDA)∗Qℓ−1, where φDA:= ` i∈I φDi,A :DA−→ A1 A\{0,1} , is an isomorphism. In order to prove that by the Sp eialization Theorem (see [Kat90℄, 8.18.2), it sues to show this for any losed geometri p oint x of HypA . As (Rm(φA)!Qℓ)−1 =m∼ =ker Rm(φXA)∗Qℓ→Rm(φDA)∗Qℓ−1 , we have to show that (Hm c(HypA,x,Qℓ))−1 =m−→ ker Hm ´et (XA,x,Qℓ)→Hm ´et (DA,x,Qℓ)−1 is an isomorphism for DA,x := ` i∈I Di,A,x . We dene the following sequene of stalks X0 A,x := XA,x , and for natural numb ers i , let Xi A,x denote the disjoint union of the irreduible omponents of the lo us, where i pairwise dierent omp onents of DA,x meet. It follows from the Weil onjetures [Del74℄ that the sp etral sequene E1=Hj ´et(Xi A,x,Qℓ)x⇒Hi+j c(UA,x,Qℓ) degenerates at E2 . Consequently, we have (Hm c(HypA,x,Qℓ))=m∼ =ker Hm ´et (XA,x,Qℓ)→Hm ´et (DA,x,Qℓ), whih proves that the map ψ is an isomorphism as laimed. So, (Rm(φHypA)∗Qℓ)−1 =m∼ =ker Rm(φXA)∗Qℓ→Rm(φDA)∗Qℓ−1 =1 2(1 −σ)ker Rm(φXA)∗Qℓ→Rm(φDA)∗Qℓ, where the last equality is by using representation theory of nite (yli) groups. It follows that φ∗K=m=φ∗(Rm(φHyp)!Qℓ)−1 =m∼ =1 2(1 −σ)ker Rm(φX)∗Qℓ→Rm(φD)∗Qℓ, as laimed. 64
5.4. Analytiation of Hm,ℓ 5.4 Analytiation of Hm,ℓ Let K b e a number eld and S⊆K a nite set. We x an emb edding K//C . This yields a ontinuous morphism ι:πtop 1(C\S)//\ πtop 1(C\S) = π´et 1(A1 C\S)−→ π´et 1(A1 K\S) . A lisse Qℓ -sheaf V on A1 K\S orresp onds by Corollary 2.2.12 to a ontinuous representation ρV:π´et 1(A1 K\S)−→ GLn(Qℓ) . Denition 5.4.1 The analytiation Van of V is the loal system of Qℓ -modules VρV◦ι on C\S orresponding to the representation ρV◦ι:πtop 1(C\S)−→ GLn(Qℓ) by Corol lary 2.2.10 . The omparison isomorphism b etween étale and singular ohomology implies (after xing an isomorphism C∼ =Qℓ of elds), that (φ∗K=m)an ∼ =1 2(1 −σ)an(ker Rm(φX)an ∗Qℓ→Rm(φD)an ∗Qℓ), as 1 2(1 −σ) is an algebrai pro jetor and hene deRham. Further we have a morphism b etween smo oth pro jetive varieties on the right hand side, whih is given by restrition of inlusions of the smo oth divisors Di . Applying [Del87℄, Prop osition 1.13. to the irreduible sheaf (φ∗K=m)an , it omes from a variation of Ho dge struture in the following sense: Remark 5.4.2 The loal system of Q -vetor spaes Gm:= 1 2(1 −σ)an |{z } =1 2(1−σan) ker (Rm(φX)an ∗Q→Rm(φD)an ∗Q) on A1 C\{0,1} is a polarized variation of Hodge struture, whih is pure of weight m , sine it is a subvariation of Rm(φX)∗Z by Remark 4.2.4 b). Moreover by Corol lary 5.3.1 and Corol lary 5.3.2 , we have Gm⊗Qℓ∼ =(φ∗K=m)an = (Hm,ℓ|A1 C\{0,1})an. Conluding we have: Theorem 5.4.3 Let Hm,ℓ be as in Theorem 3.3.1 . Then there exists a loal system of Z -modules Gm on A1 C\{0,1} underlying a polarized variation of Z -Hodge struture (Gm,F•,∇) on C\{0,1} pure of weight m suh that (i∗Hm,ℓ)an ∼ =Gm⊗Qℓ. The indued isomorphism on the stalks (i∗Hm,ℓ)an x∼ =Gm⊗Qℓx for x∈C\{0,1} is given by the 65
5. Motivi Desription of Hm,ℓ omparison isomorphism between étale ohomology and singular ohomology 1 2(1 −σ)ker Hm ´et (Xx,Qℓ)→Hm ´et (Dx,Qℓ)∼ =1 2(1 −σ)ker (Hm B(X(C)x,Z)→Hm B(D(C)x,Z)) ⊗Qℓ. Moreover, the Hodge ltration of Gm has maximal length. Proof: All laims but the last follow from Remark 5.4.2. The last laim follows from the long unip otent lo al mono dromy of Hm,ℓ using Theorem 3.3.1 and Corollary 4.3.4. In [Fal88℄, Setion 4(a) Faltings gives the onstrution of natural isomorphisms of ℓ -adi étale ohomology and deRham ohomology. Remark 5.4.4 Let v∈ΣK\{0} and ℓ= har (kv) and X a proper at OKv -sheme, then we have an isomorphism Hm ´et (X⊗Kv,Qℓ)⊗Cv ∼ −→ M p+q=m Hq(X, Ωp X/OKv)⊗Cv(−q), whih preserve up produts, GKv -ation, harateristi lasses of yles and Chern lasses of vetor bund les. This remark of Faltings has imp ortant onsequenes for the onnetion b etween τ -Ho dge-Tate numb ers of an ℓ -adi representation ρℓ and Ho dge numbers of XC , related by the ation of GK on X . Illusie arries out the essential part of the exat onnetion after [Ill94℄, Theorem 3.1.2: hj,m−j(XC) = dimQℓ(Cv⊗Hm ´et (XQℓ,Qℓ)(j))GKv. Therefore we have for Vℓ:= Hm ´et (XQℓ,Qℓ)⊗Qℓ that hv,j(ρℓ) = dimQℓ(Cv(j)⊗Vℓ)GKv=h−j,m+j(XC). Corollary 5.4.5 For X:= Gm,x as in Theorem 5.4.3 endowed with a GKv -ation, orresponding to the representation ρℓ:GKv−→ GL(Gm,x) , we get the fol lowing result: hv,j(ρℓ) = (1−m≤j≤0, 0 else . Moreover ρ= (ρℓ)ℓ prime full ls regularity in the sense of Denition 7.2.1 . By Theorem 5.4.3 these τ -Ho dge-Tate numb ers oinide with the τ -Ho dge-Tate numb ers for the system of ℓ -adi representations ρm= (ρm,ℓ)ℓ prime dened in Setion 7.2. 66
6 Irreduibility of ρm 6.1 Lifting Irreduibility Let G b e a group, n∈N and ℓ a prime numb er. Lemma 6.1.1 Let ρFℓ:G−→ GLn(Fℓ) be an irreduible representation with a long unipotent element. Then the extension ρFℓ:G−→ GLn(Fℓ), g 7→ ρFℓ(g) is irreduible (i.e. ρFℓ is absolutely irreduible). Proof: We take a minimal ρFℓ -invariant subspae {0} 6=W⊆Fℓ n . We have the omp onent-wise Galois ation of GFℓ on Fℓ n , whih maps W to an orbit of subspaes. As the representation is dened over Fℓ b oth ations ommute and therefore eah subspae in the orbit is ρFℓ -invariant. By the minimality of W these are all linearly disjoint and on the other hand xed by the long unip otent element. Eah spae xed by this element inludes the eigenvetor and therefore the orbit has just one element. As W is invariant under b oth ations, there is a subspae U⊆Fn ℓ whih is ρFℓ -invariant and for whih W=Fℓ⊗U . As ρFℓ is irreduible and {0} 6=W , we have U=Fn ℓ and therefore W=Fℓ n . The lo al ring Zℓ=x∈Qℓ|v(x)≥0 has the maximal ideal x∈Zℓ|v(x)>0 with residue eld Fℓ . In the ase of the lemma ab ove, we have hene that ρZℓ:G−→ GLn(Zℓ), g 7→ ρZℓ(g) is irreduible as well. Here we all a Zℓ[G] -mo dule irreduible if and only if it has no non-trivial Zℓ[G] -submo dules. At this p oint I want to thank Stefan Reiter and Andreas Maurishat for telling me ab out the next well-known statement. Lemma 6.1.2 If ρZℓ:G−→ GLn(Zℓ) is an irreduible representation, then the extension ρQℓ:G−→ GLn(Qℓ), g 7→ ρZℓ(g) is irreduible. Proof: Assume that ρQℓ is reduible and has the invariant subspae 06=V6=Qℓ n . Therefore the Zℓ -mo dule W:= V∩Zn ℓ is invariant and W6=Zn ℓ , b eause V6=Qℓ n . Additionally it is non-trivial as for all v∈Qℓ n there exists a λ∈Qℓ × , suh that λv ∈Zn ℓ , and we have 06=W . This is a ontradition sine ρZℓ is irreduible. This lemma an easily b e adapted to rings and their quotient elds, esp eially to valuation rings. 67
6. Irreduibility of ρm 6.2 Serre's Results on Charaters of GQ In the next part, we restate some results of Serre. The main one is that a system of homomorphisms (θℓ:GQ−→ F× ℓ)ℓ∈L fullling some ompatibility relations is the redution of the pro dut of a nite harater and a p ower of the ylotomi harater. This will b e another ingredient for the pro of of Theorem 6.4.1. Let K b e a number eld and IK the idele group of K . For a nite set S of plaes of K , we dene a modulus m with support S as a family m:= (mv)v∈S∈NS . Then we get an op en subgroup Um for eah mo dulus m , by Um:= Q v plae of K Um,v ⊆IK , where Um,v is as follows: onneted omp onent of 1 in K× v for an innite plae v6∈ S , K× v for a nite plae v6∈ S , {x∈K× v suh that v(1 −x)≥mv} for v∈S . On the other hand, we have two asso iated algebrai groups Tm and Sm over Q with an algebrai morphism Tm//Sm (for more details see [Ser68℄, Setion 2.2.). Let E b e a number eld. It is shown in [Ser68℄ how to attah a stritly ompatible system (ψℓ:GQ−→ E× λ)λ∈ΣE\{0} of one dimensional λ -adi Q -rational Galois representations to any harater ψ:Sm−→ E×. Prop osition 6.2.1 ( [Ser72℄, Prop.20 ) Let L⊆ΣQ∪ {∞} \ {0} be an innite set and let θℓ:GQ−→ F× ℓ, ℓ ∈L, be a ol letion of homomorphisms. Assume that there exists a modulus m and j∈Z suh that for al l ℓ∈L and for al l a∈Um one has θℓ(cf−1[a]) ≡a−j ℓmod ℓ, where cf : Gab Q−→ IQ/Q× is the lass eld isomorphism and aℓ is the omponent of a at ℓ. Then there exists a number eld E and a Heke harater ψ:Sm−→ E× suh that ψλ=θℓ for innitely many ℓ∈L and λ a nite plae of E above ℓ . The following result is a onsequene of [Sh88℄, Prop osition 1.4, and Serre's theory of ab elian representations [Ser68℄: Prop osition 6.2.2 Let (ψλ:GQ−→ E× λ)λ∈ΣE\{0} be a stritly ompatible system of one dimensional λ -adi E -rational Galois representations whih are assoiated to a Heke harater ψ:Sm−→ E×. Then there exists a nite harater ǫ:GQ−→ E× and an integer k∈Z suh that ψλ=ǫ·χk ℓ, where λ|ℓ. Combining b oth results, we get the following: 68
6.3. Groups of Lie Type Corollary 6.2.3 Let L⊆ΣQ∪ {∞} \ {0} be an innite set and let θℓ:GQ−→ F× ℓ, ℓ ∈L, be a ol letion of homomorphisms. Assume that there exists a modulus m and j∈Z suh that for al l ℓ∈L and for al l a∈Um one has θℓ(cf−1[a]) ≡a−j ℓmod ℓ, where cf : Gab Q−→ IQ/Q× is the lass eld isomorphism and aℓ is the omponent of a at ℓ. Then there exists a number eld E , a nite harater ǫ:GQ−→ E× and an integer k∈Z suh that ǫ·χk ℓ=θℓ for innitely many ℓ∈L. 6.3 Groups of Lie Typ e In the year 1972 Gorenstein announed a program for the omplete lassiation of nite simple groups. Many mathematiians worked on it and the last gap was lled 2004 by Ashbaher and Smith (f. [As04℄). Beside the well-known abelian nite simple groups, i.e. yli groups of prime order, there are the following p ossibilities: the alternating groups An ( n≥5 ), the nite lassial groups - that is, the linear, sympleti, unitary and orthogonal groups of nite vetor spaes, the exeptional groups of Lie type, the 26 sp oradi groups (see [Gor85℄). This shows that groups of Lie typ e play an important role in the understanding of groups and their subgroup struture. Let K b e a eld of harateristi p , q a p -p ower and n∈N , then we have the homomorphism Frobq: GLn(K)−→ GLn(K),(aij)7→ (aq ij) . If K is algebraially losed and G is a linear algebrai group over K a standard Frobenius F:G−→ G is a map suh that there exists an n∈N , an inlusion ι:G//GLn(K) and a p ower q of p suh that for all g∈G we have ι(F(g)) = Frobq(ι(g)). A homomorphism F:G−→ G is alled a Frobenius morphism , if some p ower of F is a standard Frob enius. Denition 6.3.1 Let G be a onneted redutive algebrai group over Fℓ and let F:G−→ G be a Frobenius morphism. The nite group of xed points GF is al led group of Lie typ e and sometimes also its ommutator subgroup (GF)′:= [GF, GF] and its entral quotient G/Z(G) , where Z(G) is the enter. 69
6. Irreduibility of ρm In this setting, a maximal losed onneted solvable algebrai subgroup is alled Borel subgroup and an algebrai subgroup whih ontains a Borel subgroup is alled paraboli . Remark 6.3.2 a) The standard Frobenius morphism Frobq:Fq−→ Fq, α 7→ αq , gives rise to the so al led Chevalley groups (untwisted groups of Lie type), whereas the produt of some Frobq with other automorphisms leads to twisted groups of Lie type (e.g. SUn(Fq2) = SLn(Fq)δ◦Frobq , where δ is the inverse transpose map). b) Eah group of Lie type is the quotient of Fq -points G(Fq) of an algebrai group sheme over Fq . (For the derived group of an orthogonal group, we take the spin group, f. [Wil09 ℄). Denition and remark 6.3.3 To any onneted Dynkin diagram there is an assoiated simple algebrai group G over Fq (with the exeptions A1(F2), A1(F3), A2(F3), B2(F2) and G2(F2) ). Cartan typ e Chevalley group Dynkin diagram Al(Fq) SLl+1,PGLl+1 Bl(Fq) SO2l+1 Cl(Fq) Sp2l Dl(Fq) SO2l E6(Fq) E7(Fq) E8(Fq) F4(Fq) G2(Fq) For the lassiation it is neessary to take a Dynkin diagram together with an automorphism of the graph, whih then is in orrespondene to simple group of Lie type. This leads to the fol lowing types 2Al(Fq2),2B2(F22n+1 ),2Dl(Fq2),3D4(Fq3),2E6(Fq2),2F4(F22n+1 ),2G2(F32n+1 ) . 70
6.3. Groups of Lie Type Next we will need the struture of the maximal subgroups of the ommutator subgroup Ωn(Fq) := [SOn(Fq),SOn(Fq)] for n∈N o dd and q an o dd prime p ower. In [KL90℄, we have the following denition of sets S,C of maximal subgroups of Ωn(Fq) : Denition of S A maximal subgroup H of Ωn(Fq) lies in S:= S(Ωn(Fq)) if and only if the following holds. a) The so le S of H , that is the subgroup generated by the minimal non-trivial normal subgroups of H , is a non-ab elian simple group - i.e. H has a unique minimal non-trivial normal subgroup, whih is non-ab elian and simple. b) If L is the full overing group of S , and if ρ:L−→ GL(V) is a representation of L suh that ρ(L) = S , then ρ is absolutely irreduible. ) ρ(L) annot b e realized over a prop er subeld of Fq . (rough) Denition of C For the subgroup Ωn(Fq) of SOn(Fq) we dene Ci(Ωn(Fq)) := {C∩Ωn(Fq)|C∈ Ci(SOn(Fq))} for i= 1,...,8 , and let C:= C(Ωn(Fq)) := 8 [ i=1 Ci(Ωn(Fq)). C1: stabilizers of totally singular or non-singular subspaes C2: stabilizers of deomp ositions V= t L j=1 Vj with tdimFqVj= dimFqV=n C3: stabilizers of extension elds of Fq of prime index C4: stabilizers of tensor pro dut deomp ositions V=V1⊗V2 C5: stabilizers of subelds of Fq of prime index C6: normalizers of sympleti-typ e r -groups ( r prime) in absolutely irreduible representations C7: stabilizers of deomp ositions V= t N j=1 Vj with (dimFqVj)t= dimFqV=n C8: lassial subgroups 71
6. Irreduibility of ρm The [KL90℄, Main Theorem (C) and [KL90℄, Table 3.5.D reveals the struture of Ωn(Fq) for n, q o dd: Theorem 6.3.4 ( [KL90℄, Main Theorem (C) ) Assume that n > 12 is odd an q an odd prime power. For a member H∈ C , the preise onditions under whih H is maximal in Ωn(Fq) are determined by the fol lowing table. Moreover, this table also determines the set of overgroups of H lying in C ∪S . Ci typ e onditions C1Pm1≤m≤n−1 2 Om(Fq)⊥Oǫ n−m(Fq) 1 ≤m < n , m o dd, ǫ=± C2Om(Fq)≀Stn=mt, m, t ≥2 O1(Fq)≀Snq prime C3On r(Fqr)r|n , r prime, r6=n C4Om(Fq)⊗On m(Fq)m|n , m < √n C5On(Fq0)q=qr 0 , r prime C6 do es not o ur C7Om(Fq)≀Stn=mt , (q, m)6= (3,3) C8 do es not o ur Table 6.1: The maximal subgroups of Ωn(Fq) for n, q o dd ([KL90℄, Table 3.5.D) Here Pm is a stabilizer of an m -dimensional total ly singular spae, i.e. a paraboli subgroup, Om and Oǫ m are orthogonal groups respeting some symmetri bilinear form and Sm is a symmetri group. Lemma 6.3.5 Let n∈N be a xed odd integer. For almost al l prime numbers ℓ let G(Fq)⊆SOn(Fℓ) be a group of Lie type for q a power of ℓ ontaining a long unipotent element, i.e. an element of Jordan normal form Jn(1) , and one non-trivial unipotent element with dierent Jordan normal form. Then for almost al l ℓ as above, we have the inlusion Ωn(Fℓ)⊆G(Fq) if n6= 7 and G2(Fℓ)⊆G(Fq) if n= 7 . Proof: There exists a simply onneted group of Lie typ e G(Fq) and an epimorphism G(Fq)////G(Fq) . By [Ste63℄ we have a morphism L(λ) : G(Fq)−→ Ωn(Fq)⊆GLn(Fq) of Fq -p oints of algebrai group shemes on Fq dened over Fq (where λ denotes the highest weight of the representation), if ℓ is large ompared to n . The long unip otent element u is asso iated to a ertain ro ot α in the ro ot system of G . The transp osed element ut is then asso iated to the negative ro ot −α of G . Then the group H=hu, uti is an irreduible subgroup of G of type A1 , again if ℓ is large enough. 72
6.4. Irreduibility of the mo dℓ Representation Now the existene of the non-trivial unip otent element with Jordan normal form dierent from Jn(1) implies that H is prop erly ontained in G(Fq) and that G is of dierent type then A1 (if ℓ is large enough). By Steinberg's Tensor Pro dut Theorem (f. [MT11℄, Theorem 29.6), if ℓ is large ompared to n any irreduible representation of G is given by a highest weight representation and hene was indued by a morphism of onneted group shemes over Fq . Base hange of L(λ) to Fq denes a morphism of algebrai groups and hene a representation L(λ)⊗Fq:G(Fq)−→ GLn(Fq) , whih fators over SOn(Fq) . By [SS97℄, Theorem B, any algebrai group ontaining a long unip otent element inside an underlying general linear group GL(W) dierent from GL(W) is either of typ e A1,SO(W),Sp(W), G2 or B3 . As G(Fq) denes G uniquely if the harateristi ℓ and therefore q is large enough and sine there are no non-trivial twists of o dd dimensional orthogonal groups, this implies that G lies b etween Ωn(Fℓ) and SOn(Fℓ) if n6= 7 . In the hase n= 7 , hene we have G2(Fℓ)⊆G(Fq) . 6.4 Irreduibility of the mo dℓ Representation In this setion we have K=Q , m∈N0 even and ℓ a prime numb er. Then we get the lisse Qℓ -sheaf i∗Hm,ℓ of rank m+ 1 for i:A1 Q\ {0,1}//A1 Q and Hm,ℓ like in Setion 3.3. By xing an s∈A1 Q\ {0,1} this orresp onds to a ontinuous representation of rank m+ 1 ρi∗Hm,ℓ :π1 ´et(A1 K\{0,1})−→ GL((Hm,ℓ)s) , whih fators through Zℓ and resp ets a symmetri bilinear form. This representation an b e tensored with the determinant as in Theorem 3.3.2, to obtain a ontinuous representation ρi∗Hm,ℓ ⊗det(ρi∗Hm,ℓ ) : π1 ´et(A1 Q\{0,1})−→ GL((Hm,ℓ)s), fatoring over SL and respeting a symmetri bilinear form. This representation is of weight m by Delignes' work (f. Theorem 2.3.4), i.e. maps Frobq to q−m 2 . Then we get a weight 0 representation by tensoring with the m 2 th p ower of the ylotomi harater χℓ . Now we will show that the redution mo d ℓ dρm,ℓ := ρi∗Hm,ℓ ⊗det(ρi∗Hm,ℓ )⊗χm 2 ℓ:π1 ´et(A1 Q\{0,1})−→ SOm+1(Fℓ) is irreduible for almost all ℓ . This will lead us in Setion 7.2 to the fat that ρm,ℓ is irreduible as a weakly ompatible system of Galois representations of Q . For any x∈A1 Q\{0,1} , we get the sp eialization map ιx:π´et 1({x})∼ =GQ//π´et 1(A1 Q\{0,1}) as dened in Setion 2.2. 73
7. Potential Automorphy of Sp eializations where pv is the harateristi of the residue eld kv as b efore. For v∈S the denition of Lv(π, s) is not so straightforward, but an b e obtained by the lo al Langlands orrespondene [HT01℄ and [Hen00℄. It is known that these analyti L -funtions satisfy favorable prop erties like meromorphi (mostly even holomorphi) ontinuation to the whole omplex plane and fulll funtional equations et.. On the other hand, for any irreduible, weakly ompatible system ρ= (ρℓ)ℓ prime of Galois representations ρℓ:GK−→ GL(Vℓ) (see Denition 2.1.4), we an also dene the restrited L -funtion of ρ by L(ρ, s) := Y v∈S Lv(ρ, s)·Y v∈ΣK\S det(1−p−s vFrobv,ρℓ)−1:= Y v∈S p−snv v˜ fv,ρℓ(ps v)·Y v∈ΣK\S psnv vfv,ρℓ(ps v)−1, where n is the rank for the representation and fv,ρℓ(x)∈Q[x]⊂Qℓ[x] the harateristi p olynomial of the Frob enius (see page 22) for v|ℓ . Stritly sp eaking Frobv,ρℓ is not well-dened and should b e seen as symbol for the very last pro dut on the right. In the ase v∈S , we have a preimage of the Frob enius in the deomp osition group for eah element of the inertia group Iv:= lim ←− Iw (for the system w|v ). The ation of these preimages via the Galois representation ρℓ is unique up to onjugation on the spae Vρℓ(Iv) ℓ xed by ρℓ(Iv) . Therefore the harateristi p olynomial ˜ fv,ρℓ(x) := det(1 ·x−ρℓ(Frobv)|Vρℓ(Iv) ℓ )∈Qℓ[x] is well-dened and an b e interpreted as a omplex p olynomial by ho osing an emb edding of elds ιℓ:Qℓ//C . We dene Lv(ρ, s) := det(1 −p−s vρℓ(Frobv)|Vρℓ(Iv) ℓ ) := p−snv v˜ fv,ρℓ(ps v), where nv is the dimension of Vρℓ(Iv) ℓ . Denition 7.1.1 Let ρ= (ρℓ)ℓ prime be an irreduible, weakly ompatible system of ℓ -adi representations of a number eld K , then ρ is al led automorphi , if there exists an irreduible omplex subrepresentation π of GLn(AK) , suh that L(ρ, s) = L(π, s). In this ase, we say L(ρ, s) is automorphi as wel l. If there is a nite Galois extension L|K suh that the restrition (ρℓ|GL)ℓ prime is automorphi, ρ is al led p otentially automorphi . An imp ortant onjeture whih is a part of the famous Langlands program [Lan79℄, states that any irreduible uspidal system of Galois representations is automorphi. 7.2 Mo dular Lifting Due to Barnet-Lamb, Gee, Geraghty and Taylor (f. [BLGGT10℄) one has a riterion for automorphy for whih we will have to intro due some more notation. 80
7.2. Mo dular Lifting For a numb er eld K , we have the maximal totally real subeld K+ . The innite plaes of K+ orresp ond to the emb eddings K+//R . If we x suh an innite plae v , we get an embedding of GR={1, c}//GK+ . The image of the omplex onjugation c will b e denoted by cv . Denition 7.2.1 Let K be a number eld and (ρℓ)ℓ prime a weakly ompatible system of Galois representations for K of rank n (see Denition 2.1.4 ). The system (ρℓ)ℓ prime is al led a) totally o dd, essentially onjugate self-dual in the ase K is total ly real or CM, if a weakly ompatible system (εℓ)ℓ prime of Galois representations for K+ of rank one with the fol lowing property exists. For al l prime numbers ℓ there is a non-degenerate, symmetri pairing h·,·iℓ on Qℓ n , suh that for al l σ∈GK and x, y ∈Qℓ n we have hρℓ(σ)x, ρℓ(cℓσcℓ)yiℓ=εℓ(σ)hx, yiℓ. b) regular , if for eah τ:K//Q we have n distint τ -Hodge-Tate numbers (with multipliity one). A p owerful to ol for proving the p otential automorphy of systems of ℓ -adi Galois representations is the following theorem. Theorem 7.2.2 ( [BLGGT10℄, Thm.5.3.1 ) Suppose that K is a CM (or total ly real) eld and that (ρℓ)ℓ prime is an irreduible, total ly odd, essential ly onjugate self-dual, regular, weakly ompatible system of ℓ -adi representations of K . Then there is a nite, CM (or total ly real), Galois extension L|K suh that the restrition of (ρℓ)ℓ prime to GL is automorphi. For m∈N0 and a prime numb er ℓ the Qℓ -sheaf Hm,ℓ on A1 K onstruted in 3.3 (page 47), we get a lisse Qℓ -sheaf i∗Hm,ℓ on A1 K\{0,1} by pulling bak along the inlusion i:A1 K\{0,1}//A1 K . It is by denition the tensor pro dut of a lisse Zℓ -sheaf and Qℓ over Zℓ . This orresp onds by Corollary 2.2.12 to a ontinuous representation ρi∗Hm,ℓ :π´et 1(A1 K\{0,1})−→ GL(W) on the stalk W of i∗Hm,ℓ at a hosen p oint in A1 K\{0,1} , whih fators through Zℓ . The tensor pro dut of this representation with the one dimensional representation det(ρi∗Hm,ℓ ) : π´et 1(A1 K\{0,1})−→ {±1} ⊂ Qℓ ×, whih is obtained by taking the omp osition of the representation and the determinant fators as well through SL(W) . By a hoie of x∈A1 K\{0,1} aording to the restritions of Theorem 6.4.1 81
7. Potential Automorphy of Sp eializations and sp eializing ρi∗Hm,ℓ ⊗det(ρi∗Hm,ℓ ) to x (see page 29) ρi∗Hm,ℓ ⊗det(ρi∗Hm,ℓ )◦ιx:GK ιx //π´et 1(A1 K\{0,1})−→ GL(W), we get an ℓ -adi Galois representation of K for eah prime numb er ℓ . Out of these maps we want to onstrut a weakly ompatible system by semi-simpliation (see page 39) of this system. Lemma 7.2.3 For m∈N0 , K=Q and x∈A1 Q\{0,1} , the system ρm= (ρm,ℓ)ℓ prime := ((ρι∗Hm,ℓ ⊗det(ρι∗Hm,ℓ )) ◦ιx)ss ℓ prime is a weakly ompatible system of ℓ -adi Galois representations, whih respets an orthogonal respetively sympleti form if m is even respetively odd. Let m= 6 or m even and m≥12 for x∈A1 Q\ {0,1} , suh that there exist odd prime numbers p, q satisfying νp(x)<0 but ℓ∤νp(x) and νq(x−1) >0 but ℓ∤νq(x−1) (see Theorem 6.4.1 ), this system is irreduible. Proof: By Setion 3.3 we get representations ρm,ℓ :GK−→ SOm+1(Qℓ) for m even and ρm,ℓ :GK−→ Spm+1(Qℓ) for m o dd. The rationality and ompatibility of the system ρm is a diret onsequene of [Kat96℄, Theorem 5.5.4. From Corollary 5.3.1 and Corollary 5.3.2 we have that i∗Hm,ℓ ∼ =1 2(1 −σ) ker Rm(φX)∗Qℓ→Rm(φD)∗Qℓ. We have go o d redution for almost all ℓ . The derived funtors are rystalline as ohomology and therefore the kernel is rystalline as this ommutes with morphisms. As the pro jetor 1 2(1 −σ) is algebrai this is rystalline. The laim follows from the rystalline omparison isomorphism sine for ℓ large enough, the bre W is smo oth over Zℓ if ℓ is large enough (f. [Fal89℄). Hene for v∈ΣK\{0} and ℓ equal to the harateristi kv , the representation ρℓ is deRham in v and for almost all v even rystalline. By Corollary 5.4.5 we see that the τ -Ho dge-Tate numb ers are independent of ℓ for any emb edding τ:K//Q . Combining the results of this hapter, we see that for every even m∈N0 the set of prime numb ers for whih ρm,ℓ is reduible is nite. Therefore the Dirihlet density of the other prime numb ers is 1 and the system ρm= (ρm,ℓ)ℓ prime of ℓ -adi Galois representations, dened in 7.2, is irreduible in the sense of Denition 2.1.4. 82
Theorem 7.2.4 For m= 6 or m∈N0 even, m≥12 and K=Q the irreduible, weakly ompatible system ρm= (ρm,ℓ)ℓ prime of Galois representations is potential ly automorphi. Proof: Q is a totally real eld and has exatly one innite plae ∞ , the usual absolute value, with c∞∈GQ the usual omplex onjugation. We dene (εℓ)ℓ prime as the trivial system, whih is weakly ompatible. As b efore, we have im (ρm,ℓ)⊆SOm+1(Qℓ) . The pairing hx, yi∞:= xtρm,ℓ(c∞)y for x, y ∈Qℓ m+1 is non-degenerate, symmetri and satises ρm,ℓ(c∞)t=ρm,ℓ(c∞)−1=ρm,ℓ(c−1 ∞) = ρm,ℓ(c∞). For σ∈GQ , we have hρm,ℓ(σ)x, ρm,ℓ(c∞σc∞)yi∞=xtρm,ℓ(σ)tρm,ℓ(c∞)ρm,ℓ(c∞) |{z } =1 ρm,ℓ(σ)ρm,ℓ(c∞)y =xtρm,ℓ(c∞)y= 1 ·hx, yi∞ and ρm is totally o dd, essentially onjugate self-dual. The regularity is a diret onsequene of Corollary 5.4.5. This shows that for m even we are in the situation of Theorem 7.2.2 and proves that ρm is p otentially automorphi. 83
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