From the function-sheaf dictionary to quasicharacters of p -adic tori
aa r X i v : . [ m a t h . AG ] J u l FROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERSOF p -ADIC TORI CLIFTON CUNNINGHAM AND DAVID ROE
Abstract.
We consider the rigid monoidal category of character sheaves on a smooth com-mutative group scheme G over a finite field k and expand the scope of the function-sheafdictionary from connected commutative algebraic groups to this setting. We find the group ofisomorphism classes of character sheaves on G and show that it is an extension of the groupof characters of G ( k ) by a cohomology group determined by the component group scheme of G . We also classify all morphisms in the category character sheaves on G . As an application,we study character sheaves on Greenberg transforms of locally finite type N´eron models ofalgebraic tori over local fields. This provides a geometrization of quasicharacters of p -adictori. Introduction
As Deligne explained in [13, Sommes trig.], if G is a connected commutative algebraic groupover a finite field k , then the trace of Frobenius provides a bijection between the group G ( k ) ∗ of ℓ -adic characters of G ( k ) and isomorphism classes of those rank-one ℓ -adic local systems E on G for which(1) m ∗ E ∼ = E ⊠ E , where m : G × G → G is the multiplication map. If one wishes to make a category from thisclass of local systems, one is led to consider morphisms E → E ′ of sheaves which are compatiblewith particular choices of (1) for E and E ′ . A priori , the composition
E → E ′ → E ′′ of two suchmorphisms need not be compatible with the choices of (1) for E and E ′′ . However, for connected G the isomorphism (1) is unique, if it exists, and there is no impediment to making the dictionarycategorical.If G is a commutative algebraic group over k which is not connected, however, then theisomorphism (1) need not be unique. In order to track the choice of isomorphism, consider thecategory CS ( G ) of pairs ( E , µ E ) where E is a rank-one local system on G and µ E : m ∗ E → E ⊠ E is a chosen isomorphism of local systems on G × G . In this case, the trace of Frobenius providesan epimorphism from isomorphism classes of objects in CS ( G ) to characters of G ( k ), but theepimomorphism need not be injective; consequently, every character of G ( k ) may be geometrizedas a pair ( E , µ E ) but perhaps not uniquely. Indeed, it follows from a special case of the mainresult of this paper that the kernel of the trace of Frobenius CS ( G ) → G ( k ) ∗ trivial if and onlyif the group scheme of connected components of G is cyclic. The defect in the function-sheafdictionary for characters of commutative algebraic groups over finite fields may be addressedwith the following observation: if ( E , µ E ) and ( E ′ , µ ′E ) determine the same character of G ( k ) then E ∼ = E ′ as local systems on G .Motivated by an application to quasicharacters of algebraic tori over local fields, in this paperwe extend the function-sheaf dictionary from commutative algebraic groups over finite fields to Mathematics Subject Classification.
Key words and phrases. character sheaves, p-adic tori, N´eron models, Greenberg functor, geometrization,quasicharacter sheaves. smooth commutative group schemes G over k . In order to do this, we replace the local system E on G with a Weil local system while retaining the extra structure µ E . In this way we are led tothe category CS ( G ) of character sheaves on G ( § CS ( G ) are triples ( ¯ L , µ, φ ), where¯ L is a rank-one local system on ¯ G := G × Spec( k ) Spec(¯ k ) and φ : Fr ∗ G ¯ L → ¯ L and µ : ¯ m ∗ ¯ L ∼ = ¯ L ⊠ ¯ L are isomorphisms of sheaves satisfying certain compatibility conditions; morphisms in CS ( G )are then morphisms of Weil sheaves which are compatible with the extra structure. This paperestablishes the basic properties of category CS ( G ), using the group homomorphismTr G : CS ( G ) / iso → G ( k ) ∗ provided by the trace of Frobenius to find the relation between character sheaves on G andcharacters of G ( k ). Then we return to our motivating application and use character sheaves togeometrize and categorify quasicharacters algebraic tori over local fields.We begin our study of category CS ( G ) by returning to the case when G is a connectedcommutative algebraic group over k , revisiting Deligne’s function-sheaf dictionary ( § k from local systems on G ( § H → G ( § G , in the connected case we showthat every character sheaf can be described in both of these ways ( § G : CS ( G ) → G ( k ) ∗ is an isomorphism for connected commutative algebraic groups G . We also determine the automorphism groups of character sheaves on such G . These facts arewell known.Next, we consider character sheaves on ´etale commutative group schemes G over k ( § CS ( G ) in terms of stalks ( § § W ⋉ ¯ G , where W ⊂
Gal( k/k ) is the Weil group for k . We define ( § S G from CS ( G ) / iso to the second cohomology of the totalspace of the spectral sequence E p,q := H p ( W , H q ( ¯ G, ¯ Q × ℓ )) ⇒ H p + q ( W ⋉ ¯ G, ¯ Q × ℓ ) . Paired with the short exact sequence0 → H ( W , H ( ¯ G, ¯ Q × ℓ )) → H ( E • G ) → H ( W , H ( ¯ G, ¯ Q × ℓ )) → S G allows us to show ( § G : CS ( G ) / iso → G ( k ) ∗ is surjective with kernel H ( ¯ G, ¯ Q × ℓ ) W . The necessityof using Weil local systems on G in the definition of CS ( G ) already appears here: if one wereto use local systems on G instead, the group homomorphism Tr G would not then be surjective( § § G is non-trivial in general.Having understood CS ( G ) in two extreme cases – for connected commutative algebraic groupsand for ´etale commutative group schemes – we turn to the case of smooth commutative groupschemes ( §
3) using the component group sequence0 → G → G → π ( G ) → . Using pullbacks of character sheaves we obtain a diagram0 CS ( π ( G )) / iso CS ( G ) / iso CS ( G ) / iso π ( G )( k ) ∗ G ( k ) ∗ G ( k ) ∗ . Tr π G ) Tr G Tr G ROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF p -ADIC TORI 3 We show that the rows of this diagram are exact ( §§ Theorem (Thm. 3.6) . If G is a smooth commutative group scheme over k then the trace ofFrobenius gives a short exact sequence ( π ( ¯ G ) , ¯ Q × ℓ ) W CS ( G ) / iso G ( k ) ∗ . Tr G If the component group scheme π ( ¯ G ) is cyclic, then the kernel of Tr G will be trivial and eachcharacter of G ( k ) will uniquely determine a character sheaf on G , up to isomorphism. But when π ( ¯ G ) is large (c.f. Remark 2.9), G will admit invisible character sheaves with trivial trace ofFrobenius.We also illuminate the nature of the category CS ( G ) by showing that every morphism in thiscategory is either an isomorphism or trivial, and by showing Theorem (Thm. 3.9) . If G is a smooth commutative group scheme over k then Aut( L ) ∼ = H ( π ( ¯ G ) , ¯ Q × ℓ ) W for all quasicharacter sheaves L on G . Application to quasicharacters of p -adic tori and abelian varieties. As indicated above,our interest in the function-sheaf dictionary for smooth commutative group schemes over finitefields comes from an application to p -adic representation theory, specifically to quasicharacters( § p -adic tori. However, we found that our method of passing from p -adic tori to groupschemes over k applies more generally to any local field K with finite residue field k and to anycommutative algebraic group over K that admits a N´eron model X . This class of algebraic groupsover K includes abelian varieties and unipotent K -wound groups, in addition to the algebraictori we initially considered.In this paper we show that if X is as above then quasicharacters of X ( K ) are geometrizedand categorified by character sheaves on the Greenberg transform Gr R ( X ) of the N´eron model X . Although not locally of finite type, Gr R ( X ) is a commutative group scheme over k and also aprojective limit of smooth commutative group schemes Gr Rn ( X ). This structure allows us to adaptour work on character sheaves on smooth group schemes over finite fields to construct ( § QCS ( X ) of quasicharacter sheaves for X , which are certain sheaves on Gr R ( X ) × Spec( k ) Spec(¯ k ), with extra structure. The ability to generalize the function-sheaf dictionary to non-connected group schemes plays a crucial role in this application.Having defined quasicharacter sheaves on N´eron models of commutative algebraic groups over K and character sheaves on commutative group schemes over k , we consider how these categoriesare related as K and k vary. We describe ( § G ( k ′ ) ∗ → G ( k ) ∗ and G ( k ) ∗ → G ( k ′ ) ∗ , and describe how quasicharacter sheaves behave under Weil restriction( § § X = T is the N´eron model of an algebraic torus over K ( § → H ( X ∗ ( T ) I K , ¯ Q × ℓ ) W → QCS ( T ) / iso → Hom( T ( K ) , ¯ Q × ℓ ) → , where X ∗ ( T ) I K is the group of coinvariants of the cocharacter lattice X ∗ ( T ) of the algebraictorus T K by the action of the inertia group I K of K , and where Hom( T ( K ) , ¯ Q × ℓ ) denotes thegroup of quasicharacters of T ( K ). We further show that automorphism groups in QCS ( T ) are CLIFTON CUNNINGHAM AND DAVID ROE given, for every quasicharacter sheaf F for T , byAut( F ) ∼ = ( ˇ T ℓ ) W K , where W K is the Weil group for K and ˇ T ℓ is the ℓ -adic dual torus to T . Relation to other work.
The main use of the term character sheaf is of course due to Lusztig.It is applied to certain perverse sheaves on connected reductive algebraic groups over algebraicallyclosed fields in [28, Def. 2.10] and to certain perverse sheaves on certain reductive groups over al-gebraically closed fields in the series of papers beginning with [29]. When commutative, it is notdifficult to relate Frobenius-stable character sheaves to our character sheaves (Remark 3.11).The new features that we have found pertaining to Weil sheaves and H ( π ( ¯ G ) , ¯ Q × ℓ ) W donot arise in that context because, for such groups, Weil sheaves are unnecessary ( § ( π ( ¯ G ) , ¯ Q × ℓ ) W = 0 (Remark 2.9).For a connected commutative algebraic group over a finite field, it is not uncommon to referto local systems satisfying (1) as character sheaves; see for example, [25, Intro]. Our definitionof character sheaves on smooth commutative group schemes over finite fields evolved from thisnotion, with an eye toward quasicharacters of p -adic groups. The process of creating a categoryfrom the group of quasicharacters of a p -adic torus informs our choice of the term quasicharactersheaf in this paper.We anticipate that future work on quasicharacter sheaves will make use of [32] and [31],and will clarify the relation between this project and other attempts to geometrize admissibledistributions on p -adic groups, such as [12] (limited to quasicharacters of Z × p ) and [1] (limited tocharacters of depth-zero representations). We are actively pursuing the question of how to extendthe notion of quasicharacter sheaves to provide a geometrization of admissible distributions onconnected reductive algebraic groups over p -adic fields, not just commutative ones. Acknowledgements.
We thank Pramod Achar, Masoud Kamgarpour and Hadi Salmasian forallowing us to hijack much of a Research in Teams meeting at the Banff International ResearchStation into a discussion of quasicharacter sheaves. We also thank them for their kindness,knowledge and invaluable help. We thank Takashi Suzuki for clarifying the relation betweenthis project and his recent work and for very helpful observations and suggestions, especiallyrelated to our use of the Hochschild-Serre spectral sequence. We thank Alessandra Bertrapelleand Cristian Gonz´ales-Avil´es for disabusing us of a misapprehension concerning the Greenbergrealization functor and for drawing our attention to their result on Weil restriction and theGreenberg transform. Finally, we thank Joseph Bernstein for very helpful comments.Finally, we gratefully acknowledge the financial support of the Pacific Institute for the Math-ematical Sciences and the National Science and Engineering Research Council (Canada), as wellthe hospitality of the Banff International Research Station during a Research in Teams program.
Contents
Introduction 1Application to quasicharacters of p -adic tori and abelian varieties 3Relation to other work. 4Acknowledgements. 41. Definitions and recollections 51.1. Notations 51.2. Character sheaves on commutative group schemes over finite fields 61.3. Trace of Frobenius 71.4. Descent 91.5. Discrete isogenies 10 ROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF p -ADIC TORI 5 p -adic tori 244.1. N´eron models 244.2. Quasicharacters 254.3. Review of the Greenberg transform 254.4. Quasicharacter sheaves 264.5. Quasicharacter sheaves for p -adic tori 274.6. Weil restriction and quasicharacter sheaves 294.7. Transfer of quasicharacters 30References 311. Definitions and recollections
Notations.
Throughout this paper, G is a smooth commutative group scheme over a finitefield k and m : G × G → G is its multiplication morphism.We will make use of the short exact sequence of smooth group schemes defining the componentgroup scheme for G : 0 G G π ( G ) 0 . ι π Then G is a connected algebraic group and π ( G ) is an ´etale commutative group scheme. Incontrast to the case of algebraic varieties, the component group scheme π ( G ) for G need not befinite.It follows from the smoothness of G that the structure morphism G → Spec( k ) is locally offinite type, being smooth. If the structure morphism G → Spec( k ) is also ´etale, then G is an´etale group scheme; this does not imply that π ( G ) is finite. An algebraic group over k is asmooth group scheme of finite type, in which case its component group scheme is finite.We fix an algebraic closure ¯ k of k and write ¯ G for the smooth commutative group scheme G × Spec( k ) Spec(¯ k ) over ¯ k obtained by base change from k . The multiplication morphism for ¯ G will be denoted by ¯ m .Let Fr denote the geometric Frobenius element in Gal(¯ k/k ) as well as the correspondingautomorphism of Spec(¯ k ). The Weil group W ⊂
Gal(¯ k/k ) is the subgroup generated by Fr. LetFr G := id G × Fr be the Frobenius automorphism of ¯ G = G × Spec( k ) Spec(¯ k ). CLIFTON CUNNINGHAM AND DAVID ROE
We fix a prime ℓ , invertible in k . We will work with constructible ℓ -adic sheaves [14, § k , employing the standard formalism.We also make extensive use of the external tensor product of ℓ -adic sheaves, defined as follows:if F and G are constructible ℓ -adic sheaves on schemes X and Y and p X : X × Y → X and p Y : X × Y → Y are the projections, then F ⊠ G := p ∗ X F ⊗ p ∗ Y G .For any commutative group A , we will write A ∗ for the dual group Hom( A, ¯ Q × ℓ ).1.2. Character sheaves on commutative group schemes over finite fields.Definition 1.1. A character sheaf on G is a triple L := ( ¯ L , µ, φ ) where: (CS.1) ¯ L is a rank-one ℓ -adic local system on ¯ G , by which we mean a constructible ℓ -adic sheaf on¯ G , lisse on each connected component of ¯ G , whose stalks are one-dimensional ¯ Q ℓ -vectorspaces; (CS.2) µ : ¯ m ∗ ¯ L → ¯ L ⊠ ¯ L is an isomorphism of sheaves on ¯ G × ¯ G such that the following diagramcommutes, where m := m ◦ ( m × id) = m ◦ (id × m );¯ m ∗ ¯ L ¯ m ∗ ¯ L ⊠ ¯ L ¯ L ⊠ ¯ m ∗ ¯ L ¯ L ⊠ ¯ L ⊠ ¯ L ( ¯ m × id) ∗ µ (id × ¯ m ) ∗ µ µ ⊠ idid ⊠ µ (CS.3) φ : Fr ∗ G ¯ L → ¯ L is an isomorphism of constructible ℓ -adic sheaves on ¯ G compatible with µ in the sense that the following diagram commutes.Fr ∗ G × G ¯ m ∗ ¯ L Fr ∗ G × G ( ¯ L ⊠ ¯ L )¯ m ∗ Fr ∗ G ¯ L Fr ∗ G ¯ L ⊠ Fr ∗ G ¯ L ¯ m ∗ ¯ L ¯ L ⊠ ¯ L Fr ∗ G × G µ ¯ m ∗ φ φ ⊠ φµ Morphisms of character sheaves are defined in the natural way: (CS.4) if L = ( ¯ L , µ, φ ) and L ′ = ( ¯ L ′ , µ ′ , φ ′ ) are character sheaves on G then a morphism ρ : L → L ′ is a map ¯ ρ : ¯ L → ¯ L ′ of constructible ℓ -adic sheaves on ¯ G such that the followingdiagrams both commute.Fr ∗ G ¯ L Fr ∗ G ¯ L ′ ¯ m ∗ ¯ L ¯ m ∗ ¯ L ′ ¯ L ¯ L ′ ¯ L ⊠ ¯ L ¯ L ′ ⊠ ¯ L ′ Fr ∗ G ¯ ρφ φ ′ µ ¯ m ∗ ¯ ρ µ ′ ¯ ρ ρ ⊠ ρ The category of character sheaves on G will be denoted by CS ( G ).Category CS ( G ) is a rigid monoidal category [18, § L⊗L ′ definedby ( ¯ L ⊗ ¯ L ′ , µ ⊗ µ ′ , φ ⊗ φ ′ ), with duals given by applying the sheaf hom functor H om ( − , ¯ Q ℓ ).This rigid monoidal category structure for CS ( G ) gives the set CS ( G ) / iso of isomorphism classesin CS ( G ) the structure of a group. ROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF p -ADIC TORI 7 Remark 1.2.
The category of character sheaves on G is not abelian since it is not closed underdirect sums; thus CS ( G ) is not a tensor category in the sense of [15, 0.1]. We suspect thatrequiring that µ be injective rather than an isomorphism and dropping the condition that thestalks be one-dimensional would yield an abelian category.We will describe the group CS ( G ) / iso in Theorem 3.6 and the sets Hom( L , L ′ ) in Theorem 3.9;in this way we provide a complete description of the category CS ( G ). In the meantime, we makean elementary observation about Hom( L , L ′ ). Lemma 1.3.
Let G be a smooth commutative group scheme over k . If L and L ′ are charactersheaves on G , then every ρ ∈ Hom( L , L ′ ) is either trivial (zero on every stalk) or an isomorphism.Proof. Suppose ρ ∈ Hom( L , L ′ ). We prove the lemma by considering the linear transformations¯ ρ ¯ g : ¯ L ¯ g → ¯ L ¯ g at the stalks above geometric points ¯ g on G and showing that, either each ¯ ρ ¯ g trivial or each ¯ ρ ¯ g is an isomorphism. (This idea is expanded upon in Section 2.1.) Let ¯ e bethe geometric point above the identity e for G determined by our choice of algebraic closure ¯ k of k . If ¯ ρ ¯ e = 0 then the second diagram in (CS.4) implies that ¯ ρ ¯ g = 0 for all ¯ g , in which case ρ is trivial. On the other hand, if ¯ ρ ¯ e is non-trivial then the second diagram in (CS.4) impliesthat ¯ ρ ¯ g is non-trivial for all ¯ g and thus an isomorphism, since the stalks of character sheaves areone-dimensional; in this case ρ is an isomorphism. (cid:3) Trace of Frobenius.
In this section we introduce two tools which will help us understandisomorphism classes of objects in CS ( G ): the map CS ( G ) / iso → G ( k ) ∗ given by trace of Frobeniusand the pullback functor CS ( G ) → CS ( H ) associated to a morphism H → G of smooth groupschemes over k .Let ( ¯ L , φ ) be a Weil sheaf on G . Every g ∈ G ( k ) determines a geometric point ¯ g fixed by Fr G .Together with the canonical isomorphism (Fr ∗ G ¯ L ) ¯ g ∼ = ¯ L Fr G (¯ g ) , the automorphism φ determinesan automorphism φ ¯ g of the ¯ Q ℓ -vector space ¯ L ¯ g . Let Tr( φ ¯ g ; ¯ L ¯ g ) be the trace of φ ¯ g ∈ Aut ¯ Q ℓ ( ¯ L ¯ g )and let t ( ¯ L ,φ ) : G ( k ) → ¯ Q ℓ be the function defined by(2) t ( ¯ L ,φ ) ( g ) := Tr( φ ¯ g ; ¯ L ¯ g ) , commonly called the trace of Frobenius of ( ¯ L , φ ). Note that if ( ¯ L , φ ) ∼ = ( ¯ L ′ , φ ′ ) as Weil sheaves,then t ( ¯ L ,φ ) = t ( ¯ L ′ ,φ ′ ) as functions on G ( k ).Now suppose L = ( ¯ L , µ, φ ) is a character sheaf on G . Then the isomorphism ¯ m ∗ ¯ L ∼ = ¯ L ⊠ ¯ L andthe diagram of (CS.3) guarantee that the function t ( ¯ L ,φ ) : G ( k ) → ¯ Q × ℓ is a group homomorphism,which we will also denote by t L . Moreover, this homomorphism depends only on the isomorphismclass of L , so we obtain a map Tr G : CS ( G ) / iso → G ( k ) ∗ , L 7→ t L . Since tensor products on the stalks of L induce pointwise multiplication on the trace of Frobenius,Tr G is a group homomorphism.The next two results follow easily from the definitions. Lemma 1.4. If f : H → G is a morphism of smooth commutative group schemes over k , then f ∗ : CS ( G ) → CS ( H )( ¯ L , µ, φ ) ( ¯ f ∗ ¯ L , ( ¯ f × ¯ f ) ∗ µ, ¯ f ∗ F ) CLIFTON CUNNINGHAM AND DAVID ROE defines a monoidal functor dual to f : H ( k ) → G ( k ) in the sense that CS ( G ) / iso CS ( H ) / iso G ( k ) ∗ H ( k ) ∗ f ∗ Tr G Tr H is a commutative diagram of groups. Moreover, ( f ◦ g ) ∗ = g ∗ ◦ f ∗ .Proof. Let L be a character sheaf on G . Pullback by ¯ f takes rank-one local systems to rank-onelocal systems. To see that ( ¯ f × ¯ f ) ∗ µ satisfies (CS.2), apply the functor ( ¯ f × ¯ f ) ∗ to (CS.2) for L anduse the canonical isomorphism ( ¯ f × ¯ f ) ∗ ( ¯ L ⊠ ¯ L ) ∼ = ¯ f ∗ ¯ L ⊠ ¯ f ∗ ¯ L . To show that f ∗ L satisfies (CS.3),apply the same functor to (CS.3) for L . Since f is a morphism of group schemes defined over k itprovides isomorphisms ( ¯ f × ¯ f ) ∗ Fr ∗ G × G ∼ = Fr ∗ G × G ( ¯ f × ¯ f ) ∗ and ( ¯ f × ¯ f ) ∗ ¯ m ∗ ∼ = ¯ m ∗ ¯ f ∗ between functorsof constructible sheaves.Applying ¯ f ∗ and ¯ f ∗ × ¯ f ∗ to (CS.4) defines the action of f ∗ on morphisms of character sheaves;arguing as above shows that f ∗ is a functor from CS ( G ) to CS ( H ). Since tensor products commutewith pullback in schemes, f ∗ : CS ( G ) → CS ( H ) is a monoidal functor. The diagram relating f ∗ : CS ( G ) → CS ( H ), f ∗ : G ( k ) ∗ → H ( k ) ∗ and trace of Frobenius commutes by [27, 1.1.1.2],where the ambient finite type hypothesis can be replaced by locally of finite type.Finally, the fact that ( f ◦ g ) ∗ = g ∗ ◦ f ∗ follows from the analogous statements about thepullback functor on ℓ -adic constructible sheaves. (cid:3) If G and G are smooth commutative group schemes over k then characters of ( G × G )( k )all take the form χ ⊗ χ for characters χ of G ( k ) and χ of G ( k ). The next lemma showsthat character sheaves on G enjoy an analogous property. Lemma 1.5. If G and G are smooth commutative group schemes over k then the followingdiagram commutes. CS ( G ) / iso × CS ( G ) / iso CS ( G × G ) / iso ( G )( k ) ∗ × ( G )( k ) ∗ ( G × G )( k ) ∗ Tr G × Tr G ( L , L ) ⊠ L Tr G × G ( χ ,χ ) χ ⊗ χ Moreover, every character sheaf on G × G is isomorphic to L ⊠ L for some character sheaves L on G and L on G .Proof. The only non-trivial part is the last claim, so we will only address that point here. Set G := G × G and write e and e for the identity elements of G and G . Define f : G → G × G by f ( g , g ) := ( g , e , e , g ). Observe that m ◦ f = id G . Let p , p be the projection morphismspictured below: G G × G G. p p Let r and r be the projection morphisms pictured below, with sections q and q , also mor-phisms of group schemes: G G × G G . q r r q ROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF p -ADIC TORI 9 Observe that p ◦ f = q ◦ r and p ◦ f = q ◦ r . Now, let L := ( ¯ L , µ, φ ) be a character sheaf on G and set L := q ∗ L and L := q ∗ L . By Lemma 1.4, L is a character sheaf on G and L is acharacter sheaf on G . We will obtain an isomorphism L ∼ = L ⊠ L .Applying the functor f ∗ to the isomorphism µ yields(3) f ∗ µ : f ∗ m ∗ ¯ L → f ∗ ( ¯ L ⊠ ¯ L ) . We have already seen that m ◦ f = id G , so f ∗ m ∗ ¯ L = ¯ L . Since f ∗ p ∗ ¯ L = r ∗ q ∗ ¯ L = r ∗ ¯ L and f ∗ p ∗ ¯ L = r ∗ q ∗ ¯ L = r ∗ ¯ L , we have f ∗ ( ¯ L ⊠ ¯ L ) = f ∗ p ∗ ¯ L ⊗ f ∗ p ∗ ¯ L = ¯ L ⊠ ¯ L . It follows that (3) gives an isomorphism ¯
L → ¯ L ⊠ ¯ L . It is routine to show that this morphismsatisfies (CS.4), as it applies here, from which it follows that we have exhibited an isomorphism L → L ⊠ L of characters sheaves on G × G . (cid:3) Using these results on pullbacks and products, we may prove a naturality property of Tr G . Proposition 1.6.
The homomorphism Tr G : CS ( G ) / iso → G ( k ) ∗ defines a natural transforma-tion between the two contravariant additive functors F : G
7→ CS ( G ) / iso F : G G ( k ) ∗ from the category of smooth commutative group schemes over k to the category of commutativegroups.Proof. The first part of Lemma 1.4 shows that F is a functor, while the second part showsthat Trace of Frobenius is a natural transformation T : F → F . When further combined withLemma 1.5, we see that F is an additive functor and T : F → F is a natural transformationbetween additive functors, concluding the proof of Proposition 1.6. (cid:3) Descent.
In this section we consider a category of sheaves on G obtained by replacing theWeil sheaf ( ¯ L , φ ) on ¯ G in the definition of a character sheaf with an ℓ -adic local system on G itself; these will play a role in Sections 1.6 and 3.6. Definition 1.7.
Let CS ( G ) be the category of pairs ( E , µ E ) where E is an ℓ -adic local systemon G of rank-one, equipped with an isomorphism µ E : m ∗ E → E ⊠ E satisfying the analogue of(CS.2) on G ; morphisms in CS ( G ) are defined as in the second part of (CS.4).We put a rigid monoidal structure on CS ( G ) in the same way as for CS ( G ). Proposition 1.8.
Extension of scalars defines a full and faithful functor B G : CS ( G ) → CS ( G ) . Proof.
Suppose ( E , µ E ) in an object of CS ( G ). Let b G : ¯ G → G be the pullback of Spec(¯ k ) → Spec( k ) along G → Spec( k ). Set ¯ L = b ∗ G E . The functor b ∗ G takes local systems on G to localsystems on ¯ G . The local system ¯ L comes equipped with an isomorphism φ : Fr ∗ G ¯ L → ¯ L . Theresulting functor from local systems on G to Weil local systems on ¯ G , given on objects by E 7→ ( ¯ L , φ ), is full and faithful; see [16, Expos´e XIII] and [2, Prop. 5.1.2]. The isomorphism µ := b ∗ G × G µ E satisfies (CS.2) for ¯ L and φ is compatible with µ in the sense of (CS.3). Thisconstruction defines the functor B G : CS ( G ) → CS ( G ) given on objects by ( E , µ E ) ( ¯ L , µ, φ ),as defined here. Because morphisms in CS ( G ) and CS ( G ) are morphisms of local systems on G and ¯ G , respectively, satisfying condition (CS.4), this functor is also full and faithful. (cid:3) We will say that a character sheaf
L ∈ CS ( G ) descends to G if it is isomorphic to some B G ( E , µ E ). Remark 1.9.
In fact, it is not difficult to recognize character sheaves that descend to G : theyare exactly those character sheaves L = ( ¯ L , µ, φ ) for which the action of W on ¯ L given by φ extends to a continuous action of Gal(¯ k/k ) on ¯ L ; see [16, Expos´e XIII, Rappel 1.1.3] for example.1.5. Discrete isogenies.
Here we consider character sheaves on G that are defined by discreteisogenies onto G ( § H → G of smooth group schemes over k for which theaction of Gal(¯ k/k ) on the kernel is trivial is called a discrete isogeny , inspired by [25, § Proposition 1.10.
Let f : H → G be a discrete isogeny and let A be the kernel of f . Let V be a -dimensional representation of A equipped with an isomorphism V → V ⊗ V . Let ψ : A → ¯ Q × ℓ be the character of V . Then ( f ! V H ) ψ (the ψ -isotypic component of f ! V H ) is an object of CS ( G ) .Proof. Let f , A , V and ψ be as above and set E = ( f ! V H ) ψ . Since A is abelian, E is an ℓ -adiclocal system on G of rank one. We must show that E comes equipped with an isomorphism µ E : m ∗ E → E ⊠ E . To do this we use ´etale descent to see that pullback along f gives anequivalence between ℓ -adic local systems on G and A -equivariant local systems on H cf. [3, Prop8.1.1]. In particular, f ∗ E is the A -equivariant constant sheaf V on H with character ψ . Since f is a morphism of group schemes, the functor f ∗ defines µ E : m ∗ E → E ⊠ E from the isomorphism m ∗ ψ ∼ = ψ ⊠ ψ determined by V → V ⊗ V . (cid:3) Remark 1.11.
Since V is 1-dimensional, the choice of V → V ⊗ V is exactly the choice of anisomorphism V ∼ = ¯ Q ℓ . Remark 1.12.
A descent argument similar to the one employed in the proof of Lemma 1.10 isused in [7, Lemma 1.10], though in the more restrictive case of connected algebraic groups.1.6.
Recollections on character sheaves for connected algebraic groups.
If the smoothcommutative group scheme G is of finite type, then every character sheaf descends to G ; we willsee that they are not equivalent, in general, however. Lemma 1.13. If G is a connected commutative algebraic group over k then B G : CS ( G ) → CS ( G ) is an equivalence of categories.Proof. Choose any k -rational point g on G and let ¯ g be the geometric point on G lying above g . Recall that the Weil group of G , which we will denote by W ( G, ¯ g ), is a subgroup of the ´etalefundamental group defined by the following diagram:1 π ( ¯ G, ¯ g ) W ( G, ¯ g ) W π ( ¯ G, ¯ g ) π ( G, ¯ g ) Gal(¯ k/k ) 1 . The k -rational point g under the geometric point ¯ g determines a splitting W → W ( G, ¯ g ) of W ( G, ¯ g ) → W . Since G is connected, the geometric point ¯ g determines an equivalence betweenthe category of ℓ -adic Weil local systems on G and ℓ -adic representations of W ( G, ¯ g ) [14, 1.1.12].Now let ( ¯ L , µ, φ ) be a character sheaf on G and let λ : W ( G, ¯ g ) → ¯ Q × ℓ be the characterdetermined by ( ¯ L , φ ). Composing with the splitting W → W ( G, ¯ g ) yields an ℓ -adic character λ g : W → ¯ Q × ℓ , which is the same as the Trace of Frobenius defined in Section 1.3, for every k rational point g on G : λ g (Fr) = t L ( g ) . On the other hand, we have already seen that t L : G ( k ) → ¯ Q × ℓ is a group homomorphism. Since G is an algebraic group over k , G ( k ) is finite.Therefore t L ( g ) = λ g (Fr) is a root of unity for every g ∈ G ( k ). Since W is generated by Fr ROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF p -ADIC TORI 11 and λ g : W → ¯ Q × ℓ is a character, it follows that the image of λ g is a finite group. Thus, λ g extends to an ℓ -adic character of Gal(¯ k/k ), which we will also denote λ g . We may now lift the ℓ -adic character λ g : Gal(¯ k/k ) → ¯ Q × ℓ to an ℓ -adic character π ( G, ¯ g ) → ¯ Q × ℓ using the canonicaltopological group homomorphism π ( G, ¯ g ) → Gal(¯ k/k ). The k rational point g also determinesan equivalence between the category of ℓ -adic representations of π ( G, ¯ g ) and ℓ -adic local systemson G . Let E be a local system on G in the isomorphism class determined by this ℓ -adic characterof π ( G, ¯ g ). Then b ∗ G E ∼ = ¯ L .Since b ∗ G × G is full and faithful (again, see [16, Expos´e XIII] or [2, Prop. 5.1.2]), b ∗ G × G : Hom( m ∗ E , E ⊠ E ) → Hom( ¯ m ∗ ¯ L , ¯ L ⊠ ¯ L )is a bijection (hom taken in the categories on constructible ℓ -adic sheaves on G × G and ¯ G × ¯ G respectively, in which ℓ -adic local systems sit as full subcategories). Let µ E : m ∗ E → E ⊠ E be theisomorphism matching µ : ¯ m ∗ ¯ L → ¯ L ⊠ ¯ L , the latter appearing in the definition of L . Then, asin Section 1.4, ( E , µ E ) is an object of CS ( G ) and L := ( ¯ L , µ, φ ) is isomorphic to ( b ∗ G E , b ∗ G × G µ E )in CS ( G ). Thus, the full and faithful functor B G : CS ( G ) → CS ( G ) from Section 1.4 is alsoessentially surjective, hence an equivalence. (cid:3) Using this equivalence of categories, we may give a good description of CS ( G ) when G isconnected and finite type. Proposition 1.14. If G is a connected, commutative algebraic group over k then: (1) Tr G : CS ( G ) / iso → G ( k ) ∗ is an isomorphism of groups; (2) every character sheaf on G is isomorphic to one defined by a discrete isogeny; (3) Aut( L ) = 1 , for all character sheaves L on G .Proof. By Lemma 1.13, we know that every character sheaf L on ¯ G descends to G ; let E bean object of CS ( G ) for which B G ( E ) ∼ = L . Since the functor B G : CS ( G ) → CS ( G ) is fulland faithful, Aut( L ) = Aut( E ). From here, Deligne’s function-sheaf dictionary for connectedcommutative algebraic groups over finite fields, as in [13, Sommes trig.] or [27, 1.1.3], gives usall we need for points (1) and (2), as we briefly recall.As in the proof of Proposition 1.10, use ´etale descent to see that pullback by the Lang isogenyLang : G → G defines an equivalence of categories between local systems on G and G ( k )-equivariant local systems on G . Under this equivalence, local systems E on G arising fromobjects in CS ( G ) are matched with G ( k )-equivariant constant local systems of rank-one on G ,and therefore with one-dimensional representations of G ( k ). In the same way, pullback along theisogeny Lang × Lang : G × G → G × G matches the extra structure µ E : m ∗ E → E ⊠ E with anisomorphism m ∗ V → V ⊠ V of one-dimensional representations of G ( k ) × G ( k ), which is exactlyan isomorphism V → V ⊗ V of one-dimensional representations, which is exactly the choice of anisomorphism V ∼ = ¯ Q ℓ . We see that CS ( G ) is equivalent to the category of characters of G ( k ).Let ¯ g be a geometric point above g ∈ G ( k ). If E matches ψ : G ( k ) → ¯ Q × ℓ under this equivalence,a simple calculation on stalks reveals that the action of Frobenius on E ¯ g is multiplication by ψ ( g ) − . In other words, for every E in CS ( G ), the trace of Lang ∗ E is t − E as a representation of G ( k ), proving parts (1) and (2).For part (3), suppose Lang ∗ E = V with isomorphism V → V ⊗ V . Observe that the equiva-lence above establishes a bijection between Aut( E ) and the group of automorphisms of ρ : V → V for which V VV ⊗ V V ⊗ V ρρ ⊗ ρ commutes. Since the only such isomorphism ρ is id V , it follows that Aut( E ) = 1, completing theproof. (cid:3) We have just seen that, for a connected commutative algebraic group G over k , the category ofcharacter sheaves on G is equivalent to the category of one-dimensional representations V of G ( k )equipped with an isomorphism V ∼ = ¯ Q ℓ , and therefore equivalent to the category of characters ψ of G ( k ). We have also just seen that if the character of Lang ∗ E is ψ then the canonicalisomorphism m ∗ ψ ∼ = ψ ⊠ ψ determines the isomorphism µ E : E → E ⊠ E . This fact leads (back)to a perspective on the function-sheaf dictionary common in the literature in which one considersone-dimensional local systems E on G for which there exists an isomorphism m ∗ E ∼ = E ⊠ E [25, Intro]. As a slight variation, one may also consider one-dimensional local systems ¯ L on ¯ G for which there exists an isomorphism Fr ∗ G ¯ L ∼ = ¯ L and an isomorphism ¯ m ∗ ¯ L ∼ = ¯ L ⊠ ¯ L .Although the category CS ( G ) of character sheaves on G specializes to CS ( G ) when G is offinite type ( § not sufficient when extending the dictionary to smoothcommutative group schemes, as we will see already in Section 2. In particular, for a given ¯ L and φ there may be many µ so that ( ¯ L , µ, φ ) is a character sheaf. For ´etale G , Proposition 2.7 showsthat H ( ¯ G, ¯ Q × ℓ ) W measures the possibilities for µ . We will see in Section 3 that H ( π ( ¯ G ) , ¯ Q × ℓ ) W plays an analogous role for general smooth commutative group schemes G .2. Character sheaves on ´etale commutative group schemes over finite fields
In this section we give a complete characterization of the category of character sheaves on´etale commutative group schemes over finite fields.2.1.
Stalks of character sheaves.
The equivalence G G (¯ k ), from the category of ´etalecommutative group schemes over k to the category of commutative groups equipped with acontinuous action of Gal(¯ k/k ), provides the following simple description of character sheaves. Acharacter sheaf L on an ´etale commutative group scheme G over k is: (cs.1) an indexed set of one-dimensional ¯ Q ℓ -vector spaces ¯ L x , as x runs over G (¯ k ); (cs.2) an indexed set of isomorphisms µ x,y : ¯ L x + y ∼ = −−→ ¯ L x ⊗ ¯ L y , for all x, y ∈ G (¯ k ), such that¯ L x + y + z ¯ L x + y ⊗ ¯ L z ¯ L x ⊗ ¯ L y + z ¯ L x ⊗ ¯ L y ⊗ ¯ L zµ x + y,z µ x,y + z µ x,y ⊗ idid ⊗ µ y,z commutes, for all x, y, z ∈ G (¯ k ); and (cs.3) an indexed set of isomorphisms φ x : ¯ L Fr( x ) → ¯ L x such that¯ L Fr( x )+Fr( y ) ¯ L Fr( x ) ⊗ ¯ L Fr( y ) ¯ L x + y ¯ L x ⊗ ¯ L yφ x + y µ Fr( x ) , Fr( y ) φ x ⊗ φ y µ x,y commutes, for all x, y ∈ G (¯ k ).Under this equivalence, a morphism ρ : L → L ′ of character sheaves on G is given by ROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF p -ADIC TORI 13 (cs.4) an indexed set ¯ ρ x : ¯ L x → ¯ L ′ x of linear transformations such that¯ L Fr( x ) ¯ L ′ Fr( x ) ¯ L x + y ¯ L ′ x + y ¯ L x ¯ L ′ x ¯ L x ⊗ ¯ L y ¯ L ′ x ⊗ ¯ L ′ yφ x ¯ ρ Fr( x ) φ ′ x and µ x,y ¯ ρ x + y µ ′ x,y ¯ ρ x ¯ ρ x ⊗ ¯ ρ y both commute, for all x, y ∈ G (¯ k ).We will see that Tr G : CS ( G ) / iso → G ( k ) ∗ may not provide complete information aboutisomorphism classes of character sheaves on G when G is not a connected algebraic group. Ourmain tool for understanding this phenomenon is a group homomorphism S G : CS ( G ) / iso → H ( E • G ) defined in Section 2.3, for which the next two sections are preparation.2.2. A spectral sequence.
Let G be a smooth commutative group scheme over k . The zerothpage of the Hochschild-Serre spectral sequence is a double complex E • , • defined by E i,j = C i ( W , C j ( G (¯ k ) , ¯ Q × ℓ ));see [33, § § d G : E i,j → E i,j +1 and d W : E i,j → E i +1 ,j ;we use the first as the derivative d on the zeroth page, and the second to induce d . Combiningthem also yields a derivative d = d G + ( − j d W on the total complex E nG = M i + j = n E i,j . The machinery of spectral sequences gives us a sequence of pages E i,jr , converging to a page E i,j ∞ .We summarize the key properties of this spectral sequence in the following proposition. Proposition 2.1.
In the spectral sequence defined above,(1) the second page is given by E i,j = H i ( W , H j ( G (¯ k ) , ¯ Q × ℓ )) ,(2) there is an isomorphism H n ( W ⋉ G (¯ k ) , ¯ Q × ℓ ) ∼ = H n ( E • G ) , and(3) there is a filtration H n ( W ⋉ G (¯ k ) , ¯ Q × ℓ ) = F n ⊃ · · · ⊃ F − = 0 , with F i /F i − ∼ = E i,n − i ∞ . Moreover, since
W ∼ = Z has cohomological dimension 1, E i,j = 0 for i > E i,j ∞ = E i,j . We obtain the following corollary: Corollary 2.2.
There is a short exact sequence → H ( W , H ( ¯ G, ¯ Q × ℓ )) → H ( E • G ) → H ( W , H ( ¯ G, ¯ Q × ℓ )) → . This sequence will play a key role in understanding the kernel of Tr G , as described in the nextfew sections. For this application, we need a good understanding of these maps to and from thetotal complex. Proposition 2.3.
Consider the short exact sequence in Corollary 2.2.(1) Every class [ α ⊕ β ⊕ γ ] ∈ H ( E • G ) is cohomologous to one with γ = 0 .(2) The map H ( E • G ) → H ( W , H ( ¯ G, ¯ Q × ℓ )) is given by [ α ⊕ β ⊕ [ β ] .(3) Suppose a ∈ Z ( ¯ G, ¯ Q × ℓ ) represents a class in H ( ¯ G, ¯ Q × ℓ ) fixed by Frobenius. The map H ( W , H ( ¯ G, ¯ Q × ℓ )) → H ( E • G ) is given by [ a ] [ a ⊕ ⊕ . Proof.
Since H ( W , C ( ¯ G, ¯ Q × ℓ )) = 0, we may find a γ ∈ C ( W , C ( ¯ G, ¯ Q × ℓ )) with d W γ = γ .Subtracting dγ from α ⊕ β ⊕ γ , we may assume that γ = 0.The latter two claims follow from tracing through the definition of latter pages in the spectralsequence. (cid:3) From character sheaves to the total complex.
Let G be a smooth commutative groupscheme over k . In this section we define a group homomorphism S G : CS ( G ) / iso → H ( E • G ) . Let L = ( ¯ L , µ, φ ) be a character sheaf on G . For each geometric point x ∈ ¯ G , choose a basis { v x } for ¯ L x . Through this choice, L determines functions a : ¯ G × ¯ G → ¯ Q × ℓ b : ¯ G → ¯ Q × ℓ µ x,y ( v x + y ) = a ( x, y ) v x ⊗ v y φ x ( v Fr G ( x ) ) = b ( x ) v x . Condition (CS.2) implies that(4) a ( x + y, z ) a ( x, y ) = a ( x, y + z ) a ( y, z )for all x, y, z ∈ ¯ G , so a ∈ Z ( ¯ G, ¯ Q × ℓ ). Similarly, condition (CS.3) gives(5) a (Fr G ( x ) , Fr G ( y )) a ( x, y ) = b ( x + y ) b ( x ) b ( y )for x, y ∈ ¯ G . Let α ∈ C ( W , C ( ¯ G, ¯ Q × ℓ ) be the 0-cochain corresponding to a and let β ∈ C ( W , C ( ¯ G, ¯ Q × ℓ ) be the cocycle such that β (Fr) is b . We will write both α and β additively.Then d G α = 0 , d W α = d G β, d W β = 0;in other words, α ⊕ β ∈ Z ( E • G ) . Although the cocycle α ⊕ β is not well defined by L , its class in H ( E • G ) is. To see this, let { v ′ x ∈ ¯ L × x | x ∈ ¯ G } be another choice and let α ′ ⊕ β ′ ∈ Z ( E • G ) be defined by L and this choice,as above. Now let δ ∈ C ( W , C ( ¯ G, ¯ Q × ℓ )) correspond to the function d : ¯ G → ¯ Q × ℓ defined by v ′ x = d ( x ) v x . Chasing through (CS.2) and (CS.3), we find α ′ ⊕ β ′ = α ⊕ β + dδ, so the class [ α ⊕ β ] of α ⊕ β in H ( E • G ) is independent of the choice made above. It is also easyto see that [ α ⊕ β ] = [ α ⊕ β ] when L ∼ = L , which concludes the definition of the function S G : CS ( G ) / iso → H ( E • G )[ L ] [ α ⊕ β ] . It is also easy to see that [ α ⊕ β ] + [ α ⊕ β ] = [ α ⊕ β ] when L = L ⊗ L , so S G is a grouphomomorphism. Proposition 2.4. If G is ´etale then S G : CS ( G ) / iso → H ( E • G ) is an isomorphism.Proof. Suppose [ L ] ∈ CS ( G ) / iso with S G ([ L ]) = [ α ⊕ β ] = 0, so that α ⊕ β = dσ for some σ ∈ C ( W , C ( ¯ G, ¯ Q × ℓ ) = C ( ¯ G, ¯ Q × ℓ ). For each x ∈ ¯ G , define σ x : ¯ L x → ¯ Q ℓ by σ x : v x σ ( x ). Thenthe indexed set of isomorphisms { σ x : ¯ L x → ¯ Q ℓ | x ∈ ¯ G } defines an isomorphism L → ( ¯ Q ℓ ) G .Since L = 0 ∈ CS ( G ) / iso , S G is injective.To see that S G is surjective, begin with α ⊕ β ⊕ ∈ Z ( E • G ). Since d W β = 0, we may define a = α ∈ C ( ¯ G, ¯ Q × ℓ ) and b = β (Fr) ∈ C ( ¯ G, ¯ Q × ℓ ), which are related to α and β as above. Set¯ L x = ¯ Q ℓ , define µ x,y : ¯ L x + y → ¯ L x ⊗ ¯ L y by µ x,y (1) = a ( x, y )(1 ⊗
1) and φ x : ¯ L Fr G ( x ) → ¯ L x by φ x (1) = b ( x ). Then (CS.1) holds since d G α = 0 and (CS.2) holds since d W α = d G β . Tracing ROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF p -ADIC TORI 15 the construction backward, we have defined a character sheaf L on G with S G ( L ) = [ α ⊕ β ⊕ S G is surjective. (cid:3) Objects in the ´etale case.
In this section we fit the group homomorphisms Tr G and S G into a commutative diagram, determining the kernel and cokernel of Tr G when G is an´etale commutative group scheme over k . We begin with a simple, general result relating duals,invariants and coinvariants. Lemma 2.5.
Let X be an abelian group equipped with an action of W . Then ( X ∗ ) W → ( X W ) ∗ [ f ] f | X W is an isomorphism.Proof. We can describe X W as the kernel of the map X Fr − −−−→ X ; let Y = (Fr − X be theaugmentation ideal. Dualizing the sequence0 → X W → X → Y → → Y ∗ → X ∗ → ( X W ) ∗ → Ext Z ( Y, ¯ Q × ℓ ) . Since Ext Z ( − , ¯ Q × ℓ ) vanishes, we get a natural isomorphism from the cokernel of Y ∗ Fr − −−−→ X ∗ to( X W ) ∗ . (cid:3) Proposition 2.6. If G is ´etale, then Tr G : CS ( G ) / iso → G ( k ) ∗ is surjective and split.Proof. Pick χ ∈ G ( k ) ∗ . Let [ β ] ∈ H ( W , ¯ G ∗ ) be the class corresponding to χ under Lemma 2.5.Every representative cocycle β ∈ Z ( W , ¯ G ∗ ) determines a homomorphism β (Fr) : G (¯ k ) → ¯ Q × ℓ such that β (Fr) | G ( k ) = χ . Set ¯ L x = ¯ Q ℓ for every x ∈ G (¯ k ). Define µ x,y : ¯ L x + y → ¯ L x ⊗ ¯ L y by µ x,y (1) = 1 ⊗ φ x : ¯ L Fr( x ) → ¯ L x by φ x (1) = β (Fr)( x ). Since β (Fr) : G (¯ k ) → ¯ Q × ℓ is a grouphomomorphism, condition (5) is satisfied with a = 1. So L = ( ¯ L , µ, φ ) is a character sheaf with t L = χ . This shows that Tr G is surjective.Now let β ′ ∈ Z ( W , ¯ G ∗ ) be another representative for [ β ] so β − β ′ = d W δ for some δ ∈ C ( W , ¯ G ∗ ) defining d ∈ Hom( G (¯ k ) , ¯ Q × ℓ ). Let L ′ be the character sheaf on G defined by β ′ , asabove. For each x ∈ G (¯ k ), define ¯ ρ x : L x → L ′ x by ¯ ρ x (1) = d ( x ). The collection of isomorphisms { ¯ ρ x | x ∈ G (¯ k ) } satisfies condition (CS.4), so it defines a morphism ρ : L → L ′ , which is clearlyan isomorphism. We have now defined a section of Tr G .Now suppose χ , χ ∈ G ( k ) ∗ . Pick cocycles β , β ∈ Z ( W , ¯ G ∗ ) and construct charactersheaves L and L on G as above. Since L ⊗ L is exactly the character sheaf built from thecocycle β · β , and since t L ⊗L = t L · t L , the section of Tr G defined here is a homomorphism. (cid:3) Proposition 2.7. If G is ´etale then the map S G : CS ( G ) / iso → H ( E • G ) induces an isomorphismof split short exact sequences G CS ( G ) / iso G ( k ) ∗
00 H ( W , H ( ¯ G, ¯ Q × ℓ )) H ( E • G ) H ( W , H ( ¯ G, ¯ Q × ℓ )) 0 . S G Tr G Proof.
This result follows from Propositions 2.3, 2.4 and 2.6. (cid:3)
Definition 2.8.
We call a character sheaf L on G invisible if it is nontrivial and Tr G ( L ) = 1. The proposition gives a method for determining whether a given G admits invisible charactersheaves. Remark 2.9.
Recall the K¨unneth formula in group cohomology [8, Prop. I.0.8]: if A and A ′ are groups and M and M ′ are abelian groups with M Z -free, thenH n ( A × A ′ , M ⊗ M ′ ) ∼ = M i + j = n H i ( A, M ) ⊗ H j ( A ′ , M ′ ) ⊕ M i + j = n +1 Tor Z (cid:0) H i ( A, M ) , H j ( A ′ , M ′ ) (cid:1) . Now suppose ¯ G = Z r × Q mi =1 Z /N i Z is an arbitrary finitely generated abelian group, with N i | N i +1 . Then the K¨unneth formula implies that(6) H ( ¯ G, ¯ Q × ℓ ) ∼ = (cid:0) ¯ Q × ℓ (cid:1) r ( r − / × m Y i =1 ( Z /N i Z ) m + r − i . We see that H ( ¯ G, ¯ Q × ℓ ) is trivial if and only if ¯ G is cyclic. Of course, H ( W , H ( ¯ G, ¯ Q × ℓ )) may ormay not be trivial, even when H ( ¯ G, ¯ Q × ℓ ) is non-trivial. Example 2.10.
Consider the simplest non-trivial case, where ¯ G = { , i, j, k } ∼ = Z / Z × Z / Z . Using (6), we have H ( ¯ G, ¯ Q × ℓ ) ∼ = Z / Z , on which W must act trivially, regardless of its actionon ¯ G itself. The non-trivial element corresponds to the extension(7) 1 → ¯ Q × ℓ → Q → ¯ G → , where Q = { c + c i i + c j j + c k k | c, c i , c j , c k ∈ ¯ Q ℓ with exactly one nonzero } is a subgroup of thequaternion algebra over ¯ Q ℓ . Let a be a 2-cocycle corresponding to this extension, with valuesin {± } . When Fr G acts trivially on ¯ G , any homomorphism b : ¯ G → ¯ Q × ℓ will satisfy (5), andthe corresponding α ⊕ β are non-cohomologous in H ( E • G ). When Fr G exchanges i and j , thenwe may take b (1) = 1 , b ( i ) = − b ( j ) = b ( k ) = ±
1, up to coboundaries. Finally, when Fr G cycles i , j and k , any homomorphism b : ¯ G → ¯ Q × ℓ will satisfy (5), but now the corresponding α ⊕ β are all cohomologous in H ( E • G ). In each case, we may produce an explicit character sheaffrom the listed a and b .Note that these character sheaves arise from discrete isogenies, as in Section 1.5. Let ¯ H bethe quaternion group of order 8: the subgroup of Q with c, c i , c j , c k ∈ {± } . The sequence (7) isthe pushforward of 1 → {± } → ¯ H → ¯ G → {± } ֒ → ¯ Q × ℓ . Note that these character sheaves arise from a non-commutativecover of ¯ G , justifying the inclusion of such covers in the definition of a discrete isogeny.2.5. On the necessity of working with Weil sheaves.
In this section we justify the appear-ance of Weil sheaves in Definition 1.1.
Proposition 2.11.
Let G be a commutative ´etale group scheme over k . Then the image of CS ( G ) under Tr G : CS ( G ) → G ( k ) ∗ is Hom( G ( k ) , ¯ Z × ℓ ) .Proof. Objects in CS ( G ) may be described by a small modification to the technique used inSections 2.2 and 2.3. Set F i,j := C i cts (Gal(¯ k/k ) , C j ( G (¯ k ) , ¯ Q × ℓ )). Then the results of Section 2.2adapt to give a short exact sequence in continuous Galois cohomology0 → H ( k, H ( ¯ G, ¯ Q × ℓ )) → H ( F • G ) → H ( k, H ( ¯ G, ¯ Q × ℓ )) → , for which the maps are given by the analogues of Proposition 2.3. Moreover, using [16, Expos´eXIII, Rappel 1.1.3] we see that Proposition 2.4 adapts to provide an isomorphism CS ( G ) / iso → ROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF p -ADIC TORI 17 H ( F • G ) compatible with CS ( G ) → CS ( G ) and with the the canonical map of exact sequences0 H ( k, H ( ¯ G, ¯ Q × ℓ )) H ( F • G ) H ( k, H ( ¯ G, ¯ Q × ℓ )) 00 H ( W , H ( ¯ G, ¯ Q × ℓ )) H ( E • G ) H ( W , H ( ¯ G, ¯ Q × ℓ )) 0 . In this way, Proposition 2.11 is now reduced to the claimH ( k, H ( ¯ G, ¯ Q × ℓ )) = Hom( G ( k ) , ¯ Z × ℓ ) . To see that, one may argue as follows. Pick i ∈ π ( G ) and let G i ֒ → G be the correspondingconnected component. Pick a geometric point x on G i and observe that since G i is connected as a k -scheme, G i (¯ k ) is canonically identified with the Gal(¯ k/k )-orbit of x . We remark that while G i is defined over k , the set G i ( k ) is non-empty only when G i (¯ k ) = { x } . Since H ( ¯ G, ¯ Q × ℓ ) =Hom( ¯ G, ¯ Q × ℓ ), evaluation χ χ ( x ) defines H ( k, H ( ¯ G, ¯ Q × ℓ )) → H ( k, ¯ Q × ℓ ). By continuity,H ( k, ¯ Q × ℓ ) = H ( k, ¯ Z × ℓ ). Letting i range over π ( G ) we conclude that H ( k, H ( ¯ G, ¯ Q × ℓ )) =H ( k, H ( ¯ G, ¯ Z × ℓ )). When adapted to abelian groups with continuous action of Gal(¯ k/k ), thestrategy of the proof of Lemma 2.5 gives H ( k, H ( ¯ G, ¯ Z × ℓ )) = Hom( G ( k ) , ¯ Z × ℓ ), concluding theproof. (cid:3) Proposition 2.11 reveals the necessity of working with Weil sheaves in Definition 1.1: one can-not geometrize all characters of G ( k ) using local systems on G , for general smooth commutativegroups schemes G . Proposition 2.11 is extended to all smooth commutative groups schemes inSection 3.4. Example 2.12.
Consider the case when G is the ´etale group scheme Z over k with Fr G trivial.If χ : Z → ¯ Q × ℓ is the character of G ( k ) determined by χ (1) = ℓ and if L is a character sheaf on G in the isomorphism class corresponding to χ under Proposition 2.6, then L does not descendto G , since the image of χ is not bounded. If χ ′ : Z → ¯ Q × ℓ is the character of G ( k ) determinedby χ ′ (1) = 1 + ℓ and if L ′ corresponds to χ ′ under Proposition 2.6, then L ′ does descend to G ,since the image of χ ′ is bounded. However, L ′ is not defined by a discrete isogeny ( § χ ′′ : Z → ¯ Q × ℓ is the character of G ( k ) determined by χ ′′ (1) = ζ , a root of unity in ¯ Q × ℓ , and if L ′′ corresponds to χ ′′ under Proposition 2.6, then L ′′ is defined by a discrete isogeny.2.6. Morphisms in the ´etale case.
A complete understanding of the morphisms in CS ( G )also requires a description of the automorphisms of an arbitrary character sheaf L . Proposition 2.13.
Let G be an ´etale commutative group scheme over k . If L and L ′ arecharacter sheaves on G then every ρ ∈ Hom( L , L ′ ) is either trivial or an isomorphism. Moreover,the trace map induces an isomorphism of groups Aut( L ) → Hom( G (¯ k ) W , ¯ Q × ℓ ) . Proof.
We have already seen, in Lemma 1.3, that every ρ ∈ Hom( L , L ′ ) is either trivial or anisomorphism. Now suppose ρ ∈ Aut( L ). The second diagram in (cs.4) shows that the association x ¯ ρ x is a homomorphism from G (¯ k ) to ¯ Q × ℓ and the first diagram in (cs.4) shows that it factorsthrough G (¯ k ) → G (¯ k ) W .Conversely, if ρ : G (¯ k ) W → ¯ Q × ℓ is any homomorphism, then defining ¯ ρ x as multiplication by ρ ( x ) will define a morphism ¯ L → ¯ L ′ satisfy the two diagrams in (cs.4).Composition of morphisms corresponds to pointwise multiplication in this correspondence,showing that the resulting bijection is actually a group isomorphism. (cid:3) Character sheaves on smooth commutative group schemes over finite fields
Restriction to the identity component.
Consider the short exact sequence defining thecomponent group scheme for G :(8) 0 G G π ( G ) 0 . ι π Since π ( G ) is an ´etale commutative group scheme – and thus smooth – Lemma 1.4 implies that(8) defines a sequence of functors(9) CS (0) CS ( π ( G )) CS ( G ) CS ( G ) CS (0) π ∗ ι ∗ and therefore, after passing to isomorphism classes, a sequence of abelian groups(10) 0 CS ( π ( G )) / iso CS ( G ) / iso CS ( G ) / iso . π ∗ ι ∗ Note that we found the groups CS ( π ( G )) / iso and CS ( G ) / iso in Sections 2.4 and 1.6, respectively.We will shortly see that (10) is exact. Lemma 3.1.
Every discrete isogeny to G extends to a discrete isogeny to G inducing an iso-morphism on component groups.Proof. Let π : B → G be a discrete isogeny, and set A := ker π . We will find a discrete isogeny f : H → G such that that H = B , f = π and π ( f ) : π ( H ) → π ( G ) is an isomorphism ofcomponent groups. Namely, we will fit π into the following diagram,(11) A AB H π ( H ) G G π ( G ) , π f ∼ π ( f ) where all rows and columns are exact and all maps are defined over k . We will do so by passingback and forth between group schemes over k and their ¯ k -points.Extensions of G (¯ k ) by A (¯ k ) with W -equivariant maps, such as B (¯ k ), correspond to classesin Ext Z [ W ] ( G (¯ k ) , A (¯ k )). Similarly, extensions of G (¯ k ) by A (¯ k ) with W -equivariant maps corre-spond to classes in Ext Z [ W ] ( G (¯ k ) , A (¯ k )). The map G (¯ k ) → G (¯ k ) induces a homomorphismExt Z [ W ] ( G (¯ k ) , A (¯ k )) → Ext Z [ W ] ( G (¯ k ) , A (¯ k ))fitting into the long exact sequenceExt Z [ W ] ( G (¯ k ) , A (¯ k )) → Ext Z [ W ] ( G (¯ k ) , A (¯ k )) → Ext Z [ W ] ( π ( G )(¯ k ) , A (¯ k ))derived from applying the functor Hom(— , A (¯ k )) to G (¯ k ) → G (¯ k ) → π ( G )(¯ k ). Since W ∼ = Z has cohomological dimension 1 [8, Ex. 4.3], Ext Z [ W ] ( π ( G )(¯ k ) , A (¯ k )) vanishes [9, Thm. 2.6].We therefore have the existence of diagram (11) at the level of ¯ k -points. This expresses H (¯ k )as a disjoint union of translates of B (¯ k ); by transport of structure we may take H to be a groupscheme over ¯ k . Similarly, the restriction of f to each component of H is a morphism of schemes,and thus f is as well. Finally, the whole diagram descends to a diagram of k -schemes since the¯ k -points of the objects come equipped with continuous Gal(¯ k/k )-actions and the morphisms areGal(¯ k/k )-equivariant. (cid:3) ROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF p -ADIC TORI 19 We now wish to apply the results of Section 1.6 to the identity component of G , for which wemust confirm that the identity component of G is actually an algebraic group over k . Lemma 3.2. If G is a commutative smooth group scheme over k then its identity component, G , is a connected algebraic group over k .Proof. Since G is a smooth group scheme over k , its identity component G is a connectedsmooth, group scheme of finite type over k , reduced over some finite extension of k [19, 3.17].Since k is a finite field and hence perfect, G is actually reduced over k [22, Prop 6.4.1]. Sinceevery group scheme over a field is separated [19, 3.12], it follows that G is a connected algebraicgroup. (cid:3) Proposition 3.3.
The restriction functor ι ∗ : CS ( G ) → CS ( G ) is essentially surjective.Proof. By Lemma 3.2 and Proposition 1.14, every character sheaf on G is isomorphic to ( π ! ¯ Q ℓ ) ψ for some discrete isogeny π : B → G and character ψ : ker π → ¯ Q × ℓ . So to prove the proposi-tion it suffices to show that ( π ! ¯ Q ℓ ) ψ extends to a character sheaf on G . By Lemma 3.1, thereis an extension of the discrete isogeny π : B → G to a discrete isogeny f : H → G suchthat π ( f ) : π ( H ) → π ( G ) is an isomorphism. Then ( f ! ¯ Q ℓ ) ψ is a character sheaf on G and( f ! ¯ Q ℓ ) ψ | G ∼ = ( π ! ¯ Q ℓ ) ψ . (cid:3) The component group sequence.Lemma 3.4.
The group homomorphism π ∗ : CS ( π ( G )) / iso → CS ( G ) / iso is injective.Proof. Let L be a character sheaf on π ( G ) and let ρ : π ∗ L → ( ¯ Q ℓ ) G be an isomorphism in CS ( G ).For each x ∈ π ( ¯ G ), set ¯ G x := π − ( x ). The restriction π ∗ ¯ L| ¯ G x is the constant sheaf ( ¯ L x ) ¯ G x sothe isomorphism ¯ ρ | ¯ G x : ( ¯ L x ) ¯ G x → ( ¯ Q ℓ ) ¯ G x determines an isomorphism ¯ ρ x : ¯ L x → ( ¯ Q ℓ ) x . Thecollection { ¯ ρ x | x ∈ π ( ¯ G ) } determines an isomorphism L → ( ¯ Q ℓ ) π ( G ) in CS ( π ( G )). (cid:3) Proposition 3.5.
The sequence CS ( π ( G )) / iso CS ( G ) / iso CS ( G ) / iso . π ∗ ι ∗ is exact.Proof. Exactness at CS ( G ) / iso follows from Proposition 3.3, and exactness at CS ( π ( G )) / iso from Lemma 3.4. Here we show that it is also exact at CS ( G ) / iso . First note that ι ∗ ◦ π ∗ istrivial by Lemma 1.4. So it suffices to show that if L = ( ¯ L , µ, φ ) is a character sheaf on G with L| G = ( ¯ Q ℓ ) G then L is in the essential image of π ∗ .As above, set ¯ G x := π − ( x ) for x ∈ π ( ¯ G ). Let g, g ′ be geometric points in the same geometricconnected component ¯ G x . Set a = g − g ′ and note that a is a geometric point in ¯ G . Let µ g,a :¯ L ga → ¯ L g ⊗ ¯ L a be the isomorphism of vector spaces obtained by restriction of µ : m ∗ ¯ L → ¯ L ⊠ ¯ L to the geometric point ( g, a ) on ¯ G x × ¯ G . Since L| G = ( ¯ Q ℓ ) G , the stalk of ¯ L at a is ¯ Q ℓ . In thisway the pair of geometric points g, g ′ ∈ ¯ G x determines an isomorphism ϕ g,g ′ := µ − g,a from ¯ L g to¯ L g ′ . The isomorphisms ϕ g,g ′ : ¯ L g → ¯ L g ′ are canonical in the following sense: if g, g ′ ∈ ¯ G x and h, h ′ ∈ ¯ G y then it follows from (CS.2) and (CS.3) that(12) ¯ L gh ¯ L g ′ h ′ ¯ L Fr( g ) ¯ L Fr( g ′ ) ¯ L g ⊗ ¯ L h ¯ L g ′ ⊗ ¯ L h ′ ¯ L g ¯ L g ′ ϕ gh,g ′ h ′ µ g,h µ g ′ ,h ′ and ϕ Fr( g ) , Fr( g ′ ) φ g φ g ′ ϕ g,g ′ ⊗ ϕ h,h ′ ϕ g,g ′ both commute. For each x ∈ π ( ¯ G ), pick g ( x ) ∈ ¯ G x and set ¯ E x := ¯ L g ( x ) . Let φ x : ¯ E Fr( x ) → ¯ E x be theisomorphism of ¯ Q ℓ -vector spaces obtained by composing ϕ g (Fr( x )) , Fr( g ( x )) : ¯ L g (Fr( x )) → ¯ L Fr( g ( x )) with φ g ( x ) : ¯ L Fr( g ( x )) → ¯ L g ( x ) . For each pair x, y ∈ π ( ¯ G ) let µ x,y : ¯ E x + y → ¯ E x ⊗ ¯ E y be theisomorphism of ¯ Q ℓ -vector spaces obtained by composing ϕ g ( x + y ) ,g ( x ) g ( y ) : ¯ L g ( x + y ) → ¯ L g ( x ) g ( y ) with µ g ( x ) ,g ( y ) : ¯ L g ( x ) g ( y ) → ¯ L g ( x ) ⊗ ¯ L g ( y ) . Using (12), it follows that (CS.1), (CS.2) and (CS.3)are satisfied for E := ( ¯ E x , µ x,y , φ x ), thus defining a character sheaf on π ( G ).The pullback π ∗ ( E ) of E along π : G → π ( G ) is constant on geometric connected components,with stalks given by ( π ∗ E ) g = E x for all g ∈ ¯ G x . Thus both π ∗ E and L are constant on geometricconnected components of G . The choices above define isomorphisms ¯ L| ¯ G x → ( ¯ E x ) ¯ G x for each x ∈ π ( ¯ G ). The resulting isomorphism ¯ L → π ∗ ¯ E satisfies (CS.4), thus defining an isomorphism L → π ∗ E in CS ( G ). (cid:3) The dictionary.
We saw in Proposition 1.6 that Tr G : CS ( G ) / iso → G ( k ) ∗ is a functorialgroup homomorphism. In this section we find the image and kernel of Tr G . Theorem 3.6. If G is a smooth commutative group scheme over k then Tr G : CS ( G ) / iso → G ( k ) ∗ is surjective and has kernel canonically isomorphic to H ( π ( ¯ G ) , ¯ Q × ℓ ) W : ( π ( ¯ G ) , ¯ Q × ℓ ) W CS ( G ) / iso G ( k ) ∗ . Tr G is an exact sequence.Proof. Let(13) 0 CS ( π ( G )) / iso CS ( G ) / iso CS ( G ) / iso π ( G )( k ) ∗ G ( k ) ∗ G ( k ) ∗ Tr π G ) Tr G Tr G be the commutative diagram of abelian groups obtained by applying Lemma 1.4 to (8). Thesequence of abelian groups1 G ( k ) G ( k ) π ( G )( k ) 0 , is exact since H ( k, G ) = 0 by Lemma 3.2 and Lang’s theorem on connected algebraic groupsover finite fields [26]. Since ¯ Q × ℓ is divisible, Hom( − , ¯ Q × ℓ ) is exact and thus the dual sequenceof character groups in (13) is exact. The upper row in (13) is exact by Proposition 3.5. NowLemma 3.2 and Proposition 1.14 imply that ker Tr G = 0 and coker Tr G = 0, while Proposi-tion 2.7 gives ker Tr π ( G ) ∼ = H ( W , H ( π ( ¯ G ) , ¯ Q × ℓ )) and coker Tr π ( G ) = 0. It now follows from ROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF p -ADIC TORI 21 the snake lemma(14) 0 ker Tr π ( G ) ker Tr G ker Tr G = 00 CS ( π ( G )) / iso CS ( G ) / iso CS ( G ) / iso π ( G )( k ) ∗ G ( k ) ∗ G ( k ) ∗ π ( G ) = 0 coker Tr G coker Tr G = 0 Tr π G ) Tr G Tr G that coker Tr G = 0 and ker Tr π ( G ) → ker Tr G is an isomorphism. This gives the promised shortexact sequence 0 H ( π ( ¯ G ) , ¯ Q × ℓ ) W CS ( G ) / iso G ( k ) ∗ . Tr G (cid:3) Remark 3.7.
Although Tr π ( G ) is split and Tr G is an isomorphism, we do not know if Tr G is split, in general. Surjectively of Tr G shows that every ℓ -adic character of G ( k ) admits ageometrization, but without a splitting for Tr G we do not know how to make this geometrizationcanonical.3.4. Descent, revisited.
We now extend Proposition 2.11 to all smooth commutative groupschemes over k . Proposition 3.8.
Let G be a smooth commutative group scheme over k . Then L ∈ CS ( G ) descends to G if and only if t L : G ( k ) → ¯ Q × ℓ has bounded image.Proof. By Lemma 3.2, the identity component G is a connected algebraic group over k . Itfollows from Proposition 1.14 that the restriction of L to G descends to G . Also, since G ( k )is finite, the image of t L : G ( k ) → ¯ Q × ℓ is a finite subgroup and therefore has bounded image. If χ ∈ G ( k ) ∗ then there is some finite-image character χ with the same restriction to G ( k ) since G ( k ) is lies inside the torsion part of the finitely generated abelian group G ( k ). Therefore χ isbounded if and only if χ · χ − is bounded. But χ · χ − descends to a character of π ( G ). Thus, itis enough to prove Corollary 3.8 for ´etale group schemes G , which is done in Proposition 2.11. (cid:3) Proposition 3.8 shows that the full subcategory CS ( G ) ⊂ CS ( G ) is not an equivalence, forgeneral smooth commutative group schemes G . Again we see the necessity of working with Weilsheaves in Definition 1.1.3.5. Morphisms of character sheaves.Theorem 3.9.
Let G be a smooth commutative group scheme over k . There is a canonicalisomorphism Aut( L ) ∼ = Hom( π ( ¯ G ) W , ¯ Q × ℓ ) . Proof.
Fix L = ( ¯ L , µ, φ ) and consider the group homomorphism from Aut( L ) to Hom( ¯ G W , ¯ Q × ℓ )defined in the proof of Proposition 2.13. This homomorphism is injective because morphismsof sheaves are determined by the linear transformations induced on stalks. Homomorphismsin the image of Aut( L ) → Hom( ¯ G W , ¯ Q × ℓ ) are continuous when ¯ G is viewed as the base of the espace ´etal´e attached to ¯ L . Since ℓ is invertible in k , it follows that the image of Aut( L ) → Hom( ¯ G W , ¯ Q × ℓ ) is contained in Hom( π ( ¯ G W ) , ¯ Q × ℓ ). We also have π ( ¯ G W ) = π ( ¯ G ) W . To seethat Aut( L ) → Hom( π ( ¯ G ) W , ¯ Q × ℓ ) is surjective, begin with θ ∈ Hom( π ( ¯ G ) W , ¯ Q × ℓ ) and, for each[ x ] ∈ π ( ¯ G ) W define ¯ ρ y : ¯ L y → ¯ L y by scalar multiplication by θ ([ x ]) ∈ ¯ Q × ℓ for each y ∈ [ x ]. Thisdefines an isomorphism ¯ ρ : ¯ L → ¯ L of local systems on ¯ G compatible with µ and φ , and thus anisomorphism ρ : L → L which maps to θ under Aut( L ) → Hom( π ( ¯ G ) W , ¯ Q × ℓ ). (cid:3) The dictionary for commutative algebraic groups over finite fields.
Having ex-tended the function–sheaf dictionary from connected commutative algebraic groups over k tosmooth commutative group schemes G over k , we look back briefly to the case when G is a com-mutative algebraic group. Although Weil sheaves are not necessary in that case, the dictionaryis still not perfect, generally. Corollary 3.10.
Let G be a commutative algebraic group over k . All character sheaves on G descend to G : CS ( G ) is equivalent to CS ( G ) ( § ( π ( ¯ G ) , ¯ Q × ℓ ) W CS ( G ) / iso G ( k ) ∗ . Tr G The group H ( π ( ¯ G ) , ¯ Q × ℓ ) W need not be trivial.Proof. Since G ( k ) is finite, the first statement follows from Propositions 1.8 and 3.8. The secondstatement is then a consequence of Theorem 3.6. The third statement is justified by Example 2.10. (cid:3) Remark 3.11.
Suppose G is a commutative algebraic group over k appearing in the series ofpapers starting with [29]. Then G is a commutative reductive algebraic group over k with cycliccomponent group scheme; such groups are extensions of finite cyclic groups by algebraic tori.Every Frobenius-stable character sheaf on G , in the sense of [29], is a character sheaf on G , inour sense, and vice versa. Moreover, since H ( π ( ¯ G ) , ¯ Q × ℓ ) = 0 by Remark 2.10, it follows thateach Frobenius-stable character sheaf on G as in [29], is determined by its trace of Frobenius, upto isomorphism.3.7. Base change.
When using character sheaves to study characters, it is useful to understandhow character sheaves behave under change of fields. Let k ′ be a finite extension of k . Then k ֒ → k ′ induces a group homomorphism i k ′ /k : G ( k ) ֒ → G ( k ′ ) and thus a homomorphism i ∗ k ′ /k : G ( k ′ ) ∗ → G ( k ) ∗ χ χ ◦ i k ′ /k . We can interpret this operation on characters in terms of character sheaves:
Proposition 3.12.
Set G k ′ := G × Spec( k ) Spec( k ′ ) and let CS (Res k ′ /k ( G k ′ )) ι ∗ −→ CS ( G ) be the functor obtained by pullback along the canonical closed immersion ι : G ֒ → Res k ′ /k ( G k ′ ) ROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF p -ADIC TORI 23 of k -schemes. The following diagram commutes: CS (Res k ′ /k ( G k ′ )) / iso CS ( G ) / iso G ( k ′ ) ∗ G ( k ) ∗ . ι ∗ Tr Res k ′ /k ( Gk ′ ) Tr G i ∗ k ′ /k Proof.
The closed immersion ι : G ֒ → Res k ′ /k ( G k ′ ) is given by [6, § k ′ /k ( G k ′ )( k ) ∼ = G k ′ ( k ′ ) ∼ = G ( k ′ )from the definitions of Weil restriction and base change. (cid:3) In the opposite direction, let Nm : G ( k ′ ) → G ( k ) be the norm map and consider the grouphomomorphism: Nm ∗ : G ( k ) ∗ → G ( k ′ ) ∗ χ χ ◦ Nm . We can also interpret this operation in terms of character sheaves.If L := ( ¯ L , µ, φ ) is a character sheaf on G , we define L ′ := ( ¯ L , µ, φ k ′ ) on the base change G k ′ of G to k ′ by setting φ k ′ := φ ◦ Fr ∗ G ( φ ) ◦ · · · ◦ (Fr n − G ) ∗ ( φ ) . The commutativity of the diagram (CS.3) for φ k ′ follows from the fact that Fr G k ′ = Fr nG . Notethat we may also think about the construction of φ k ′ from φ as restricting the action ϕ of W k on ¯ L , defined in Section 1.2, to W k ′ . Proposition 3.13.
With notation above, the rule ν k ′ /k : ( ¯ L , µ, φ ) ( ¯ L , µ, φ k ′ ) defines amonoidal functor CS ( G ) → CS ( G k ′ ) such that the following diagram commutes: CS ( G ) / iso CS ( G k ′ ) / iso G ( k ) ∗ G ( k ′ ) ∗ . ν k ′ /k Tr G Tr Gk ′ Nm ∗ Proof.
Let L := ( ¯ L , µ, φ ) ∈ CS ( G ) and write F for Fr G . For any x ∈ G ( k ′ ), we may com-pute the value of Tr G k ′ ( ν k ′ /k L )( x ) = t ν k ′ /k L ( x ) as the trace of φ k ′ on ¯ L x , and the value ofNm ∗ (Tr G ( L ))( x ) as the trace of φ on ¯ L Nm( x ) . Applying (CS.3) to the stalk of ¯ L ⊠ n at the point( x, Fr( x ) , . . . , Fr n − ( x )) yields a diagram¯ L Nm( x ) ¯ L F ( x ) ⊗ ¯ L F ( x ) ⊗ · · · ⊗ ¯ L x ¯ L Nm( x ) ¯ L x ⊗ ¯ L F ( x ) ⊗ · · · ⊗ ¯ L F n − ( x ) . φ Nm( x ) φ x ⊗ ( F ∗ φ ) x ⊗···⊗ (( F n − ) ∗ φ ) x Choose a basis vector v for ¯ L Nm( x ) and write v ⊗ v ⊗ · · · ⊗ v n − for the image of v under thebottom map, for v i ∈ ¯ L Fr i ( x ) . By (CS.2), v maps to v ⊗ v ⊗ · · · ⊗ v along the top of thediagram. Let α i ∈ ¯ Q × ℓ represent (( F i ) ∗ φ ) x with respect to these bases and let α be the trace of φ Nm( x ) . We may now equate the trace α of φ on ¯ L Nm( x ) with the product α · · · α n − , which isthe trace of φ k ′ on ¯ L x . (cid:3) Finally, let G ′ be a smooth commutative group scheme over k ′ . We explain how to geometrizethe canonical isomorphism between characters of G ′ ( k ′ ) and of (Res k ′ /k G ′ )( k ). We may decom-pose the base change (Res k ′ /k G ′ ) k ′ of Res k ′ /k G ′ to k ′ into a product of copies of G ′ , indexed byelements of Gal( k ′ /k ): (Res k ′ /k G ′ ) k ′ ∼ = Y Gal( k ′ /k ) G ′ . Since products and coproducts agree for group schemes we have a natural inclusion of k ′ -schemes G ′ ֒ → (Res k ′ /k G ′ ) k ′ , mapping G ′ into the summand corresponding to 1 ∈ Gal( k ′ /k ). Composing ν k ′ /k from Proposi-tion 3.13 with pullback along this map yields a functor ρ : CS (Res k ′ /k G ′ ) → CS ( G ′ ) . Proposition 3.14.
Let k ′ /k be a finite extension and let G ′ be a smooth commutative groupscheme over k ′ . Then the functor ρ : CS (Res k ′ /k G ′ ) → CS ( G ′ ) , defined above, induces CS (Res k ′ /k G ′ ) / iso CS ( G ′ ) / iso G ′ ( k ′ ) ∗ G ′ ( k ′ ) ∗ , Tr Res k ′ /k G ′ ρ Tr G ′ where the bottom map is the identity.Proof. By Lemma 1.4 the pullback part of the definition of ρ corresponds to the map(Res k ′ /k G ′ )( k ′ ) ∗ → G ′ ( k ′ ) ∗ induced by g ( g, , . . . , k ′ /k ) on(Res k ′ /k G ′ ) k ′ ∼ = Y Gal( k ′ /k ) G ′ is given by permuting coordinates, composition with the norm map yields the identity on G ′ ( k ′ ). (cid:3) Quasicharacter sheaves for p -adic tori Let K be a local field with ring of integers R and finite residue field k ; in this section wedenote the group W by W k . We continue to assume that ℓ is invertible in k .4.1. N´eron models.
We will consider connected commutative algebraic groups over K thatadmit a N´eron model, by which we mean a locally finite type N´eron model. By [6, § K that containno subgroup isomorphic to G a . Write N K for the full subcategory of the category of algebraicgroups consisting of such objects. This category is additive, and includes all algebraic tori over K , abelian varieties over K and unipotent K -wound groups. We write N for the category ofN´eron models that arise in this way; in particular, N is a full subcategory of the category ofsmooth commutative group schemes over R . ROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF p -ADIC TORI 25 Example 4.1. If T K = G m ,K , then a N´eron model can be obtained by gluing copies of G m ,R (one for each n ∈ Z ) along their generic fibers, via the gluing morphisms Spec( Z [ t, t − ]) → Spec( Z [ t, t − ]) defined by t π n t [6, § K ′ /K is a quadratic extension and T K = U ( K ′ /K ) is the unitary group. When K ′ /K is unramified, the N´eron model of T K is a form of the N´eron model for G m ,K , with thenontrivial automorphism σ ∈ Gal( K ′ /K ) mapping ( x, n ) to ( σ ( x ) , − n ) for x ∈ R ′× and n ∈ Z specifying the copy of G m ,R ′ . This example illustrates the compatibility between N´eron modelsand unramified base change [6, § K ′ = K ( √ π ) is totally ramified over K then the N´eron model of U ( K ′ /K ) is affine, namely Spec( R [ x, y ] / ( x − πy − K ′ /K is any finite extension of local fields and X ′ is a N´eron modelfor X K ′ then Res R ′ /R ( X ′ ) is a N´eron model for Res K ′ /K ( X K ′ ) [6, § Quasicharacters.
Write m for the maximal ideal of R and set R n = R/ m n +1 for everynon-negative integer n . Suppose X ∈ N . Note that X ( K ) = X ( R ). A quasicharacter of X ( K )is a group homomorphism X ( K ) → ¯ Q × ℓ that factors through X ( R ) → X ( R n ) for some non-negative integer n . We note that this definition is compatible with [10, Ch XV, § X ( K ) will be denoted by Hom( X ( K ) , ¯ Q × ℓ ) and the subgroup of those thatfactor through X ( R n ) will be denoted by Hom n ( X ( K ) , ¯ Q × ℓ ). In this section we will see how togeometrize and categorify quasicharacters of X ( K ) using character sheaves.4.3. Review of the Greenberg transform.
Let K , R and R n be as above. For each n ∈ N ,the Greenberg functor maps schemes over R n to schemes over k . See [4] for the definition andfundamental properties of the Greenberg functor as we use it; other useful references include[20], [21], [17, V, §
4, no. 1], [6, Ch. 9, §
6] and [30, § n we willwrite Gr Rn : Sch /R → Sch /k for the functor produced by composing pullback along Spec( R n ) → Spec( R ) with the Greenbergfunctor. This functor respects open immersions, closed immersions, ´etale morphisms, smoothmorphisms and geometric components. Moreover, there is a canonical isomorphismGr Rn ( X )( k ) ∼ = X ( R n )for any scheme X over R .For any n ≤ m , the surjective ring homomorphism R m → R n determines a natural transfor-mation ̺ Rn ≤ m : Gr Rm → Gr Rn between additive functors. Crucially, ̺ Rn ≤ m ( X ) : Gr Rm ( X ) → Gr Rn ( X ) is an affine morphism of k -schemes, for every R -scheme X and every n ≤ m [4, Prop 4.3]. This observation is key to theproof that, for any scheme X over R , the projective limitGr R ( X ) := lim ←− n ∈ N Gr Rn ( X ) , taken with respect to the surjective morphisms ̺ Rn ≤ m ( X ) : Gr Rm ( X ) → Gr Rn ( X ), exists in thecategory of group schemes over k ; see [23, § Greenbergtransform : Gr R : Sch /R → Sch /k . By construction, the k -scheme Gr R ( X ) comes equipped with morphisms ̺ Rn ( X ) : Gr R ( X ) → Gr Rn ( X ) , ∀ n ∈ N . Quasicharacter sheaves.
Set S = Spec( R ) and S n = Spec( R n ); note that S = Spec( k )is the special fibre of S . Let X be a smooth commutative group scheme over S . For every non-negative integer n , the Greenberg transform Gr Rn ( X ) is a smooth commutative group schemeover S . The Greenberg transform Gr R ( X ) of X is a commutative group scheme over k with k -rational points X ( R ). The morphism of k -schemes ̺ Rn ( X ) : Gr R ( X ) → Gr Rn ( X ) induces afunctor ̺ Rn ( X ) ∗ : CS (Gr Rn ( X )) → CS (Gr R ( X )) , as in Lemma 1.4. Definition 4.2.
Let X be a smooth group scheme over R . A quasicharacter sheaf of X is atriple F := ( n, {F i } n ≤ i , { α i ≤ j } n ≤ i ≤ j ) , where n is a non-negative integer, each F i is a character sheaf on Gr Ri ( X ) and each α i ≤ j : L j → ̺ Ri ≤ j ( X ) ∗ L i is an isomorphism; here α i ≤ i is the identity and the α i ≤ j are compatible with each other. If F := ( n, {F i } , { α i ≤ j } ) and F ′ := ( m, {F ′ i } , { α ′ i ≤ j } ) are objects then Hom( F , F ′ ) is the set ofequivalence classes of pairs ( k, { β i } k ≤ i ), where n, m ≤ k and the β i : F i → F ′ i are morphisms ofcharacter sheaves so that F j ̺ Ri ≤ j ( X ) ∗ F i F ′ j ̺ Ri ≤ j ( X ) ∗ L ′ iα j β i f ∗ i ≤ j β j α j commutes for all k ≤ i ≤ j ; we identify two such pairs ( k, { β i } ) and ( l, { γ i } ) if β i = γ i forsufficiently large i . Identities and composites are defined in the natural way. Let QCS ( X ) denotethe category of quasicharacter sheaves for X . Remark 4.3. If ̺ Rn ( X ) ∗ : CS (Gr Rn ( X )) → CS (Gr R ( X )) is full then the construction above canbe improved by forming QCS ( X ) from the essential images of the functors ̺ Rn ( X ) ∗ ; however, wedo not know if ̺ Rn ( X ) ∗ is full. Remark 4.4.
We offer the following alternate construction of
QCS ( X ). As above, let X bea smooth group scheme over R . Although Gr R ( X ) is not locally of finite type and thereforenot smooth, let us consider the rigid monoidal category CS (Gr R ( X )) as defined in Section 1.2,though without insisting that the commutative group k -scheme G is smooth. A quasicharactersheaf for X is an object of the following rigid monoidal subcategory of CS (Gr R ( X )), denoted by QCS ( X ):(1) objects in QCS ( X ) are the ℓ -adic sheaves ̺ Rn ( X ) ∗ L , for n ∈ N and L ∈ CS (Gr Rn ( X ));(2) morphisms ̺ Rn ( X ) ∗ L → ̺ Rm ( X ) ∗ L ′ in QCS ( X ) are those morphisms in CS (Gr R ( X ))which take the form ̺ Rm ( X ) ∗ ρ for ρ ∈ Hom( ̺ Rn ≤ m ( X ) ∗ L , L ′ ) when n ≤ m , and ̺ Rn ( X ) ∗ ρ for ρ ∈ Hom( L , ̺ Rm ≤ n ( X ) ∗ L ′ ) when m ≤ n . Theorem 4.5.
Let K be a local field with residue field k , in which ℓ is invertible; let R be thering of integers of K . (0) The trace of Frobenius provides a natural transformation between the additive functors X
7→ QCS ( X ) / iso and X Hom( X ( K ) , ¯ Q × ℓ ) as functors from N to the category of commutative groups.Regarding this natural transformation, for every X ∈ N : ROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF p -ADIC TORI 27 (1) there is a canonical short exact sequence of commutative groups → H ( π ( X ) ¯ k , ¯ Q × ℓ ) W k → QCS ( X ) / iso → Hom( X ( K ) , ¯ Q × ℓ ) → (2) for all quasicharacter sheaves F , F ′ on Gr R ( X ) , and for every ρ ∈ Hom( F , F ′ ) , either ρ is trivial or ρ is an isomorphism; (3) for all quasicharacter sheaves F for X , there is a canonical isomorphism Aut( F ) ∼ = Hom(( π ( X ) ¯ k ) W k , ¯ Q × ℓ ) Proof.
To prove (0), use: Proposition 1.6 with G = Gr Rn ( X ); the fact that N´eron models areunique up to isomorphism; the fact that every CS (Gr Rn ( X )) is a full subcategory of QCS ( X ); andthe observation that every object in QCS ( X ) is in the essential image of CS (Gr Rn ( X )) for some n . To prove (1), use Theorem 3.6 with G = Gr Rn ( X ) and then argue as in part (0). To prove (2),argue as in the proof of Lemma 1.3. To prove (3), use: the fact that the component group ofGr Rn ( X ) is independent of n ; Theorem 3.9 with G = Gr Rn ( X ), in which case π ( G ) = π ( X × S S )and π ( ¯ G ) = π ( X ) ¯ k ; then argue as in part (0). (cid:3) Remark 4.6.
In Section 4.5 we see that the ´etale site on Gr R ( X ) is rich enough to geometrizeall quasicharacters of X ( K ) as ℓ -adic local systems on Gr R ( X ), where X is a N´eron model foran algebraic torus or an abelian variety over a local field K . It is natural to ask if the ´etalesite on the generic fibre X K would have sufficed. This seems unlikely, since the geometric ´etalefundamental group of G m ,K is ˆ Z ; however, limited results in this direction were established in[12] when K = Q p .4.5. Quasicharacter sheaves for p -adic tori. As we explained in the Introduction, our origi-nal motivation for this paper was to find a geometrization of quasicharacters of p -adic tori. Thisis now provided by the following adaptation of Theorem 4.5 in the case when T ∈ N is a N´eronmodel for an algebraic torus over K . Corollary 4.7.
Let T be a N´eron model for an algebraic torus over K . The following is acommutative diagram of exact sequences. ( X ∗ ( T ) I K , ¯ Q × ℓ ) W k H ( X ∗ ( T ) I K , ¯ Q × ℓ ) W k QCS ( π ( T )) / iso QCS ( T ) / iso QCS ( T ) / iso
00 Hom( π ( T )( k ) , ¯ Q × ℓ ) Hom( T ( K ) , ¯ Q × ℓ ) Hom( T ( R ) , ¯ Q × ℓ ) 00 0 0 π ∗ Tr π T ) ι ∗ Tr Gr R ( T ) Tr Gr R ( T )0 inf ∼ = ∼ = res ∼ = Proof.
The horizontal sequence of groups coming from categories of quasicharacter sheaves isexact by Proposition 3.5, together with the observation that the functors π ∗ and ι ∗ preservelimits. It is elementary that the horizontal sequence of quasicharacters is exact. Accordingly,by Theorem 4.5, the kernel of the is H ( π ( T ) ¯ k , ¯ Q × ℓ ) W . By [5, Eq 3.1], the special fibre of thecomponent group scheme for T is given by π ( T ) ¯ k = X ∗ ( T ) I K , where X ∗ ( T ) is the cocharacter lattice of T K and I K is the inertia group for K . Thus,H ( π ( T ) ¯ k , ¯ Q × ℓ ) W k = H ( X ∗ ( T ) I K , ¯ Q × ℓ ) W k . Thus, the middle vertical sequence is exact. Since T and π ( T ) do not lie in N , we cannot useTheorem 4.5 to determine the image and trace of Frobenius for these schemes. Instead, we observethat T and π ( T ) are smooth commutative group schemes over R , so Definition 4.2 gives meaningto categories QCS ( T ) and QCS ( π ( T )), and, moreover, that π (Gr Rn ( T )) = π ( T ) k = 1 and π (Gr Rn ( π ( T )) = π ( T ) k are both independent of n . It follows that the vertical sequencesthrough QCS ( T ) / iso and QCS ( π ( T )) / iso are exact by Theorem 3.6 and Definition 4.2. Thediagram commutes by Lemma 1.4. (cid:3) Example 4.8.
When T K = G m ,K or T K = U ( K ′ /K ), the geometric component group π ( T ) ¯ k is cyclic, so Tr Gr R ( T ) is an isomorphism. Conversely, when T K = G m ,K thenH ( W k , H ( π ( T ) , ¯ Q × ℓ )) = ¯ Q × ℓ , and there are uncountably many invisible quasicharacter sheaves for T .We may also give examples of tori whose N´eron models have component groups appearing inExample 2.10. Let L = K ′ ( √ π ) be a quadratic ramified extension of K ′ . When K = K ′ and T K = U ( L/K ) × U ( L/K ), the component group π ( T ) ¯ k is Z / Z × Z / Z with trivial Frobeniusaction. When K ′ /K is an unramified quadratic extension and T K = Res K ′ /K U ( L/K ′ ), then π ( T ) ¯ k is Z / Z × Z / Z with Frobenius exchanging the direct factors. Finally, let K ′ /K be acubic unramified extension and S K = Res K ′ /K U ( L/K ′ ). If T K is the subtorus with characterlattice X ∗ ( S K ) / h (1 , , i , then π ( T ) ¯ k is Z / Z × Z / Z with Frobenius of order 3. Each of thesetori will have one invisible quasicharacter sheaf.We may also extract information about the automorphism groups of quasicharacter sheavesfrom Theorem 4.5. Corollary 4.9.
Let T be a N´eron model for an algebraic torus over K . For E ∈ QCS ( π ( T )) , F ∈ QCS ( T ) and F ∈ QCS ( T ) , there are canonical isomorphisms Aut( E ) ∼ = ( ˇ T ℓ ) W K , Aut( F ) ∼ = ( ˇ T ℓ ) W K , Aut( F ) ∼ = 1 , where ˇ T ℓ := Hom( X ∗ ( T ) , ¯ Q × ℓ ) , the ℓ -adic dual torus to T K .Proof. We already know Aut( E ) = 1 from Proposition 1.14, part (3). By Theorem 4.5,Aut( F ) ∼ = Hom(( π ( T ) ¯ k ) W k , ¯ Q × ℓ ) . By [5, Eq 3.1] again, Hom(( π ( T ) ¯ k ) W k , ¯ Q × ℓ ) ∼ = Hom( X ∗ ( T ) W K , ¯ Q × ℓ ) . But Hom( X ∗ ( T ) W K , ¯ Q × ℓ ) ∼ = Hom( X ∗ ( T ) , ¯ Q × ℓ ) W K . So, for any quasicharacter sheaf F for T ,Aut( F ) ∼ = ( ˇ T ℓ ) W K , canonically. The case X = π ( T ) is handled by the same argument, replacing Theorem 4.5with Theorem 3.6 and Definition 4.2, as in the proof of Corollary 4.7, after observing that π ( π ( T ) ¯ k ) = π ( T ) ¯ k . (cid:3) Remark 4.10.
Since π ( T ) ¯ k = X ∗ ( T ) I K by [5, Eq 3.1], we haveHom( π ( T )( k ) , ¯ Q × ℓ ) = Hom(( X ∗ ( T ) I K ) W k , ¯ Q × ℓ ) = Hom( X ∗ ( T ) I K , ¯ Q × ℓ ) W k = H ( W k , ˇ T I K ℓ ) . By the Langlands correspondence for p -adic tori [35],Hom( T ( K ) , ¯ Q × ℓ ) ∼ = H ( W K , ˇ T ℓ ) , ROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF p -ADIC TORI 29 where we refer to continuous cohomology, since Hom( T ( K ) , ¯ Q × ℓ ) refers to continuous grouphomomorphisms T ( K ) → ¯ Q × ℓ . It now follows from the inflation-restriction exact sequence thatthe following diagram commutes:0 Hom( π ( T )( k ) , ¯ Q × ℓ ) Hom( T ( K ) , ¯ Q × ℓ ) Hom( T ( R ) , ¯ Q × ℓ ) 00 H ( W k , ˇ T I K ℓ ) H ( W K , ˇ T ℓ ) H ( I K , ˇ T ℓ ) W k . inf ∼ = ∼ = res ∼ =inf res Combining this with Corollary 4.7 produces the following commutative diagram of exact se-quences:(15) 0 0H ( X ∗ ( T ) I K , ¯ Q × ℓ ) W k H ( X ∗ ( T ) I K , ¯ Q × ℓ ) W k QCS ( π ( T )) / iso QCS ( T ) / iso QCS ( T ) / iso
00 H ( W k , ˇ T I K ℓ ) H ( W K , ˇ T ℓ ) H ( I K , ˇ T ℓ ) W k
00 0 0 π ∗ ι ∗ inf res It is natural to ask if the vertical surjections can be defined directly, without making use of localclass field theory, for which the results of [32] and [31] may be helpful. The case T K = G m ,K isalready very interesting, in which case (15) becomes(16) 0 QCS ( Z ) / iso QCS ( G Neron m ,K ) / iso QCS ( G m ,R ) / iso
00 Hom( W k , ¯ Q × ℓ ) Hom( W K , ¯ Q × ℓ ) Hom( I K , ¯ Q × ℓ ) 0 π ∗ ∼ = ι ∗ ∼ = ∼ =inf res We suspect that the general case of (15), where K is any local field and T K is any torus over K ,may be deduced from (16). In Section 4.6 we develop a tool for further work in that direction.4.6. Weil restriction and quasicharacter sheaves.
Let K ′ /K be a finite Galois extensionof local fields and let k ′ /k be the corresponding extension of residue fields. Let R and R ′ be therings of integers of K and K ′ , respectively. Suppose X ∈ N and set X K ′ := X K × Spec( K ) Spec( K ′ )and let X ′ be a N´eron model for X K ′ . Proposition 4.11.
The canonical closed immersion X K ֒ → Res K ′ /K X K ′ of K -group schemes induces a map of k -group schemes f : Gr R ( X ) → Res k ′ /k Gr R ′ ( X ′ ) which, through quasicharacter sheaves, induces Hom( X K ( K ′ ) , ¯ Q × ℓ ) Hom( X K ( K ) , ¯ Q × ℓ ) . χ χ | X ( K ) Proof.
By the N´eron mapping property, the canonical closed immersion X K ֒ → Res K ′ /K ( X K ′ )extends uniquely to a morphism(17) X → Res R ′ /R ( X ′ )of smooth R -group schemes. Applying the functor Gr Rn to (17) and using [4, Thm 1.1] definesthe morphism of smooth group schemes(18) f n : Gr Rn − ( X ) → Res k ′ /k Gr R ′ en − ( X ′ ) , where e is the ramification index of K ′ /K . Using Lemma 1.4, (18) induces a functor(19) f ∗ n : CS (Res k ′ /k Gr R ′ en − ( X ′ )) → CS (Gr Rn − ( X )) . Since (cid:16)
Res k ′ /k Gr R ′ en − ( X ′ ) (cid:17) ( k ) = (cid:16) Gr R ′ en − ( X ′ ) (cid:17) ( k ′ ) , it follows from Lemma 1.4 that the pullback functor (19) actually inducesHom en − ( X ′ ( R ′ ) , ¯ Q × ℓ ) → Hom n − ( X ( R )) , ¯ Q × ℓ )Since X ′ is a N´eron model for X K ′ and X is a N´eron model for X K , this may be re-written asHom en − ( X K ′ ( K ′ ) , ¯ Q × ℓ ) = Hom en − ( X K ( K ′ ) , ¯ Q × ℓ ) → Hom n − ( X K ( K )) , ¯ Q × ℓ )Passing to limits now definesHom( X K ( K ′ ) , ¯ Q × ℓ ) → Hom( X K ( K ) , ¯ Q × ℓ )Argue as in Proposition 3.12 to see that this is indeed restriction of characters. (cid:3) Transfer of quasicharacters.
Let K and L be local fields with rings of integers O K and O L , respectively. Pick uniformizers ̟ K and ̟ L for O K and O L , respectively; what follows willnot depend on these choices. Suppose ℓ is invertible in the residue fields of K and L .We begin with X K ∈ N K with N´eron model X and Y L ∈ N L with N´eron model Y . Suppose m is a positive integer such that O K /̟ mK O K ∼ = O L /̟ mL O L . Suppose also that(20) X × Spec( O K ) Spec( O K /̟ mK O K ) ∼ = Y × Spec( O L ) Spec( O L /̟ mL O L )as smooth group schemes over O K /̟ mK O K . ThenGr O K m − ( X ) ∼ = Gr O L m − ( Y )as smooth group schemes over k . Accordingly, by Lemma 1.4, the isomorphism above determinesan equivalence of categories(21) CS (Gr O K m − ( X )) ∼ = CS (Gr O L m − ( Y ))which induces an isomorphism(22) Hom m − ( X ( K ) , ¯ Q × ℓ ) ∼ = Hom m − ( Y ( L ) , ¯ Q × ℓ ) . ROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF p -ADIC TORI 31 The isomorphism (22) is an instance of transfer of (certain) quasicharacters between X ( K ) and Y ( L ). We now recognize this transfer of quasicharacters as a consequence of the equivalence ofcategories of quasicharacter sheaves (21).The isomorphism (20) can indeed exist between quasicharacters of algebraic tori over localfields, even when the characteristics of K and L differ. Suppose T K and M L are tori overlocal fields K and L , splitting over K ′ and L ′ , respectively. Then T K and M L are said to be n -congruent [11, §
2] if there are isomorphisms α : O K ′ /̟ nK O K ′ → O L ′ /̟ nL O L ′ β : Gal( K ′ /K ) → Gal( L ′ /L ) φ : X ∗ ( T K ) → X ∗ ( M L )satisfying the conditions(1) α induces an isomorphism O K /̟ nK O K → O L /̟ nL O L ,(2) α is Gal( K ′ /K )-equivariant relative to β , and(3) φ is Gal( K ′ /K )-equivariant relative to β .If T K and M L are n -congruent then α , β and φ determine an isomorphism(23) Hom n − ( T K ( K ) , ¯ Q × ℓ ) ∼ = Hom n − ( M L ( L ) , ¯ Q × ℓ ) . Note that if T K and M L are n -congruent, then they are m -congruent for every m ≤ n .Now let T be a N´eron model for T K and let M be a N´eron model for M L . One of the mainresults of [11] gives an isomorphism of group schemes T × Spec( O K ) Spec( O K /̟ mK O K ) ∼ = M × Spec( O L ) Spec( O L /̟ mL O L )assuming that T K and M L are sufficiently congruent. They define a quantity h (the smallestinteger so that ̟ h lies in the Jacobian ideal associated to a natural embedding of T K into aninduced torus [11, § n > h and T K and M L are n -congruent then there is acanonical isomorphism of smooth group schemes Gr n − h − ( T ) → Gr n − h − ( M ) determined by α, β and φ [11, Thm. 8.5]. Combining this with the paragraph above gives the following instanceof the geometrization of the transfer of quasicharacters. Proposition 4.12.
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