Automorphic functions as the trace of Frobenius
D. Arinkin, D. Gaitsgory, D. Kazhdan, S. Raskin, N. Rozenblyum, Y. Varshavsky
aa r X i v : . [ m a t h . AG ] F e b AUTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS
D. ARINKIN, D. GAITSGORY, D. KAZHDAN, S. RASKIN, N. ROZENBLYUM, Y. VARSHAVSKY
Abstract.
We prove that the trace of the Frobenius endofunctor of the category of automorphicsheaves with nilpotent singular support maps isomorphically to the space of unramified automor-phic functions, settling a conjecture from [AGKRRV1]. More generally, we show that traces ofFrobenius-Hecke functors produce shtuka cohomologies.
Contents
Introduction 20.1. Overview 20.2. Formulation of the main result 30.3. Insertion of Hecke functors 50.4. Outline of the proof 70.5. Organization of the paper 80.6. Notations and conventions 90.7. Acknowledgements 101. The Hecke action 101.1. Hecke functors 101.2. The ULA property of the Hecke action 111.3. Hecke action on Shv
Nilp (Bun G ) 121.4. The category Rep( ˇ G ) Ran G ) Ran restrˇ G ( X ) 172.1. The (pre)stack LocSys restrˇ G ( X ) 182.2. Description of the category QCoh(LocSys restrˇ G ( X )) 192.3. Localization 202.4. Beilinson’s spectral projector 213. The reciprocity law for shtuka cohomology 233.1. Functorial shtuka cohomology 233.2. Proof of Theorem 3.1.3 and relation to the usual shtuka cohomology 233.3. Drinfeld’s sheaf 243.4. Some remarks 254. Calculating the trace 264.1. Traces of Frobenius-Hecke operators 264.2. Proof of Theorem 4.1.2 284.3. Self-duality for Shv Nilp (Bun G ) 284.4. Proof of Theorem 4.2.2 294.5. Interpretation as enhanced trace 305. Local terms 315.1. Formulation of the problem 315.2. Naive and true local terms 325.3. Serre local term 34 Date : February 17, 2021.
Serre and LT
Sht R true and LT Serre
Introduction
Overview.
This work is part of a series, following [AGKRRV1] and [AGKRRV2], attemptingto understand the (unramified, function field) arithmetic Langlands conjectures via geometric andcategorical techniques. We begin with an overview of the problems considered here.0.1.1. Some of the most striking applications of geometric representation theory pass through thesheaves-functions dictionary of Grothendieck–Deligne.Recall the setting: one has an algebraic stack Y over F q , assumed to be defined over F q , and an ℓ -adic Weil sheaf F on Y . For a rational point y ∈ Y ( F q ), one takes the trace of Frobenius on the stalk y ∗ ( F ) to obtain an element of Q ℓ ; this defines a function funct( F ) : Y ( F q ) → Q ℓ .One finds that many functions of interest in harmonic analysis over finite fields arise by this proce-dure, and that the perspective offered by sheaf theory provides deep insights into function theory.For example, this is the case in the theory of automorphic functions (for function fields), whoserealizations via ℓ -adic sheaves exhibit explicit constructions of Langlands’s conjectures.0.1.2. In this paper, we establish a higher categorical analogue of the sheaves-functions dictionary.Instead of passing from sheaves to functions, we pass from categories (of sheaves) to vector spaces (offunctions).As in the previous setting, we decategorify using trace of Frobenius. However, whereas before weconsidered the trace of a Frobenius endomorphism of a vector space (and thus produced a scalar), wenow consider the trace of a Frobenius endofunctor of a category (and thus produce a vector space).Unlike the usual Grothendieck–Deligne paradigm, where one may take a general algebraic stack Y , our results are specialized to spherical automorphic functions. The results of this paper establishconjectures from [AGKRRV1], which specify a relation between the category of automorphic sheavesand the vector space of automorphic functions via the trace of Frobenius.0.1.3. We give a precise formulation of our main results below. However, by way of motivation, wehighlight the following application, which illustrates how sheaf-theoretic considerations provide insightsinto the classical theory of automorphic functions.In [AGKRRV1], we introduced a version of the geometric Langlands conjecture suitable for ℓ -adicsheaves. From this conjecture, combined with the Trace Conjecture , proved in this paper, we deducedthat the space of compactly supported spherical automorphic functions (denoted Funct c (Bun G ( F q )) inthe body of this text) can be described as(0.1) Funct c (Bun G ( F q )) ≃ Γ(LocSys arthmˇ G ( X ) , ω LocSys arthmˇ G ( X ) ) . UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 3
In the above formula, X is the smooth projective curve corresponding to our choice of function fieldand LocSys arthmˇ G ( X ) is the algebraic stack over Q ℓ , defined in [AGKRRV1], parametrizing (unramified)Langlands parameters. In the everywhere unramified function field setting, this conjecture provides an interesting alternativeto Langlands’s original perspective: for general reductive G , the above conjecture yields a competedescription of the space of automorphic functions in terms of Galois data, whereas classical Langlandsconjectures only concern L -packets. Remark . The work [VLaf] of V. Lafforgue and its extension [Xue1] by C. Xue provide a decom-position of the space of (not necessarily unramified!) automorphic functions in terms of Langlandsparameters. It should come as no surprise that our results are closely related to their work.First, our work is also based on considering cohomologies of shtukas.And second, our constructions show that Funct c (Bun G ( F q )) arises as global sections of some quasi-coherent sheaf Drinf arithm on LocSys arthmˇ G ( X ). (The conjecture expressed by formula (0.1) says thatDrinf arithm is the dualizing sheaf of LocSys arthmˇ G ( X ).) The existence of Drinf arithm recovers the spectraldecomposition of Funct c (Bun G ( F q )) along the set of classical Langlands parameters, see Remark 3.4.2.We refer to [AGKRRV1, Sect. 16] for further discussion of the relation to V. Lafforgue’s work. Remark . The principle that geometric methods enrich Langlands’s conjectures is an old one,dating (at least) to [De] and [Dr]. But precise refinements of Langlands’s conjectures using geometricideas have emerged recently. The present paper provides one example. Similarly, the Fargues–Scholzegeometrization program (see [Fa]) aims to produce a more robust form of the local Langlands conjec-tures.0.2.
Formulation of the main result.
We now proceed to the statement of our main result.0.2.1.
Notation.
Throughout the paper, we use algebraic geometry over the two fields k := F q and e := Q ℓ (where ℓ ∈ F × q ).When we work over F q , we generally work with algebraic stacks Y that are assumed to be defined over F q ; we abuse notation somewhat in letting Y ( F q ) denote the groupoid of F q -points of the correspondingstack. We let Frob Y : Y → Y denote the geometric ( q -)Frobenius morphism, whose stack of fixed-points Y Frob is a discrete, and identifies with (the ´etale sheafification of) the groupoid Y ( F q ).Let X be a smooth projective curve over F q , but assumed defined over F q . Let G/ F q be a reductivegroup, considered over F q via its split form. Let Bun G denote the moduli stack of principal G -bundleson X .We refer to Sect. 0.6 for further details on our conventions.0.2.2. Categorical trace.
Throughout this paper, all DG categories are enriched over the field e . Inparticular, Vect = Vect e (the DG category of chain complexes of e -vector spaces), and so on.We remind that the category DGCat of cocomplete DG categories is equipped with a canonicalsymmetric monoidal structure ⊗ , the Lurie tensor product . On general grounds, this means one mayspeak about categorical duals and traces as follows.If C ∈ DGCat is dualizable, there is another DG category C ∨ equipped with canonical unit andcounit maps u C : Vect → C ⊗ C ∨ and ev C : C ⊗ C ∨ → Vect . For an endofunctor Φ : C → C of C , we have Tr(Φ , C ) ∈ Vect defined asTr(Φ , C ) := ev (cid:0) (Φ ⊗ Id)(u C ) (cid:1) . An alternative construction of LocSys arthmˇ G ( X ) due to P. Scholze and X. Zhu, may be found in [Zhu]. Theirconstruction proceeds along very different lines, but conjecturally produces an equivalent object. D. ARINKIN, D. GAITSGORY, D. KAZHDAN, S. RASKIN, N. ROZENBLYUM, Y. VARSHAVSKY
We refer to [GKRV] for further discussion.0.2.3.
Categories of sheaves.
We consider the category Shv(Bun G ) of automorphic sheaves . Precisely,Shv(Bun G ) is the category of ind-constructible Q ℓ -sheaves on Bun G . As in [AGKRRV1], we alsoconsider its full subcategory(0.2) Shv Nilp (Bun G ) ⊂ Shv(Bun G )of objects with singular support in the global nilpotent cone.The categories Shv(Bun G ) and Shv Nilp (Bun G ) have favorable finiteness properties: by [AGKRRV1,Theorems 10.1.4 and 10.1.6], they are compactly generated and therefore dualizable in DGCat. More-over, the embedding Shv Nilp (Bun G ) → Shv(Bun G ) preserves compactness.0.2.4. Pushforward with respect to the geometric Frobenius endomorphism (see Sect. 0.6.2) definesan auto-equivalence (Frob Bun G ) ∗ of Shv(Bun G ), which preserves the subcategory Shv Nilp (Bun G ).Hence, it makes sense to consider the categorical traceTr((Frob Bun G ) ∗ , Shv
Nilp (Bun G )) ∈ Vect . As we describe below, the goal of this paper is to show that this vector space maps isomorphically ontothe space automorphic functions via the sheaves-functions dictionary.0.2.5. For any quasi-compact algebraic stack Y over F q and defined over F q , there is a canonical localterm map (see [AGKRRV1, Sect. 15.2]),LT Y : Tr((Frob Y ) ∗ , Shv( Y )) → Funct( Y ( F q )) , where Funct( − ) stands for the (classical) vector space e -valued functions.In addition, in loc. cit ., we extended this construction to non-quasi-compact stacks (such as Bun G ).In this setting, the above map takes the formLT Y : Tr((Frob Y ) ∗ , Shv( Y )) → Funct c ( Y ( F q )) , where Funct c ( Y ( F q )) ⊂ Funct( Y ( F q )) is the subspace of compactly supported e -valued functions. Remark . By functoriality of traces, any (possibly lax) Weil sheaf( F ∈ Shv( Y ) c , α : F → (Frob Y ) ∗ ( F ))on Y defines an element cl( F , α ) ∈ Tr((Frob Y ) ∗ , Shv( Y )) . According to [AGKRRV1, Sect. 15.2], the value of the map LT Y on cl( F , α ) produces the correspondingGrothendieck–Deligne function, denoted funct( F ).0.2.7. By the above, we obtain a map(0.3) Tr((Frob Bun G ) ∗ , Shv
Nilp (Bun G )) → Tr((Frob
Bun G ) ∗ , Shv(Bun G )) LT Bun G → Funct c (Bun G ( F q )) , where we note that the vector space Funct c (Bun G ( F q )) is the space of compactly supported unramifiedautomorphic functions. Main Theorem 0.2.8.
The composition (0.3) is an isomorphism.
This result was proposed as a conjecture in [AGKRRV1], where it was termed the
Trace Conjecture . Remark . Informally, one can view Theorem 0.2.8 as saying that “there are enough weak Weilsheaves on Bun G with nilpotent singular support to recover all automorphic functions, and any relationsbetween the automorphic functions defined by such sheaves have categorical origins.”Of course, the fact that we are considering sheaves with nilpotent singular support is crucial here.If we did not have the singular support condition, we would obviously have enough sheaves to recoverall functions. However, the relations imposed by sheaves would not match the relations on functions. UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 5
Remark . The phrase in quotation makes in the previous remark would be a correct assertion ifthe phrase inside the quotation marks was understood in the derived sense.More precisely, for a (compactly generated) category C with an endofunctor Φ, the trace objectTr(Φ , C ) ∈ Vect is computed as the geometric realization of a certain canonically defined simplicialobject of Vect; let us denote it Tr(Φ , C ) • .In particular, we have a map Tr(Φ , C ) → Tr(Φ , C ) (here the superscript 0 denotes the space of0-simplices), and hence a map H (Tr(Φ , C ) ) → H (Tr(Φ , C )) . Now, the image of the latter map is the span of the classescl( c , α ) , c ∈ C c , α : c → Φ( c ) . However, it is not true, in general, that H (Tr(Φ , C )) is spanned by the above classes: highercohomologies of higher simplices can also contribute.0.3. Insertion of Hecke functors.
So far, we have not mentioned Langlands duality. Remarkably,it plays an essential role in the proof of Theorem 0.2.8.In fact, our proof of Theorem 0.2.8 constructs isomorphisms between two families of objects, whichare, roughly speaking, indexed by Hecke functors. As we explain below, this amounts to a proof of[AGKRRV1, Conjecture 15.5.7], the so-called
Shtuka Conjecture of loc. cit Moreover, our method of proof relies in an essential way on consideration of all
Hecke functorssimultaneously.We presently explain what these objects are, and what our results about them assert.0.3.1. Let ˇ G denote the Langlands dual group of G considered as an algebraic group over the coefficientfield e := Q ℓ . We let Rep( ˇ G ) denote the (symmetric monoidal) DG category of representations of ˇ G .0.3.2. First, recall the formalism of Hecke functors in geometric Langlands. The key feature is thatthey may be indexed by tuples of moving, possibly colliding points.Precisely, given a finite set I and an object V ∈ Rep( ˇ G ) ⊗ I , we can consider the Hecke functor(0.4) H ( V, − ) : Shv(Bun G ) → Shv(Bun G × X I ) . Restricting to Shv
Nilp (Bun G ) ⊂ Shv(Bun G ), we obtain a functor that lands in the subcategoryShv Nilp (Bun G ) ⊗ QLisse( X ) ⊗ I ⊂ Shv(Bun G × X I )(see Corollary 1.3.5), where QLisse( X ) denotes the category of lisse sheaves on X , see Sect. 0.6.4.0.3.3. As we recall in Sect. 1.4.1, there is a symmetric monoidal category Rep( ˇ G ) Ran , the
Ran version of the category Rep( ˇ G ), that is the universal source of Hecke functors.More precisely, there is an action of Rep( ˇ G ) Ran on Shv(Bun G ) by integral Hecke functors , preservingthe subcategory Shv Nilp (Bun G ). By design, this action encodes the Hecke actions for varying finite sets I . We refer to Sect. 1.5.1 for the construction.We denote the monoidal product on Rep( ˇ G ) Ran by ⋆ and its monoidal unit by Rep( ˇ G ) Ran . We draw the reader’s attention to our conventions regarding the cohomological shift in the Shtuka Conjecture, seeRemarks 1.3.7 and 3.2.6. In particular, in [AGKRRV1, Conjecture 15.5.7], the embedding QLisse( X I ) ֒ → Shv( X I )must be understood in the sense of (1.6) of the present paper. D. ARINKIN, D. GAITSGORY, D. KAZHDAN, S. RASKIN, N. ROZENBLYUM, Y. VARSHAVSKY S : Rep( ˇ G ) Ran → Vect.By the definition of Rep( ˇ G ) Ran , such functors amount to compatible systems of functors S I : Rep( ˇ G ) ⊗ I → Shv( X I )defined for I ∈ fSet.In examples, the functors S I tend to be more familiar avatars of the functor S , so it is convenient toreference them.0.3.5. We have a functor Sht Tr : Rep( ˇ G ) Ran → Vectconstructed as the compositionRep( ˇ G ) Ran → End
DGCat (Shv
Nilp (Bun G )) −◦ (Frob Bun G ) ∗ → End
DGCat (Shv
Nilp (Bun G )) Tr → Vect . In other words, we take V ∈ Rep( ˇ G ) Ran , form the corresponding Hecke functor, compose with Frobenius,and take the trace of the resulting endofunctor of Shv
Nilp (Bun G ).By construction, we haveSht Tr ( Rep( ˇ G ) Ran ) = Tr((Frob
Bun G ) ∗ , Shv
Nilp (Bun G )) . Remark . To make the above more explicit, we describe the functors Sht Tr I : Rep( ˇ G ) ⊗ I → Shv( X I )for a finite set I .Precomposing (0.4) with (Frob Bun G ) ∗ , we obtain a functor H ( V, − ) ◦ (Frob Bun G ) ∗ : Shv Nilp (Bun G ) → Shv
Nilp (Bun G ) ⊗ QLisse( X ) ⊗ I , and we can consider its parameterized traceTr( H ( V, − ) ◦ (Frob Bun G ) ∗ , Shv
Nilp (Bun G )) ∈ QLisse( X ) ⊗ I . Unwinding the constructions, the resulting functorRep( ˇ G ) ⊗ I → QLisse( X ) ⊗ I , followed by the embedding QLisse( X ) ⊗ I ֒ → Shv( X I ) , is our Sht Tr I .0.3.7. On the other hand, following [VLaf], to the data ( I ∈ fSet , V ∈ Rep( ˇ G ) ⊗ I ) we can attach the“shtuka cohomology”, which is an objectSht I ( V ) ∈ Shv( X I ) , see Sect. 3.1 in the main body of the paper. These functors satisfy the requisite compatibilities neededto define a functor Sht : Rep( ˇ G ) Ran → Vect . Example . For I = ∅ , the functor Sht ∅ amounts to a map Vect → Vect, i.e., a vector space, whichcorresponds to Sht( Rep( ˇ G ) Ran ). As is standard, this vector space is Funct c (Bun G ( F q )). UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 7
Main Theorem 0.3.10.
There is a canonical equivalence
Sht Tr ≃ Sht of functors
Rep( ˇ G ) Ran → Vect . Moreover, the resulting map
Tr((Frob
Bun G ) ∗ , Shv
Nilp (Bun G )) = Sht Tr ( Rep( ˇ G ) Ran ) ∼ → Sht( Rep( ˇ G ) Ran ) = Funct c (Bun G ( F q )) is the local term map. Clearly this result refines Theorem 0.2.8.It follows that the functors Sht Tr I and Sht I canonically coincide. This is exactly the assertion of theShtuka Conjecture [AGKRRV1, Conjecture 15.5.7].0.4. Outline of the proof.
We now give an overview of the proof of Theorem 0.3.10. It requires afew additional objects and some relations between them.0.4.1. First, we recall the (non-algebraic) stackLocSys restrˇ G ( X )over e introduced in [AGKRRV1], the stack of local systems with restricted variation .We have a naturally defined symmetric monoidal localization functorLoc : Rep( ˇ G ) Ran → QCoh(LocSys restrˇ G ( X )) , see Sect. 2.3.0.4.2. We use Beilinson’s spectral projector , which is a certain object R ∈ Rep( ˇ G ) Ran related to LocSys restrˇ G ( X ). It has the following properties:(i) Loc( R ) ≃ O LocSys restrˇ G ( X ) .(ii) The functor Sht Tr is canonically isomorphic to Sht( R ⋆ − ), i.e., act by R and then apply Sht.The above Property (i) is one of the key results of the paper [AGKRRV1] (it follows from what isstated in this paper as Theorem 2.4.6, which is in turn [AGKRRV1, Theorem 9.1.3]).Property (ii) is a straightforward calculation using the main result of the paper [AGKRRV2], andappears here as Theorem 4.2.2.0.4.3. Using the result of [Xue2] mentioned above, we show in Corollary 3.1.4 that Sht factors canon-ically as a composition Rep( ˇ G ) Ran Loc −→ QCoh(LocSys restrˇ G ( X )) Sht
Loc −→ Vectfor a certain functor Sht
Loc .At this point, the result follows from Properties (i) and (ii) above: we haveSht Tr ≃ Sht
Loc ◦ Loc( R ⋆ − ) , and because Loc is monoidal and sends R to the unit of QCoh(LocSys restrˇ G ( X )), we haveLoc( R ⋆ − ) ≃ Loc , so combining we obtain Sht Tr = Sht Loc ◦ Loc( R ⋆ − ) ≃ Sht
Loc ◦ Loc ≃ Sht , which is the assertion of Theorem 0.3.10. as desired. D. ARINKIN, D. GAITSGORY, D. KAZHDAN, S. RASKIN, N. ROZENBLYUM, Y. VARSHAVSKY G ) Ran , the isomorphismSht Tr ≃ Sht is given by the local term map.This is the content of Theorem 5.1.3, which requires some additional ideas.0.5.
Organization of the paper.
We now describe how the present paper is structured.0.5.1. In Sect. 1 we review the formalism of Hecke functors acting on Shv(Bun G ).In particular, we introduce the Ran version of the category Rep( ˇ G ), denoted Rep( ˇ G ) Ran , which is amonoidal category that acts on Shv(Bun G ) by integral Hecke functors .This section does not contain any results original to this paper.0.5.2. In Sect. 2 we review some notions associated with the stack of local systems with restrictedvariation , introduced in [AGKRRV1], and denoted LocSys restrˇ G ( X ).Apart from the definition of LocSys restrˇ G ( X ), the main points are:(i) Description of the category QCoh(LocSys restrˇ G ( X )) as the category of compatible collections of func-tors Rep( ˇ G ) ⊗ I → QLisse( X ) ⊗ I , I ∈ fSet;(ii) The localization functor Loc : Rep( ˇ G ) Ran → QCoh(LocSys restrˇ G ( X ));(iii) Construction of Beilinson’s spectral projector, which is an explicit object R ∈ Rep( ˇ G ) Ran , one ofwhose main properties is the isomorphismLoc( R ) ≃ O LocSys restrˇ G ( X ) . (iv) Corollary 2.3.3, which asserts that a functor S : Rep( ˇ G ) Ran → Vect factors (uniquely) through thelocalization functor exactly when the functors S I are valued in QLisse( X ) ⊗ I ⊂ Shv( X I ).The entirety of the material of this section is a reformulation of the results in Parts I and II of[AGKRRV1].0.5.3. In Sect. 3 we review the shtuka construction and some of its variants.First, we introduce the functor Sht : Rep( ˇ G ) Ran → Vect.We then quote the main result from [Xue2], which we interpret as saying that the functors Sht I takevalues in QLisse( X ) ⊗ I ⊂ Shv( X I ). Using (iv) above, this yields the existence of the functor Sht Loc from above.By duality, the functor Sht
Loc yields an object of QCoh(LocSys restrˇ G ( X )), which we denote by Drinf.0.5.4. In Sect. 4 we formulate and prove the main result of this paper, Theorem 4.1.2, which is theassertion of Theorem 0.3.10 modulo compatibility with local terms. As particular cases, this statementcontains both the (unrefined) Trace Conjecture and the Shtuka Conjecture.The argument is the one outlined above.0.5.5. In Sects. 5 and 6 we show that the isomorphismTr(Frob ∗ , Shv
Nilp (Bun G )) ≃ Funct c (Bun G ( F q ))from Theorem 4.1.2 induces the local term map (0.3), as in the statement of Theorem 0.3.10. UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 9 non-standard self-duality of the category Shv
Nilp (Bun G ),studied in [AGKRRV2] and reviewed in Sect. 4.3.In the Appendix we attempt to re-run the calculation of Tr(Frob ∗ , Shv
Nilp (Bun G )) by relying on theusual Verdier self-duality of Shv Nilp (Bun G ). In the process we encounter a new object that we name“co-shtukas”.0.6. Notations and conventions.
The notations in this paper largely follow those of [AGKRRV1]and [AGKRRV2].0.6.1.
Algebraic geometry.
There will be disjoint “two algebraic geometries” at play in this paper: oneon the automorphic side, and another on the spectral side.On the automorphic side, our algebraic geometry will be over the ground field k , which in this paperis F q . Our algebro-geometric objects will be either schemes or algebraic stacks locally of finite typeover k . In this paper we will not need more general prestacks. Moreover, the algebraic geometry thatwe consider over k is classical (i.e. not derived).On the spectral side, our algebraic geometry will be over the field of coefficients e := Q ℓ , see below.We will consider just one algebro-geometric object over e –the (pre)stack LocSys restrˇ G ( X ) (see Sect. 2),but it will play quite a prominent role. Importantly, the algebraic geometry we consider over e is derived : by default, all schemes, stacks, etc. over e are derived.0.6.2. Frobenius endomorphism.
Let Y be an algebraic stack over F q , but defined over F q . In this case,we can consider the geometric Frobenius endomorphism of Y , denotedFrob Y : Y → Y . Thus, whenever we refer to the Frobenius endomorphism of Y , we will assume that Y is defined over F q . This is the case of our curve X , the reductive group G , and the stack Bun G of principal G -bundleson X .0.6.3. Higher algebra.
We will work with DG categories over the field of coefficients e := Q ℓ .All our conventions and notations regarding DG categories are imported from [AGKRRV2, Sects.0.5.2-0.5.3].There will be two kinds of sources that feed into higher algebra, i.e., the sources of DG categories.One will be various categories produced out of ℓ -adic sheaves on the automorphic side. Another willbe categories of quasi-coherent sheaves on the spectral side (specifically, the category of quasi-coherentsheaves on LocSys restrˇ G ( X )).0.6.4. Sheaves.
For a scheme S of finite type, we let Shv( S ) constr denote the category of constructible Q ℓ -adic sheaves on S , viewed as a (small) DG category over the field of coefficients e = Q ℓ .We let Shv( S ) denote the (cocomplete) DG category Ind(Shv( S ) constr ). We extend the assignment S Shv( S )from schemes to algebraic stacks by the procedure of [AGKRRV2, Sect. 1.1.1].For a given stack Y , we will denote by e Y , ω Y ∈ Shv( Y )the constant and dualizing sheaves, respectively. Singular support.
Let Y be an algebraic stack and N a conical Zariski-closed subset of T ∗ ( Y ).We will denote by Shv N ( Y ) ⊂ Shv( Y )the corresponding full subcategory, defined as in [AGKRRV1, Sects. B.4.1 and C.5.1].If Y is smooth and N is the zero-section, usually denoted { } , we will also use the notation QLisse( Y )for Shv { } ( Y ).0.6.6. Functors (co)defined by kernels.
In a few places in this paper we will make reference to functors defined or codefined by a kernel. We refer the reader to [AGKRRV2, Sects. A.1 and C.2.1], where thesenotions are introduced.Following loc. cit. , given a functor Shv( Y ) → Shv( Y ), defined or codefined by a kernel F , we willdenote by Id ⊗ F : Shv( Z × Y ) → Shv( Z × Y )for an algebraic stack Z (thought of as a stack of parameters).0.7. Acknowledgements.
We are grateful to C. Xue for communicating to us her results from [Xue2]before making them publicly available. We are also grateful to A. Beilinson and V. Lafforgue forinspiring discussions.The project was supported by David Kazhdan’s ERC grant No 669655. The work of D.K. and Y.V.was supported BSF grant 2016363. The work of D.A. was supported by NSF grant DMS- 1903391. Thework of D.G. was supported by NSF grant DMS-2005475. The work of D.K. was partially supportedby ISF grant 1650/15. The work of Y.V. was partially supported by ISF grant 822/17.1.
The Hecke action
In this section we will recall the pattern of Hecke acton of the category Rep( ˇ G ) on Shv(Bun G ), andsome related formalism. The section contains no original material.1.1. Hecke functors. I and every algebraic stack Z (thought of as a stack of parameters):(1.1) H ! : Rep( ˇ G ) ⊗ I ⊗ Shv( Z × Bun G ) → Shv( Z × Bun G × X I ) . For a fixed V ∈ Rep( ˇ G ) ⊗ I , we will denote by H ! ( V, − ) the resulting functorShv( Z × Bun G ) → Shv( Z × Bun G × X I ) . G ) ⊗ I ⊗ Rep( ˇ G ) ⊗ I ⊗ Shv( Z × Bun G ) mult −−−−−→ Rep( ˇ G ) ⊗ I ⊗ Shv( Z × Bun G ) Id ⊗ H ! y Rep( ˇ G ) ⊗ I ⊗ Shv( Z × Bun G × X I ) y H ! H ! y Shv( Z × Bun G × X I × X I ) (Id × ∆) ! −−−−−→ Shv( Z × Bun G × X I ) , where mult : Rep( ˇ G ) ⊗ I ⊗ Rep( ˇ G ) ⊗ I → Rep( ˇ G ) ⊗ I is the tensor product functor and ∆ X I : X I → X I × X I is the diagonal embedding. The data ofassociativity of (1.1) comes additionally with higher coherence for higher powers of Rep( ˇ G ) ⊗ I . UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 11
We can rephrase that as saying that the category Shv( Z × Bun G × X I ) is a module category for themonoidal category Rep( ˇ G ) ⊗ I , and the action is Shv( X I )-linear (i.e. it is a module category for themonoidal category Rep( ˇ G ) ⊗ I ⊗ Shv( X I )).1.1.3. The functors (1.1) are naturally compatible with maps between finite sets. Namely, for a map ψ : I → J , we have a data of commutativity for the diagram(1.2) Rep( ˇ G ) ⊗ I ⊗ Shv( Z × Bun G ) H ! −−−−−→ Shv( Z × Bun G × X I ) Id ⊗ mult ψ y y (Id × ∆ ψ ) ! Rep( ˇ G ) ⊗ J ⊗ Shv( Z × Bun G ) H ! −−−−−→ Shv( Z × Bun G × X J )where mult ψ : Rep( ˇ G ) ⊗ I → Rep( ˇ G ) ⊗ J is the functor given by the symmetric monoidal structure on Rep( ˇ G ), and∆ ψ : X J → X I the diagonal map defined by ψ .The above data of commutativity are endowed with a homotopy coherent system of compatibilitiesfor compositions of maps of finite sets.Moreover, this data is compatible with the associativity described in Sect. 1.1.2. Namely, the functor(Id × ∆ ψ ) ! : Shv( Z × Bun G × X I ) → Shv( Z × Bun G × X J )is a functor of Shv( X I ) ⊗ Rep( ˇ G ) ⊗ I -module categories.1.1.4. A feature of the functors H ! ( V, − ) is that they are functors that are both defined and codefinedby kernels ; see Sect. 0.6.6 for what this means. In practical terms, this implies that for a map Z → Z between algebraic stacks, we have a datum of commutativity for the diagramsShv( Z × Bun G ) H ! ( V, − ) −−−−−→ Shv( Z × Bun G × X I ) y y Shv( Z × Bun G ) H ! ( V, − ) −−−−−→ Shv( Z × Bun G × X I ) , where the vertical arrows are given by either *- or !- pushforwards, and also for the diagramsShv( Z × Bun G ) H ! ( V, − ) −−−−−→ Shv( Z × Bun G × X I ) x x Shv( Z × Bun G ) H ! ( V, − ) −−−−−→ Shv( Z × Bun G × X I ) , where the vertical arrows are given by either !- or *- pullbacks.Moreover, this datum of commutativity is functorial in V ∈ Rep( ˇ G ) ⊗ I , and is compatible with thedatum of commutativity of the diagrams (1.2).1.2. The ULA property of the Hecke action. H ! is that for F ∈ Shv( Z × Bun G ) c and V ∈ (Rep( ˇ G ) ⊗ I ) c ,the object H ! ( V, F ) ∈ Shv( Z × Bun G × X I )is ULA with respect to the projection Z × Bun G × X I → X I . H ∗ the functorRep( ˇ G ) ⊗ I ⊗ Shv( Z × Bun G ) → Shv( Z × Bun G × X I )defined as H ∗ ( V, F ) := H ! ( V, F ) ! ⊗ p !3 ( e X I ) . Note that for a fixed I , the difference between H ! and H ∗ amounts to a cohomological shift by 2 | I | since e X I ≃ ω X I [ − | I | ].1.2.3. The ULA property of the objects H ! ( V, F ) implies that we have canonical isomorphisms(1.3) H ! ( V, F ) ! ⊗ p !3 ( M ) ≃ H ∗ ( V, F ) ∗ ⊗ p ∗ ( M ) , M ∈ Shv( X I ) . Furthermore, for a map of finite sets ψ : I → J , we have a data of commutativity for the diagram(1.4) Rep( ˇ G ) ⊗ I ⊗ Shv( Z × Bun G ) H ∗ −−−−−→ Shv( Z × Bun G × X I ) Id ⊗ mult ψ y y (Id × ∆ ψ ) ∗ Rep( ˇ G ) ⊗ J ⊗ Shv( Z × Bun G ) H ∗ −−−−−→ Shv( Z × Bun G × X J ) , endowed with a homotopy coherent system of compatibilities for compositions of maps of finite sets.1.2.4. Thus, we can regard the functors H ∗ ( V, − ) also as defined and codefined by kernels, and theyhave the formal properties parallel to those of the functors H ! ( V, − ).1.3. Hecke action on
Shv
Nilp (Bun G ) . Nilp (Bun G ) ⊂ Shv(Bun G ) . The following result, essentially due to [NY], describes the behavior of this subcategory under theHecke functors (this is stated as [AGKRRV1, Theorem 10.2.3]):
Theorem 1.3.2.
The Hecke functor H ! for I = {∗} sends the full subcategory Rep( ˇ G ) ⊗ Shv
Nilp (Bun G ) ⊂ Rep( ˇ G ) ⊗ Shv(Bun G ) to the full subcategory Shv
Nilp ×{ } (Bun G × X ) ⊂ Shv(Bun G × X ) . Y and a conicalhalf-dimensional N ⊂ T ∗ ( Y ), the (a priori fully faithful) functorShv N ( Y ) ⊗ QLisse( X ) → Shv N ×{ } ( Y × X )is an equivalence.Thus, from Theorem 1.3.2 we obtain: Corollary 1.3.4.
The Hecke functor H ! for I = {∗} sends the full subcategory Rep( ˇ G ) ⊗ Shv
Nilp (Bun G ) ⊂ Rep( ˇ G ) ⊗ Shv(Bun G ) to the full subcategory Shv
Nilp (Bun G ) ⊗ QLisse( X ) ⊂ Shv(Bun G × X ) . Iterating, from Corollary 1.3.4 we further obtain:
Corollary 1.3.5.
The Hecke functors H ! map the full subcategory Rep( ˇ G ) ⊗ I ⊗ Shv
Nilp (Bun G ) ⊂ Rep( ˇ G ) ⊗ I ⊗ Shv(Bun G ) to the full subcategory Shv
Nilp (Bun G ) ⊗ QLisse( X ) ⊗ I ⊂ Shv(Bun G × X I ) . UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 13 Y , we can consider QLisse( Y ) as a full subcategory of Shv( Y )in two different ways.One is the tautological embedding (1.5) QLisse( Y ) ֒ → Shv( Y ) , L L ;it endows QLisse( Y ) with a symmetric monoidal structure induced by the !-tensor product on Shv( Y ).We also have a different embedding:(1.6) QLisse( Y ) ֒ → Shv( Y ) , L L ! ⊗ e Y ;it endows QLisse( Y ) with a symmetric monoidal structure induced by the *-tensor product on Shv( Y ).However, it follows tautologically that the two symmetric monoidal structures on QLisse( Y ) coincide.Moreover, the operationsQLisse( Y ) ⊗ Shv( Y ) (1.5) ⊗ Id −→ Shv( Y ) ⊗ Shv( Y ) ! ⊗ → Shv( Y )and QLisse( Y ) ⊗ Shv( Y ) (1.6) ⊗ Id −→ Shv( Y ) ⊗ Shv( Y ) ∗ ⊗ → Shv( Y )define the same monoidal action of QLisse( Y ) on Shv( Y ).We will apply this discussion to Y = X I . Remark . Note also that when Y is smooth of dimension n , the embeddings (1.5) and (1.6) differby the cohomological shift [2 n ]. Yet they should not be confused.1.3.8. An assertion parallel to Corollary 1.3.4 holds for the H ∗ functors. Namely, these functors sendRep( ˇ G ) ⊗ I ⊗ Shv
Nilp (Bun G ) ⊂ Rep( ˇ G ) ⊗ I ⊗ Shv(Bun G )to Shv Nilp (Bun G ) ⊗ QLisse( X ) ⊗ I ⊂ Shv(Bun G × X I ) . where we will think of the embedding QLisse( X ) ⊗ I ֒ → Shv( X I ) as given by (1.6).1.3.9. Thus, we can think of the Hecke action on Shv Nilp (Bun G ) either by means of the functors(1.7) H ! : Rep( ˇ G ) ⊗ I ⊗ Shv
Nilp (Bun G ) → Shv
Nilp (Bun G ) ⊗ QLisse( X ) ⊗ I , when we think of QLisse( X I ) as embedded into Shv( X I ) via (1.5), or, equivalently, as(1.8) H ∗ : Rep( ˇ G ) ⊗ I ⊗ Shv
Nilp (Bun G ) → Shv
Nilp (Bun G ) ⊗ QLisse( X ) ⊗ I , when we think of QLisse( X I ) as embedded into Shv( X I ) via (1.6).The functors (1.7) and (1.8) are canonically isomorphic. Thus, in what follows we will not distin-guish notationally between H ! and H ∗ , when applied to objects from Shv Nilp (Bun G ), and denote thecorresponding functors simply by(1.9) H : Rep( ˇ G ) ⊗ I ⊗ Shv
Nilp (Bun G ) → Shv
Nilp (Bun G ) ⊗ QLisse( X ) ⊗ I . For a map of finite sets ψ : I → J , we have a data of commutativity for the diagram(1.10) Rep( ˇ G ) ⊗ I ⊗ Shv
Nilp (Bun G ) H −−−−−→ Shv(Bun G ) ⊗ QLisse( X ) ⊗ I Id ⊗ mult ψ y y Id × mult ψ Rep( ˇ G ) ⊗ J ⊗ Shv
Nilp (Bun G ) H −−−−−→ Shv(Bun G ) ⊗ QLisse( X ) ⊗ J , where in the right vertical arrow the functormult ψ : QLisse( X ) ⊗ I → QLisse( X ) ⊗ J Recall that our conventions are such that the default pullback functor is !-pullback, and therefore, by definition,lisse sheaves are those that are locally !-pulled back from a point. As explained below, it will be important to alsoconsider the usual notion of lisse sheaves, i.e. those which are locally ∗ -pulled back from a point. is given by the symmetric monoidal structure on QLisse( X ).These data of commutativity are endowed with a homotopy coherent system of compatibilities forcompositions of maps of finite sets.1.4. The category
Rep( ˇ G ) Ran . We will now introduce a device that allows us to express the Heckeaction on Shv(Bun G ) in terms of a single monoidal category.1.4.1. Let C be a symmetric monoidal DG category. We define a new symmetric monoidal DG category C Ran by the following construction.Let TwArr(fSet) be the category of twisted arrows on fSet, see [GKRV, Sect. 1.2.2].The category C Ran is the colimit over TwArr(fSet) of the functor(1.11) TwArr(fSet) → DGCatthat sends ( I → J ) C ⊗ I ⊗ Shv( X J ) . Here for a map(1.12) I −−−−−→ J φ I y x φ J I −−−−−→ J , in TwArr(fSet), the corresponding functor C ⊗ I ⊗ Shv( X J ) → C ⊗ I ⊗ Shv( X J )is given by the tensor product functor along the fibers of φ I (1.13) mult φ I : C ⊗ I → C ⊗ I and the functor(1.14) (∆ φ J ) ! : Shv( X J ) → Shv( X J ) , where ∆ φ J : X J → X J is the diagonal map induced by φ J .1.4.2. The functor (1.11) is naturally right-lax symmetric monoidal. Therefore, the colimit C Ran carries a natural symmetric monoidal structure. Explicitly, this symmetric monoidal structure can bedescribed as follows. For V ⊗ M ∈ C ⊗ I ⊗ Shv( X J ) and V ⊗ M ∈ C ⊗ I ⊗ Shv( X J ) , the tensor product of their images in C Ran is the image of the object( V ⊗ V ) ⊗ ( M ⊠ M ) ∈ C ⊗ ( I ⊔ I ) ⊗ Shv( X J ⊔ J ) . We will denote the resulting monoidal operation on C Ran by V , V V ⋆ V . We denote the unit object by C Ran . It arises from (Id : ∅ → ∅ ) ∈ TwArr(fSet) and the correspondingmap Vect = C ⊗∅ ⊗ Shv( X ∅ ) → C Ran . ψ : I → J ) ∈ TwArr(fSet) be given.We denote by ins ψ : C ⊗ I ⊗ Shv( X J ) → C Ran the corresponding functor.In the important special case ψ = Id I : I → I , we use the notation ins I in place of ins Id I . UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 15 C = Rep( ˇ G ).We denote the resulting (symmetric monoidal) category by Rep( ˇ G ) Ran .1.5.
A Ran version of the Hecke action. G ) Ran on Shv( Z × Bun G ).Namely, for ( I ψ → J ) ∈ TwArr(fSet) and V ⊗ M ∈ Rep( ˇ G ) ⊗ I ⊗ Shv( X J ) , we let the corresponding endofunctor of Shv( Z × Bun G ) be the composition(1.15) Shv( Z × Bun G ) H ! ( V, − ) −→ Shv( Z × Bun G × X I ) − ! ⊗ p !3 ((∆ ψ ) ∗ ( M )) −→→ Shv( Z × Bun G × X I ) ( p , ) ∗ −→ Shv( Z × Bun G ) . Z × Bun G ) H ∗ ( V, − ) −→ Shv( Z × Bun G × X I ) − ∗ ⊗ p ∗ ((∆ ψ ) ∗ ( M )) −→→ Shv( Z × Bun G × X I ) ( p , ) ! −→ Shv( Z × Bun G ) . f : Z → Z . And the interpretation of the Hecke action via (1.16) impliesthat it commutes with *-pullbacks and !-pushforwards along maps f : Z → Z .This implies that for a given V ∈ Rep( ˇ G ) Ran its Hecke action is a functor both defined and codefinedby a kernel , (see Sect. 0.6.6 for what this means).We will denote the resulting endofunctor of Shv(Bun G ) by F V ⋆ F and of Shv( Z × Bun G ) by Id ⊗ ( V ⋆ − ) (see Sect. 0.6.6 for the ⊗ notation).We will refer to endofunctors of Shv(Bun G ) (or, more generally, Shv( Z × Bun G )) that arise in thisway as integral Hecke functors .1.5.4. For later use, we introduce the following notation. For V ∈ Rep( ˇ G ) Ran we let K V ∈ Shv(Bun G × Bun G )denote the object equal to (Id ⊗ ( V ⋆ − ))((∆ Bun G ) ! ( e Bun G )) . G ) Ran on Shv(Bun G ) preserves the full subcategoryShv Nilp (Bun G ) ⊂ Shv(Bun G ) . The dual category of
Rep( ˇ G ) Ran . For what follows we will need to recall some constructionspertaining to duality on Rep( ˇ G ) Ran . We will do so in the general setting of Sect. 1.4.Thus, we take C to be a general symmetric monoidal DG category.1.6.1. Assume that C is dualizable, and that for every ( I ψ → J ) ∈ TwArr(fSet), the functormult ψ : C ⊗ I → C ⊗ J is such that the dual functor ( C ⊗ J ) ∨ → ( C ⊗ I ) ∨ admits a left adjoint.In this case, one shows that the category C Ran is dualizable (see, e.g., [GR, Chapter 1, Proposition6.3.4]). C Ran ) ∨ is the category of continuous functors C Ran → Vect, and hence itcan be described as(1.17) lim ( I ψ → J ) ∈ TwArr(fSet) op ( C ⊗ I ⊗ Shv( X J )) ∨ , where the limit is formed using the functors dual to the ones used in the formation of the colimit inSect. 1.4.1.Using the Verdier self-duality on Shv( X J ), we can rewrite(1.18) ( C Ran ) ∨ ≃ lim ( I ψ → J ) ∈ TwArr(fSet) op Maps
DGCat ( C ⊗ I , Shv( X J )) , where Maps
DGCat ( − , − ) stands for the DG category of continuous e -linear functor between two objectsof DGCat.Explicitly, the transition functors in (1.18) are defined as follows. For a morphism in TwArr(fSet)given by (1.12), the corresponding functor Maps
DGCat ( C ⊗ I , Shv( X J )) → Maps
DGCat ( C ⊗ I , Shv( X J ))is given by precomposition (1.13) and postcomposition with(∆ φ J ) ! : X J → X J , which is the functor dual to (1.14), under the Verdier self-duality of Shv( X ? ). Remark . Suppose for a moment that C is rigid. In this case, we have a natural identification C ∨ ≃ C , and we can further rewrite the right-hand side in (1.18) aslim ( I ψ → J ) ∈ TwArr(fSet) op C ⊗ I ⊗ Shv( X J ) , where the limit is formed using the functors right adjoint to the ones used in the formation of thecolimit in Sect. 1.4.1. Hence, the above limit is isomorphic to the colimitcolim ( I ψ → J ) ∈ TwArr(fSet) C ⊗ I ⊗ Shv( X J )(see [GR, Chapter 1, Proposition 2.5.7]), i.e., to C Ran itself.This implies that for C rigid, the category C Ran is naturally self-dual. However, we will not use thisself-duality for the purposes of this paper.1.6.4. Consider the ( ∞ , fSet := Funct(fSet , DGCat) . There will be several DG categories of interest in this paper that will arise as
Maps
DGCat fSet ( C , C )for some particular C , C ∈ DGCat fSet .Concretely, an object of
Maps
DGCat fSet ( C , C ) is a collection of functors between DG categories C ( I ) → C ( I ) , I ∈ fSetthat make the diagrams C ( I ) −−−−−→ C ( I ) C ( ψ ) y y C ( ψ ) C ( J ) −−−−−→ C ( J )commute for I ψ → J , along with a homotopy coherent system of higher compatibilities. UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 17 fSet that we will need.One is the object denoted C ⊗ fSet , and defined by I ∈ fSet C ⊗ I ∈ DGCat , where the functoriality is furnished by the symmetric monoidal structure on C .Another object, denoted Shv ! ( X fSet ), is defined by I ∈ fSet Shv( X I ) ∈ DGCat , where for I φ → J , the corresponding functor Shv( X I ) → Shv( X J ) is (∆ φ ) ! .1.6.6. Consider the category Maps
DGCat fSet ( C ⊗ fSet , Shv ! ( X fSet )) . Note that this category identifies with the limit (1.18) (see e.g. [GKRV, Lemma 1.3.12]).1.6.7. Thus, to summarize, we obtain a canonical equivalence(1.19)
Maps
DGCat fSet ( C ⊗ fSet , Shv ! ( X fSet )) ≃ ( C Ran ) ∨ . Explicitly, given S : C Ran → Vect , the corresponding system of functors S I : C ⊗ I → Shv( X I )is recovered as follows:We precompose S with ins I to obtain a functor C ⊗ I ⊗ Shv( X I ) → Vect . By Verdier duality, the datum of the latter functor is equivalent to the datum of a functor S I : S ◦ ins I ( c ⊗ M ) = C · ( X I , S I ( c ) ! ⊗ M ) , c ∈ C ⊗ I , M ∈ Shv( X I ) . C Ran ⊗ Maps
DGCat fSet ( C ⊗ fSet , Shv ! ( X fSet )) → Vectis explicitly given as follows:For an object { S I } ∈ Maps
DGCat fSet ( C ⊗ fSet , Shv ! ( X fSet )), the corresponding functor S : C Ran → Vectis such that for ( I ψ → J ) ∈ TwArr(fSet), the resulting functor C ⊗ I ⊗ Shv( X J ) ins ψ −→ C Ran S → Vect , equals C ⊗ I ⊗ Shv( X J ) mult ψ ⊗ Id −→ C ⊗ J ⊗ Shv( X J ) S J ⊗ Id −→ Shv( X J ) ⊗ Shv( X J ) → Vect , where the last arrow is the Verdier duality pairing on Shv( X J ), i.e.,Shv( X J ) ⊗ Shv( X J ) ∆ ! XJ → Shv( X J ) C · ( X J , − ) → Vect . Quasi-coherent sheaves on
LocSys restrˇ G ( X )Although the statement of the Trace Conjecture does not involve Langlands duality, we will needsome of its ingredients for the proof. Indeed, one of the key tools in the proof will be the categoryof quasi-coherent sheaves on the (pre)stack LocSys restrˇ G ( X ), classifying local systems with restrictedvariation with respect to the Langlands dual group ˇ G of G . The (pre)stack
LocSys restrˇ G ( X ) . We start by recalling the definition of the prestackLocSys restrˇ G ( X ) , following [AGKRRV1, Sect. 1.3].2.1.1. For a test affine (derived) scheme S , we let Maps( S, LocSys restrˇ G ( X )) be the space of right t-exactsymmetric monoidal functors Rep( ˇ G ) → QCoh( S ) ⊗ QLisse( X ) . According to [AGKRRV1, Theorem 1.3.2], the prestack LocSys restrˇ G ( X ) can be written as the quotient Z / ˇ G , where Z is a disjoint union of formal affine schemes locally almost of finite type (over the fieldof coefficients e ).2.1.2. The main results of this paper will be based on considering the (symmetric monoidal) categoryQCoh(LocSys restrˇ G ( X )) . We will now explain a certain feature that this category possesses, which is a consequence of prop-erties of LocSys restrˇ G ( X ) as a prestack.(i) First, according to [AGKRRV1, Lemma 5.3.2], the diagonal map∆ LocSys restrˇ G ( X ) : LocSys restrˇ G ( X ) → LocSys restrˇ G ( X ) × LocSys restrˇ G ( X )is affine, so the functor(∆ LocSys restrˇ G ( X ) ) ∗ : QCoh(LocSys restrˇ G ( X )) → QCoh(LocSys restrˇ G ( X ) × LocSys restrˇ G ( X ))is continuous.(ii) Second, according to [AGKRRV1, Corollary 5.7.5], the category QCoh(LocSys restrˇ G ( X )) is dualizable.By [GR, Chapter 3, Proposition 3.1.7], this implies that for any prestack Y over e , the functor of externaltensor product QCoh(LocSys restrˇ G ( X )) ⊗ QCoh( Y ) → QCoh(LocSys restrˇ G ( X ) × Y )is an equivalence.In particular, we can view (∆ LocSys restrˇ G ( X ) ) ∗ ( O LocSys restrˇ G ( X ) )as an object of QCoh(LocSys restrˇ G ( X )) ⊗ QCoh(LocSys restrˇ G ( X )) . (iii) And third, according to [AGKRRV1, Corollary 5.7.10], the above object(∆ LocSys restrˇ G ( X ) ) ∗ ( O LocSys restrˇ G ( X ) ) ∈ QCoh(LocSys restrˇ G ( X )) ⊗ QCoh(LocSys restrˇ G ( X ))defines the unit of a self-duality on QCoh(LocSys restrˇ G ( X )). UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 19
The functor of “sections with scheme-theoretic support”.
We can view the structure sheaf O LocSys restrˇ G ( X ) ∈ QCoh(LocSys restrˇ G ( X )) as defining a functor(2.1) Vect → QCoh(LocSys restrˇ G ( X )) . Note that the object O LocSys restrˇ G ( X ) ∈ QCoh(LocSys restrˇ G ( X )) is not compact, so the functor of globalsections Γ(LocSys restrˇ G ( X ) , − ) : QCoh(LocSys restrˇ G ( X )) → Vect , the right adjoint to (2.1), is not continuous.However, due to the self-duality of QCoh(LocSys restrˇ G ( X )), we can consider the functor dual to (2.1),which is a functor(2.2) QCoh(LocSys restrˇ G ( X )) → Vect , which we will denote by Γ ! (LocSys restrˇ G ( X ) , − ), and refer to as the functor of sections with scheme-theoretic support . Remark . The terminology for Γ ! (LocSys restrˇ G ( X ) , − ) is explained by the following assertion (see[AGKRRV1, Corollary 5.7.8 and Sect. 5.7.11]): Proposition . (a) The functor colim S ∈ (Sch aft ) / LocSysrestrˇ G ( X ) QCoh( S ) is an equivalence, where the transition functors in the family are f : S → S f ∗ : QCoh( S ) → QCoh( S ) . (b) In terms of the above equivalence, the functor Γ ! (LocSys restrˇ G ( X ) , − ) corresponds to the functor colim S ∈ (Sch aft ) / LocSysrestrˇ G ( X ) QCoh( S ) → Vect given by the compatible family of functors Γ( S, − ) : QCoh( S ) → Vect . The tautological objects.
For a finite set I and an object V ∈ Rep( ˇ G ) ⊗ I , letEv( V ) ∈ QCoh(LocSys restrˇ G ( X )) ⊗ QLisse( X ) ⊗ I be the corresponding tautological object:For S → LocSys restrˇ G ( X ) corresponding to a symmetric monoidal functorΦ S : Rep( ˇ G ) → QCoh( S ) ⊗ QLisse( X ) , the pullback of Ev( V ) to S , viewed as an object in QCoh( S ) ⊗ QLisse( X ) ⊗ I equals the value on V ofthe functor Rep( ˇ G ) ⊗ I Φ ⊗ IS → (QCoh( S ) ⊗ QLisse( X )) ⊗ I → QCoh( S ) ⊗ QLisse( X ) ⊗ I , where the second arrow is given by the I -fold tensor product functorQCoh( S ) ⊗ I → QCoh( S ) . Description of the category
QCoh(LocSys restrˇ G ( X )) . The prestack LocSys restrˇ G ( X )) was definedusing the symmetric monoidal categories Rep( ˇ G ) and QLisse( X ). Therefore, it would not be verysurprising to have a description of the category QCoh(LocSys restrˇ G ( X )) purely in terms of functorsbetween the above two categories.In this subsection, we will provide such a description, following [AGKRRV1]. fSet , see Sect. 1.6.4, and the objectQLisse( X ) ⊗ fSet ∈ DGCat fSet , see Sect. 1.6.5.Note that we could also consider the object QLisse( X fSet ): I ∈ fSet QLisse( X I ) ∈ DGCat . We have a naturally defined map in DGCat fSet (2.3) QLisse( X ) ⊗ fSet → QLisse( X fSet ) . However, from [AGKRRV1, Theorem B.5.8 and Corollary B.3.7], we obtain:
Lemma 2.2.2.
The map (2.3) is an isomorphism.
Maps
DGCat fSet (Rep( ˇ G ) ⊗ fSet , QLisse( X ) ⊗ fSet ), and the followingfunctor, to be denoted coLoc:(2.4) QCoh(LocSys restrˇ G ( X )) → Maps
DGCat fSet (Rep( ˇ G ) ⊗ fSet , QLisse( X ) ⊗ fSet ) . Namely, coLoc sends F ∈ QCoh(LocSys restrˇ G ( X )) to the collection of functors F I : Rep( ˇ G ) ⊗ I → QLisse( X ) ⊗ I defined by F I ( V ) := Γ ! (LocSys restrˇ G ( X ) , F ⊗ Ev( V )) , where Γ ! is as in Sect. 2.1.3.The following is one of the main results of the paper [AGKRRV1] (see Theorem 6.2.11 and Sect.9.2.3 in loc. cit. ): Theorem 2.2.4.
The functor coLoc is an equivalence.Remark . We note that the results of this subsection and the previous one hold more generally:instead of a curve X one can take any scheme of finite type (not necessarily proper).The only place were properness was used was Lemma 2.2.2, but Theorem 2.2.4 does not rely on it.2.3. Localization.
Recall the category Rep( ˇ G ) Ran (see Sect. 1.4.1). In this paper, we will use it as adevice to relate the phenomenon of Hecke action and the category QCoh(LocSys restrˇ G ( X )).In this subsection we will see how Rep( ˇ G ) Ran is related to QCoh(LocSys restrˇ G ( X )).2.3.1. Consider the symmetric monoidal category Rep( ˇ G ) Ran , see Sect. 1.4.1. We are going to con-struct a symmetric monoidal functor(2.5) Loc : Rep( ˇ G ) Ran → QCoh(LocSys restrˇ G ( X ))that plays a key role in this work.For an individual ( I ψ → J ) ∈ TwArr(fSet) , the corresponding functorLoc I ψ → J : Rep( ˇ G ) ⊗ I ⊗ Shv( X J ) → QCoh(LocSys restrˇ G ( X ))sends V ∈ Rep( ˇ G ) ⊗ I to the functor Shv( X J ) → QCoh(LocSys restrˇ G ( X )) equal toShv( X J ) Ev( V ) ⊗ Id −→ QCoh(LocSys restrˇ G ( X )) ⊗ QLisse( X ) ⊗ I ⊗ Shv( X J ) Id ⊗ mult ψ ⊗ Id −→→ QCoh(LocSys restrˇ G ( X )) ⊗ QLisse( X ) ⊗ J ⊗ Shv( X J ) −→→ QCoh(LocSys restrˇ G ( X )) ⊗ Shv( X J ) Id ⊗ C · c ( X J , − ) −→ QCoh(LocSys restrˇ G ( X )) , UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 21 where the third arrow using the canonical action of QLisse( X ) ⊗ J ≃ QLisse( X J ) on Shv( X J ), seeSect. 1.3.6.It is easy to see that the functors Loc I ψ → J indeed combine to define a functor, to be denote Loc, asin (2.5). Moreover, this functor carries a naturally defined symmetric monoidal structure.2.3.2. Consider the dual functorLoc ∨ : QCoh(LocSys restrˇ G ( X )) → (Rep( ˇ G ) Ran ) ∨ . Recall now that the category QCoh(LocSys restrˇ G ( X )) is self-dual (see Sect. 2.1.2), and that thecategory (Rep( ˇ G ) Ran ) ∨ can be described as Maps
DGCat fSet ( C ⊗ fSet , Shv ! ( X fSet )) (see (1.19)).Thus, we can view Loc ∨ as a functorQCoh(LocSys restrˇ G ( X )) → Maps
DGCat fSet ( C ⊗ fSet , Shv ! ( X fSet )) . Unwinding the definitions, we obtain that Loc ∨ identifies with the compositionQCoh(LocSys restrˇ G ( X )) coLoc → Maps
DGCat fSet ( C ⊗ fSet , QLisse( X ) ⊗ fSet ) ≃≃ Maps
DGCat fSet ( C ⊗ fSet , QLisse( X fSet )) → Maps
DGCat fSet ( C ⊗ fSet , Shv ! ( X fSet )) , where the last arrow is given by the (fully faithful) functor (1.5).Hence, combining with Theorem 2.2.4, we obtain: Corollary 2.3.3. (a)
The functor
Loc ∨ is fully faithful. (b) An object S ∈ (Rep( ˇ G ) Ran ) ∨ lies in the essential image of Loc ∨ if and only if the correspondingfamily of functors { S I } Rep( ˇ G ) ⊗ I → Shv( X I ) takes values in QLisse( X I ) ⊂ Shv( X I ) . G ) Ran ) ∨ QLisse ⊂ (Rep( ˇ G ) Ran ) ∨ be the full subcategory that under the equivalence (1.19) corresponds to Maps
DGCat fSet ( C ⊗ fSet , QLisse( X fSet )) ⊂ Maps
DGCat fSet ( C ⊗ fSet , Shv ! ( X fSet )) , where the embedding QLisse( X fSet ) ֒ → Shv ! ( X fSet ) is (1.5).We obtain that Corollary 2.3.3 can be reformulated as follows: Corollary 2.3.5.
The functor
Loc ∨ defines an equivalence QCoh(LocSys restrˇ G ( X )) → (Rep( ˇ G ) Ran ) ∨ QLisse . Beilinson’s spectral projector.
In this subsection we will introduce a certain object R ∈ Rep( ˇ G ) Ran , that will play a crucial role in the proof of the main results in this paper. R ˇ G ∈ Rep( ˇ G ) ⊗ Rep( ˇ G )denote the regular representation.For ( I ψ → J ) ∈ TwArr(fSet) let R I ψ → J LocSys restrˇ G ( X ) be the object of the tensor product categoryQCoh(LocSys restrˇ G ( X )) ⊗ (cid:16) Rep( ˇ G ) ⊗ I ⊗ Shv( X J ) (cid:17) , equal to the image of ( R ˇ G ) ⊗ I ∈ Rep( ˇ G ) ⊗ I ⊗ Rep( ˇ G ) ⊗ I along the functorRep( ˇ G ) ⊗ I ⊗ Rep( ˇ G ) ⊗ I Ev( − ) ⊗ Id −→ QCoh(LocSys restrˇ G ( X )) ⊗ QLisse( X ) ⊗ I ⊗ Rep( ˇ G ) ⊗ I Id ⊗ mult ψ ⊗ Id −→→ QCoh(LocSys restrˇ G ( X )) ⊗ QLisse( X ) ⊗ J ⊗ Rep( ˇ G ) ⊗ I →→ QCoh(LocSys restrˇ G ( X )) ⊗ Shv( X J ) ⊗ Rep( ˇ G ) ⊗ I = QCoh(LocSys restrˇ G ( X )) ⊗ Rep( ˇ G ) ⊗ I ⊗ Shv( X J ) . G ) ⊗ I ⊗ Shv( X J ) to Rep( ˇ G ) Ran , the assignment( I ψ → J ) R I ψ → J LocSys restrˇ G ( X ) naturally extends to a functor(2.6) TwArr(fSet) → QCoh(LocSys restrˇ G ( X )) ⊗ Rep( ˇ G ) Ran R LocSys restrˇ G ( X ) ∈ QCoh(LocSys restrˇ G ( X )) ⊗ Rep( ˇ G ) Ran to be the colimit of the functor (2.6) over TwArr(fSet).
Remark . Recall that the category Rep( ˇ G ) Ran is canonically self-dual as a DG category, see Remark1.6.3. Let u
Rep( ˇ G ) Ran ∈ Rep( ˇ G ) Ran ⊗ Rep( ˇ G ) Ran be the unit of this self-duality.One can show that we have a canonical isomorphism R LocSys restrˇ G ( X ) ≃ (Loc ⊗ Id)(u
Rep( ˇ G ) Ran ) , as objects of QCoh(LocSys restrˇ G ( X )) ⊗ Rep( ˇ G ) Ran .2.4.5. Consider the object(2.7) (Id ⊗ Loc)( R LocSys restrˇ G ( X ) ) ∈ QCoh(LocSys restrˇ G ( X )) ⊗ QCoh(LocSys restrˇ G ( X )) . The following is [AGKRRV1, Theorem 9.1.3]:
Theorem 2.4.6.
The image of (Id ⊗ Loc)( R LocSys restrˇ G ( X ) ) under QCoh(LocSys restrˇ G ( X )) ⊗ QCoh(LocSys restrˇ G ( X )) ∼ → QCoh(LocSys restrˇ G ( X ) × LocSys restrˇ G ( X )) identifies canonically with (∆ LocSys restrˇ G ( X ) ) ∗ ( O LocSys restrˇ G ( X ) ) . R := (Γ ! (LocSys restrˇ G ( X ) , − ) ⊗ Id)( R LocSys restrˇ G ( X ) ) ∈ Rep( ˇ G ) Ran , where Γ ! (LocSys restrˇ G ( X ) , − ) is as in Sect. 2.1.3.From Theorem 2.4.6 we obtain: Corollary 2.4.8.
The object
Loc( R ) ∈ QCoh(LocSys restrˇ G ( X )) identifies canonically with O LocSys restrˇ G ( X ) . UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 23 The reciprocity law for shtuka cohomology
The main result of this section is Corollary 3.1.4, which asserts that the functor Sht introducedbelow factors through the localization functor Loc. We will deduce it from a theorem of C. Xue onlisseness of shtuka cohomology.As an application, we construct an object Drinf ∈ QCoh(LocSys restrˇ G ) that encodes the cohomologyof shtuka moduli spaces.3.1. Functorial shtuka cohomology.
In this subsection we will interpret shtuka cohomology as afunctor Sht : Rep( ˇ G ) Ran → Vect , i.e., an object of the category (Rep( ˇ G ) Ran ) ∨ .3.1.1. Recall the functor V ∈ Rep( ˇ G ) Ran K V ∈ Shv(Bun G × Bun G ) , see Sect. 1.5.4.Let Graph Frob
Bun G : Bun G → Bun G × Bun G be the graph of Frobenius, i.e., the mapBun G ∆ Bun G −→ Bun G × Bun G Frob
Bun G × Id −→ Bun G × Bun G . We define the functor Sht : Rep( ˇ G ) Ran → Vect as the compositionRep( ˇ G ) Ran V K V −→ Shv(Bun G × Bun G ) (Graph FrobBun G ) ∗ −→ Shv(Bun G ) C · c (Bun G , − ) −→ Vect . Theorem 3.1.3.
The object
Sht ∈ (Rep( ˇ G ) Ran ) ∨ belongs to the full subcategory (Rep( ˇ G ) Ran ) ∨ QLisse ⊂ (Rep( ˇ G ) Ran ) ∨ . Applying Corollary 2.3.5, from Theorem 3.1.3, we obtain:
Corollary 3.1.4.
The functor
Sht : Rep( ˇ G ) Ran → Vect factors as
Sht
Loc ◦ Loc for a uniquely definedfunctor
Sht
Loc : QCoh(LocSys restrˇ G ( X )) → Vect . Proof of Theorem 3.1.3 and relation to the usual shtuka cohomology.
Once we relatethe functor Sht to shtuka cohomology, the proof of Theorem 3.1.3 will be almost immediate from arecent theorem of C. Xue [Xue2].3.2.1. For a finite set I , consider the functorRep( ˇ G ) ⊗ I → Shv( X I ) , to be denoted Sht I , that sends V ∈ Rep( ˇ G ) ⊗ I to(3.1) Sht I ( V ) := ( p ) ! ◦ (Graph Frob
Bun G ) ∗ ◦ H ∗ ( V, − ) ◦ ∆ ! ( e Bun G ) . The above functor Sht I is the usual functor of (compactly supported) shtuka cohomology, studiedby [VLaf].3.2.2. We now quote the following crucial result of [Xue2]: Theorem 3.2.3.
The functor
Sht I takes values in QLisse( X I ) ⊂ Shv( X I ) . X is proper), we obtain that for I ∈ fSet, the functorRep( ˇ G ) ⊗ I ⊗ Shv( X I ) ins I → Rep( ˇ G ) Ran Sht → Vectis given by sending V ∈ Rep( ˇ G ) ⊗ I , M ∈ Shv( X I ) C · (cid:18) X I , (Sht I ( V ) ∗ ⊗ ω X I ) ! ⊗ M (cid:19) . Thus, the object of
Maps
DGCat fSet (Rep( ˇ G ) ⊗ fSet , Shv ! ( X fSet )) corresponding to Sht is given by V ∈ Rep( ˇ G ) ⊗ I Sht I ( V ) ∗ ⊗ ω X I ≃ Sht I ( V )[2 | I | ] , see Sect. 1.6.7.This object belongs to QLisse( X I ) by Theorem 3.2.3, as required. Remark . Note that by Theorem 3.2.3, the expression C · (cid:18) X I , (Sht I ( V ) ∗ ⊗ ω X I ) ! ⊗ M (cid:19) , whichappears above, can be also rewritten asC · (cid:16) X I , Sht I ( V ) ∗ ⊗ M (cid:17) , see Sect. 1.2.3. Remark . Along with the object Shv ! ( X fSet ) ∈ DGCat fSet , we can consider the object, denotedShv ∗ ( X fSet ), whose value on I is again the category Shv( X I ), but now for I ψ → J we use the functor(∆ ψ ) ∗ : Shv( X I ) → Shv( X J ) . As in Sect. 1.3.6, we have the fully faithful embeddingsShv ∗ ( X fSet ) ← ֓ QLisse( X fSet ) ֒ → Shv ! ( X fSet ) , given by (1.5) and (1.6).By its construction, the system of functors { Sht I } is naturally an object of the category Maps
DGCat fSet (Rep( ˇ G ) ⊗ fSet , Shv ∗ ( X fSet )), and we can view Theorem 3.2.3 as saying that it actuallybelongs to the essential image of the functor Maps
DGCat fSet (Rep( ˇ G ) ⊗ fSet , QLisse( X fSet )) ֒ → Maps
DGCat fSet (Rep( ˇ G ) ⊗ fSet , Shv ∗ ( X fSet )) . Now, the object of
Maps
DGCat fSet (Rep( ˇ G ) ⊗ fSet , Shv ! ( X ) fSet ) corresponding to Sht equals the imageof the resulting object of Maps
DGCat fSet (Rep( ˇ G ) ⊗ fSet , QLisse( X fSet )) under Maps
DGCat fSet (Rep( ˇ G ) ⊗ fSet , QLisse( X fSet )) ֒ → Maps
DGCat fSet (Rep( ˇ G ) ⊗ fSet , Shv ! ( X fSet )) . Thus, we denote by the same symbol { Sht I } the above objects of the categories Maps
DGCat fSet (Rep( ˇ G ) ⊗ fSet , Shv ∗ ( X fSet )) ← ֓ Maps
DGCat fSet (Rep( ˇ G ) ⊗ fSet , QLisse( X fSet )) ֒ → ֒ → Maps
DGCat fSet (Rep( ˇ G ) ⊗ fSet , Shv ! ( X fSet )) . Example . We have a canonical isomorphismSht( Rep( ˇ G ) Ran ) ≃ Sht ∅ ( e ) = C c (cid:16) Bun G , (Graph Frob
Bun G ) ∗ ◦ (∆ Bun G ) ! ( e Bun G ) (cid:17) ≃ Funct c (Bun G ( F q ))where the last isomorphism is by base-change.3.3. Drinfeld’s sheaf.
UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 25
Loc : QCoh(LocSys restrˇ G ( X )) → Vect, i.e.,Sht
Loc ∈ QCoh(LocSys restrˇ G ( X )) ∨ . As QCoh(LocSys restrˇ G ( X )) is canonically self-dual (cf. Sect. 2.1.2), there is a corresponding object Drinf ∈ QCoh(LocSys restrˇ G ( X )) . c (LocSys restrˇ G ( X ) , Drinf) ≃ Sht
Loc ( O LocSys restrˇ G ( X ) ) . by duality.As O LocSys restr = Loc( Rep( ˇ G ) Ran ) , we deduce from Example 3.2.7 that there is a canonical isomorphism(3.2) Γ c (LocSys restrˇ G ( X ) , Drinf) = Sht( Rep( ˇ G ) Ran ) = Funct c (Bun G ( F q )) . F ∈ QCoh(LocSys restrˇ G ( X )) and a finite set I , one defines a functor F I : Rep( ˇ G ) ⊗ I → QLisse( X I ) . By construction, for F = Drinf, the functor Drinf I coincides with Sht I . In this manner, we see thatDrinf encodes cohomology of shtuka moduli spaces.3.4. Some remarks.
We now provide some additional remarks on the object Drinf.
Remark . Recall (see [AGKRRV1, Sect. 16.1]) that the stack LocSys arithmˇ G ( X ) is defined as(LocSys restrˇ G ( X )) Frob . Let ι denote the forgetful map LocSys arithmˇ G ( X ) → LocSys restrˇ G ( X ) . One can show that the object Drinf can be a priori obtained as ι ∗ (Drinf arithm ) for a canonicallydefined object Drinf arithm ∈ QCoh(LocSys arithmˇ G ( X )) . This additional structure on Drinf encodes the equivariance of the objectsSht I ( V ) ∈ Shv( X I ) , V ∈ Rep( ˇ G ) ⊗ I with respect to the partial Frobenius maps acting on X I . See also Sect. 4.5.6. Remark . The object Drinf arithm allows to recover the spectral decomposition of the space ofautomorphic functions along classical Langlands parameters, established in [VLaf] for the cuspidalsubspace and extended in [Xue1] to the entire space.Namely, by (3.2), we have:Γ(LocSys arithmˇ G ( X ) , Drinf arithm ) ≃ Funct c (Bun G ( F q )) . Set A := Γ(LocSys arithmˇ G ( X ) , O LocSys arithmˇ G ( X ) );this is is a commutative DG algebra over e that lives in non-positive cohomological degrees. SetLocSys arithm , coarseˇ G ( X ) := Spec( A );this is an affine (derived) scheme over e . This object is named after V. Drinfeld, since it was his idea, upon learning about V. Lafforgue’s work, that shtukacohomology should be encoded by a quasi-coherent sheaf on the stack of Langlands parameters.
Let A denote the 0-th cohomology of A , so that Spec( A ) is the classical affine scheme cl LocSys arithm , coarseˇ G ( X ) underlying LocSys arithm , coarseˇ G ( X )Now, one can show that the set of e -points of Spec( A ) is in bijection with isomorphism classes ofsemi-simple Frobenius-equivariant ˇ G -local systems on X , see [AGKRRV1, Corollary 2.4.8] . I.e., wecan view cl LocSys arithm , coarseˇ G ( X ) as the scheme of classical Langlands parameters.By construction, A acts on the space of global actions of any object of QCoh(LocSys arithmˇ G ( X )). Inparticular, we obtain an action of A on Funct c (Bun G ( F q )). However, since Funct c (Bun G ( F q )) sits incohomologcal degree 0, this action factors through an action of A on Funct c (Bun G ( F q )).Thus, we can view Funct c (Bun G ( F q )) as global sections of a canonically defined quasi-coherentsheaf on cl LocSys arithm , coarseˇ G ( X ). This indeed may be viewed as a spectral decomposition ofFunct c (Bun G ( F q )) over classical Langlands parameters.Furthermore, one can show that A is a quotient of V. Lafforgue’s algebra of excursion operators. Sothe above action of A recovers the action of the excursion algebra on Funct c (Bun G ( F q )), establishedin [Xue1]. Remark . The above construction of the object Drinf (resp., Drinf arithm ) was specific to the everywhere unramified situation. In a subsequent publication, we will show this construction can begeneralized to allow for level structure.I.e., given a divisor D ⊂ X define over F q , one can construct objectsDrinf D ∈ LocSys restrˇ G ( X − D ) and Drinf arithm D ∈ LocSys arithmˇ G ( X − D )that encode shtuka cohomology with level structure.What is for now a far-fetched goal is to interpret Drinf arithm D also as categorical trace, see Sect. 4.5.7for what we mean by that. 4. Calculating the trace
In this section we will prove the main result of this paper, Corollary 4.1.4, which asserts that thespace of (compactly supported) automorphic functions can be obtained as the (categorical) trace ofFrobenius on the category of automorphic sheaves with nilpotent singular support.4.1.
Traces of Frobenius-Hecke operators. G ) Ran action on Shv(Bun G ) preserves its subcategory Shv Nilp (Bun G ).Therefore, we obtain a functorRep( ˇ G ) Ran → Maps
DGCat (Shv
Nilp (Bun G ) , Shv
Nilp (Bun G ))sending V to the functor V ⋆ − .Note also that the subcategory Shv Nilp (Bun G ) ⊂ Shv(Bun G ) is preserved by the endofunctor(Frob Bun G ) ∗ , see [AGKRRV1, Sect. 15.3.3].We define a functor Sht Tr : Rep( ˇ G ) Ran → Vectas the functor V Tr (cid:0) ( V ⋆ − ) ◦ (Frob Bun G ) ∗ , Shv
Nilp (Bun G ) , (cid:1) . In other words, we compose V ⋆ − with pushforward along the geometric Frobenius Frob Bun G endomor-phism of Bun G and form the trace (as an endofunctor of the (dualizable) DG category Shv Nilp (Bun G ).Our main theorem asserts: Theorem 4.1.2.
There is a canonical isomorphism of functors
Sht ≃ Sht Tr : Rep( ˇ G ) Ran → Vect . We will prove this result in Sect. 4.2. The quoted result of [AGKRRV1] applies to LocSys restrˇ G ( X ), but the case of LocSys arithmˇ G ( X ) is similar. UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 27 Tr ( Rep( ˇ G ) Ran ) = Tr((Frob
Bun G ) ∗ , Shv
Nilp (Bun G )) . On the other hand, by Example 3.2.7 we haveSht( Rep( ˇ G ) Ran ) = Funct c (Bun G ( F q )) . Hence, we obtain:
Corollary 4.1.4.
There exists a canonical isomorphism in
VectTr((Frob
Bun G ) ∗ , Shv
Nilp (Bun G )) ≃ Funct c (Bun G ( F q )) . Tr corresponds to a system of functors { Sht Tr I } (4.1) Sht Tr I : Rep( ˇ G ) ⊗ I → Shv( X I ) . From Theorem 4.1.2 we immediately obtain:
Corollary 4.1.6.
For an individual finite set I , the functors Sht I and Sht Tr I Rep( ˇ G ) ⊗ I → QLisse( X ) ⊗ I are canonically isomorphic. Let us describe the functors Sht Tr I explicitly.4.1.7. We will use the following construction.Recall that if C is a dualizable DG category and T : C → C ⊗ D is a functor, we can consider the relative trace object Tr( T, C ) ∈ D defined as the composition Vect u C → C ∨ ⊗ C Id ⊗ T → C ∨ ⊗ C ⊗ D ev C ⊗ Id → D . I be a finite set. Recall that we have a canonically defined functor H : Rep( ˇ G ) ⊗ I ⊗ Shv
Nilp (Bun G ) → Shv
Nilp (Bun G ) ⊗ QLisse( X I ) , see (1.9).In particular, for V ∈ Rep( ˇ G ) ⊗ I we can consider the objectTr( H ( V, − ) ◦ (Frob Bun G ) ∗ , Shv
Nilp (Bun G )) ∈ QLisse( X I ) , and this operation defines a functor(4.2) Rep( ˇ G ) ⊗ I → QLisse( X I ) . Unwinding the definitions, it is easy to see that the functor Sht Tr I of (4.1) identifies with the com-position of (4.2) and the embedding QLisse( X I ) → Shv( X I ) of (1.5).4.1.9. Hence, from Corollary 4.1.6 we obtain: Corollary 4.1.10.
For an individual finite set I , the functor Sht I : Rep( ˇ G ) ⊗ I → QLisse( X I ) of (3.1) identifies canonically with the functor V Tr( H ( V, − ) ◦ (Frob Bun G ) ∗ , Shv
Nilp (Bun G )) . Corollary 4.1.10 is the
Shtuka Conjecture from [AGKRRV1] (Conjecture 15.5.7 in loc.cit. ). Tr I take values in QLisse( X I ). Therefore, by Corol-lary 2.3.5, Sht Tr factors uniquely as Sht TrLoc ◦ Loc for a uniquely defined functorSht
TrLoc : QCoh(LocSys restrˇ G ( X )) → Vect . As in Sect. 3.3, by self-duality of QCoh(LocSys restrˇ G ( X )), to Sht TrLoc there corresponds an objectDrinf Tr ∈ QCoh(LocSys restrˇ G ( X )).From Theorem 4.1.2, we obtain: Corollary 4.1.12.
The objects
Drinf and
Drinf Tr of QCoh(LocSys restrˇ G ) are canonically isomorphic. Proof of Theorem 4.1.2. R ∈ Rep( ˇ G ) Ran of Sect. 2.4. The proof of Theorem 4.1.2 will be based on thefollowing result.
Theorem 4.2.2.
There is a canonical isomorphism
Sht Tr ≃ Sht( R ⋆ − ) as functors Rep( ˇ G ) Ran → Vect .Remark . As the proof of Theorem 4.2.2 will show, we will rather construct an isomorphismSht Tr ≃ Sht( − ⋆ R ) , and we will swap Sht( − ⋆ R ) for Sht( R ⋆ − ) using the fact that the category Rep( ˇ G ) Ran is symmetricmonoidal .Our preference of one over the other is purely notational.4.2.4. We postpone the proof of Theorem 4.2.2 to Sect. 4.4. It is essentially a calculation using theresults of [AGKRRV2], which we recall in Sect. 4.3.For the present, we assume Theorem 4.2.2 and deduce Theorem 4.1.2 from it.4.2.5. The argument is straightforward at this point using Theorem 3.1.3. Recall that loc. cit . providesa factorization Sht = Sht Loc ◦ Loc.Now recall that Loc is monoidal and sends R to the structure sheaf by Corollary 2.4.8. Therefore,Loc( R ⋆ − ) ≃ Loc( − )as functors Rep( ˇ G ) Ran → QCoh(LocSys restrˇ G ( X )).By Theorem 4.2.2, we obtainSht Tr ≃ Sht( R ⋆ − ) ≃ Sht
Loc ◦ Loc( R ⋆ − ) ≃ Sht
Loc ◦ Loc( − ) ≃ Shtas desired.4.3.
Self-duality for
Shv
Nilp (Bun G ) . To prove Theorem 4.2.2, we use the explicit description of thedual of Shv
Nilp (Bun G ) from [AGKRRV2]. We review these results below.4.3.1. Consider the object K R ∈ Shv(Bun G × Bun G ) , where R ∈ Rep( ˇ G ) Ran was defined in Sect. 2.4, and where the notation K − is an in Sect. 1.5.4.A priori, it is defined as an object of the category Shv(Bun G × Bun G ). However, we have thefollowing key result, which follows from [AGKRRV2, Sect. 2.1.3 and Proposition 2.1.10]: Theorem 4.3.2.
The object K R lies in the essential image of the fully faithful functor Shv
Nilp (Bun G ) ⊗ Shv
Nilp (Bun G ) → Shv(Bun G × Bun G ) . UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 29 Y let ev ∗ Y : Shv( Y ) ⊗ Shv( Y ) → Vectbe the functor given by F , F C · c ( Y , F ∗ ⊗ F ) . Warning : in general, the pairing ev ∗ Y is not perfect.4.3.4. Take Y = Bun G , and we restrict the pairing ev ∗ Bun G toShv Nilp (Bun G ) ⊗ Shv
Nilp (Bun G ) ⊂ Shv(Bun G ) ⊗ Shv(Bun G ) . We now quote the following result (see [AGKRRV2, Theorem 2.1.5]):
Theorem 4.3.5.
The object K R ∈ Shv
Nilp (Bun G ) ⊗ Shv
Nilp (Bun G ) together with the pairing ev ∗ Bun G : Shv Nilp (Bun G ) ⊗ Shv
Nilp (Bun G ) define an identification Shv
Nilp (Bun G ) ∨ ≃ Shv
Nilp (Bun G ) . Proof of Theorem 4.2.2. C and functors T, S : C → C .In this case, recall that we can tautologically compute Tr( S ◦ T, C ) ∈ Vect as the compositionVect u C −→ C ∨ ⊗ C T ∨ ⊗ S −→ C ∨ ⊗ C ev C ⊗ Id −→ Vect . V ∈ Rep( ˇ G ) Ran , we deduce from Theorem 4.3.5 and the above that we can compute Sht Tr ( V )as the compositionVect K R −→ Shv
Nilp (Bun G ) ⊗ Shv
Nilp (Bun G ) ((Frob Bun G ) ∗ ) ∨ ⊗ ( V ⋆ − ) −→→ Shv
Nilp (Bun G ) ⊗ Shv
Nilp (Bun G ) − ∗ ⊗− −→ Shv(Bun G ) C c (Bun G , − ) −→ Vect . Here ((Frob
Bun G ) ∗ ) ∨ is the dual functor to (Frob Bun G ) ∗ with respect to the duality of Theorem 4.3.5.4.4.3. To proceed further, we need to compute ((Frob Bun G ) ∗ ) ∨ . Lemma 4.4.4.
In the above notation, the functor ((Frob
Bun G ) ∗ ) ∨ : Shv Nilp (Bun G ) → Shv
Nilp (Bun G ) is canonically isomorphic to (Frob Bun G ) ∗ .Proof. For an algebraic stack Y , the geometric Frobenius morphism Frob Y is finite, so we have acanonical isomorphism(4.3) (Frob Y ) ! ≃ (Frob Y ) ∗ . Therefore, we need to compute the dual to (Frob
Bun G ) ! : Shv Nilp (Bun G ) → Shv
Nilp (Bun G ).Note that (Frob Bun G ) ! is an auto-equivalence of Shv Nilp (Bun G ). Moreover, the unit and counitof the duality are equivariant with respect to this autoequivalence. It follows formally under suchcircumstances that the dual to (Frob Bun G ) ! is its inverse.Observe that the inverse to (Frob Bun G ) ! ≃ (Frob Bun G ) ∗ is its left adjoint (Frob Bun G ) ∗ , completingthe argument. (This argument shows we could also take (Frob Bun G ) ! in the assertion of the lemma.) (cid:3) V ∈ Rep( ˇ G ) Ran , we deduce that we can compute Sht Tr ( V ) as the composition:Vect K R −→ Shv
Nilp (Bun G ) ⊗ Shv
Nilp (Bun G ) (Frob Bun G ) ∗ ⊗ ( V ⋆ − ) −→→ Shv
Nilp (Bun G ) ⊗ Shv
Nilp (Bun G ) − ∗ ⊗− −→ Shv(Bun G ) C c (Bun G , − ) −→ Vect . Note that (Id ⊗ ( V ⋆ − ))( K R ) ≃ K V ⋆ R ≃ K R ⋆ V . Therefore, we can compute Sht Tr ( V ) as the compositionVect K R ⋆ V −→ Shv
Nilp (Bun G ) ⊗ Shv
Nilp (Bun G ) (Frob Bun G ) ∗ ⊗ Id −→→ Shv
Nilp (Bun G ) ⊗ Shv
Nilp (Bun G ) ∆ ∗ Bun G ( − ⊠ − ) −→ Shv(Bun G ) C c (Bun G , − ) −→ Vect . This coincides withVect K R ⋆ V −→ Shv
Nilp (Bun G ) ⊗ Shv
Nilp (Bun G ) (Graph FrobBun G ) ∗ −→ Shv(Bun G ) C c (Bun G , − ) −→ Vect . By definition, this composition is Sht( R ⋆ V ), concluding the argument.4.5. Interpretation as enhanced trace.
The contents of this section are an extended remark andare not necessary for the rest of the paper.4.5.1. Recall (see [AGKRRV1, Theorem 10.5.2]) that the category Shv
Nilp (Bun G ) carries a monoidalaction of QCoh(LocSys restrˇ G ( X )).Moreover, the action of (Frob Bun G ) ∗ is compatible with the action of the monoidal automorphismof Frob ∗ of QCoh(LocSys restrˇ G ( X )), where Frob is the automorphism of LocSys restrˇ G ( X ), induced byFrobenius endomorphism Frob X of X .In this case, following [GKRV, Sect. 3.8.2], to the pair (Shv Nilp (Bun G ) , (Frob Bun G ) ∗ ) we can attachedits enhanced trace ,Tr enhQCoh(LocSys restrˇ G ( X )) ((Frob Bun G ) ∗ , Shv
Nilp (Bun G )) ∈ HH • (QCoh(LocSys restrˇ G ( X )) , Frob ∗ ) , where HH • (QCoh(LocSys restrˇ G ( X )) , Frob ∗ ) is the (symmetric monoidal) category of Hochschild chainson the (symmetric) monoidal category QCoh(LocSys restrˇ G ( X )) with respect to the (symmetric) monoidalendofunctor Frob ∗ .Furthermore, we haveHH • (QCoh(LocSys restrˇ G ( X )) , Frob ∗ ) ≃ QCoh((LocSys restrˇ G ( X )) Frob ) , see [GKRV, Sect. 3.7.3].4.5.2. Recall that we denote LocSys arithmˇ G ( X ) := (LocSys restrˇ G ( X )) Frob , and by ι the forgetful map LocSys arithmˇ G ( X ) → LocSys restrˇ G ( X )) , see [AGKRRV1, Sect. 16.1].Thus, we can interpret the above object Tr enhQCoh(LocSys restrˇ G ( X )) ((Frob Bun G ) ∗ , Shv
Nilp (Bun G )) as anobject, denoted Drinf Tr , arithm ∈ LocSys arithmˇ G ( X ) . We have(4.4) Drinf Tr ≃ ι ∗ (Drinf Tr , arithm );this assertion is essentially [GKRV, Theorem 4.4.4]. UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 31 V ) ∈ QCoh(LocSys restrˇ G ( X )) ⊗ QLisse( X ) ⊗ I . By construction, the corresponding objects( ι ∗ ⊗ Id)(Ev( V )) ∈ QCoh(LocSys arithmˇ G ( X )) ⊗ QLisse( X ) ⊗ I carry a structure of equivariance with respect to the partial Frobenius endmorphisms acting along X I .By (4.4) and the projection formula, we haveSht Tr I ( V ) ≃ (Γ(LocSys arithmˇ G ( X ) , − ) ⊗ Id) (cid:16) ( ι ∗ ⊗ Id)(Ev( V )) ⊗ Drinf Tr , arithm ) (cid:17) . This interpretation shows that the the objectsSht Tr I ( V ) ∈ QLisse( X ) ⊗ I ⊂ Shv( X I )carry a natural structure of equivariance with respect to the partial Frobenius endomorphisms on X I .4.5.4. Having proved Theorem 4.1.2, and hence Corollary 4.1.6, we obtain that the objectsSht I ( V ) ∈ Shv( X I )also carry a natural structure of equivariance with respect to the partial Frobenius endomorphisms on X I .However, it is not difficult to show that this structure identifies with the of partial Frobenius equiv-ariance on shtukas (see in [VLaf]). The matching between the two is is essentially [GKRV, Lemma4.5.4].4.5.5. Now, given an object F ∈ QCoh(LocSys restrˇ G ( X )), the datum required to exhibit it as ι ∗ ( F arithm )for some F arithm ∈ LocSys arithmˇ G ( X ) is equivalent to a compatible collection of structure of partialFrobenius equivariance on the associated functors F I : Rep( ˇ G ) ⊗ I → QLisse( X ) ⊗ I , V (Γ(LocSys restrˇ G ( X ) , − ) ⊗ Id)(Ev( V ) ⊗ F ) . I , thereby producing an objectDrinf arithm ∈ QCoh(LocSys arithmˇ G ( X )) . Thus, the assertion in Sect. 4.5.4 can interpreted as an isomorphismDrinf arithm ≃ Drinf Tr , arithm , which induces the isomorphism Drinf ≃ Drinf Tr of Theorem 4.1.2 by applying ι ∗ .4.5.7. Furthermore, as was mentioned in Remark 3.4.3, the corresponding structure of partial Frobe-nius equivariance can be constructed also on the functorsSht I,D : Rep( ˇ G ) ⊗ I → QLisse( X − D ) ⊗ I that encode the cohomology of shtukas with level structure, thereby producing an objectDrinf arithm D ∈ QCoh(LocSys restrˇ G ( X − D )) . What we do not have at the moment is the interpretation of this object Drinf arithm D as enhancedcategorical trace (nor of the functors Sht I,D as just categorical traces).5.
Local terms
Formulation of the problem. naive , LT true , LT Serre , LT Sht : Tr((Frob
Bun G ) ∗ , Shv
Nilp (Bun G )) → Funct c (Bun G ( F q ))that we refer to as local term morphisms.At this point, we have only encountered the last of these four morphisms: by definition, LT Sht isthe isomorphism of Corollary 4.1.4.5.1.2. Assuming the construction of the other three morphisms, we can state the main result of thissection and the next:
Theorem 5.1.3.
The four morphisms LT naive , LT true , LT Serre , and LT Sht are equal.Remark . By Corollary 4.1.4, the source and target of each local term map is in Vect ♥ , so equality(as opposed to homotopy) is the relevant notion.5.1.5. As LT Sht is an isomorphism, from Theorem 5.1.3 we deduce:
Corollary 5.1.6.
Each of the four local term morphisms in an isomorphism.
In the case of LT naive (see below), this corollary recovers the Trace Conjecture as formulated in[AGKRRV1, Conjecture 15.3.5].We now proceed to the construction of the local term morphisms.5.2.
Naive and true local terms.
In what follows, we fix an algebraic stack Y , which plays the roleof Bun G . We will add certain additional assumptions to Y as we proceed.The material in this subsection closely follows [AGKRRV1, Sect. 15.1-15.2], to which we refer thereader for more detail.5.2.1. Naive local term.
Any algebraic stack Y has the property that Shv( Y ) is compactly generatedand every compact object is !-extended from some quasi-compact open in Y (see [AGKRRV1, Sect.C.1.1]).Suppose y ∈ Y ( F q ) is given. In this case, the functor y ∗ : Shv( Y ) → Vect preserves compact objectsand intertwines (Frob Y ) ∗ with the identity functor (by (4.3)). Therefore, functoriality of traces yieldsan induced map Tr((Frob Y ) ∗ , Shv( Y )) → Tr(Id , Vect) = e . In the quasi-compact case, this yields a mapLT naive Y : Tr((Frob Y ) ∗ , Shv( Y )) → Funct( Y ( F q )) . Remark . By definition, this construction satisfies the compatibility referenced in Remark 0.2.6.5.2.3.
True local term.
Suppose first that Y is quasi-compact. Suppose in addition that Y is locally ofthe form Z/H , where Z is a scheme of finite type and H is an affine algebraic group.As in [AGKRRV2, Sects. 1.1.3], these assumptions imply that Shv( Y ) is canonically self-dual (viaVerdier duality), with pairing, denoted ev ! Y :Shv( Y ) ⊗ Shv( Y ) ⊂ Shv( Y × Y ) ∆ ! Y → Shv( Y ) C · N ( Y , − ) → Vect . The unit for this duality, denoted u
Shv( Y ) ∈ Shv( Y ) ⊗ Shv( Y )is obtained by applying the right adjoint to the embedding(5.1) Shv( Y ) ⊗ Shv( Y ) ֒ → Shv( Y × Y )to (∆ Y ) ∗ ( ω Y ). UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 33
In what follows we will not distinguish notationally between u
Shv( Y ) and its image under the fullyfaithful functor (5.1). Thus, by adjunction we obtain a mapu Shv( Y ) → (∆ Y ) ∗ ( ω Y ) . From here, we obtain a mapTr((Frob Y ) ∗ , Shv( Y )) ≃ C · N (cid:0) Y , ∆ ! Y ◦ ((Frob Y ) ∗ ⊗ Id)(u
Shv( Y ) ) (cid:1) ≃ C · N (cid:0) Y , ∆ ! Y ◦ (Frob Y × Id) ∗ (u Shv( Y ) ) (cid:1) →→ C · N (cid:0) Y , ∆ ! Y ◦ (Frob Y × Id) ∗ ◦ (∆ Y ) ∗ ( ω Y )) (cid:1) ≃ C · N ( Y Frob , ω Y Frob ) ≃ Funct( Y ( F q ))whose composition we denote byLT true Y : Tr((Frob Y ) ∗ , Shv( Y )) → Funct( Y ( F q )) . Theorem 5.2.5. (a)
The maps LT true Y and LT naive Y are canonically homotopic. (b) For a schematic map f : Y → Y , the diagram Tr((Frob Y ) ∗ , Shv( Y )) LT true Y −−−−−→ Funct( Y ( F q )) y y Tr((Frob Y ) ∗ , Shv( Y )) LT true Y −−−−−→ Funct( Y ( F q )) is commutative, where the left vertical arrow is induced by the functor f ! : Shv( Y ) → Shv( Y ) , and theright vertical arrow is given by pushforward. (c) The commutative diagram in point (b) is compatible with the identification of point (a) with the(tautologically) commutative diagram
Tr((Frob Y ) ∗ , Shv( Y )) LT naive Y −−−−−→ Funct( Y ( F q )) y y Tr((Frob Y ) ∗ , Shv( Y )) LT naive Y −−−−−→ Funct( Y ( F q )) . Y be not necessarily quasi-compact. We will consider the poset of quasi-compact opensubstacks U j ֒ → Y , and the corresponding functors j ! : Shv( U ) → Shv( Y ).The category Shv( Y ) is compactly generated by the essential images of j ! | Shv( U ) c . Furthermore, theinduced map colim U Tr((Frob U ) ∗ , Shv( U )) → Tr((Frob Y ) ∗ , Shv( Y ))is an isomorphism (see [AGKRRV1, Sect. 15.1.10]).Using the commutative diagrams in Theorem 5.2.5(b),(c), this allows to define the mapsLT naive Y and LT true Y from Tr((Frob Y ) ∗ , Shv( Y )) to colim U Funct( U ( F q )) ≃ Funct c ( Y ( F q )) . Moreover, by Theorem 5.2.5(a), these two maps are canonically homotopic.
The case of
Bun G . We now specialize to the case of Bun G . We consider the full subcategory(5.2) Shv Nilp (Bun G ) ֒ → Shv(Bun G ) . As was mentioned earlier, it is preserved by the endofunctor (Frob
Bun G ) ∗ . Furthermore, it is gen-erated by objects that are compact in the ambient category Shv(Bun G ), see [AGKRRV1, Theorem10.1.6].In particular, the category Shv Nilp (Bun G ) is itself compactly generated, and the embedding (5.2)preserves compactness. Thus, we have a well-defined mapTr((Frob Bun G ) ∗ , Shv
Nilp (Bun G )) → Tr((Frob
Bun G ) ∗ , Shv(Bun G )) . Composing with the maps LT naiveBun G and LT trueBun G , we obtain two mapsTr((Frob Bun G ) ∗ , Shv
Nilp (Bun G )) ⇒ Funct c (Bun G ( F q )) , that we will denote LT naive and LT true , respectively.However, the homotopy between LT naiveBun G and LT trueBun G (see Sect. 5.2.6) gives rise to a homotopybetween LT naive and LT true . Remark . Let us explain the practical implication of the equality LT naive = LT
Sht , stated inTheorem 5.1.3.Let F be an an object of Shv Nilp (Bun G ) c equipped with a weak Weil structure, i.e., a map(5.3) α : F → (Frob Bun G ) ∗ ( F ) . To such a pair ( F , α ), we can attach its classcl( F , α ) ∈ Tr((Frob
Bun G ) ∗ , Shv
Nilp (Bun G ))(see [GKRV, Sect. 3.4.3]). Thus, using Corollary 4.1.4, to ( F , α ) we can attach an element ofFunct c (Bun G ( F q )) , i.e., a compactly supported automorphic function.Now, the content of Theorem 5.1.3 is that the above element of Funct c (Bun G ( F q )) equals the functionattached to F , viewed as a weak Weil sheaf via α , by the usual sheaf-function correspondence, i.e., bytaking pointwise traces of the Frobenius.5.3. Serre local term.
In this subsection we will define the last remaining map in Sect. 5.1.3, denotedLT
Serre .5.3.1. Let Y be an algebraic stack, and let N ⊂ T ∗ ( Y ) be a conical Zariski-closed subset. Consider the(fully faithful) embedding(5.4) Shv N ( Y ) ⊗ Shv N ( Y ) → Shv( Y × Y ) . Denote ps-u Y := ∆ ! ( e Y ) ∈ Shv( Y × Y ) , and let ps-u Y , N ∈ Shv N ( Y ) ⊗ Shv N ( Y )be obtained by applying to ps-u Y the right adjoint to the functor (5.4). We will not distinguishnotationally between ps-u Y , N and its image along (5.4).The counit of the adjunction defines a map(5.5) ps-u Y , N → ps-u Y . UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 35 ∗ Y ⊗ Id) ◦ (Id ⊗ ps-u Y , N ) = ((C · c ( Y , − ) ◦ ∆ ∗ Y ) ⊗ Id) ◦ (Id ⊗ ps-u Y , N ) →→ ((C · c ( Y , − ) ◦ ∆ ∗ Y ) ⊗ Id) ◦ (Id ⊗ ps-u Y ) ≃ Idas endofunctors of Shv N ( Y ) (see Sect. 4.3.3 for the ev ∗ Y notation).Recall, following [AGKRRV2, Sect. 3.3], that the pair ( Y , N ) is said to be Serre if the naturaltransformation (5.6) is an isomorphism. If this is the case, then the data ofps-u Y , N ∈ Shv N ( Y ) ⊗ Shv N ( Y ) and ev ∗ Y : Shv N ( Y ) ⊗ Shv N ( Y ) → Vectdefine the unit and counit of a self-duality on Shv N ( Y ).5.3.3. Suppose that ( Y , N ) is Serre. As with the true local term morphism, we defineLT Serre Y , N : Tr((Frob Y ) ∗ , Shv N ( Y )) → Funct c ( Y ( F q ))as the compositionTr((Frob Y ) ∗ , Shv( Y )) ≃ C · c (cid:0) Y , ∆ ∗ Y ◦ ((Frob Y ) ∗ ⊗ Id)(ps-u Y , N ) (cid:1) ≃≃ C · c (cid:0) Y , ∆ ∗ Y ◦ ((Frob Y ) ! ⊗ Id)(ps-u Y , N ) (cid:1) (5.5) −→ C · c (cid:0) Y , ∆ ∗ Y ◦ ((Frob Y ) ! ⊗ Id)(ps-u Y ) (cid:1) == C · c (cid:0) Y , ∆ ∗ Y ◦ (Graph Frob Y ) ! ( e Y ) (cid:1) ≃ C · c ( Y Frob , e Y Frob ) ≃ Funct c ( Y ( F q )) . Y = Bun G and N = Nilp. We will use the notation P Nilp as a short-hand for theendofunctor R ⋆ − of Shv(Bun G ). Similarly, for a stack Z , we will write (Id ⊗ P Nilp ) instead of Id ⊗ ( R ⋆ − ).Using this notation, we have K R := (Id ⊗ P Nilp )(ps-u
Bun G ) ∈ Shv(Bun G × Bun G ) . The following is [AGKRRV2, Theorem 1.3.6]:
Theorem 5.3.5. (a)
For a stack Z , the endofunctor of Shv( Z × Bun G ) given by Id ⊗ P Nilp is the idempotent correspondingto the embedding of the full subcategory (5.7) Shv T ∗ ( Z ) × Nilp ( Z × Bun G ) ֒ → Shv( Z × Bun G ) , preceded by its right adjoint. (b) The fully faithful functor
Shv( Z ) ⊗ Shv
Nilp (Bun G ) → Shv T ∗ ( Z ) × Nilp ( Z × Bun G ) is an equivalence. K R equals the value of theright adjoint to(5.8) Shv Nilp (Bun G ) ⊗ Shv
Nilp (Bun G ) ֒ → Shv(Bun G × Bun G )on ps-u Bun G .Hence, in the notations of Sect. 5.3.1 above, we can writeps-u Bun G , Nilp ≃ K R . Finally, we note that Theorem 4.3.5 exactly says that the pair (Bun G , Nilp) is Serre.5.3.7. Thus, we obtain a well-defined mapLT
Serre := LT
SerreBun G , Nilp . naive = LT true .The rest of this section is devoted to the proof of the equality LT Serre = LT
Sht in the rest of thisSect. 5.4.Finally, we show LT
Serre = LT true in Sect. 6.5.4.
Comparison of LT Serre and LT Sht . f LT Serre , f LT Sht : Sht Tr → Shtof functors Rep( ˇ G ) Ran → Vectwith the property that when we evaluate either of these natural transformations on Rep( ˇ G ) Ran , weobtain the relevant local term mapLT ? : Tr((Frob Bun G ) ∗ , Shv
Nilp (Bun G )) = Sht Tr ( Rep( ˇ G ) Ran ) → Sht( Rep( ˇ G ) Ran ) ≃ Funct c (Bun G ( F q )) . f LT Sht (in fact, an isomorphism) has been already defined: this isthe isomorphism coming from Theorem 4.1.2.5.4.3. We will now construct f LT Serre .Recall the isomorphism Sht Tr ≃ Sht( R ⋆ − )of Theorem 4.2.2. Thus, we can interpret the sought-for map f LT Serre as a natural transformation(5.9) Sht( R ⋆ − ) → Sht . R ⋆ − ) send V ∈ Rep( ˇ G ) Ran to the vector spaceobtained by applyingC · c (Bun G , (Graph Frob
Bun G ) ∗ ( − )) : Shv(Bun G × Bun G ) → Vectto the objects (Id ⊗ ( V ⋆ − ))(ps-u Bun G ) and (Id ⊗ ( R ⋆ V ⋆ − ))(ps-u Bun G ) , respectively.Recall the notation P Nilp (resp., (id ⊗ P Nilp )), see Sect. 5.3.4. Let ε denote the counit of the adjunction(5.10) P Nilp → id , and similarly for (id ⊗ P Nilp ).Now, the sought-for natural transformation (5.9) is induced by the map(Id ⊗ ( R ⋆ V ⋆ − ))(ps-u Bun G ) ≃ (id ⊗ P Nilp ) ◦ (Id ⊗ ( V ⋆ − ))(ps-u Bun G ) ε → (Id ⊗ ( V ⋆ − ))(ps-u Bun G ) . Theorem 5.4.6.
There is a canonical isomorphism f LT Serre ≃ f LT Sht of natural transformations
Sht Tr → Sht . Clearly, Theorem 5.4.6 implies the isomorphism LT
Serre ≃ LT Sht . The proof of Theorem 5.4.6 willbe given in Sect. 5.6.5.5.
An algebra structure on R . For the proof of Theorem 5.4.6 we need to digress and discusssome structures related to the object R ∈ Rep( ˇ G ) Ran . UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 37 R LocSys restrˇ G ( X ) ∈ QCoh(LocSys restrˇ G ( X )) ⊗ Rep( ˇ G ) Ran (see Sect. 2.4.3) carries a naturally defined structure of (non-unital) commutative algebra.Since the functor Loc is symmetric monoidal, it carries commutative algebras to commutative alge-bras. The following results from the construction of the isomorphism of Theorem 2.4.6:
Proposition 5.5.2.
The isomorphism (Id ⊗ Loc)( R LocSys restrˇ G ( X ) ) ≃ ∆ ∗ ( O LocSys restrˇ G ( X ) ) of Theorem 2.4.6 is compatible with commutative algebra structures. ! (LocSys restrˇ G ( X ) , − ) : QCoh(LocSys restrˇ G ( X )) → Vect is canonicallyright-lax symmetric monoidal.I.e., for F , F ∈ QCoh(LocSys restr ( X )), there are canonical, homotopy coherent mapsΓ ! (LocSys restrˇ G ( X ) , F ) ⊗ Γ ! (LocSys restrˇ G ( X ) , F ) → Γ ! (LocSys restrˇ G ( X ) , F ⊗ O LocSysrestrˇ G ( X ) F ) . In particular, Γ ! (LocSys restrˇ G ( X ) , − ) carries (non-unital) commutative algebras in the categoryQCoh(LocSys restrˇ G ( X )) to (non-unital) commutative algebras in Vect.5.5.4. Thus, we obtain that the object R := (Γ ! (LocSys restrˇ G ( X ) , − ) ⊗ Id)( R LocSys restrˇ G ( X ) ) ∈ Rep( ˇ G ) Ran carries a (non-unital) commutative algebra structure.From Proposition 5.5.2 we obtain:
Corollary 5.5.5.
The isomorphism
Loc( R ) ≃ O LocSys restrˇ G ( X ) of Corollary 2.4.8 is compatible with commutative algebra structures. R .Let Z be an algebraic stack. Consider the endofunctor (Id ⊗ P Nilp ) of Shv( Z × Bun G ), see Sect. 5.3.4.On the one hand, the algebra structure on R yields a map(5.11) m : (Id ⊗ P Nilp ) ◦ (Id ⊗ P Nilp ) → (Id ⊗ P Nilp ) . On the other hand, we have the map(5.12) (Id ⊗ P Nilp ) ◦ (Id ⊗ P Nilp ) ε ◦ (Id ⊗ P Nilp ) −→ (Id ⊗ P Nilp ) , where ε is as in (5.10). (The map (5.12) is in fact an isomorphism and equals the structure on (Id ⊗ P Nilp )of idempotent endofunctor.)
Proposition 5.5.7.
The maps (5.11) and (5.12) are canonically homotopic. V ∈ Rep( ˇ G ) Ran we have a tautological identification(5.13) (Id ⊗ P Nilp )( K V ) ≃ K R ⋆ V as objects of Shv(Bun G × Bun G ).For W ∈ Rep( ˇ G ) Ran , let ε W : (Id ⊗ P Nilp )( K W ) → K W denote the morphism, given by the counit of the adjunction corresponding to the embedding (5.4). Corollary 5.5.9.
For V ∈ Rep( ˇ G ) Ran , the diagram (Id ⊗ P Nilp ) ◦ (Id ⊗ P Nilp )( K V ) m −−−−−→ (Id ⊗ P Nilp )( K V ) (5.13) y ∼ (5.13) y ∼ (Id ⊗ P Nilp )( K R ⋆ V ) ε R ⋆ V −−−−−→ K R ⋆ V commutes. Proof of Proposition 5.5.7.
Both the source and the target functor vanish on ker(Id ⊗ P Nilp ).Hence, it is sufficient to establish the commutativity when evaluated on the full subcategoryShv(Bun G ) ⊗ Shv
Nilp (Bun G ) ⊂ Shv(Bun G × Bun G ) . Since all the functors involved act only on the second factor, it is sufficient to show that the mapId ≃ ( P Nilp ◦ P Nilp ) | Shv
Nilp (Bun G ) m → P Nilp | Shv
Nilp (Bun G ) ≃ Idis the identity map.This follows from [AGKRRV1, Theorem 10.5.2]. This theorem asserts that the action of Rep( ˇ G ) Ran on Shv
Nilp (Bun G ) factors uniquely through an action of QCoh(LocSys restrˇ G ( X )) via the localizationfunctor.Now the result follows from Corollary 5.5.5, as Loc( R ), considered as a non-unital (commutative)algebra, is canonically identified with the unit object in QCoh(LocSys restrˇ G ( X )). (cid:3) [Proposition 5.5.7]5.6. Proof of Theorem 5.4.6.
Proof of Theorem 5.4.6, Step 0.
We begin by introducing some notation.Throughout the argument, we will replace Sht Tr with Sht( R ⋆ − ). In particular, we consider ournatural transformations as mapping f LT Serre , f LT Sht : Sht( R ⋆ − ) → Sht . For a fixed object V ∈ Rep( ˇ G ) Ran , we denote the corresponding maps by f LT Serre V , f LT Sht V : Sht( R ⋆ V ) → Sht( V ) . To simplify notation, we will use − ⊗ − to denote − ⊗ O LocSysrestrˇ G ( X ) − and O to denote O LocSys restrˇ G ( X ) . UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 39
Proof of Theorem 5.4.6, Step 1.
Observe that we have a commutative diagram
Maps
DGCat (QCoh(LocSys restrˇ G ( X ) , Vect) F F ( O ⊗− ) −−−−−−−−→ Maps
DGCat (QCoh(LocSys restrˇ G ( X ) , Vect)
Loc ∨ y y Loc ∨ Maps
DGCat (Rep( ˇ G ) Ran , Vect) F F ( R ⋆ − ) −−−−−−−−→ Maps
DGCat (Rep( ˇ G ) Ran , Vect) . Here we have used the isomorphism Loc( R ) ≃ O . The vertical arrows are fully faithful by Corol-lary 2.3.3(a), and the top horizontal arrow is tautologically isomorphic to the identity. Therefore, thebottom arrow is fully faithful when restricted to the essential image of Loc ∨ .The (isomorphic!) functors Sht and Sht( R ⋆ − ) lie in the essential image of Loc ∨ by Theorem 3.1.3.Therefore, it suffices to identify the natural transformations f LT Serre and f LT Sht after precomposing with R ⋆ − .That is, it suffices to identify the two induced natural transformationsSht( R ⋆ ( R ⋆ − )) → Sht( R ⋆ − ) , i.e., the two maps f LT Serre R ⋆ V , f LT Sht R ⋆ V : Sht( R ⋆ ( R ⋆ V )) → Sht( R ⋆ V ) , V ∈ Rep( ˇ G ) Ran . Proof of Theorem 5.4.6, Step 2.
Let m : R ⋆ R → R denote the multiplication for the algebrastructure on R .We obtain a map m V : Sht( R ⋆ R ⋆ V ) → Sht( R ⋆ V ) . We claim that there is a natural identification(5.14) m V ≃ f LT Sht R ⋆ V of morphisms Sht( R ⋆ R ⋆ V ) → Sht( R ⋆ V ).Indeed, by Corollary 5.5.5, we have a commutative diagramLoc( R ) ⊗ Loc( R ⋆ V ) ∼ −−−−−→ Loc( R ⋆ R ⋆ V ) ∼ y y Loc( m⋆ Id V ) Loc( R ⋆ V ) Id −−−−−→ Loc( R ⋆ V )where the left vertical arrow is the canonical isomorphism obtained by identifying Loc( V ) ≃ O . ApplyingSht Loc and the definition yields the claim.5.6.4.
Proof of Theorem 5.4.6, Step 3.
By the above, it suffices to show that there are natural identi-fications(5.15) m V ≃ f LT Serre R ⋆ V of morphisms Sht( R ⋆ R ⋆ V ) → Sht( R ⋆ V ).Now, (5.15) is obtained by applying C · c (Bun G , (Graph Frob
Bun G ) ∗ ( − )) to the commutative diagramof Corollary 5.5.9. 6. Comparison of LT true and LT Serre
Statement of the result. Y be a quasi-compact algebraic stack, and let N be a conical Zariski-closed subset of T ∗ ( Y ).We will assume that the subcategory(6.1) Shv N ( Y ) ֒ → Shv( Y )is generated by objects that are compact in Shv( Y ) (in [AGKRRV1, Sect. C.5] this property of ( Y , N )was termed “renormalization-adapted and constraccessible”).Assume that N is Frobenius-invariant, so the endofunctor (Frob Y ) ∗ of Shv( Y ) preserves the subcat-egory Shv N ( Y ) ⊂ Shv( Y ), see [AGKRRV1, Sect. 15.3.1 and Lemma 15.3.2].In this case, the embedding (6.1) induces a map(6.2) Tr((Frob Y ) ∗ , Shv N ( Y )) → Tr((Frob Y ) ∗ , Shv( Y )) . Let us denote by LT true Y , N : Tr((Frob Y ) ∗ , Shv N ( Y )) → Funct( Y ( F q ))the composition of (6.2) with the mapLT true Y : Tr((Frob Y ) ∗ , Shv( Y )) → Funct( Y ( F q )) , defined in Sect. 5.2.3.6.1.2. Assume now that the pair ( Y , N ) is Serre (see Sect. 5.3.2 for what this means).Recall that in this case, we also have the mapLT Serre Y , N : Tr((Frob Y ) ∗ , Shv N ( Y )) → Funct( Y ( F q )) . U , see [AGKRRV2, Sect. 3.4]. Following loc. cit. ,we will say that the pair N is miraculous-compatible if the endofunctor Mir U preserves the subcategory(6.1).The main result of this section reads: Theorem 6.1.4.
Assume that N is miraculous-compatible. Then the maps LT true Y , N and LT Serre Y , N Tr((Frob Y ) ∗ , Shv N ( Y )) ⇒ Funct( Y ( F q )) are canonically homotopic.Remark . The assertion of Theorem 6.1.4 is far from tautological: it says that two ways to iden-tify Tr((Frob Y ) ∗ , Shv N ( Y )) with Funct( Y ( F q )), corresponding to two different self-dualities on Shv N ( Y )coincide.A somewhat analogous problem arises when we calculate the trace of the identity endofunctor on thecategory QCoh( Z ), where Z is a smooth proper scheme. There are two ways to calculate the trace thatcorrespond to two choices of self-duality data on QCoh( Z ): the naive self-duality and Serre self-duality.Each calculation yields Hodge cohomology of Z , i.e., ⊕ i Γ( Z, Ω i ( Z ))[ i ] . However, the resulting two identifications are different, and the difference is given by the Todd classof Z . This observation lies at the core of a proof of the Grothendieck-Riemann-Roch theorem viacategorical traces, see [KP]. UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 41 true = LT
Serre (the issue here is thefact that Bun G is not quasi-compact).Let Y be a not necessarily quasi-compact algebraic stack, and let N ⊂ T ∗ ( Y ) be a conical Zariski-closed subset. We will recall some definitions from [AGKRRV2, Sect. A.5].An open substack U j ֒ → Y is said to be cotruncative if for every quasi-compact open U ′ ⊂ Y , theopen embedding U ∩ U ′ j ֒ → U ′ , is such that the functor j ∗ : Shv( U ∩ U ′ ) → Shv( U ′ )admits a right adjoint as a functor defined by a kernel .An open substack U j ֒ → Y is said to be N - cotruncative if it is cotruncative, and for every stack Z ,the functor (id × j ) ! : Shv( Z × U ) → Shv( Z × Y )sends Shv T ∗ ( Z ) × N ( Z × U ) ⊂ Shv( Z × U ) to Shv T ∗ ( Z ) × N ( Z × Y ) ⊂ Shv( Z × Y ).Recall that Y is said to be truncatable if we can write Y as a union of quasi-compact cotruncativeopen substacks. Finally, recall that Y is is said to be N - truncatable if we can write Y as a union ofquasi-compact N -cotruncative open substacks.6.1.7. Assume that N is Frobenius-invariant. By Sect. 5.2.6 we have a well-defined mapLT true Y , N : Tr((Frob Y ) ∗ , Shv N ( Y )) ⇒ Funct c ( Y ( F q )) . Let us make the following assumptions: • Y is N -truncatable; • For every quasi-compact N -cotruncative open substack U ⊂ Y ,we have: – The pair ( U , N | U ) is renormalization-adapted and constraccessible; – The pair ( U , N | U ) is Serre; – N | U is miraculous-compatible.It follows that in this case, the subcategory Shv N ( Y ) is generated by objects that are compact inShv( Y ), and that the pair ( Y , N ) is Serre. So, the mapLT Serre Y , N : Tr((Frob Y ) ∗ , Shv N ( Y )) → Funct c ( Y ( F q ))is also well-defined by Sect. 5.3.3. Moreover, we have a commutative diagramcolim U Tr((Frob U ) ∗ , Shv N ( U )) colim U LT Serre U , N −−−−−−−−−→ colim U Funct( U ( F q )) y y ∼ Tr((Frob Y ) ∗ , Shv N ( Y )) LT Serre Y , N −−−−−→ Funct c ( Y ( F q )) . It now follows formally from Theorems 6.1.4 and 5.2.5(b) that the maps LT true Y , N and LT Serre Y , N arecanonically homotopic.6.1.8. We apply the above discussion to Y = Bun G and Nilp = N . The conditions in Sect. 6.1.7 aresatisfied by [AGKRRV1, Theorems 10.1.4 and 10.1.6] and [AGKRRV2, Corollaries 3.6.6 and 1.6.9],respectively. This implies the desired equalityLT true = LT Serre . A geometric local term theorem. Y be a quasi-compact algebraic stack. We start by constructing a natural transformation(6.3) C · c ( Y , ∆ ∗ Y ◦ (Id ⊗ Mir Y )( − )) → C · N ( Y , ∆ ! Y ( − )) , as functors Shv( Y × Y ) ⇒ Vect . Q ∈ Shv( Y × Y ).We note that the map C · N ( Y , ∆ ! Y ( Q )) → C · ( Y , ∆ ! Y ( Q ))is an isomorphism for Q . So we need to construct a map(6.4) C · c ( Y , ∆ ∗ Y ◦ (Id ⊗ Mir Y )( Q )) → C · ( Y , ∆ ! Y ( Q )) , functorial in Q ∈ Shv( Y × Y ) c .In what follows we will use the notationu Y := (∆ Y ) ∗ ( ω Y ) ∈ Shv( Y × Y ) . We start with the map Q ⊠ D Verdier ( Q ) → u Y × Y , given by Verdier duality. Applying the transposition σ , , we interpret it as a map(6.5) ( Q ⊠ D Verdier ( Q )) σ , → u Y ⊠ u Y . By definition(6.6) (Id ⊗ Mir Y )(u Y ) ≃ ps-u Y Applying the functor Id ⊗ Id ⊗ Mir Y ⊗ Id to the map (6.5), we obtain a map(6.7) ((Id ⊗ Mir Y )( Q ) ⊠ D Verdier ( Q )) σ , → u Y ⊠ ps-u Y . Applying the functor( p , ) ! ◦ (∆ Y × id × id) ∗ ◦ σ , : Shv( Y × Y × Y × Y ) → Shv( Y × Y ) , from (6.7) we obtain a mapC · c ( Y , ∆ ∗ Y ◦ (Id ⊗ Mir Y )( Q )) ⊗ D Verdier ( Q ) → u Y . Applying Verdier duality again, we obtain the desired map (6.4).6.2.3. Let us take Q := (Graph Frob Y ) ∗ ( ω Y ) ∈ Shv( Y × Y ).Then the right-hand side in (6.3) identifies withC · N ( Y Frob , ω Y Frob ) ≃ C · ( Y Frob , ω Y Frob ) . Since Frob Y is a finite map,(Id ⊗ Mir Y )((Graph Frob Y ) ∗ ( ω Y )) ≃ (Graph Frob Y ) ! ( e Y ) . Hence, the left-hand side in (6.3) identifies withC · c ( Y Frob , e Y Frob ) . UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 43
Theorem 6.2.5.
The diagram (6.8) C · c ( Y , ∆ ∗ Y ◦ (Id ⊗ Mir Y ) ◦ (Graph Frob Y ) ∗ ( ω Y )) (6.3) −−−−−→ C · N ( Y , ∆ ! Y ◦ (Graph Frob Y ) ∗ ( ω Y )) ≃ y y ≃ C · c ( Y Frob , e Y Frob ) C · ( Y Frob , ω Y Frob ) ≃ y y ≃ Funct( Y ( F q )) id −−−−−→ Funct( Y ( F q )) commutes. The proof will be given in Sect. 6.4. We will presently show how Theorem 6.2.5 implies Theo-rem 6.1.4.6.3.
Proof of Theorem 6.1.4. N ( Y ) is generated by objects compact in the ambient category Shv( Y )implies that the pairing ev ! Y induces a perfect pairingShv N ( Y ) ⊗ Shv N ( Y ) → Vect . The unit of this duality, to be denoted u Y , N is obtained by applying to u Y the right adjoint to thefully faithful embedding(6.9) Shv N ( Y ) ⊗ Shv N ( Y ) ֒ → Shv( Y × Y ) . In what follows, we will not distinguish notationally between u Y , N and its image along (6.9). Thecounit of the adjunction gives rise to a map(6.10) u Y , N → u Y . N is miraculous-compatible tautologically implies that the endofunctorId ⊗ Mir Y of Shv( Y × Y ) preserves the subcategory(6.11) Shv N ( Y ) ⊗ Shv N ( Y ) ֒ → Shv( Y × Y ) . By [AGKRRV2, Corollary 3.4.7], the assumption to the pair ( Y , N ) is Serre implies that theabove functor Id ⊗ Mir Y intertwines the duality data given by the pair (u Y , N , ev ! Y ) with one given by(ps-u Y , N , ev ∗ Y ).In particular, we have a canonical isomorphism(6.12) ps-u Y , N ≃ (Id ⊗ Mir Y )(u Y , N )and a datum of commutativity for the diagramShv N ( Y ) ⊗ Shv N ( Y ) Id ⊗ Mir Y −−−−−−→ Shv N ( Y ) ⊗ Shv N ( Y ) ev ! Y y y ev ∗ Y Vect Id −−−−−→ Vect , i.e., an isomorphism of functors(6.13) ev ∗ Y ◦ (Id ⊗ Mir Y ) ≃ ev ! Y , Shv N ( Y ) ⊗ Shv N ( Y ) ⇒ Vect . F of Shv N ( Y ), we have a commutative diagram(6.14) Tr( F, Shv N ( Y )) id −−−−−→ Tr( F, Shv N ( Y )) ∼ y y ∼ ev ∗ Y ◦ ( F ⊗ id)(ps-u Y , N ) ∼ −−−−−→ (6.12) ev ∗ Y ◦ ( F ⊗ Mir Y )(u Y , N ) ∼ −−−−−→ (6.13) ev ! Y ◦ ( F ⊗ id)(u Y , N )6.3.4. We now use the following two observations:(i) The diagram ps-u Y , N (6.12) −−−−−→ ∼ (Id ⊗ Mir Y )(u Y , N ) (5.5) y y (6.10) ps-u Y −−−−−→ (6.6) (Id ⊗ Mir Y )(u Y )commutes. This follows tautologically from the constructions.(ii) The isomorphism (6.13) is canonically homotopic to the restriction of the natural transformation(6.3) along the embedding (6.11). This follows from [AGKRRV2, Proposition 4.1.4].Concatenating, we obtain a commutative diagramev ∗ Y ◦ ((Frob Y ) ∗ ⊗ id)(ps-u Y , N )) (6.13) ◦ (6.12) −−−−−−−−→ ev ! Y ◦ ((Frob Y ) ∗ ⊗ id)(u Y , N )) y (5.5) y (6.10) C · c ( Y , − ) ◦ (∆ Y ) ∗ ◦ (Frob Y × id) ∗ (ps-u Y ) (6.3) ◦ (6.6) −−−−−−−→ C · N ( Y , − ) ◦ (∆ Y ) ! ◦ (Frob Y × id) ∗ (u Y ) . F = (Frob Y ) ∗ , we obtain a commutative diagramTr((Frob Y ) ∗ , Shv N ( Y )) id −−−−−→ Tr((Frob Y ) ∗ , Shv N ( Y )) ∼ y y ∼ ev ∗ Y ◦ ((Frob Y ) ∗ ⊗ id)(ps-u Y , N )) (6.13) ◦ (6.12) −−−−−−−−→ ev ! Y ◦ ((Frob Y ) ∗ ⊗ id)(u Y , N )) y (5.5) y (6.10) C · c ( Y , − ) ◦ (∆ Y ) ∗ ◦ (Frob Y × id) ∗ (ps-u Y ) (6.3) ◦ (6.6) −−−−−−−→ C · N ( Y , − ) ◦ (∆ Y ) ! ◦ (Frob Y × id) ∗ (u Y )6.3.6. Finally, we note that the commutative diagram (6.8) can be rephrased asC · c ( Y , − ) ◦ (∆ Y ) ∗ ◦ (Frob Y × id) ∗ (ps-u Y ) (6.3) ◦ (6.6) −−−−−−−→ C · N ( Y , − ) ◦ (∆ Y ) ! ◦ (Frob Y × id) ∗ (u Y ) ∼ y y ∼ C · c ( Y Frob , e Y Frob ) C · ( Y Frob , ω Y Frob ) ≃ y y ≃ Funct( Y ( F q )) id −−−−−→ Funct( Y ( F q )) , UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 45
Thus, concatenating with the commutative diagram in Sect. 6.3.5 above, we obtain a commutativediagram Tr((Frob Y ) ∗ , Shv N ( Y )) id −−−−−→ Tr((Frob Y ) ∗ , Shv N ( Y )) ∼ y y ∼ ev ∗ Y ◦ ((Frob Y ) ∗ ⊗ id)(ps-u Y , N )) (6.13) ◦ (6.12) −−−−−−−−→ ev ! Y ◦ ((Frob Y ) ∗ ⊗ id)(u Y , N )) y (5.5) y (6.10) C · c ( Y , − ) ◦ (∆ Y ) ∗ ◦ (Frob Y × id) ∗ (ps-u Y ) (6.3) ◦ (6.6) −−−−−−−→ C · N ( Y , − ) ◦ (∆ Y ) ! ◦ (Frob Y × id) ∗ (u Y ) ∼ y y ∼ C · c ( Y Frob , e Y Frob ) C · ( Y Frob , ω Y Frob ) ≃ y y ≃ Funct( Y ( F q )) id −−−−−→ Funct( Y ( F q )) , in which the left composite vertical arrow is LT Serre Y , N , and the right composite vertical arrow is LT true Y , N .This provides the sought-for homotopy between LT Serre Y , N and LT true Y , N .6.4. Proof of Theorem 6.2.5. Y be a quasi-compact algebraic stack, and let F ∈ Shv( Y ) be a constructible sheaf, equippedwith a weak Weil structure, i.e., a morphism α : F → (Frob Y ) ∗ ( F ) , or equivalently, a morphism α L : Frob ∗ Y ( F ) → F . On the one hand, we attach to the pair ( F , α L ) a function funct( F ) naive ∈ Funct( Y ( F q )) by thestandard procedure of taking the trace of Frobenius on *-fibers of F at F q -points of F . I.e., for aFrobenius-invariant point pt i y → Y , we consider the endomorphism i ∗ y ( F ) ≃ (Frob Y ◦ i y ) ∗ ( F ) ≃ i ∗ y ◦ Frob ∗ Y ( F ) α L → i ∗ y ( F ) , and we set the value of funct( F ) naive at y ∈ Y ( F q ) to be the trace of the above endomorphism.On the other hand, we can attach to ( F , α ) a function funct( F ) true ∈ Funct( Y ( F q )), defined asfollows.Consider the canonical maps F ⊠ D Verdier ( F ) → (∆ Y ) ∗ ( ω Y ) and e Y → F ! ⊗ D Verdier ( F ) . From the first of these maps we produce the map(6.15) F ⊠ D Verdier ( F ) α ⊠ id −→ (Frob Y × id) ∗ ( F ⊠ D Verdier ( F )) → (Frob Y × id) ∗ ( ω Y ) ≃ (Graph Frob Y ) ∗ ( ω Y ) . The function funct( F ) true , viewed as an element ofC · ( Y Frob , ω Y Frob ) ≃ C · ( Y , ∆ ! Y ◦ (Graph Frob Y ) ∗ ( ω Y )) , corresponds to the map e Y → F ! ⊗ D Verdier ( F ) = ∆ ! Y ( F ⊠ D Verdier ( F )) (6.15) −→ ∆ ! Y ◦ (Graph Frob Y ) ∗ ( ω Y ) . The local Grothendiek-Lefschetz trace formula says:
Theorem 6.4.2.
The functions funct( F ) naive and funct( F ) true are equal.Remark . When F is compact , the assertion of Theorem 6.4.2 is a particular case of that ofTheorem 5.2.5. Namely, the functions funct( F ) naive and funct( F ) true are the values of the maps LT naive Y and LT true Y on the element cl( F , α ) ∈ Tr((Frob Y ) ∗ , Shv( Y )) , respectively.For F which is constructible but not compact, the assertion of Theorem 6.4.2 can be obtained byproving a version of Theorem 5.2.5 for the renormalized version of the category Shv( Y ), namely, oneobtained as the ind-completion of the constructible subcategory of Shv( Y ).6.4.4. We precede the proof of Theorem 6.2.5 by the following observation.Let Q be a constructible object of Shv( Y × Y ). Note that the procedure in Sect. 6.2.2 defines a map(6.16) ψ : C · c ( Y , ∆ ∗ Y ◦ (Id ⊗ Mir Y )( Q )) → C · ( Y , ∆ ! Y ( Q )) . It is easy to see that the map (6.16) equals the composition of the value of the natural transformation(6.3), followed by the canonical map(6.17) C · N ( Y , ∆ ! Y ( Q )) → C · ( Y , ∆ ! Y ( Q )) . Q := (Graph Frob Y ) ∗ ( ω Y ) . Note that in this case, the map (6.17) is an isomorphism, as the corresponding map identifies withC · N ( Y Frob , ω Y Frob ) → C · ( Y Frob , ω Y Frob ) . We need to show that a certain mapFunct( Y ( F q )) = C · c ( Y Frob , e Y Frob ) → C · ( Y Frob , ω Y Frob ) ≃ Funct( Y ( F q ))equals the identity, where the middle arrow is the result of the construction in Sect. 6.2.2 applied tothe above choice of Q .We interpret the above map as a functional(6.18) Funct( Y ( F q )) ⊗ Funct( Y ( F q )) ≃ C · c ( Y Frob , e Y Frob ) ⊗ C · c ( Y Frob , e Y Frob ) → e , and we wish to show that this functional is given by f , f Σ y ∈ Y ( F q ) f ( y ) · f ( y ) . Q . This is a map(6.19) (cid:0) (Graph Frob Y ) ! ( e Y ) ⊠ (Graph Frob Y ) ! ( e Y ) (cid:1) σ , → (∆ Y ) ∗ ( ω Y ) ⊠ (∆ Y ) ! ( e Y ) . We apply to this map the functor( p , ) ! ◦ (∆ Y × id × id) ∗ ◦ σ , : Shv( Y × Y × Y × Y ) → Shv( Y × Y ) , and we obtain a map C · c ( Y Frob , e Y Frob ) ⊗ (Graph Frob Y ) ! ( e Y ) → (∆ Y ) ∗ ( ω Y ) . We apply to the latter map the adjunctionC · c ( Y , − ) ◦ ∆ ∗ Y : Shv( Y × Y ) ⇄ Vect : (∆ Y ) ∗ ( ω Y ) , and we obtain the desired pairingC · c ( Y Frob , e Y Frob ) ⊗ C · c ( Y Frob , e Y Frob ) → e . UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 47 Y ) ! ( e Y ) ⊠ (∆ Y ) ∗ ( ω Y ) → (cid:0) (Graph Frob Y ) ∗ ( ω Y ) ⊠ (Graph Frob Y ) ∗ ( ω Y ) (cid:1) σ , . From this morphism we obtain an element ofC · ( Y Frob , ω Y Frob ) ⊗ C · ( Y Frob , ω Y Frob )by the following procedure.We apply to (6.20) the functor( p , ) ∗ ◦ (∆ Y × id × id) ! ◦ σ , : Shv( Y × Y × Y × Y ) → Shv( Y × Y ) , and we obtain a map (∆ Y ) ! ( e Y ) → C · ( Y Frob , ω Y Frob ) ⊗ (Graph Frob Y ) ∗ ( ω Y ) . Applying the adjunction (∆ Y ) ! ( e Y ) : Vect ⇄ Shv( Y × Y ) : C · ( Y , − ) ◦ ∆ ! Y , we obtain the desired element of(6.21) C · ( Y Frob , ω Y Frob ) ⊗ C · ( Y Frob , ω Y Frob ) ≃ Funct( Y ( F q )) ⊗ Funct( Y ( F q )) . We wish to show that the resulting function is the characteristic function of the diagonal, i.e., itsvalue on a ( y , y ) ∈ Y ( F q ) × Y ( F q ) equals the cardinality of the set of isomorphisms between thecorresponding two points of the groupoid Y ( F q ).6.4.8. However, unwinding the definitions, we obtain that the map (6.20) identifies with the map(6.15) for F = (∆ Y ) ! ( e Y ) and α being the tautological map α taut (∆ Y ) ! ( e Y ) ≃ (Frob Y × Y ) ! ◦ (∆ Y ) ! ( e Y ) ≃ (Frob Y × Y ) ∗ ◦ (∆ Y ) ! ( e Y ) . From here, we obtain that the element in (6.21) constructed above equalsfunct true ((∆ Y ) ! ( e Y ) , α taut ) . Applying Theorem 6.4.2, we obtain that the above element equalsfunct naive ((∆ Y ) ! ( e Y ) , α taut ) . Now, the classical Grothendiek-Lefschetz trace formula about the compatibility of the assignment( F , α ) funct naive ( F , α )with the !-pushforward functor implies that the above function equals the direct image (=sum alongthe fibers) of the constant function along the map Y ( F q ) → ( Y × Y )( F q ) ≃ Y ( F q ) × Y ( F q ) , as required. Appendix A. Co-shtukas
Our calculation of Tr(Frob ∗ , Shv
Nilp (Bun G )) used as an ingredient the non-standard self-duality onShv Nilp (Bun G ), for which the counit was ev ∗ Bun G .In this Appendix we will try a different path: we will rely on the usual Verdier self-duality ofShv Nilp (Bun G ), and try to re-run the arguments in this paper. However, as we shall see, instead ofshtukas, we will have to deal with another object, we call them “co-shtukas”.We do not have at our disposal the analogs of the results from [Xue2] that pertain to co-shtukas,so we cannot really complete the trace calculation following this path. Instead, we state a numberof (mutually equivalent) conjectures, which we rather give the status of “questions”, that would haveallowed us to do so.A.1. Kernels on truncatable stacks.
A.1.1. Let Y be a (not necessarily) quasi-compact algebraic stack. Following [AGKRRV2, Sect. A.5.6],we introduce the category Shv( Y ) co , as the ind-completion of the full subcategory of Shv( Y ) formed byobjects of the form j ∗ ( F U ), where U j ֒ → Y is a quasi-compact open substack, and F U is a compact object in Shv( U ).Alternatively, we can write(A.1) Shv( Y ) co ≃ colim U Shv( U ) , where U runs over the poset of quasi-compact open substacks of Y , and for j , : U ֒ → U , the transitionfunctor Shv( U ) → Shv( U ) is ( j , ) ∗ .By [GR, Chapter 1, Proposition 2.5.7], we can also writeShv( Y ) co ≃ lim U Shv( U ) , where for j , : U ֒ → U as above, the transition functorShv( U ) → Shv( U )is ( j , ) ? , the right adjoint of ( j , ) ∗ .The operation of !-tensor product naturally defines an action of Shv( Y ) on Shv( Y ) co . Remark
A.1.2 . Recall for comparison that the usual category Shv( Y ) can be described asShv( Y ) co ≃ colim U Shv( U ) , with transition functors ( j , ) ! and also asShv( Y ) co ≃ lim U Shv( U ) , with transition functors ( j , ) ∗ .A.1.3. We denote by C · N ( Y , − ) : Shv( Y ) co → Vectthe functor that corresponds in terms of (A.1) to the compatible family of functorsC · N ( U , − ) : Shv( U ) → Vect . Explicitly, for an object F ∈ Shv( Y ) co , we haveC · N ( Y , F ) ≃ colim U C · N ( U , j ? ( F )) , where j ? : Shv( Y ) co → Shv( U )is the right adjoint to the tautological functor j ∗ , co : Shv( U ) → Shv( Y ) co . Remark
A.1.4 . We have a tautologically defined functor Shv( Y ) co → Shv( Y ), denoted Id naive , anddetermined by the requirement Id naive ( j ∗ , co )( F U ) = j ∗ ( F U )for a quasi-compact U j → Y and F U ∈ Shv( U ).We also have the functorsC · , bad ( Y , − ) : Shv( Y ) → Vect and C · , bad N ( Y , − ) : Shv( Y ) → Vect . Namely, for F ∈ Shv( Y ), we setC · , bad ( Y , − ) := lim U C · ( U , j ∗ ( F )) and C · , bad N ( Y , − ) := lim U C · N ( U , j ∗ ( F )) , respectively. UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 49
We have the natural transformationsC · N ( Y , − ) → C · , bad N ( Y , − ) ◦ Id naive → C · , bad ( Y , − ) ◦ Id naive as functors Shv( Y ) co → Vect, and they become isomorphisms when evaluated on compact objects ofShv( Y ) co , but they are not at all isomorphisms on all of Shv( Y ) co . In other words, the latter twofunctors are in general discontinuous.A.1.5. Assume now that Y is truncatable (see Sect. 6.1.6). Then in the the above limits and colimitswe can replace the index set of quasi-compact open substacks U by a cofinal subset consisting of those U that are cotruncative .By definition, for a pair j , : U ֒ → U of cotruncative open substacks, the corresponding functor( j , ) ? is defined by a kernel.A.1.6. We continue to assume that Y is truncatable, and let Z be another algebraic stack. As wasexplained in [AGKRRV2, Sect. A.6.1], in this case, we can consider the categoryShv( Y × Z ) co Y , which can be defined as colim U Y , U Z Shv( U Y × U Z ) , where the index category is that of pairs of a contruncative quasi-compact open U Y ⊂ Y and a quasi-compact open U Z ⊂ Z , with transition functors( j Y , , ) ∗ ⊗ ( j Z , , ) ! : Shv( U , Y × U , Z ) → Shv( U , Y × U , Z ) , which are well-defined because the embeddings j Y , , : U , Y → U , U = Y are cotruncative, and so the corresponding functors ( j Y , , ) ∗ are codefined by kernels.The operation of !-tensor product naturally defines an action of Shv( Y × Z ) on Shv( Y × Z ) co Y .The construction of the functor C · N ( Y , − ) extends to a functor( p ) N : Shv( Y × Z ) co Y → Shv( Z ) . An object Q ∈ Shv( Y × Z ) co Y gives rise to a functor F Q : Shv( Y ) → Shv( Z ) , F Q ( F ) := ( p ) N ◦ ( p !1 ( F ) ! ⊗ Q ) . A.1.7. Let us take Z = Y . In this case we will denote the corresponding category Shv( Y × Z ) co Y byShv( Y × Y ) co .Note that u Y := (∆ Y ) ∗ ( ω Y ) can be naturally regarded as an object of Shv( Y × Y ) co . The corre-sponding functor F u Y of Shv( Y ) is the identity.In addition, we note that ∆ ! Y naturally defines a functorShv( Y × Y ) co → Shv( Y ) co . A.1.8. For Q ∈ Shv( Y × Y ) co we defineTr geom ( F Q , Shv( Y )) := C · N ( Y , ∆ ! Y ( Q )) . As in Sect. 5.2.3, we have a naturally defined mapTr( F Q , Shv( Y )) → Tr geom ( F Q , Shv( Y )) . A.1.9.
Example.
Let Q := (Graph Frob Y ) ∗ ( ω Y ) := (Frob Y × id) ∗ ◦ (∆ Y ) ∗ ( ω Y ) ∈ Shv( Y × Y ) co , so that the corresponding endofunctor F Q of Shv( Y ) is (Frob Y ) ∗ .We claim that Tr geom ((Frob Y ) ∗ , Shv( Y )) ≃ Funct c ( Y ( F q )) . Indeed, we can write ω Y ≃ colim U j ! ( ω U ) , and hence (∆ Y ) ∗ ( ω Y ) ≃ colim U ( j ∗ ⊗ j ! ) ◦ (∆ U ) ∗ ( ω U ) , where we let U run over the poset of cotruncative quasi-compact open substacks of Y .This implies that∆ ! Y ◦ (Frob Y × id) ∗ ◦ (∆ Y ) ∗ ( ω Y ) ≃ colim U j ∗ ◦ ∆ ! U ◦ (Graph Frob U ) ∗ ( ω U ) , so Tr geom ((Frob Y ) ∗ , Shv( Y )) ≃ colim U Funct( U ( F q )) , and it follows from Theorem 5.2.5(b) that the transition maps in this colimitFunct( U ( F q )) → Funct( U ( F q ))are given by push-forward, so the colimit indeed identifies with Funct c ( Y ( F q )).Note that the composite mapTr( F Q , Shv( Y )) → Tr geom ( F Q , Shv( Y )) ≃ Funct c ( Y ( F q ))is the map LT true Y from Sect. 5.2.6.A.2. The definition of co-shtukas.
A.2.1. For V ∈ Rep( ˇ G ) Ran , the endofunctor ( V ⋆ − ) of Shv(Bun G ) is defined by a kernel, namely by(A.2) co- K V := (Id ⊗ ( V ⋆ − ))((∆ Bun G ) ∗ ( ω Bun G )) ∈ Shv(Bun G × Bun G ) co . Consider now the endofunctor ( V ⋆ − ) ◦ (Frob Bun G ) ∗ , which is also given by a kernel, namely(Id × Frob
Bun G ) ! (co- K V ) ∈ Shv(Bun G × Bun G ) co . A.2.2. We define the functor co-Sht : Rep( ˇ G ) Ran → Vectto send V to Tr geom (( V ⋆ − ) ◦ (Frob Bun G ) ∗ , Shv(Bun G )) , which is by definitionC · N (Bun G , (∆ Bun G ) ! ◦ (Frob Bun G × id) ! (co- K V )) ≃ C · N (Bun G , (Graph Frob
Bun G ) ! (co- K V )) . UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 51
A.2.3. By Sect. 1.6.7, we can interpret the functor co-Sht as a family of functorsco-Sht I : Rep( ˇ G ) ⊗ I → Shv( X I ) , compatible under the functors (∆ φ ) ! : Shv( X I ) → Shv( X J ) , I φ → J. Explicitly, the functors co-Sht I can be described as follows. For V ∈ Rep( ˇ G ) ⊗ I , consider the object(id × Frob
Bun G × id) ! ◦ (Id ⊗ H ! ( V, − ))((∆ Bun G ) ∗ ( ω Bun G )) ∈ Shv(Bun G × (Bun G × X I )) co Bun G . We apply to it the functor(∆
Bun G × id) ! : Shv(Bun G × (Bun G × X I )) co Bun G → Shv(Bun G × X I ) co Bun G , and follow it by ( p ) N : Shv(Bun G × X I ) co Bun G → Shv( X I ) . A.2.4. Thus, we regard co-Sht as an object of (Rep( ˇ G ) Ran ) ∨ . We propose: Question A.2.5.
Does the object co-Sht belong to the subcategory (Rep( ˇ G ) Ran ) ∨ QLisse of (Rep( ˇ G ) Ran ) ∨ ? Explicitly, the above question is asking whether the functorsco-Sht I : Rep( ˇ G ) ⊗ I → Shv( X I )take values in QLisse( X I ) ⊂ Shv( X I ) . Remark
A.2.6 . In addition to co-Sht, one could consider two more functorsRep( ˇ G ) Ran → Vect . Namely, we can send V ∈ Rep( ˇ G ) Ran toC · , bad (Bun G , Id naive ◦ (Graph Frob
Bun G ) ! (co- K V )) and C · , bad N (Bun G , Id naive ◦ (Graph Frob
Bun G ) ! (co- K V )) , see Remark A.1.4. Denote these functors byco-Sht bad and co-Sht bad N , respectively.We have the natural transformationsco-Sht → co-Sht bad N → co-Sht bad , but they are not at all isomorphisms.For example, the values of both co-Sht bad N and co-Sht bad on the unit object of Rep( ˇ G ) Ran yieldsFunct(Bun G ( F q )), the space of not necessarily compactly supported functions on Bun G ( F q ).A.3. Relation to trace.
A.3.1. Let us be again in the general setting of Sect. A.1. Let N ⊂ T ∗ ( Y ) be a conical Zariski-closedsubset. We will assume that Y is N -truncatable and that N is Frobenius-invariant.We consider the full subcategory Shv N ( Y ) ⊂ Shv( Y ) , and let us define the full subcategory Shv N ( Y ) co ⊂ Shv( Y ) co as colim U Shv N ( U ) , where the index category consists of N -cotruncative quasi-compact open substacks of Y and the colimitis taken with respect to the *-pushforward functors. A.3.2. Assume that for every N -cotruncative quasi-compact open substack U ⊂ Y , the pair ( U , N | U )is renormalization-adapted and constraccessible (see Sect. 6.1.1).It follows formally that the restriction ofev ! Y := C · N ◦ ! ⊗ : Shv( Y ) co ⊗ Shv( Y ) → Vectto Shv N ( Y ) co ⊗ Shv N ( Y ) ⊂ Shv( Y ) co ⊗ Shv( Y )defines perfect pairing.By construction, the corresponding contravariant equivalence((Shv N ( Y ) co ) c ) op → (Shv N ( Y )) c sends j ∗ , co ( F U ) j ! ( D Verdier ( F U )) , F U ∈ Shv N ( U ) c . Denote by u Y , N ∈ Shv N ( Y ) co ⊗ Shv N ( Y )the unit object for this duality.As in Sect. 6.3.1, it is easy to see that u Y , N equals the image of u U := (∆ Y ) ∗ ( ω Y ) under the rightadjoint to the fully faithful embeddingShv N ( Y ) co ⊗ Shv N ( Y ) ֒ → Shv( Y × Y ) co . In what follows we will not notationally distinguish between u Y , N and its image under the aboveembedding. When viewed as such, it is equipped with a tautologically defined mapu Y , N → u Y . A.3.3. We take Y = Bun G and N = Nilp. It follows from Theorem 5.3.5(a) that the endofunctor(Id ⊗ P Nilp ) of Shv(Bun G × Bun G ) co identifies with the composition of the fully faithful embeddingShv T ∗ (Bun G ) × Nilp (Bun G × Bun G ) co ֒ → Shv(Bun G × Bun G ) co , preceded by its right adjoint.We now quote the following result from [AGKRRV2, Corollary 1.5.5], which is analog of Theo-rem 4.3.2. Theorem A.3.4.
The object (Id ⊗ P Nilp )(u
Bun G ) ∈ Shv T ∗ (Bun G ) × Nilp (Bun G × Bun G ) co belongs to the full subcategory Shv
Nilp (Bun G ) co ⊗ Shv
Nilp (Bun G ) ⊂ Shv T ∗ (Bun G ) × Nilp (Bun G × Bun G ) co . As a consequence, we obtain that the object (Id ⊗ P Nilp )(u
Bun G ) identifies canonically withu Bun G , Nilp . Moreover, this isomorphism is compatible with the natural maps of both objects to u
Bun G .A.3.5. Hence, arguing as in Sect. 4.4, we obtain: Proposition A.3.6.
There exists a canonical isomorphism of functors
Rep( ˇ G ) Ran → Vectco-Sht( R ⋆ − ) ≃ Sht Tr . UTOMORPHIC FUNCTIONS AS THE TRACE OF FROBENIUS 53
A.3.7. Recall now (see Sect. A.1.8 above) that we have a canonical natural transformation(A.3) Sht Tr → co-Sht . In addition, as in Sect. 5.4.4, the natural transformation ε : (Id ⊗ P Nilp ) → Iddefines a natural transformation(A.4) co-Sht( R ⋆ − ) → co-Sht . It follows as in Sect. 5.6 that the isomorphism of Proposition A.3.6 is compatible with these maps.A.3.8. We propose:
Question A.3.9.
Are the maps (A.3) and (A.4) isomorphisms?
A.3.10. We claim, however, that Questions A.2.5 and A.3.9 are equivalent.Indeed, an affirmative answer to Question A.3.9 immediately implies thatco-Sht ∈ (Rep( ˇ G ) Ran ) ∨ QLisse because Sht Tr has this property.The inverse implication is obtained by following the logic of the proofs of Theorems 3.1.3 and 4.1.2.A.4. Shtukas vs coshtukas.
A.4.1. Let us recall, following [AGKRRV2, Sect. A.6.4], that for a not necessarily quasi-compact stack Y , we have a well-defined miraculous functorMir Y : Shv( Y ) co → Shv( Y ) . Assume that Y is truncatable. Then more generally, for a stack Z , we have a functor(Mir Y ⊗ Id) : Shv( Y × Z ) co Y → Shv( Y × Z ) . A.4.2.
Example.
Take Z = Y . Then (Mir Y ⊗ Id)(u Y ) ≃ ps-u Y . A.4.3. Let us return to the setting of Sect. A.1.7. Given an object Q ∈ Shv( Y × Y ) co , as in Sect. 6.2, we construct a map(A.5) C · c ( Y , ∆ ∗ Y ◦ (Mir Y ⊗ Id)( Q )) → C · N ( Y , ∆ ! Y ( Q )) . A.4.4. Take Y = Bun G . For V ∈ Rep( ˇ G ) Ran , recall the notationco- K V ∈ Shv(Bun G × Bun G ) co , see Sect. A.2.By Sect. A.4.2 and [AGKRRV2, Lemma 1.6.4], we have(Mir Bun G ⊗ Id)(co- K V ) ≃ K V . Hence, the map (A.5) gives rise to a natural transformation(A.6) Sht → co-Sht , as functors Rep( ˇ G ) Ran → Vect.A.4.5. We propose:
Question A.4.6.
Is the natural transformation (A.6) an isomorphism?
A.4.7. We claim, however, that Question A.4.6 is equivalent to Question A.3.9. Indeed, we have a(tautologically) commutative diagramSht( R ⋆ − ) (A.6) −−−−−→ co-Sht( R ⋆ − ) ∼ y y Sht (A.6) −−−−−→ co-Sht , and we also have a diagram Sht Tr id −−−−−→
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