1.1.1

We work over a base scheme $S$.

1.1.2

Suppose that $G$ is an étale sheaf of groups over $S$. We denote by $B_S(G)$ the classifying stack of $G$ sheafified in the étale topology. The term $G$-torsor over an $S$-scheme $X$ refers to a section of $B_S(G)$ over $X$.

If the base scheme $S$ is clear from the context, we shall suppress it from the notation.

1.1.3

When $G$ is furthermore abelian, we write $B^n(G)$ for the $n$-fold delooping as an étale stack of $\infty$-groupoids for any integer $n\ge 1$, see [Reference ZhaoZha22, § 1].

For an $S$-scheme $X$, the $\infty$-groupoid of sections $\mathrm {Maps}(X, B^n(G))$ has homotopy groups described by the étale cohomology groups of $X$ valued in $G$:

\begin{align*} \pi_i\mathrm{Maps}(X, B^n(G)) \cong H^{n-i}(X, G). \end{align*}

The term $G$-gerbe over an $S$-scheme $X$ refers to a section of $B^2(G)$ over $X$.

The $2$-groupoid of $G$-gerbes over $X$ carries a natural $\mathbb {E}_{\infty }$-monoidal structure, and we use $\otimes$ to denote the product operation.

1.1.5

Suppose that $X$ is a connected $S$-scheme equipped with a geometric point $\bar x$. Let $\pi _1(X, \bar x)$ denote the profinite fundamental group.

Let $Z$ be a locally constant étale sheaf of finite abelian groups over $X$. Its geometric fiber $Z_{\bar x}$ is thus equipped with a continuous action of $\pi _1(X, \bar x)$.

Denote by $\mathscr Z^2(\pi _1(X, \bar x), Z_{\bar x})$ the groupoid of short exact sequences of profinite groups:

\begin{align*} 1 \rightarrow Z_{\bar x} \rightarrow \Pi \rightarrow \pi_1(X, \bar x) \rightarrow 1, \end{align*}

such that the conjugation action of $\Pi$ on $Z_{\bar x}$ factors through the natural $\pi _1(X, \bar x)$-action.

1.1.6

To interpret $Z$-gerbes in terms of $\pi _1(X, \bar x)$, the following condition on $X$ is needed:

(1.1)\begin{equation} \begin{matrix} \text{For any finite abelian group $A$,}\\ \text{any class of $H^2(X, A)$ vanishes over a} \, finite \, \unicode{x00E8}\text{tale cover of} X. \end{matrix} \end{equation}

Examples of such $X$ include spectra of fields (automatic) and Henselian local rings [Sta18, 09ZI] as well as all connected affine $\mathbb {F}_p$-schemes [Reference AchingerAch17, Theorem 1.1.1], although we are only interested in the case where $X$ is an affine curve over a finite field, where (1.1) can be verified directly.

An example of an $S$-scheme failing condition (1.1) is the projective line.

1.1.7

Let $X$, $\bar x$, and $Z$ be as in § 1.1.5. Denote by $\mathrm {Maps}_{\bar x}(X, B^2(Z))$ the (1-)groupoid of $A$-gerbes over $X$ rigidified along $\bar x$.

If $X$ satisfies condition (1.1), the usual comparison between étale and Galois cohomology lifts to a canonical equivalence of Picard groupoids:

(1.2)\begin{equation} \mathrm{Maps}_{\bar x}(X, B^2(Z)) \cong \mathscr Z^2(\pi_1(X, \bar x), Z_{\bar x}). \end{equation}

Following [Reference WeissmanWei18, § 19], we refer to the image of a rigidified $A$-gerbe $(\mathscr G, \bar g)$ under (1.2) as the ‘fundamental group’ of $(\mathscr G, \bar g)$ and denote it as follows:

(1.3)\begin{equation} 1 \rightarrow Z_{\bar x} \rightarrow \pi_1(\mathscr G, \bar g) \rightarrow \pi_1(X, \bar x) \rightarrow 1. \end{equation}

Here, the rigidification $\bar g$ of $\mathscr G$ along $\bar x$ may equivalently be viewed as a geometric point of $\mathscr G$ lifting $\bar x$.

Proof Construction of (1.2)

Let $\widetilde X$ denote the universal cover of $(X, \bar x)$: it is the pro-object of the category of finite étale $X$-schemes which co-represents the fiber functor $(p : X_1 \rightarrow X) \mapsto p^{-1}(\bar x)$. In particular, $\bar x$ canonically lifts to $\widetilde X$, so we may view it as a geometric point of $\widetilde X$.

Let $\mathscr G$ be a $Z$-gerbe over $X$. Condition (1.1) implies that the pullback $\widetilde {\mathscr G}$ of $\mathscr G$ to $\widetilde X$ is constant. Thus, for each $\sigma \in \pi _1(X, \bar x)$, we obtain an automorphism:

(1.4)\begin{equation} \widetilde{\mathscr G}_{\bar x} \cong \widetilde{\mathscr G}_{\sigma(\bar x)} \cong \widetilde{\mathscr G}_{\bar x}, \end{equation}

where the first map is provided by the constancy of $\widetilde {\mathscr G}$ and the second map is the descent datum of $\widetilde {\mathscr G}$ along $\widetilde X \rightarrow X$.

Any rigidification $\bar g$ of $\mathscr G$ along $\bar x$ may be viewed as a section of $\widetilde {\mathscr G}_{\bar x}$. The automorphism (1.4) acts on $\bar g$ as multiplication by a (set-theoretic) $Z_{\bar x}$-torsor $t_{\sigma }$.

The association $\sigma \mapsto t_{\sigma }$ is multiplicative in the following sense: for two elements $\sigma _1, \sigma _2 \in \pi _1(X, \bar x)$, there is an identification of $Z_{\bar x}$-torsors

(1.5)\begin{equation} (\sigma_2)^*(t_{\sigma_1}) \otimes t_{\sigma_2} \cong t_{\sigma_1\sigma_2}, \end{equation}

where $(\sigma _2)^*(t_{\sigma _1})$ is the $Z_{\bar x}$-torsor induced from $t_{\sigma _1}$ along $\sigma _2^{-1}$ (i.e. $z\in Z_{\bar x}$ acts through $\sigma _2$). The identification (1.5) satisfies the natural cocycle condition. Furthermore, $t_e$ is canonically trivialized, satisfying the unit condition with respect to (1.5).

Therefore, the union

\begin{align*} \pi_1(\mathscr G, \bar g) := \bigsqcup_{\sigma \in \pi_1(X, \bar x)} t_{\sigma} \end{align*}

defines an extension of $\pi _1(X, \bar x)$ by $Z_{\bar x}$, with multiplication induced from (1.5). This concludes the definition of the functor (1.2) in the forward direction. It is symmetric monoidal with respect to the natural symmetric monoidal structures on both sides.

To show that (1.2) is an equivalence, we observe that it induces the isomorphism between $H^2(X, Z)$ and $H^2(\pi _1(X, \bar x), Z_{\bar x})$ on $\pi _0$ (owing to condition (1.1)). On $\pi _1$, it induces the isomorphism of abelian groups between $\mathrm {Maps}_{\bar x}(X, B(Z))$ and maps $f : \pi _1(X, \bar x) \rightarrow Z_{\bar x}$ satisfying $\sigma _2^{-1}(f(\sigma _1))f(\sigma _2) = f(\sigma _1\sigma _2)$ for each $\sigma _1, \sigma _2 \in \pi _1(X, \bar x)$.

Remark 1.1.8 If $Z = \underline A$ is the constant étale sheaf with values in a finite abelian group $A$, the isomorphism $\underline {A}_{\bar x} \cong A$ induces a retract:

(1.6)\begin{equation} \mathrm{Maps}(X, B^2(\underline A)) \rightarrow \mathrm{Maps}_{\bar x}(X, B^2(\underline A)),\quad \mathscr G\mapsto \mathscr G\otimes \mathscr G_{\bar x}^{\otimes -1}. \end{equation}

On the other hand, the groupoid $\mathscr Z^2(\pi _1(X, \bar x), \underline A_{\bar x})$ is equivalent to that of central extensions of $\pi _1(X, \bar x)$ by $A$, which we denote by $\mathrm {CExt}(\pi _1(X, \bar x), A)$. Hence, the composition of (1.6) with (1.2) is a functor of Picard groupoids:

(1.7)\begin{equation} \mathrm{Maps}(X, B^2(\underline A)) \rightarrow \mathrm{CExt}(\pi_1(X, \bar x), A). \end{equation}

The functor (1.7) induces an equivalence after $1$-truncation.

1.2.1

We continue to work over a base scheme $S$. We fix a prime $\ell$ invertible on $S$ and an algebraic closure $\mathbb {Q}_{\ell }\subset \overline {\mathbb {Q}}{}_{\ell }$.

Let $E$ be an intermediate field $\mathbb {Q}_{\ell }\subset E\subset \overline {\mathbb {Q}}_{\ell }$ and $A\subset E^{\times }$ be a finite subgroup whose order is also invertible on $S$.

1.2.2

For any $S$-scheme $X$, we have the $\infty$-category $\mathrm {Shv}(X, E)$ of constructible complexes of $E$-sheaves on $X$. This is the $\infty$-category denoted by $D_{\mathrm {cons}}(X, E)$ in [Reference Hemo, Richarz and ScholbachHRS23] where $E$ is equipped with the $\ell$-adic topology.

1.2.3

In the presence of an $A$-gerbe $\mathscr G$ on $X$, there is a variant: the $\infty$-category $\mathrm {Shv}_{\mathscr G}(X)$ of $\mathscr G$-twisted constructible complexes of $E$-sheaves on $X$.

To define it, consider the abelian category $\mathscr A(X)$ of proétale $E$-sheaves on $X$. The association $X_1\mapsto \mathscr A(X_1)$ is a stack of abelian categories on the small étale site of $X$, to be denoted (temporarily) by $\underline {\mathscr A}$.

Since $A$ is a subgroup of $E^{\times }$, it acts on the identity endofunctor of $\underline {\mathscr A}$. The construction of [Reference ZhaoZha22, Appendix A] then yields a stack $\underline {\mathscr A}{}_{\mathscr G}$. Its global section is an abelian category $\mathscr A_{\mathscr G}(X)$. Then we may form its derived $\infty$-category $D_{\mathscr G}(X)$ and $\mathrm {Shv}_{\mathscr G}(X) \subset D_{\mathscr G}(X)$ is the full $\infty$-subcategory characterized by the constructibility condition of [Reference Hemo, Richarz and ScholbachHRS23, Definition 1.1].

1.2.4

For brevity, we shall call an object of $\mathrm {Shv}_{\mathscr G}(X)$ a ‘$\mathscr G$-twisted $E$-sheaf’ on $X$.

Such an object is said to be lisse or a $\mathscr G$-twisted $E$-local system if it is dualizable and belongs to the heart $\mathscr A_{\mathscr G}(X)$ of $D_{\mathscr G}(X)$. They form an abelian category $\mathrm {Lis}_{\mathscr G}(X)$.

1.2.6

It is convenient to consider the $2$-category of pairs $(X, \mathscr G)$, where $X$ is an $S$-scheme (or, more generally, an $S$-prestack) and $\mathscr G$ is an $A$-gerbe over $X$.

A morphism $(X_1, \mathscr G_1)\rightarrow (X_2, \mathscr G_2)$ consists of a morphism $f : X_1\rightarrow X_2$ and an isomorphism $\alpha : \mathscr G_1\xrightarrow {\sim } f^*(\mathscr G_2)$.

A $2$-morphism $(f_1, \alpha _1) \rightarrow (f_2, \alpha _2)$ is an equality $f_1 = f_2 : X_1\rightarrow X_2$ together with a $2$-morphism $\alpha _1 \rightarrow \alpha _2$ in the $2$-category of $A$-gerbes on $X_1$. All $2$-morphisms are invertible.

1.2.7

Given a morphism $(X_1, \mathscr G_1) \rightarrow (X_2, \mathscr G_2)$ as in § 1.2.6, there is a pullback functor $f^* : \mathrm {Shv}_{\mathscr G_2}(X_2) \rightarrow \mathrm {Shv}_{\mathscr G_1}(X_1)$.

If $f$ is separated and of finite presentation, we also have a functor $f_! : \mathrm {Shv}_{\mathscr G_1}(X_1) \rightarrow \mathrm {Shv}_{\mathscr G_2}(X_2)$. (By our convention, functors are derived unless otherwise stated.)

The adjunctions $(f^*, f_*)$, $(f^!, f_!)$ exist on the twisted categories of $E$-sheaves whenever they exist on the untwisted ones.

Indeed, these functors are constructed from the usual functors by étale descent, in view of Remark 1.2.5.

1.3.1

We remain in the context of § 1.2.1.

1.3.2

Let $G$ be an étale sheaf of groups over $S$. There is an equivalence of groupoids between $A$-gerbes on $B(G)$ rigidified along $e : S\rightarrow B(G)$ (simply called ‘rigidified $A$-gerbes’ in the following) and monoidal morphisms $G \rightarrow B(\underline A)$.

In turn, monoidal morphisms $G\rightarrow B(\underline A)$ are equivalent to multiplicative $A$-torsors on $G$. They induce character $E$-local systems on $G$ along the inclusion $A\subset E^{\times }$.

1.3.3

Suppose that $G$ acts on an $S$-scheme $X$. Let $\mathscr G$ be a rigidified $A$-gerbe on $B(G)$. Denote by $\chi _{\mathscr G}$ the induced character $E$-local system on $G$.

By abuse of notation, we write $\mathrm {Shv}_{\mathscr G}(X/G)$ for the $\infty$-category of $E$-sheaves on $X/G$ twisted by the pullback of $\mathscr G$ along $X/G \rightarrow B(G)$.

The $\infty$-category $\mathrm {Shv}_{\mathscr G}(X/G)$ admits a more concrete description: it is the $\infty$-category of $E$-sheaves on $X$ equipped with $G$-equivariance structures against the character $E$-local system $\chi _{\mathscr G}$.

1.3.4

Let $X$ be a separated $S$-scheme of finite presentation. We write $\Gamma (X, -)$ (respectively, $\Gamma _c(X, -)$) for the direct image functors $p_*(-)$ (respectively, $p_!(-)$) along the structure morphism $p : X\rightarrow S$. Similar conventions apply to $H^i(X, -)$ and $H^i_c(X, -)$.

The following lemma is a variant of the standard fact that an $E$-sheaf equivariant against a nontrivial character on the stabilizer group must vanish.

Lemma 1.3.5 Let $G$ be a separated group $S$-scheme of finite presentation acting on $X$. Let $\mathscr G$ be a rigidified $A$-gerbe on $B(G)$, with induced character $E$-local system $\chi _{\mathscr G}$ on $G$. If $H^0(G_{\bar s}, \chi _{\mathscr G}) = 0$ for all geometric points $\bar s$ of $S$, then any object $\mathscr F \in \mathrm {Shv}_{\mathscr G}(X/G)$ satisfies

(1.8)\begin{equation} H^i_c(X, \mathscr F) = 0 \quad \text{for all }i\ge 0. \end{equation}

(In particular, we find $\mathrm {Shv}_{\mathscr G}(B(G)) = 0$ by setting $X = S$.)

Proof. The vanishing (1.8) can be verified at geometric points of $S$, so we may assume that $S$ is the spectrum of an algebraically closed field.

Write $f : X \rightarrow X/G$ for the quotient in the étale topology. It suffices to show $f_!(\mathscr F) = 0$.

Consider the Cartesian diagram:

(1.9)

where $\mathrm {act}$ (respectively, $\mathrm {pr}$) denotes the action (respectively, projection) map.

We view $\mathscr F$ as an $E$-sheaf on $X$ which is $G$-equivariant against $\chi _{\mathscr G}$. Base change along (1.9), the equivariance structure, and projection formula imply an isomorphism:

(1.10)\begin{equation} f^*f_!(\mathscr F) \cong \mathscr F\boxtimes \Gamma_c(G, \chi_{\mathscr G}). \end{equation}

It remains to show $\Gamma _c(G, \chi _{\mathscr G}) = 0$.

For $X = G$ and $\mathscr F = \chi _{\mathscr G}$, (1.10) reads

(1.11)\begin{equation} E\boxtimes\Gamma_c(G, \chi_{\mathscr G}) \cong \chi_{\mathscr G}\boxtimes \Gamma_c(G, \chi_{\mathscr G}). \end{equation}

In particular, any nonzero $H^i_c(G, \chi _{\mathscr G})$ implies the existence of a nonzero section of $\chi _{\mathscr G}$.

1.3.6

Let us discuss an example for which the condition of Lemma 1.3.5 is satisfied.

Suppose that $G = \mathbb {G}_{m, S}$ is the multiplicative group over $S$. For each integer $n\ge 0$ invertible over $S$, the degree-$n$ Kummer cover of $\mathbb {G}_{m, S}$ defines a character $\mu _n$-torsor on $\mathbb {G}_{m, S}$.

In particular, we have a monoidal morphism:

(1.12)\begin{equation} \Psi : \mathbb{G}_{m, S} \rightarrow \lim_{\substack{n\ge 0 \\ \text{invertible}}} B(\mu_n). \end{equation}

For each section $a$ of the étale sheaf $\underline A(-1)$, we thus obtain a monoidal morphism $a_*(\Psi ) : \mathbb {G}_{m, S} \rightarrow B(\underline A)$. We also write $\Psi ^a := a_*(\Psi )$ and keep the same notation for the induced character $E$-local system on $\mathbb {G}_{m, S}$.

When $a$ is nowhere vanishing on $S$, there holds $H^0(\mathbb {G}_{m, \bar s}, \Psi ^a) = 0$ at all geometric points $\bar s$ of $S$. Indeed, this follows from the vanishing of $\pi _1(\mathbb {G}_{m, \bar s}, e)$-invariants of the corresponding $1$-dimensional character.

1.4.1

Suppose that the base scheme $S = \operatorname {Spec}(\textsf k)$, where $\textsf k$ is a finite field of cardinality $q$. Let $A$ be a finite abelian group whose order is coprime to $q$.

For any $\textsf k$-scheme $X$, we write $\mathrm {Fr}_X : X \rightarrow X$ for the absolute Frobenius endomorphism: it acts as identity on the topological space $|X|$ and the $q$th power map on $\mathscr O_X$.

1.4.2

Consider the special case $x = \operatorname {Spec}(\textsf k)$. Then any $A$-torsor $t$ on $x$ defines a character $\mathrm {Gal}(\bar {\textsf k}/\textsf k)\rightarrow A$ for any algebraic closure $\textsf k\subset \bar {\textsf k}$, and the image of the geometric Frobenius $\varphi _x \in \mathrm {Gal}(\bar {\textsf k}/\textsf k)$ may be called the trace-of-Frobenius of $t$.

When a coefficient field $E$ with $A\subset E^{\times }$ is supplied, this is indeed the trace of $\varphi _x$ on the $1$-dimensional representation induced from $t$ along $A\subset E^{\times }$.

1.4.3

We shall describe an analogous construction for $A$-gerbes on a $\textsf k$-scheme (or $\textsf k$-stack) $X$. It is helpful to perform this construction in two steps:

(1.13)

In other words, we shall first extract an étale $A$-torsor $\operatorname {Tr}(\mathrm {Fr}\mid \mathscr G)$ over the $\mathrm {Fr}_X$-fixed point locus $X^{\mathrm {Fr}}\subset X$, defined to be the fiber product:

(1.14)

and then set $\operatorname {Tr}(\mathrm {Fr}\mid \mathscr G)(\textsf k)$ to be its set (or groupoid) of $\textsf k$-points.

1.4.4

Recall that for any $\textsf k$-scheme $X$, the endofunctor $\mathrm {Fr}_X^*$ on the $2$-groupoid of $A$-gerbes over $X$ is naturally isomorphic to the identity (the ‘baffling theorem’ [Sta18, 03SN]).

Let us explicitly describe the value of this natural isomorphism at an $A$-gerbe $\mathscr G$:

(1.15)\begin{equation} \mathrm{Fr}_X^*(\mathscr G) \cong \mathscr G. \end{equation}

For any étale morphism $f : X_1\rightarrow X$, the groupoid $\mathrm {Fr}_X^*(\mathscr G)(X_1)$ is the filtered colimit of $\mathscr G(U)$ over étale morphisms $u : U \rightarrow X$ through which $\mathrm {Fr}_X\circ f$ factors. This index category has an initial object, given by $(U, u) = (X_1, f)$ and the factorization $\mathrm {Fr}_X \circ f = f\circ \mathrm {Fr}_{X_1}$. The colimit is thus identified with $\mathscr G(X_1)$.

Let us now give two constructions of $\operatorname {Tr}(\mathrm {Fr}\mid \mathscr G)$.

Proof Construction 1

Since $\mathrm {Fr}_X$ restricts to the identity map on $X^{\mathrm {Fr}}$, we obtain a ‘tautological’ identification $\mathrm {Fr}_X^*(\mathscr G) \cong \mathscr G$ of $A$-gerbes over $X^{\mathrm {Fr}}$.

The $A$-torsor $\operatorname {Tr}(\mathrm {Fr}_X\mid \mathscr G)$ over $X^{\mathrm {Fr}}$ is defined so that the action by it renders the following diagram of $A$-gerbes over $X^{\mathrm {Fr}}$ commutative:

(1.16)

where $\mathrm {taut}$ refers to the tautological identification.

Proof Construction 2

The identification (1.15) yields an isomorphism between the endomorphism $\mathrm {Fr}_{B^2(\underline A)}$ of $B^2(\underline A)$ and the identity.

Hence, we find an isomorphism:

(1.17)\begin{equation} B^2(\underline A)^{\mathrm{Fr}} \cong B^2(\underline A) \times B(\underline A),\quad (\mathscr G, \alpha) \mapsto (\mathscr G, \mathscr G^{-1}\otimes\alpha), \end{equation}

where the isomorphism $\alpha : \mathscr G \cong \mathrm {Fr}_X^*(\mathscr G)$ is viewed as an automorphism of $\mathscr G$ via the identification (1.15), so $\mathscr G^{-1}\otimes \alpha$ is an automorphism of the trivial $A$-gerbe, i.e. an $A$-torsor.

The $A$-torsor $\operatorname {Tr}(\mathrm {Fr}_X\mid \mathscr G)$ over $X^{\mathrm {Fr}}$ is set to be the composition:

\begin{align*} X^{\mathrm{Fr}} \rightarrow B^2(\underline A)^{\mathrm{Fr}} \rightarrow B(\underline A), \end{align*}

where the second map is the projection of (1.17) onto its second factor.

1.4.6

Let us argue that $\operatorname {Tr}(\mathrm {Fr}\mid \mathscr G)(\textsf k)$ is indeed a set-theoretic $A$-torsor over $X(\textsf k)$.

Namely, we must show that its fiber over any $x\in X(\textsf k)$ is nonempty. To see this, it suffices to note that $H^2(x, A) = 0$, so any $A$-gerbe over $x$ admits a section. The choice of any such section trivializes $\operatorname {Tr}(\mathrm {Fr}\mid \mathscr G)$ over $x$.

Remark 1.4.8 Suppose that $\textsf k\subset \textsf k_1$ is a finite extension. Let $X_1$ be a $\textsf k_1$-scheme, whose restriction of scalars along $\textsf k\subset \textsf k_1$ is denoted by $X := \mathrm {res}(X_1)$. To each $A$-gerbe $\mathscr G_1$ over $X_1$, we may associate an $A$-gerbe $\operatorname {Nm}(\mathscr G_1)$ over $X$. To define it, we restrict $\mathscr G_1$ along the counit map $X_{\textsf k_1} := X\times \operatorname {Spec}(\textsf k_1) \rightarrow X_1$ and take its image under the norm map:

\begin{align*} \Gamma(X_{\textsf k_1}, A[2]) \rightarrow \Gamma(X, A[2]), \end{align*}

which exists thanks to $X_{\textsf k_1} \rightarrow X$ being finite étale.

The trace-of-Frobenius construction may be performed for $\mathscr G_1$ (using the $|\textsf k_1|$th power Frobenius) as well as $\operatorname {Nm}(\mathscr G_1)$ (using the $|\textsf k|$th power Frobenius). They yield canonically isomorphic set-theoretic $A$-torsors:

(1.18)

1.4.9

Let us bring in the coefficient field $E$ as in § 1.2.1 and assume that $A\subset E^{\times }$.

Let $X$ be a $\textsf k$-scheme locally of finite type equipped with an $A$-gerbe $\mathscr G$. Denote by $\widetilde X$ the set-theoretic $A$-torsor $\operatorname {Tr}(\mathrm {Fr}\mid \mathscr G)(\textsf k)$ over $X(\textsf k)$.

Let $\operatorname {Fun}_c(\widetilde X, A\subset E^{\times })$ denote the $E$-vector space of compactly supported functions $f : \widetilde X \rightarrow E$ such that $f(\tilde x\cdot a) = f(\tilde x)\cdot a$ for each $a\in A$. Elements of $\operatorname {Fun}_c(\widetilde X, A\subset E^{\times })$ are called genuine functions over $\widetilde X$ with respect to the inclusion $A\subset E^{\times }$.

In this set-up, there is a canonical isomorphism of $E$-vector spaces:

(1.19)\begin{equation} \operatorname{Fun}_c(\widetilde X, A\subset E^{\times}) \cong H^0_c(X^{\mathrm{Fr}}, \operatorname{Tr}(\mathrm{Fr}_X \mid \mathscr G)^{\otimes -1}), \end{equation}

where $H^0_c(X^{\mathrm {Fr}}, -)$ denotes the colimit of functors $H^0_c(U, -)$ over quasi-compact open subschemes $U\subset X^{\mathrm {Fr}}$.

1.4.10

We are now in a position to explain how genuine functions arise from twisted $E$-local systems as defined in § 1.2: this is the mechanism by which the cohomology of Shtukas will define genuine automorphic forms.

Consider the $A$-gerbe $\mathscr G\boxtimes (\mathscr G^{\otimes -1})$ over $X\times X$. In reference to (1.14), its restriction along $\Delta$ is canonically trivial, as is its restriction along $(\mathrm {id}, \mathrm {Fr}_X)$ by the isomorphism (1.15).

In particular, the restriction $\mathscr G_0$ of $\mathscr G\boxtimes (\mathscr G^{\otimes -1})$ to $X^{\mathrm {Fr}}$ admits two sections, corresponding to the two circuits of (1.14). If we write $g \in \mathscr G_0(X^{\mathrm {Fr}})$ for the section induced from the lower circuit, then the section induced from the upper is identified with $g \cdot \operatorname {Tr}(\mathrm {Fr}_X\mid \mathscr G)$.

Now, we let $\mathscr F$ be the $\mathscr G_0$-twisted $E$-local system on $X^{\mathrm {Fr}}$, which is identified with $\underline E$ using the section $g$ (see Remark 1.2.5). The same $\mathscr F$ is identified with $\operatorname {Tr}(\mathrm {Fr}_X\mid \mathscr G)^{\otimes -1}$ using the section $g\cdot \operatorname {Tr}(\mathrm {Fr}_X\mid \mathscr G)$.

In other words, taking $H^0_c(X^{\mathrm {Fr}}, \mathscr F)$ using the section of $\mathscr G_0$ induced from the upper circuit of (1.14) yields the $E$-vector space $\operatorname {Fun}_c(\widetilde X, A\subset E^{\times })$.

1.5.1

Let $S$ be a scheme and $A$ be a finite abelian group whose order is invertible on $S$.

Suppose that $X$ is an $S$-scheme and $G\rightarrow X$ is a smooth affine group scheme.

1.5.2

An étale metaplectic cover of $G\rightarrow X$ with values in $A$ is defined to be a section of $B^4\underline A(1)$ over $B_X(G)$ equipped with a rigidification along $e : X\rightarrow B_X(G)$.

Recall that the subscript in $B_X$ means taking the classifying stack of $G$ relative to $X$ (as opposed to the base scheme $S$).

Note that a rigidified section of $B^4\underline A(1)$ over $B_X(G)$ may be equivalently viewed as a morphism of $\mathbb {E}_1$-monoidal stacks $G \rightarrow B^3_X\underline A(1)$ over $X$.

1.5.4

Suppose that $X$ is the spectrum of a local nonarchimedean field $F_x$. Then an étale metaplectic cover $\mu$ defines a central extension of topological groupsFootnote ²:

(1.20)\begin{equation} 1 \rightarrow A \rightarrow \widetilde G_x \rightarrow G(F_x) \rightarrow 1. \end{equation}

If $X$ is instead the spectrum of the rings of integers $\mathscr O_x \subset F_x$, then the extension (1.20) produced by restricting $(G, \mu )$ to $\operatorname {Spec}(F_x)$ admits a canonical splitting over $G(\mathscr O_x)$.

1.5.5

If $X$ is the spectrum of a global field $F$ without real places, then an étale metaplectic cover $\mu$ defines a central extension of topological groups:

(1.21)\begin{equation} 1 \rightarrow A \rightarrow \widetilde G_F \rightarrow G(\mathbb{A}_F) \rightarrow 1, \end{equation}

where $\mathbb {A}_F$ denotes the topological ring of adèles of $F$. Furthermore, (1.21) is equipped with a canonical splitting over $G(F)$.

For each nonarchimedean place $x$ of $X$, the restriction of (1.21) along the inclusion $G(F_x) \subset G(\mathbb {A}_F)$ recovers the central extension (1.20).

When $A$ occurs as a subgroup of $E^{\times }$ for a field $E$, we have the $E$-vector space

(1.22)\begin{equation} \operatorname{Fun}(G(F)\backslash\widetilde G_F, A\subset E^{\times}) \end{equation}

of $G(F)$-invariant locally constant functions $f : \widetilde G_F \rightarrow E$ satisfying $f(\tilde x\cdot a) = f(\tilde x)\cdot a$ for every $\tilde x\in \widetilde G_F$ and $a\in A$. They are called genuine automorphic forms on $\widetilde G_F$.

Roughly speaking, the Langlands program for étale metaplectic covers seeks to understand the decomposition of various subspaces of (1.22) according to ‘spectral data’, defined in terms of an L-group associated to $(G, \mu )$.

1.6.1

We keep the notation of § 1.5.1. Furthermore, we bring in the coefficient field $E$ as in § 1.2.1 and assume that $A\subset E^{\times }$.

1.6.2

We also assume that $G\rightarrow X$ is a reductive group scheme. Let $\Lambda$ (respectively, $\check {\Lambda }$) be the étale sheaf of cocharacters of the universal Cartan $T\rightarrow X$ of $G$.

The based root data of $G$ consist of a sheaf of coroots (respectively, simple coroots) $\Phi$ (respectively, $\Delta$) with $\Delta \subset \Phi \subset \Lambda$, a sheaf of roots (respectively, simple roots) $\check {\Phi }$ (respectively, $\check {\Delta }$) with $\check {\Delta }\subset \check {\Phi }\subset \check {\Lambda }$, and an isomorphism $\Phi \cong \check {\Phi }$. The image of $\alpha \in \Phi$ under this isomorphism is denoted by $\check {\alpha }$.

1.6.3

To each étale metaplectic cover $\mu$ of $G\rightarrow X$ with values in $A$, the recipe of [Reference ZhaoZha22, § 6] defines its metaplectic dual data $(H, \nu )$, where:

1. (1) $H$ is a locally constant étale sheaf over $X$ of pinned split reductive groups over $E$;

2. (2) $\nu : \hat Z_H \rightarrow B^2_X(\underline A)$ is an $\mathbb {E}_{\infty }$-monoidal morphism.

Here, $Z_H$ denotes the center of $H$, $\hat Z_H$ the abelian group of its characters, viewed as a locally constant étale sheaf of abelian groups over $X$.

We partially recall the construction of $(H, \nu )$ in the following.

1.6.5

Given an étale metaplectic cover $\mu$ of $G\rightarrow X$, we first extract a triplet of invariants $(Q, \nu ^{\sharp }, \varphi )$, where:

1. (1) $Q$ is an $\underline A(-1)$-valued quadratic form on $\Lambda$;

2. (2) $\nu ^{\sharp } : \Lambda ^{\sharp }\rightarrow B^2_X(\underline A)$ is an $\mathbb {E}_{\infty }$-monoidal morphism or, equivalently, an extension of stacks of Picard groupoids over $X$,

   \begin{align*} B_X(\underline A) \rightarrow \widetilde{\Lambda}^{\sharp} \rightarrow \Lambda^{\sharp}; \end{align*}

3. (3) $\varphi$ is an $\mathbb {E}_{\infty }$-monoidal trivialization of $\nu ^{\sharp }$ over $\Lambda ^{\sharp, r} \subset \Lambda ^{\sharp }$.

Here, $\Lambda ^{\sharp } \subset \Lambda$ denotes the kernel of the symmetric form $b$ associated to $Q$ and $\Lambda ^{\sharp, r}\subset \Lambda$ the $\mathbb {Z}$-span of the set:

\begin{align*} \mathrm{ord}(Q(\alpha))\cdot \alpha\quad \text{for each }\alpha\in\Phi. \end{align*}

The fact that $\Lambda ^{\sharp, r}$ belongs to $\Lambda ^{\sharp }$ follows from the equality satisfied by $Q$:

(1.23)\begin{equation} b(\alpha, \lambda) = Q(\alpha)\langle\check{\alpha}, \lambda\rangle,\quad\text{for each }\alpha\in\Phi,\ \lambda\in\Lambda. \end{equation}

1.6.6

Let us sketch the construction of $(Q, \nu ^{\sharp }, \varphi )$ and provide pointers to [Reference ZhaoZha22]. The construction is performed étale locally on $X$ using a Borel subgroup $B\subset G$, but it turns out to be independent of this choice [Reference ZhaoZha22, § 5.2] and, thus, globalizes.

First, we restrict $\mu$ along $B(B) \rightarrow B(G)$, which canonically descends to an étale metaplectic cover $\mu _T$ of $T$.

The quadratic form $Q$ is the unique discrete invariant of $\mu _T$, in view of the isomorphism between $H^4(BT, A(1))$ and quadratic forms on $\Lambda$ (see [Reference ZhaoZha22, § 4.3]).

Next, the restriction $\mu _{T^{\sharp }}$ of $\mu _T$ to $B(T^{\sharp })$ acquires a canonical $\mathbb {E}_{\infty }$-monoidal structure [Reference ZhaoZha22, § 4.6]). Taking rigidified sections of $\mu _{T^{\sharp }}$ over $B(\mathbb {G}_m)$, we find a morphism of sheaves of $\mathbb {E}_{\infty }$-monoids over $X$:

(1.24)\begin{equation} \Lambda^{\sharp} \rightarrow \underline{\Gamma}{}_e(B\mathbb{G}_m, B^4\underline A(1)). \end{equation}

However, étale metaplectic covers of $\mathbb {G}_m$ admit a $\mathbb {Z}$-linear splitting into its $\mathbb {Z}$-linear component and the associated quadratic form [Reference ZhaoZha22, Remark 4.2.8]:

(1.25)\begin{equation} \underline{\Gamma}{}_e(B\mathbb{G}_m, B^4\underline A(1)) \cong B^2_X(\underline A) \oplus \underline A(-1). \end{equation}

Set $\nu ^{\sharp }$ to be the composition of (1.24) with the projection onto the first factor in (1.25).

Finally, the trivialization $\varphi$ arises from a calculation with $\mathrm {SL}_2$ (see [Reference ZhaoZha22, § 6.1.5]).

Remark 1.6.7 By construction, the composition of (1.24) with the projection onto the second factor in (1.25) equals the restriction of $Q$ to $\Lambda ^{\sharp }$, which is a $\mathbb {Z}$-linear map $\Lambda ^{\sharp } \rightarrow \underline A(-1)$ taking values in the subsheaf of $2$-torsion elements, because $b$ vanishes over $\Lambda ^{\sharp }$.

The restriction of $Q$ to $\Lambda ^{\sharp, r}$ vanishes, so we obtain a map:

(1.26)\begin{equation} Q : \Lambda^{\sharp}/\Lambda^{\sharp, r} \rightarrow \underline A(-1). \end{equation}

1.6.9

Let us now construct the metaplectic dual data $(H, \nu )$ from the triple $(Q, \nu ^{\sharp }, \varphi )$.

Proof Construction

Let us write $\check {\Lambda }^{\sharp }$ for the $\mathbb {Z}$-linear dual of $\Lambda ^{\sharp }$. For each $\alpha \in \Phi$, we set

\begin{gather*} \alpha^{\sharp} = \mathrm{ord}(Q(\alpha))\cdot \alpha \in \Lambda^{\sharp}; \\ \check{\alpha}^{\sharp} = \mathrm{ord}(Q(\alpha))^{-1} \cdot \check{\alpha} \in \check{\Lambda}^{\sharp}. \end{gather*}

Let $\Phi ^{\sharp }$ (respectively, $\check {\Phi }^{\sharp }$) be the span of $\alpha ^{\sharp }$ (respectively, $\check {\alpha }^{\sharp }$) and $\Delta ^{\sharp }$ (respectively, $\check {\Delta }^{\sharp }$) its subset corresponding to $\alpha \in \Delta$ (respectively, $\check {\alpha }\in \check {\Delta }$). Then the collection $\Delta ^{\sharp }\subset \Phi ^{\sharp }\subset \Lambda ^{\sharp }$, $\check {\Delta }^{\sharp }\subset \check {\Phi }^{\sharp } \subset \check {\Lambda }^{\sharp }$, with bijection $\Phi ^{\sharp }\cong \check {\Phi }^{\sharp }$, $\alpha ^{\sharp }\mapsto \check {\alpha }^{\sharp }$, defines a locally constant sheaf of based root data over $X$.

The sheaf $H$ is defined to be the associated sheaf of pinned split reductive groups over $E$: it has characters in $\Lambda ^{\sharp }$, roots in $\Phi ^{\sharp }$, simple roots in $\Delta ^{\sharp }$, etc.

By this definition, we have a canonical isomorphism of sheaves of abelian groups:

(1.27)\begin{equation} \hat Z_H \cong \Lambda^{\sharp}/\Lambda^{\sharp, r}, \end{equation}

so the data $(\nu ^{\sharp }, \varphi )$ of § 1.6.5 may be interpreted as an $\mathbb {E}_{\infty }$-monoidal morphism $\hat Z_H \rightarrow B^2_X(\underline A)$. This concludes the definition of $(H, \nu )$ alluded to in § 1.6.3.

1.6.10

In order to define the L-group, we need to pass to the component of $\nu$ which is $\mathbb {Z}$-linear, i.e. corresponding to a morphism of complexes $\hat Z_H \rightarrow \underline A[2]$ of étale sheaves of abelian groups over $X$ (see [Reference ZhaoZha22, § 6.2]). This is because $\nu$ can be nontrivial even over a geometric point of $X$, whereas the L-group does not capture this information.

Let us perform this construction in a more abstract setting: $M$ denotes an étale sheaf of abelian groups over $X$. An $\mathbb {E}_{\infty }$-monoidal morphism $\nu : M \rightarrow B^2_X(\underline A)$ corresponds to a symmetric monoidal extension $\widetilde M$ of $M$ by $B_X(\underline A)$.

Associating to each $m \in \widetilde M$ the commutativity constraint of $m\otimes m$ defines a character of $M$ valued in the subsheaf $\underline A{}_{[2]} \subset \underline A$ of $2$-torsion elements. This character vanishes $\Leftrightarrow$ $\widetilde M$ is strictly commutative $\Leftrightarrow$ $\nu$ is $\mathbb {Z}$-linear.

Therefore, we have a fiber sequence:

(1.28)\begin{equation} \mathrm{Maps}_{\mathbb{Z}}(M, B^2(\underline A)) \rightarrow \mathrm{Maps}_{\mathbb{E}_{\infty}}(M, B^2(\underline A)) \rightarrow \operatorname{Hom}(M, \underline A{}_{[2]}). \end{equation}

1.6.11

The fiber sequence (1.28) canonically splits.

Proof Construction

If $A_{[2]} \neq 0$, there is nothing to construct. If $A_{[2]} = 0$, the inclusion $A\subset E^{\times }$ identifies $A_{[2]}$ with $\mathbb {Z}/2$.

Given a homomorphism $\epsilon : M \rightarrow \underline A_{[2]}$, we define the $\mathbb {E}_{\infty }$-monoidal morphism ${}^{\epsilon }\nu : M \rightarrow B^2(\underline A)$ to be the trivial $\mathbb {E}_1$-monoidal morphism, whose $\mathbb {E}_{\infty }$-monoidal structure is defined by the commutativity constraint:

\begin{align*} (\widetilde M \ni m_1, m_2) \mapsto (-1)^{\epsilon(m_1)\epsilon(m_2)} \end{align*}

on the associated monoidal extension $\widetilde M$ of $M$ by $B(\underline A)$. The association $\epsilon \mapsto {}^{\epsilon }\nu$ provides the desired splitting of (1.28).

1.6.12

Let us now return to the metaplectic dual data $(H, \nu )$.

Along the split fiber sequence (1.28) with $M := \hat Z_H$, the $\mathbb {E}_{\infty }$-monoidal morphism $\nu : \hat Z_H \rightarrow B^2_X(\underline A)$ has a $\mathbb {Z}$-linear component, to be viewed as a morphism of complexes:

(1.29)\begin{equation} {}^0\nu : \hat Z_H \rightarrow \underline A[2]. \end{equation}

This datum is equivalent to a global section of the complex $(\hat Z_H)^* \otimes \underline A[2]$ over $X$, where $(\hat Z_H)^*$ denotes the $\mathbb {Z}$-linear dual of $\hat Z_H$ as a complex.

Inducing along the inclusion $A\subset E^{\times }$ and replacing $E$ by a finite extension if necessary, ${}^0\nu$ determines a global section of

\begin{align*} (\hat Z_H)^*\otimes \underline E^{\times}[2] \cong Z_H(E)[2], \end{align*}

i.e. an étale $Z_H(E)$-gerbe over $X$.

1.6.15

Let us now assume that $X$ is connected, Noetherian, geometrically unibranch, and satisfies condition (1.1). Fix a geometric point $\bar x$ of $X$.

By taking the fiber at $\bar x$, we obtain a pinned split reductive group $H_{\bar x}$ over $E$ equipped with a $\pi _1(X, \bar x)$-action preserving the pinning.

Since $\hat Z_H$ is finitely generated, ${}^0\nu$ is trivial over a finite étale cover of $X$. Fixing a rigidification $\bar g$ of ${}^0\nu$ along $\bar x$ and applying the construction of § 1.1.7 to finite subgroups of $E^{\times }$, we find an extension of topological groups:

(1.30)\begin{equation} 1 \rightarrow Z_{H, \bar x}(E) \rightarrow \pi_1({}^0\nu, \bar g) \rightarrow \pi_1(X, \bar x) \rightarrow 1, \end{equation}

where the $\pi _1({}^0\nu, \bar g)$-action on $Z_{H, \bar x}(E)$ factors through the given $\pi _1(X, \bar x)$-action.

Inducing (1.30) along the $\pi _1(X, \bar x)$-equivariant inclusion $Z_{H, \bar x}(E)\subset H_{\bar x}(E)$, we obtain an extension of topological groups, to be referred to as the L-group of $(G, \mu )$ over $X$:

(1.31)\begin{align} 1 \rightarrow H_{\bar x}(E) \rightarrow {}^LH_X \rightarrow \pi_1(X, \bar x) \rightarrow 1. \end{align}

1.6.17

Let $\mathscr L$ be a line bundle over $X$, viewed as a morphism of complexes $\mathbb {Z} \rightarrow \mathbb {G}_m[1]$. Tensoring with $\underline A(-1)$ and composing with the Kummer isomorphism $\Psi : \underline A(-1)\otimes \mathbb {G}_m \cong \underline A[1]$, we obtain a morphism $\Psi _*(\mathscr L) : \underline A(-1) \rightarrow \underline A[2]$.

We denote the composition of (1.26) with $\Psi _*(\mathscr L)$ by

(1.32)\begin{equation} \mathscr L^Q : \hat Z_H \rightarrow \underline A[2]. \end{equation}

Since (1.26) factors through the subsheaf of $2$-torsion elements, (1.32) is trivialized by any choice of a square root of $\mathscr L$.

1.6.18

In the particular case where $X$ is a smooth curve over a field, we may take $\mathscr L := \omega _X$ in (1.32) to arrive at a morphism of complexes:

(1.33)\begin{equation} \omega_X^Q : \hat Z_H \rightarrow \underline A[2]. \end{equation}

We use the notation $\nu _{\vartheta }$ (respectively, ${}^0\nu _{\vartheta }$) for the product of $\nu$ (respectively, ${}^0\nu$) with $\omega _X^Q$. The subscript is interpreted as a ‘twist by the gerbe of $\vartheta$-characteristics.’

After fixing a geometric point $\bar x$ of $X$ and a rigidification of (1.33) along $\bar x$, the $\mathbb {Z}$-linear morphism ${}^0\nu _{\vartheta }$ defines an extension:

(1.34)\begin{equation} 1 \rightarrow H_{\bar x}(E) \rightarrow {}^LH_{X, \vartheta} \rightarrow \pi_1(X, \bar x) \rightarrow 1. \end{equation}

It is also induced along $Z_{H, \bar x}(E)\subset H_{\bar x}(E)$ from the Baer sum of (1.30) and the central extension defined by (1.33).

These constructions have obvious analogues when $X$ is replaced by the spectra of its field of fractions $F$, its local fields $F_x$, or rings of integers $\mathscr O_x$.

1.7.1

We work over a base scheme $S$, a finite abelian group $A$, and a coefficient field $E$ as in § 1.2.1. Let $X$ be an $S$-scheme.

Let $H\rightarrow X$ be a locally constant étale sheaf of pinned split reductive groups over $E$. Write $Z_H\subset H$ for its center and $\hat Z_H$ its character group, viewed as an étale sheaf of abelian groups over $X$. Suppose that we are supplied with an $\mathbb {E}_{\infty }$-monoidal morphism $\nu : \hat Z_H \rightarrow B^2_X(\underline A)$.

In this context, we define an étale stack of tensor categories $\mathrm {Rep}^{\{1\}}_{H, \nu }$ on $X$. By ‘tensor category’, we mean an $E$-linear symmetric monoidal abelian category.

1.7.2

Denote by $\mathrm {Lis}_X$ the tensor category of lisse $E$-sheaves on $X$.

Note that $\mathscr O_H$ may be viewed as a Hopf algebra in $\mathrm {Ind}(\mathrm {Lis}_X)$. In particular, there is the notion of an $H$-representation on a lisse $E$-sheaf: it is an object $\mathscr F \in \mathrm {Lis}_X$ equipped with a morphism $\mathscr F\rightarrow \mathscr F\otimes \mathscr O_H$ in $\mathrm {Ind}(\mathrm {Lis}_X)$ satisfying the axioms defining a coaction.

Let $\mathrm {Rep}_H^{\{1\}}$ denote the tensor category of $H$-representations on lisse $E$-sheaves over $X$. The forgetful functor $\mathrm {Rep}_H^{\{1\}} \rightarrow \mathrm {Lis}_X$ is $E$-linear and symmetric monoidal.

1.7.3

The construction $X\mapsto \mathrm {Rep}_H^{\{1\}}$ being of étale local nature, we obtain a stack of tensor categories $\underline {\mathrm {Rep}}{}_H^{\{1\}}$ on the étale site of $X$. It admits a decomposition according to the weights of the $Z_H$-action, compatible with the tensor structure:

(1.35)\begin{equation} \underline{\mathrm{Rep}}{}_H^{\{1\}} \cong \bigoplus_{\lambda\in\hat Z_H} \underline{\mathrm{Rep}}{}_H^{\{1\}, \lambda}. \end{equation}

Since $A$ acts by automorphisms of the identity endofunctor of $\underline {\mathrm {Rep}}{}_H^{\{1\}}$, we may form the $\nu$-twisted stack of tensor categories $\underline {\mathrm {Rep}}{}_{H, \nu }^{\{1\}}$ as in [Reference ZhaoZha22, Appendix A].

Finally, we define $\mathrm {Rep}_{H, \nu }^{\{1\}}$ to be the global section of $\underline {\mathrm {Rep}}{}_{H, \nu }^{\{1\}}$.

Remark 1.7.4 There is a decomposition inherited from (1.35):

(1.36)\begin{align} \mathrm{Rep}_{H, \nu}^{\{1\}} \cong \bigoplus_{\lambda\in\check Z_H} (\mathrm{Rep}_H^{\{1\}, \lambda})_{\nu(\lambda)}, \end{align}

where each summand is the $\nu (\lambda )$-twist of the abelian $E$-linear categories $\mathrm {Rep}_H^{\{1\}, \lambda }$.

The symmetric monoidal structure on $\mathrm {Rep}_{H, \nu }^{\{1\}}$ is induced from that of $\mathrm {Rep}_H^{\{1\}}$ and the $\mathbb {E}_{\infty }$-monoidal structure of $\nu$. Concretely, the monoidal product is given by

\begin{align*} \big(\mathrm{Rep}_H^{\{1\}, \lambda_1}\big)_{\nu(\lambda_1)} \times \big(\mathrm{Rep}_H^{\{1\}, \lambda_2}\big)_{\nu(\lambda_2)} &\rightarrow \big(\mathrm{Rep}_H^{\{1\}, \lambda_1 + \lambda_2}\big)_{\nu(\lambda_1) \otimes \nu(\lambda_2)}\\ & \cong \big(\mathrm{Rep}_H^{\{1\}, \lambda_1 + \lambda_2}\big)_{\nu(\lambda_1 + \lambda_2)}. \end{align*}

Remark 1.7.5 Suppose that $X$ is connected, Noetherian, geometrically unibranch, and satisfies condition (1.1). Suppose also that $\nu$ is $\mathbb {Z}$-linear. (In practice, $\nu$ will be one of the objects ${}^0\nu$, ${}^0\nu _{\vartheta }$ defined in § 1.6.)

We fix a geometric point $\bar x$ of $X$ and a rigidification $\bar g$ of $\nu$ along $\bar x$. Then the L-group (1.31) can be used to give a ‘hands-on’ description of $\mathrm {Rep}_{H, \nu }^{\{1\}}$.

Indeed, given a profinite group $\Gamma$ and an extension ${}^LH$ of $\Gamma$ by $H_{\bar x}(E)$, we write $\mathrm {Rep}_{{}^LH}^{\mathrm {alg}}$ for the category of finite-dimensional continuous representations of ${}^LH$ whose restriction along $H_{\bar x}(E)\subset {}^LH$ lifts to an algebraic representation of $H_{\bar x}$.

Applied to ${}^LH_X$, this construction yields a tensor category canonically equivalent to that of $\nu$-twisted $H$-representations:

(1.37)\begin{equation} \mathrm{Rep}^{\mathrm{alg}}_{{}^LH_X} \cong \mathrm{Rep}_{H, \nu}^{\{1\}}. \end{equation}

Under the equivalence (1.37), the underlying (algebraic) $H_{\bar x}$-representation of an object $V\in \mathrm {Rep}^{\mathrm {alg}}_{{}^LH_X}$ is isomorphic to the fiber of the corresponding object in $\mathrm {Rep}_{H, \nu }^{\{1\}}$ at $\bar x$, by passing through the rigidification $\bar g$.

1.7.6

The construction of $\mathrm {Rep}_{H, \nu }^{\{1\}}$ has a multiple-point generalization, which justifies the superscript in the notation.

Indeed, for a nonempty finite set $I$, we have an $\mathbb {E}_{\infty }$-monoidal morphism of étale sheaves over $X^I$ out of an external direct sum of copies of $\hat Z_H$:

(1.38)\begin{equation} \nu^I : \hat Z_H^I \rightarrow B^2_{X^I}(\underline A),\quad (\lambda_i)_{i\in I} \mapsto \boxtimes_{i\in I} \nu(\lambda_i). \end{equation}

Viewing $H^I$ as a locally constant étale sheaf over $X^I$ of pinned split reductive groups over $E$, the corresponding étale sheaf $\hat Z_{H^I}$ is identified with $\hat Z_H^I$. The construction of § 1.7.3, applied to $X^I$, $H^I$, and $\nu ^I$, yields a stack of tensor categories $\underline {\mathrm {Rep}}{}_{H^I, \nu ^I}^I$ over $X^I$, and we set $\mathrm {Rep}_{H^I, \nu ^I}^I$ to be its global section.

1.7.7

The $\mathbb {E}_{\infty }$-monoidal structure on $\nu ^I$ (for varying $I$) induces its compatibility data with respect to restrictions along the diagonals.

More precisely, given a surjection of nonempty finite sets $p : I\twoheadrightarrow J$, giving rise to the diagonal immersion $\Delta ^p : X^J \rightarrow X^I$, we obtain an isomorphism $(\Delta ^p)^*\nu ^I \cong \nu ^J$. These isomorphisms are compatible with compositions in the obvious sense.

Correspondingly the association $(I\neq \varnothing ) \mapsto \mathrm {Rep}_{H^I, \nu ^I}^I$ defines a functor from the category of nonempty finite sets with surjections to the $2$-category of tensor categories. It carries a surjection $p : I\twoheadrightarrow J$ to the composition:

(1.39)\begin{align} \mathrm{Rep}_{H^I, \nu^I}^I \rightarrow \mathrm{Rep}_{(\Delta^p)^*(H^I), \nu_J}^J \rightarrow \mathrm{Rep}_{H^J, \nu_J}^J,\end{align}

where the first functor is the restriction along $\Delta ^p$, whereas the second functor is the restriction of the action along the diagonal $H^J \rightarrow (\Delta ^p)^*(H^I)$.

2.0.1

Throughout this section, we work over a field $\textsf k$. The letter $S$ is reserved for arbitrary (‘test’) affine $\textsf k$-schemes.

Let $\ell$ be a prime invertible in $\textsf k$ and fix an algebraic closure $\mathbb {Q}_{\ell } \subset \overline {\mathbb {Q}}{}_{\ell }$. The coefficient field $E$ will be an intermediate field $\mathbb {Q}_{\ell }\subset E \subset \overline {\mathbb {Q}}{}_{\ell }$.Footnote ³ Let $A\subset E^{\times }$ be a finite subgroup whose order is invertible in $\textsf k$.

Let $X$ be a smooth curve over $\textsf k$ and $G \rightarrow X$ be a smooth affine group scheme. Let $\mu$ be an $A$-valued étale metaplectic cover of $G$, i.e. a rigidified section of $B^4\underline A(1)$ over $B_X(G)$.

2.1.1

Let $I$ be a nonempty finite set.

For an $S$-point $x^I$ of $X^I$, we write $\Gamma _{x^I}$ for the scheme-theoretic union of the graphs of $x^i : S\rightarrow X$ (over $i\in I$), $D_{x^I}$ for the formal completion of $S\times X$ along $\Gamma _{x^I}$, and $\mathring D_{x^I}$ the open subscheme $D_{x^I} - \Gamma _{x^I}$. We call $D_{x^I}$ (respectively, $\mathring D_{x^I}$) the formal disk (respectively, formal punctured disk) around $\Gamma _{x^I}$.

We shall define a number of étale stacks over $X^I$. Their groupoids of lifts of an $S$-point $x^I$ of $X^I$ are tabulated as follows:

(2.1)\begin{align} {r | l} L_+^I(G) \quad & \quad a\,section\,of G\,over\,D_{x^I} \\ L^I(G) \quad & \quad a\,section\,of\,G\,over\,\mathring D_{x^I} \\ \mathrm{Gr}^I_G \quad & \quad a\,G-torsor\,P\,over\,D_{x^I}\,equipped\,with\,\alpha : P^0\overset{x^I}{\sim} P \\ \mathrm{Hec}^I_G \quad & \quad G-torsors\,P_0,\,P_1\,over\,D_{x^I}\,equipped\,with\,\alpha : P_0\overset{x^I}{\sim} P_1 \end{align}

Here, $P^0$ denotes the trivial $G$-torsor over $D_{x^I}$, and the notation $P_0\overset {x^I}{\sim } P_1$ for two $G$-torsors over $D_{x^I}$ means an isomorphism of them off $\Gamma _{x^I}$.

The étale sheaves $L_+^I(G)$ and $L^I(G)$ are valued in groups. The étale sheaf $\mathrm {Gr}_G^I$ is ind-schematic of ind-finite type.

If $G\rightarrow X$ is reductive, $\mathrm {Gr}_G^I$ is furthermore ind-proper.

2.1.3

Given a surjection of nonempty finite sets $p : I \twoheadrightarrow J$, we write $I_j := f^{-1}(j)$ and view $p$ as an unordered partition of $I$. Denote by $X^p \subset X^I$ the ‘disjoint locus’, i.e. the open subscheme where $\Gamma _{x^{I_{j_1}}}\!\cap \Gamma _{x^{I_{j_2}}} = \varnothing$ if $j_1\neq j_2\in J$.

We have a canonical isomorphism of étale stacks over $X^p$ via restriction to each $D_{X^{I_j}}$:

(2.2)\begin{equation} \varphi^p : \mathrm{Hec}_G^I \times_{X^I} X^p \cong \bigg(\prod_{j\in J} \mathrm{Hec}_G^{I_j}\bigg) \times_{X^I} X^p. \end{equation}

The isomorphism (2.2) is compatible with refinements of partitions. Namely, given two composable surjections of nonempty finite sets $p : I \twoheadrightarrow J$, $q : J\twoheadrightarrow K$, we have containments $X^p \subset X^{q\cdot p} \subset X^I$. For each $k\in K$, $p$ restricts to the partition $p_k : I_k := (q\cdot p)^{-1}(k) \twoheadrightarrow J_k$. There is a commutative diagram as follows.

(2.3)

Furthermore, given three composable surjections of nonempty finite sets, the isomorphism of arrows rendering (2.3) commutative satisfies the evident cocycle condition.

The isomorphism (2.2) and the commutativity (2.3) satisfying the cocycle condition are referred to as the factorization structure of $\mathrm {Hec}_G^I$ (for varying $I$).

2.1.5

Given an ordered partition $I = I_1\sqcup \cdots \sqcup I_k$ as nonempty finite sets, we shall consider another étale stack $\widetilde {\mathrm {Hec}}{}_G^{I_1,\ldots, I_k}$ over $X^I$ whose lift of an $S$-point $x^I$ of $X^I$ consists of $G$-torsors $P_0, \ldots, P_k$ together with isomorphisms:

(2.4)\begin{equation} (P_0\overset{x^{I_1}}{\sim} P_1 \overset{x^{I_2}}{\sim}\cdots \overset{x^{I_k}}{\sim} P_k). \end{equation}

The functors $p_a$ of remembering each segment $(x^{I_a}, P_{a-1}\overset {x^{I_a}}{\sim } P_a)$ (over $1\le a\le k$) and the functor $m$ of remembering their composition $(x^I, P_0\overset {x^I}{\sim } P_k)$ define two morphisms as follows:

(2.5)

Over the disjoint locus $X^{I_1,\ldots, I_k}$, both maps in (2.5) restrict to isomorphisms and their composition is identified with the factorization isomorphism (2.2) associated to the underlying unordered partition $I \rightarrow \{1,\ldots, k\}$.

This construction has an obvious analogue for the affine Grassmannian, namely an ind-scheme $\widetilde {\mathrm {Gr}}{}_G^{I_1,\ldots, I_k}\rightarrow X^I$ classifying the same data (2.4) as $\widetilde {\mathrm {Hec}}{}^{I_1,\ldots, I_k}_G$, together with an additional trivialization of $P_0$.

2.1.6

Given a nonempty finite set $I$, the association

(2.6)\begin{equation} ([k] \in \Delta^{\mathrm{op}}) \mapsto \widetilde{\mathrm{Hec}}{}_G^{I, [k]} := \widetilde{\mathrm{Hec}}{}_G^{I_1,\ldots, I_k} \times_{X^{I_1\sqcup\cdots\sqcup I_k}} X^I, \end{equation}

with $I_1 = \cdots = I_k := I$ defines a simplicial étale stack over $X^I$.

By convention, we set $\widetilde {\mathrm {Hec}}{}^{I, [0]}_G := B_{X^I}(L^I_+(G))$, i.e. the stack classifying a $G$-torsor $P$ over $D_{x^I}$ with no modifications. Morphisms in $\Delta ^{\mathrm {op}}$ are carried to compositions of the corresponding segment of modifications in (2.4).

This simplicial étale stack is canonically identified with the Čech nerve of the morphism

(2.7)\begin{equation} B_{X^I}(L_+^I(G)) \rightarrow B_{X^I}(L^I(G)). \end{equation}

Evidently, its value at $[1]$ is $\mathrm {Hec}_G^I$. In particular, the simplicial system (2.6) may be viewed as an additional structure on $\mathrm {Hec}_G^I$ which we call the convolution structure.

2.2.1

Let $I$ be a nonempty finite set.

We use the étale metaplectic cover $\mu$ to define an $A$-gerbe $\mathscr G^I$ on $\mathrm {Hec}_G^I$. Furthermore, $\mathscr G^I$ (for varying $I$) will come equipped with canonical compatibility data with respect to the factorization and convolution structures of $\mathrm {Hec}_G^I$.

The $A$-gerbe $\mathscr G^I$ is not new: it features prominently in [Reference ReichRei12] and [Reference Gaitsgory and LysenkoGL18], and we explain in Remark 2.2.8 how their approach is related to ours.

2.2.2

We start by defining an $\mathbb {E}_1$-monoidal section of $B^2(\underline A)$ over $L^I(G)$, trivialized as such over $L_+^I(G)$, i.e. the following commutative diagram of $\mathbb {E}_1$-monoidal morphisms.

(2.8)

Proof Construction

For an $S$-point $x^I$ of $X^I$, we give names to the natural morphisms in the following diagram as displayed.

Viewing $\mu$ as an $\mathbb {E}_1$-monoidal morphism $G \rightarrow B^3_X\underline A(1)$, we obtain an $\mathbb {E}_1$-monoidal morphism by taking its section over $\mathring D_{x^I}\rightarrow X$:

(2.9)\begin{equation} \mu_* : \Gamma(\mathring D_{x^I}, G) \rightarrow \Gamma(\mathring D_{x^I}, B^3_X\underline A(1)). \end{equation}

Note that the target is the $\infty$-groupoid associated to the connective truncation of the complex $\Gamma (D_{x^I}, \hat j_* \underline A(1)[3])$.

Using the Cousin complex and the Gabber–Fujiwara formal base change theorem [Reference FujiwaraFuj95, Corollary 6.6.4] (see [Reference Bhatt and MathewBM21, Theorem 6.11] for a proof avoiding the Noetherian hypothesis), we find morphisms of complexes:

(2.10)\begin{align} \Gamma(D_{x^I}, \hat j_*\underline A(1)[3]) &\rightarrow \Gamma(\Gamma_{x^I}, \hat i^!\underline A(1)[4])\nonumber\\ & \cong \Gamma(\Gamma_{x^I}, i^!\underline A(1)[4]) \cong \Gamma(\Gamma_{x^I}, (p\cdot i)^!\underline A[2]) \rightarrow \Gamma(S, \underline A[2]), \end{align}

where the last two maps use the smoothness of $p$ and the properness of $p\cdot i$, respectively.

Composing (2.9) with the morphism on underlying $\infty$-groupoids of (2.10), we obtain an $\mathbb {E}_1$-monoidal morphism:

(2.11)\begin{align} \Gamma(\mathring D_{x^I}, G) \rightarrow \Gamma(S, B^2(\underline A)), \end{align}

trivialized as such over $\Gamma (D_{x^I}, G)$. The construction being functorial in the $S$-point $x^I$, we obtain the commutative diagram (2.8).