On the multiplicity formula for discrete automorphic representations of disconnected tori

Yi Luo

Abstract

Kaletha extended the local Langlands conjectures to a certain class of disconnected groups and proved them for disconnected tori. Our first main result is a reinterpretation of the local Langlands correspondence for disconnected tori. Our second, and the central objective of this paper, is to establish an automorphic multiplicity formula for disconnected tori.

1 Introduction

Let $F$ be a number field and $\mathbb{A}$ be the adele ring of $F$ . Let $G$ be a connected reductive group defined over $F$ with centre $Z_{G}$ . Let $[G]:=A_{G}G(F)\backslash G(\mathbb{A})$ be its adelic quotient, where $A_{G}$ is the identity component of $\mathbb{R}$ -points of the largest $\mathbb{Q}$ -split subtorus of $\mathrm{Res}_{F/\mathbb{Q}}Z_{G}$ . Then there is a right regular representation of $G(\mathbb{A})$ on the Hilbert space $L^{2}([G])$ .

In the theory of automorphic representations, it is a crucial question to understand the decomposition of the discrete part $L_{\mathrm{disc}}^{2}([G])$ and the multiplicities of its irreducible constituents. The work of Labesse and Langlands ([labesse1979indistinguishability],[langlands1982debuts]) and the work of Kottwitz ([kottwitz1984stable]) suggest a conjectural answer for tempered automorphic representations. They expect that an admissible tempered discrete homomorphism $\phi:L_{F}\to{}^{L}G$ gives rise to an adelic $L$ -packet $\Pi_{\phi}$ of tempered representations of $G(\mathbb{A})$ and conversely, each tempered discrete automorphic representation belongs to some $L$ -packet. The topological group $L_{F}$ above is usually called the (hypothetical) Langlands group of $F$ and it should have the Weil group $W_{F}$ as a quotient. Then it is expected that the multiplicity of $\pi$ (which can be $0$ ) in $L_{\mathrm{disc}}^{2}([G])$ is the sum of $m_{\pi,\phi}$ with $\phi$ running over the equivalence classes of such parameters, where we have denoted by $m_{\pi,\phi}$ the $\phi$ -contribution towards the total multiplicity of $\pi$ . Moreover, it is conjectured that $m_{\pi,\phi}$ is given by

\displaystyle m_{\pi,\phi}=\frac{1}{|\mathcal{S}_{\phi}|}\sum_{x\in\mathcal{S}_{\phi}}\langle x,\pi\rangle,

(1.1)

where $\mathcal{S}_{\phi}$ is a finite group related to the centraliser of $\phi$ in $\hat{G}$ , and $\langle\cdot,\cdot\rangle:\mathcal{S}_{\phi}\times\Pi_{\phi}\to\mathbb{C}^{\times}$ is a complex-valued pairing relying on the local Langlands conjectures. In [kaletha2018global], after introducing the global Galois gerbes, Kaletha makes the definitions of $\mathcal{S}_{\phi}$ and the pairing $\langle\cdot,\cdot\rangle$ precise for general connected reductive groups $G$ .

In [kaletha2022local], Kaletha initiated an extension of the (refined) local Langlands conjectures to disconnected groups. To be precise, he treats the inner forms of “quasi-split” disconnected reductive groups, which are of the form $G\rtimes A$ , where $G$ is a connected quasi-split reductive group and $A$ acts on $G$ by preserving an $F$ -pinning. Kaletha extends the (refined) local Langlands conjectures to this setting and proves the conjectures in the case of disconnected tori.

1.1 Main results

Based on Kaletha’s pioneering work in the local aspects, the principal goal of this thesis is to explore the global aspects for the disconnected tori and establish a multiplicity formula. It turns out that the formula we obtain in this setting takes a similar form as (1.1).

First, we introduce the disconnected tori we treat, which are the main players of this thesis, and state the multiplicity formula on the automorphic side. Let $\tilde{T}$ be a quasi-split disconnected torus defined over $F$ with identity component a torus $T$ , that is $\tilde{T}=T\rtimes A$ is a semidirect product of $T$ with some finite group $A$ defined over $F$ . Given $z\in Z^{1}(F,T)$ , we twist the rational structure of $\tilde{T}$ via $z$ through inner automorphisms, and obtain a pure inner form $\tilde{T}_{z}$ . Regarding its rational points, there is a short exact sequence

\displaystyle 1\to T(F)\to\tilde{T}_{z}(F)\to A(F)^{[z]}\to 1,

(1.2)

where $A(F)^{[z]}$ is the stabiliser of $[z]$ in $A(F)$ . Moreover, we see that $A(F)$ acts on the set of Hecke characters of $T$ by conjugation. When $\chi$ is a Hecke character of $T$ , we denote the stabiliser of $\chi$ in $A(F)^{z}$ by $A(F)^{[z],\chi}$ .

To ease the exposition, in this introduction, we assume that $T$ is anisotropic. Under this assumption, $[\tilde{T}_{z}]:=\tilde{T}_{z}(F)\backslash\tilde{T}_{z}(\mathbb{A})$ is compact. Let $\eta$ be an irreducible constituent of $L^{2}([\tilde{T}_{z}])$ . By an argument involving the Clifford Theory, the multiplicity of $\eta$ in $L^{2}([\tilde{T}_{z}])$ can be determined by eventually passing $\eta$ to a finite group and calculating its the character. To be precise, we have (Theorem LABEL:automult)

\displaystyle m_{\eta}=\sum_{\chi}\frac{1}{|A(F)^{[z],\chi}|}\sum_{a\in A(F)^{[z],\chi}}\mathrm{tr}\bar{\eta}|_{A(F)^{[z],\chi}}(a),

where $\chi$ runs over the $\tilde{T}_{z}(F)$ -orbits (or equivalently, $A^{[z]}$ -orbits) of Hecke characters of $T$ .

Before entering the dual side, we need to review the local aspects first. We briefly recall the local Langlands correspondence for disconnected torus in terms of pure inner forms. Let $v$ be a place of $F$ , then $\tilde{T}$ is a quasi-split disconnected torus over $F_{v}$ under base change. We can consider the local pure inner form $\tilde{T}_{z_{v}}$ . It is noteworthy that the rational points of $\tilde{T}_{z}$ can be related to certain $1$ -hypercocycles in an obvious manner. Let $\phi_{v}:W_{F_{v}}\to{}^{L}T$ be a local $L$ -parameter for $T$ . By suitably enlarging the $L$ -group, one can extend the group of self-equivalences of $\phi_{v}$ from $S_{\phi_{v}}$ to $\tilde{S}_{\phi_{v}}$ , and define $\mathrm{Irr}(\pi_{0}(\tilde{S}_{\phi_{v}}),[z_{v}])$ as the set of irreducible representations of $\pi_{0}(\tilde{S}_{\phi_{v}})$ whose restrictions to $\pi_{0}({S}_{\phi_{v}})$ contains $[z_{v}]$ (which is a character of $\pi_{0}({S}_{\phi_{v}})$ in Kottwitz’s sense). On the other hand, by the local Langlands correspondence for tori, $\phi_{v}$ gives rises to a character of $T(F_{v})$ . One can define $\mathrm{Irr}(\tilde{T}_{z}(F_{v}),\chi_{v})$ as the set of irreducible representations of $\tilde{T}_{z}(F_{v})$ whose restrictions to $T(F_{v})$ contain $[\phi_{v}]$ .

Now the local Langlands correspondence for disconnected tori asserts there is a natural bijection between $\mathrm{Irr}(\tilde{T}_{z}(F_{v}),\chi_{v})$ and $\mathrm{Irr}(\pi_{0}(\tilde{S}_{\phi_{v}}),[z_{v}])$ , which satisfies the character identities. Kaletha constructed the bijection and verified the character identities. One of the main ingredients in his construction is the Tate-Nakayama pairing (or duality) for hypercohomology introduced in [kottwitz1999foundations], which organically combines the (classical) Tate-Nakayama duality and the local Langlands correspondence for (connected) tori. After passing both the group side and the dual side to certain extensions of $A^{[\phi_{v}],[z_{v}]}$ by $\mathbb{C}^{\times}$ , Kaletha shows that there is a canonical isomorphism between them, which induces the desired bijection. The slight drawback of this approach is that the isomorphism is given in a less transparent manner, due to the need of choosing sections for the extensions.

The first main result of this work is an intrinsic reinterpretation of the Kaletha’s LLC for disconnected tori. To be precise, we are able to define a simple relation (LABEL:reinterrelation) to characterise the 1-1 correspondence between $\mathrm{Irr}(\tilde{T}_{z}(F_{v}),\chi_{v})$ and $\mathrm{Irr}(\pi_{0}(\tilde{S}_{\phi_{v}}),[z_{v}])$ intrinsically, in the sense that there is no choice of “coordinate systems” involved.

Now we return to the global aspect. Let $\phi:W_{F}\to{}^{L}T$ be a global $L$ -parameter and let $\chi=\otimes_{v}\chi_{v}$ be the associated Hecke character of $T$ under the global Langlands correspondence. One can define the adelic $L$ -packet $\Pi_{\phi}$ associated to $\phi$ . Essentially, it consists of irreducible representations of $\tilde{T}_{z}(\mathbb{A})$ that are unramified almost everywhere and whose local component at any place $v$ contains $\chi_{v}$ . Then we define a complex-valued pairing (LABEL:thepairing) $\langle\cdot,\cdot\rangle:A(F)^{[z],\chi}\times\Pi_{\phi}\to\mathbb{C}$ given by

\displaystyle\langle a,\eta\rangle:=\prod_{v}\langle(\phi_{v}^{-1},a^{-1}(s_{v}^{-1})),(z_{v}^{-1},t)\rangle^{-1}_{\mathrm{TN}}\cdot\mathrm{tr}\left[\iota_{v}(\bar{\eta}_{v})(s_{v},a)\right],

where $(s_{v},a)\in\tilde{S}_{\phi_{v}}^{[z_{v}]}$ and $(t,a)\in\tilde{T}_{z}(F)^{\chi}$ are arbitrarily chosen. Despite the fact that the choices of local data appearing in the above pairing have great flexibility, we can show the well-definedness of the pairing and its independence on the choices of the local data.

At this point, a key input is our reinterpretation of the LLC for disconnected tori, which enables us to find that this pairing has its counterpart on the automorphic side. In fact, we have

\displaystyle\langle a,\eta\rangle=\mathrm{tr}\left[\bar{\eta}|_{A(F)^{[z],\chi}}(a)\right].

(1.3)

After a comparison with the (actual) multiplicity obtained from the automorphic side, we reach the desired multiplicity formula

\displaystyle m_{\eta}=\sum_{[[\phi]]}\frac{1}{|A(F)^{[z],\chi}|}\sum_{a\in A(F)^{[z],\chi}}\langle a,\eta\rangle,

where $[[\phi]]$ runs over the near $A(F)^{[z]}$ -equivalence classes of global $L$ -parameters, which are in 1-1 correspondence to the $A(F)^{[z]}$ -orbits (or equivalently $\tilde{T}_{z}(F)$ -orbits) of Hecke characters of $T$ .

1.2 Structure of the paper

In Chapter 2, we review the (local and global) Tate-Nakayama duality for hypercohomology introduced in [kottwitz1999foundations, Appendix A.3]. We aim to explicitly elucidate the isomorphisms on the chain level.

In Chapter 3, we recall the conventions (following [kaletha2022local]) on the disconnected groups considered in Kaletha’s framework. In particular, we give some examples of disconnected tori.

Chapter 4 and Chapter 5 are devoted to the local aspects. In Chapter 4, we closely follow [kaletha2022local] and review the statement of the local Langlands correspondence for disconnected tori in terms of pure inner forms. In Chapter 5, we give a new construction of the LLC, which can be eventually proved to coincide with Kaletha’s construction.

The rest of the chapters are focused on establishing the multiplicity formula. In Chapter 6, we lay the necessary foundation for the discussion of global aspects. In Chapter 7, we aim to calculate the multiplicity on the automorphic side (which means the actual multiplicity of an automorphic representation). For this purpose, we first extract a dense smooth subspace $\mathcal{A}(\tilde{T}_{z})$ from the $L^{2}$ -space. After a preliminary decomposition, we are able to break the calculation of multiplicity into a sum of $\chi$ -contribution with $\chi$ running over the $\tilde{T}_{z}(F)$ -orbits of Hecke characters. Eventually, the multiplicity can be calculated through some arguments in Clifford theory. In Chapter 8, we enter the dual side. After defining the pairing $\langle\cdot,\cdot\rangle:A(F)^{[z],\chi}\times\Pi_{\phi}\to\mathbb{C}$ , we apply our reinterpretation of the LLC so as to conclude that the pairing has an incarnation on the automorphic side. Finally, the multiplicity formula is established after a straightforward comparison.

Acknowledgements.

The author is grateful to Wee Teck Gan for his invaluable insights, as well as for his guidance and proof-reading. The author also appreciates Tasho Kaletha for his interest in this project and helpful suggestions. The author gratefully acknowledges the support of NUS Research Scholarship.

2 The Tate-Nakayama Duality for hypercohomology

In this section, we review the local and global duality between the hypercohomology of the Galois group and that of the Weil group. These results are due to Kottwitz-Shelstad [kottwitz1999foundations]. We will see that the duality is able to encompass the LLC for (connected) tori and Tate-Nakayama duality simultaneously. We refer the reader to Appendix LABEL:appA for the notion and basic properties of group hypercohomology, Appendix LABEL:appLanglands for the (local and global) Langlands correspondences for connected tori, and Appendix LABEL:appTN for the (local and global) Tate-Nakayama dualities.

2.1 Convention

Whenever we say hypercohomology of profinite groups in a (complex of) discrete module(s), we always refer to the continuous version.

Whenever we consider (hyper-)cochains/cocycles/cohomology of Weil group $W_{F}$ or relative Weil groups $W_{K/F}$ , we always assume continuity, unless otherwise stated or indicated as the abstract version by the subscript “abs”. In contrast, in this work, all the (hyper-)chains/cycles/homology of (relative) Weil groups are always in the abstract sense.

2.2 Local Tate-Nakayama duality for hypercohomology

In this part, we review the duality for the hypercohomology of the Galois group and the Weil group. These results were established by Kottwitz-Shelstad [kottwitz1999foundations]. We will see that the duality is able to encompass the LLC for tori and Tate-Nakayama duality (the case $n=-1$ in the previous section) simultaneously. We refer the reader to Appendix LABEL:appA for the notion and basic properties of group hypercohomology.

Let $F$ be a local field of characteristic zero. Let $T$ and $U$ be $F$ -tori with cocharacter groups $X$ and $Y$ . Let $f:T\to U$ be a morphism defined over $F$ . Let $f_{*}:X\to Y$ and $\hat{f}:\hat{U}\to\hat{T}$ be the maps induced by $f$ . We aim to define a pairing between $H^{1}(W_{F},\hat{U}\xrightarrow{\hat{f}}\hat{T})$ and $H^{1}(F,T\xrightarrow{f}U)$ . For convenience, we fix a finite Galois extension $K$ of $F$ such that both $T$ and $U$ split over $K$ .

Throughout this chapter, once and for all, we fix a section $s:\mathrm{Gal}(K/F)\to W_{K/F}$ with $s(1)=1$ , so that we have maps $\phi$ and $\psi$ on the chain level with explicit formulae (LABEL:themapphi_formula) and (LABEL:themappsi), respectively. We will eventually show that the choice of section $s$ has no effect on the (co)homology level (see Proposition 2.3).

2.2.1 Step 1

We quickly remind the reader that, whenever talking about (hyper)homology of (relative) Weil groups, we ignore the topology and regard them as abstract groups. First, we note that the image of the differential

\displaystyle C_{1}(W_{K/F},X)\xrightarrow{\partial}C_{0}(W_{K/F},X)

lies in $C_{0}(W_{K/F},X)_{0}$ , the subgroup of norm- $0$ elements in $C_{0}(W_{K/F},X)=X$ . Indeed, given $x\in C_{1}(W_{K/F},X)$ , we have

	$\displaystyle N_{K/F}(\partial x)=\sum_{\sigma\in\mathrm{Gal}(K/F)}\sigma(\partial x)$	$\displaystyle=\sum_{\sigma\in\mathrm{Gal}(K/F)}\sigma\left[\sum_{w\in W_{K/F}}(w^{-1}x_{w}-x_{w})\right]$
		$\displaystyle=\sum_{w\in W_{K/F}}\left[\sum_{\sigma\in\mathrm{Gal}(K/F)}(\sigma w^{-1}x_{w}-\sigma x_{w})\right].$

Now for each fixed $w\in W_{K/F}$ , the inner sum vanishes and it follows that $\partial x$ has norm $0$ .

Next, we define a modified (norm- $0$ ) hyperhomology group $H_{0}(W_{K/F},X\to Y)_{0}$ as a subgroup of $H_{0}(W_{K/F},X\to Y)$ . Given $X\xrightarrow{f_{*}}Y$ , we consider the following complex used in defining group hyperhomology:

\displaystyle\cdots\to C_{1}(W_{K/F},X)\oplus C_{2}(W_{K/F},Y)\xrightarrow{\alpha}C_{0}(W_{K/F},X)\oplus C_{1}(W_{K/F},Y)\xrightarrow{\beta}C_{0}(W_{K/F},Y),

(2.1)

where $\alpha(x,y)=(\partial x,f_{*}(x)-\partial y)$ and $\beta(x,y)=f_{*}(x)-\partial y$ . We recall from Appendix LABEL:appA that $H_{0}(W_{K/F},X\to Y)$ is defined as the quotient $\mathrm{ker}(\beta)/\mathrm{im}(\alpha)$ . We define $(\mathrm{ker}\beta)_{0}$ to be the subgroup of pairs $(x,y)$ in $\mathrm{ker}\beta$ with $x\in C_{0}(W_{K/F},X)_{0}$ . According to the observation made above, we have $\mathrm{im}\alpha\subseteq(\mathrm{ker}\beta)_{0}$ . Now we define the modified hyperhomology group

\displaystyle H_{0}(W_{K/F},X\to Y)_{0}:=(\mathrm{ker}\beta)_{0}/\mathrm{im}\alpha.

(2.2)

Analogous to Fact LABEL:Exactforhyperho, we have an exact sequence

\displaystyle H_{1}(W_{K/F},X)\to H_{1}(W_{K/F},Y)\to H_{0}(W_{K/F},X\to Y)_{0}\to H_{0}(W_{K/F},X)_{0}\to H_{0}(W_{K/F},Y)_{0}

The main goal of Step 1 is to define a natural isomorphism

\displaystyle H_{0}(W_{K/F},X\to Y)_{0}\xrightarrow{\sim}H^{1}(\mathrm{Gal}(K/F),T(K)\to U(K)).

For this purpose, we recall from (LABEL:themapphi) and (LABEL:themappsi) that we have defined

		$\displaystyle\phi:C_{1}(W_{K/F},X)\to C^{0}(K/F,T)$
and
		$\displaystyle\psi:C_{0}(W_{K/F},X)_{0}\to Z^{1}(K/F,T).$

The key observation is that $\phi$ and $\psi$ sit in the following diagram to make it commute:

(2.3)

For brevity, we dropped the (Galois/Weil) groups concerned in each term in the above diagram, but we remind the reader that the chains are of $W_{K/F}$ , while the cochains and cocycles are of $\mathrm{Gal}(K/F)$ .

Lemma 2.1.

The diagram (2.3) commutes.

Proof.

We recall that, by definition, $\phi=\sim_{D}\circ\mathrm{Res}$ . The restriction is a chain map (see (LABEL:reschain)), thus we have $\phi\circ\partial=\sim_{D}\circ\mathrm{Res}\circ\partial=\sim_{D}\circ\partial\circ\mathrm{Res}$ . Now we notice that the map $\sim_{D}$ (on the chain level)

	$\displaystyle\sim_{D}:C_{1}(K^{\times},X)$	$\displaystyle\to T(K)$
	$\displaystyle x$	$\displaystyle\mapsto\prod_{a\in K^{\times}}x_{a}(a)^{-1}$

is trivial on $1$ -boundaries, which implies $\sim_{D}\circ\partial=0$ . Thus, the commutativity of the first square follows.

Now we check the commutativity of the second square. Let $x\in C_{1}(W_{K/F},X)$ . On the one hand, we find that $\psi\circ\partial(x):=\psi(\sum_{w}w^{-1}x_{w}-y_{w})$ is an element in $Z^{1}(K/F,T)$ sending $\rho\in\mathrm{Gal}(K/F)$ to

\displaystyle\prod_{\sigma\in\mathrm{Gal}(K/F)}\langle\rho\sigma(\sum_{w}w^{-1}x_{w}-x_{w}),a_{\rho,\sigma}\rangle.

We let $w=as(\tau)$ and alter the domain over which the sum is taken accordingly. Then the above expression becomes

		$\displaystyle\prod_{\sigma,\tau,a}\langle\rho\sigma(\tau^{-1}x_{as(\tau)}-x_{as(\tau)}),a_{\rho,\sigma}\rangle$
	$\displaystyle=$	$\displaystyle\prod_{\sigma,\tau,a}\langle\rho\sigma\tau^{-1}(x_{as(\tau)}),a_{\rho,\sigma}\rangle\prod_{\sigma,\tau,a}\langle\rho\sigma(x_{as(\tau)}),a_{\rho,\sigma}\rangle^{-1}.$		(2.4)

On the other hand, through an elementary computation involving certain changes of variables, we find that $\partial\circ\phi(x)\in Z^{1}(K/F,T)$ sends $\rho\in\mathrm{Gal}(K/F)$ to

	$\displaystyle\prod_{\sigma,\tau,a}\langle\sigma(x_{as(\tau)}),a_{\sigma,\tau}\sigma(a)\rangle\rho(\prod_{\sigma,\tau,a}\langle\sigma(x_{as(\tau)}),a^{-1}_{\sigma,\tau}\sigma(a)^{-1}\rangle)$
$\displaystyle=$	$\displaystyle\prod_{\sigma,\tau,a}\langle\sigma(x_{as(\tau)}),a_{\sigma,\tau}\rangle\prod_{\sigma,\tau,a}\langle\rho\sigma(x_{as(\tau)}),\rho(a^{-1}_{\sigma,\tau})\rangle$
$\displaystyle=$	$\displaystyle\prod_{\sigma,\tau,a}\langle\sigma(x_{as(\tau)}),a_{\sigma,\tau}\rangle\prod_{\sigma,\tau,a}\langle\rho\sigma(x_{as(\tau)}),a_{\rho\sigma,\tau}^{-1}a_{\rho,\sigma\tau}a_{\rho,\sigma}^{-1}\rangle$
$\displaystyle=$	$\displaystyle\prod_{\sigma,\tau,a}\langle\rho\sigma(x_{as(\tau)}),a_{\rho,\sigma\tau}a_{\rho,\sigma}^{-1}\rangle,$	(2.5)

where we have used the $2$ -cocyle relation $\rho(a_{\sigma,\tau})^{-1}=a_{\rho\sigma,\tau}^{-1}a_{\rho,\sigma\tau}a_{\rho,\sigma}^{-1}$ in the second to last equality.

Finally, we can see immediately that (2.4) coincides with (2.5), after a change of variable. Hence the commutativity of the second square follows. ∎

We consider the following diagram with the first row a modification of (2.1) ( $0$ th-chains modified to norm- $0$ subgroups), and the second row mixed with cochains and cocycles of $\mathrm{Gal}(K/F)$ :

(2.6)

where $\gamma(t)=(\partial t,f(t))$ and $\delta(t,u)=f(t)-\partial u$ , hence the cohomology of the second row at $Z^{1}(T)\oplus C^{0}(U)$ is nothing but $H^{1}(F,T\xrightarrow{f}U)$ . And the vertical maps are indicated in the diagram. Using the commutativity of the diagram (2.3), one can immediately check:

Fact 2.2.

The diagram (2.6) commutes.

At this point, the commutativity of diagram (2.6) immediately suggests that the middle vertical map $\psi\oplus\phi$ actually passes to (co)homology:

\displaystyle\mathcal{H}:H_{0}(W_{K/F},X\to Y)_{0}\to H^{1}(\mathrm{Gal}(K/F),T(K)\to U(K)),

(2.7)

We quickly check the well-definedness of $\mathcal{H}$ :

Proposition 2.3.

$\mathcal{H}$ does not depend on the choice of section $s:\mathrm{Gal}(K/F)\to W_{K/F}$ .

Proof.

Suppose $s^{\prime}$ is another section, $a^{\prime}$ is the corresponding $2$ -cocycle given by (LABEL:2-cocycle), and $\phi^{\prime}$ and $\psi^{\prime}$ are the chain-level maps (LABEL:themapphi) and (LABEL:themappsi) defined in terms of $s^{\prime}$ and $a^{\prime}$ . Let $(x,y)\in\mathrm{ker}\beta$ . We hope to show that $(\psi(x),\phi(y))$ and $(\psi^{\prime}(x),\phi^{\prime}(y))$ differ by a $1$ -hypercoboundary. We consider the element

\displaystyle t:=\prod_{\sigma\in\mathrm{Gal}(K/F)}\langle\sigma(x),s^{\prime}(\sigma)s(\sigma)^{-1}\rangle

in $T(K)$ . Then elementary compuations suggest that, for any $\rho\in\mathrm{Gal}(K/F)$ , we have

	$\displaystyle t^{-1}\rho(t)$	$\displaystyle=\prod_{\sigma}\langle\rho\sigma(x),s(\rho\sigma)s^{\prime}(\rho\sigma)^{-1}\rho(s^{\prime}(\sigma)s(\sigma)^{-1})\rangle$
		$\displaystyle=\prod_{\sigma}\langle\rho\sigma(x),s(\rho\sigma)s^{\prime}(\rho\sigma)^{-1}s^{\prime}(\rho)s^{\prime}(\sigma)s(\sigma)^{-1}s^{\prime}(\rho)^{-1}\rangle$
		$\displaystyle=\prod_{\sigma}\langle\rho\sigma(x),a^{\prime}_{\rho,\sigma}a_{\rho,\sigma}^{-1}\rangle\prod_{\sigma}\langle\rho\sigma(x),s(\rho)s^{\prime}(\rho)^{-1}\rangle$
		$\displaystyle=\psi^{\prime}\psi^{-1}(x)(\rho).$

We note that we have used the fact that $x$ has trivial norm in the last equality above. Since $(x,y)$ lies in $\mathrm{ker}\beta$ , we have $f_{*}(x)=\partial y$ . Using this and some elementary manipulations, one can find that

	$\displaystyle f(t)$	$\displaystyle=\prod_{\sigma,\tau,a}\langle\sigma\tau^{-1}(y_{as(\tau)})-\sigma(y_{as(\tau)}),s^{\prime}(\sigma)s(\sigma)^{-1}\rangle$
		$\displaystyle=\prod_{\sigma,\tau,a}\langle\sigma(y_{as(\tau)}),s^{\prime}(\sigma\tau)s(\sigma\tau)^{-1}s(\sigma)s^{\prime}(\sigma)^{-1}\rangle$

coincides with

	$\displaystyle\phi^{\prime}(y)\phi(y)^{-1}$	$\displaystyle=\prod_{\sigma,\tau,a}\langle\sigma(y_{as^{\prime}(\tau)}),{a}_{\sigma,\tau}^{\prime-1}\sigma(a)^{-1}\rangle\prod_{\sigma,\tau,a}\langle\sigma(y_{as(\tau)}),a_{\sigma,\tau}^{-1}\sigma(a)^{-1}\rangle^{-1}$
		$\displaystyle=\prod_{\sigma,\tau,a}\langle\sigma y_{as(\tau)},a_{\sigma,\tau}^{\prime-1}a_{\sigma,\tau}\sigma(s^{\prime}(\tau)s(\tau)^{-1})\rangle$
		$\displaystyle=\prod_{\sigma,\tau,a}\langle\sigma y_{as(\tau)},a_{\sigma,\tau}^{\prime-1}a_{\sigma,\tau}s^{\prime}(\sigma)s^{\prime}(\tau)s(\tau)^{-1}s^{\prime}(\sigma)^{-1}\rangle,$

where the products are taken over triples

\displaystyle(\sigma,\tau,a)\in\mathrm{Gal}(K/F)\times\mathrm{Gal}(K/F)\times K^{\times}.

We conclude that the difference between $(\psi(x),\phi(y))$ and $(\psi^{\prime}(x),\phi^{\prime}(y))$ is a $1$ -hypercoboundary. ∎

One can further show:

Proposition 2.4.

The natural map

\displaystyle\mathcal{H}:H_{0}(W_{K/F},X\to Y)_{0}\to H^{1}(\mathrm{Gal}(K/F),T(K)\to U(K))

(2.8)

is an isomorphism.

Proof.

We consider the following diagram with rows the long exact sequences associated to hyper(co)homology (see LABEL:LESforhyper):

(2.9)

In the above diagram, $\mathcal{D}$ is the key isomorphism (LABEL:keyisomodified) induced by $\phi$ in Deligne’s convention, $TN$ is the Tate-Nakayama isomorphism (LABEL:-1TN) induced by $\psi$ , and $\mathcal{H}$ is the natural map induced by $\phi\oplus\psi$ .

We can see that the diagram (2.9) commutes. Indeed, on the one hand, we recall from LABEL:LESforhyper that the two arrows in the middle of each row are induced by an inclusion and a projection on the chain level. Hence the commutativity of the two squares in the middle is clear. On the other hand, the first and the last squares commute due to functoriality of $\mathcal{D}$ and the Tate-Nakayama isomorphism.

It follows that $\mathcal{H}$ is an isomorphism by the five lemma. ∎

2.2.2 Step 2

In this part, we will proceed as in Section LABEL:Step_2 to produce a pairing between certain hypercohomology and hyperhomology groups.

Let $B_{\bullet}$ be the bar resolution of $\mathbb{Z}$ . Then the defining complex of $H_{0}(W_{K/F},X\to Y)_{0}$ :

\displaystyle\cdots\to C_{1}(X)\oplus C_{2}(Y)\to C_{0}(X)\oplus C_{1}(Y)\to C_{0}(Y)

is nothing but

\displaystyle\cdots\to B_{1}\otimes_{\mathbb{Z}W_{K/F}}X\oplus B_{2}\otimes_{\mathbb{Z}W_{K/F}}Y\to B_{0}\otimes_{\mathbb{Z}W_{K/F}}X\oplus B_{1}\otimes_{\mathbb{Z}W_{K/F}}Y\to B_{0}\otimes_{\mathbb{Z}W_{K/F}}Y,

(2.10)

which we denote by $\mathcal{P}_{\bullet}$ .

Similar to the argument in Section LABEL:Step_2, we note that

\displaystyle\mathrm{Hom}(B_{i}\otimes_{\mathbb{Z}W_{K/F}}X,\mathbb{C}^{\times})=\mathrm{Hom}_{\mathbb{Z}W_{K/F}}(B_{i},\mathrm{Hom}(X,\mathbb{C}^{\times})),

which also holds with $X$ replaced by $Y$ .

We note that $\mathbb{C}^{\times}$ is an injective abelian group, hence after applying the functor $\mathrm{Hom}(-,\mathbb{C}^{\times})$ to complex $\mathcal{P}_{\bullet}$ (2.10) and taking cohomology, we obtain

\displaystyle\mathrm{Hom}(H_{\bullet}(\mathcal{P}_{\bullet}),\mathbb{C}^{\times})=H^{\bullet}(\mathrm{Hom}(\mathcal{P}_{\bullet},\mathbb{C}^{\times})).

Hence we have

\displaystyle\mathrm{Hom}(H_{0}(W_{K/F},X\to Y),\mathbb{C}^{\times})=H^{1}_{\mathrm{abs}}(W_{K/F},\widehat{U}\to\widehat{T}).

The right-hand side is the abstract cohomology group, ignoring the topology on $W_{K/F}$ and $\mathrm{Hom}(X,\mathbb{C}^{\times})\cong\hat{T}$ . Similarly, for hyper(co)homology we have canonical isomorphism

\displaystyle\mathrm{Hom}(H_{0}(W_{K/F},X\to Y),\mathbb{C}^{\times})\cong H^{1}_{\mathrm{abs}}(W_{K/F},\hat{U}\to\hat{T}).

Explicitly, given a 0-hypercycle $(x,y_{w})$ in $H_{0}(W_{K/F},X\to Y)$ , i.e. $f_{*}(x)=\partial y_{w}=\sum(w^{-1}y_{w}-y_{w})$ , and given a 1-hypercocycle $(u_{w},t)$ with $u_{w}\in Z^{1}_{\mathrm{abs}}(W,\hat{U})$ and $t\in\hat{T}$ such that $f_{*}(u_{w})=\partial t=t^{-1}w(t)$ . The pairing between $(x,y_{w})$ and $(u_{w},t)$ is

\displaystyle\langle(x,y_{w}),(u_{w},t)\rangle=\langle x,t\rangle\prod_{w\in W_{K/F}}\langle y(w),u(w)\rangle^{-1}.

(2.11)

Remark 2.5.

We need to point out some subtlety here. On the one hand, we have regarded $X\to Y$ as a complex concentrated at degrees $0$ and $-1$ . Hence the cohomology of dual complex $\mathrm{Hom}(H_{\bullet}(\mathcal{P}_{\bullet}),\mathbb{C}^{\times})$ actually gives hypercohomology of $\widehat{U}\to\widehat{T}$ with $\widehat{U}$ and $\widehat{T}$ placed at degrees $-1$ and $0$ , respectively. However, whenever we write $H^{1}_{\mathrm{abs}}(W_{K/F},\hat{U}\to\hat{T})$ , we are always placing $\widehat{U}$ and $\widehat{T}$ at degrees $0$ and $1$ . Hence, the inverse we add in (2.11) is due to this discrepancy.

So far, we have obtained a pairing between $H_{0}(W_{K/F},X\to Y)$ and $H^{1}_{\mathrm{abs}}(W_{K/F},\widehat{U}\to\widehat{T})$ . We recall from (2.2) that we defined a subgroup $H_{0}(W_{K/F},X\to Y)_{0}$ of $H_{0}(W_{K/F},X\to Y)$ . And we can restrict the pairing to subgroups

	$\displaystyle H_{0}(W_{K/F},X\to Y)_{0}$	$\displaystyle\subseteq H_{0}(W_{K/F},X\to Y)$
and
	$\displaystyle H^{1}(W_{K/F},\widehat{U}\to\widehat{T})$	$\displaystyle\subseteq H^{1}_{\mathrm{abs}}(W_{K/F},\widehat{U}\to\widehat{T}).$

Here, we recall that $H^{1}(W_{K/F},\hat{U}\to\hat{T})$ denotes the continuous cohomology group.

Furthermore, in view of the isomorphism $H_{0}(W_{K/F},X\to Y)_{0}\cong H^{1}(K/F,T\to U)$ we established in Step 2, we have obtained a pairing

\displaystyle H^{1}(K/F,T\to U)\times H^{1}(W_{K/F},\hat{U}\to\hat{T})\to\mathbb{C}^{\times}.

(2.12)

Suppose $K^{\prime}\supseteq K$ is another Galois extension of $F$ . Then again, we have a pairing with $K$ replaced by $K^{\prime}$ above. More precisely, we have the following diagram:

(2.13)

Proposition 2.6.

The pairing (2.12) is compatible with inflations.

Proof (Sketch)..

The desired compatibility follows from the compatibility of

(2.14)

and the commutativity of

(2.15)

The compatibility in (2.14) is obvious, since the deflation between homology is dual to the inflation between cohomology. The commutativity of the diagram (2.15) can be shown in the same manner as the proof of Proposition 2.3. Details can be found on pp. 136-137 of [kottwitz1999foundations]. ∎

Therefore, after passing to colimits, we have the desired functorial pairing

\displaystyle H^{1}(F,T\to U)\times H^{1}(W_{F},\hat{U}\to\hat{T})\to\mathbb{C}^{\times}.

(2.16)

Proposition 2.7.

The pairing (2.16) is compatible with the Langlands pairing (LABEL:Langlands-pairing) (in Langlands’ convention) and the Tate-Nakayama pairing (LABEL:TN-pairing). Precisely speaking, if we let the long exact sequences on the group side and dual side pair with each other

(2.17)

then we have

	$\displaystyle\langle i(u),\hat{z}\rangle$	$\displaystyle=\langle u,\hat{p}(\hat{z})\rangle$
and
	$\displaystyle\langle z,\hat{i}(\hat{t})\rangle$	$\displaystyle=\langle p(z),\hat{t}\rangle,$

for each $u\in H^{0}(F,U)$ , $\hat{z}\in H^{1}(W_{F},\hat{U}\to\hat{T})$ , $z\in H^{1}(F,T\to U)$ and $\hat{t}\in H^{0}(W_{F},\hat{T})$ .

Proof.

Both compatibilities follow from the commutativity of Diagram (2.9) and the definition of the pairing (2.11). Compatibility with Tate-Nakayama pairing is immediate. As for compatibility with the LLC in Langlands’ convention, we first notice that $\mathcal{H}$ is compatible with $\mathcal{D}$ , which differs from $\mathcal{L}$ by a sign. Then we compare the pairing (2.11) with the pairing (LABEL:co-hopairingllc), and find that they differ by a sign as well. Therefore, the two minus signs cancel and the desired compatibility follows. ∎

2.2.3 Step 3

In this final step, we take continuity into account. But first we need to endow $H^{1}(F,T\xrightarrow{f}U)$ with a topology. Recall that we have the long exact sequence

\displaystyle\cdots\to T(F)\to U(F)\xrightarrow{i}H^{1}(F,T\xrightarrow{f}U)\to H^{1}(F,T)\to H^{1}(F,U)\to\cdots.

Then we topogise $H^{1}(F,T\xrightarrow{f}U)$ by stipulating that

\displaystyle U(F)\xrightarrow{i}H^{1}(F,T\xrightarrow{f}U)

is a continuous open map. We quickly note that $i$ induces an isomorphism of topological groups between the quotient group $U(F)/f(T(F))$ and the image of $i$ . Since $H^{1}(F,T)$ is finite, $i(U(F))$ is an open subgroup of finite index.

In particular, a character of $H^{1}(F,T\xrightarrow{f}U)$ is continuous if and only if it is continuous on the image $i(U(F))$ . According to Fact 2.7, the pairing (2.16) is continuous on $i(U(F))$ indeed, giving us a map

\displaystyle H^{1}(W_{F},\hat{U}\to\hat{T})\to\mathrm{Hom}_{\mathrm{cts}}(H^{1}(F,T\xrightarrow{f}U),\mathbb{C}^{\times}).

In virtue of Proposition 2.7, the following diagram commutes:

(2.18)

In the above diagram, the first row consists of (hyper)cohomology groups of $\mathrm{Gal}_{F}$ applied by $\mathrm{Hom}_{\mathrm{cts}}(-,\mathbb{C}^{\times})$ . We denote $\mathrm{Hom}_{\mathrm{cts}}(-,\mathbb{C}^{\times})$ by ${}^{\prime}$ for brevity. The second row consists of continuous (hyper)cohomology groups of $W_{F}$ . The two vertical maps to the right are the LLC map for tori, and the two vertical maps to the left are induced by the Tate-Nakayama pairing (LABEL:TN-pairing). The isomorphism (LABEL:TN-Kottwitz) implies that the two vertical maps to the left are surjective, with kernels $(\hat{U}^{\Gamma})^{\circ}$ and $(\hat{T}^{\Gamma})^{\circ}$ , respectively.

We define the quotient

\displaystyle H^{1}(W_{F},\hat{U}\to\hat{T})_{\mathrm{red}}:=\frac{H^{1}(W_{F},\hat{U}\to\hat{T})}{\hat{i}((\hat{T}^{\Gamma})^{\circ})},

and modify the above commutative diagram to

(2.19)

By the five lemma, the vertical map in the middle must be an isomorphism, because all the other four are so. We have thus shown

Proposition 2.8.

The pairing (2.16) induces a functorial isomorphism

\displaystyle H^{1}(W_{F},\hat{U}\to\hat{T})_{\mathrm{red}}\cong\mathrm{Hom}_{\mathrm{cts}}(H^{1}(F,T\xrightarrow{f}U),\mathbb{C}^{\times}).

2.3 A cohomological lemma

For later convenience, we record an interesting result. This will play an important role in our reinterpretation of the local Langlands correspondence for disconnected tori (Theorem LABEL:reinterthm) and its proof (especially the proof of Lemma LABEL:lem:homo).

Theorem 2.9.

Let $T$ , $U$ and $V$ be $F$ -tori, and consider $F$ -morphisms $f$ and $g$ :

Let $\hat{f}$ and $\hat{g}$ be the morphisms between the dual tori induced by $f$ :

We consider the natural map induced by $g$

sending the class of $(z,t)$ to that of $(z,g(t))$ , and the natural map induced by $\hat{f}$

sending the class of $(\phi,s)$ to that of $(\phi,\hat{f}(s))$ . Then the Tate-Nakayama pairing between the image of $g_{*}$ and the image of $\hat{f}_{*}$ vanishes:

\displaystyle\langle g_{*}(z,t),\hat{f}_{*}(\phi,s)\rangle=1.

Proof.

We go back to the definition of Tate-Nakayama pairing. First we fix a finite Galois extension $K/F$ such that $T$ , $U$ and $V$ split over $K$ . Under the isomorphism, we let $(\lambda,\mu)\in Z_{0}(W_{K/F},X\xrightarrow{f_{*}}Y)_{0}$ correspond to $(z,t)$ . Then $(\lambda,f_{*}(\mu_{w}))$ corresponds to $(z,f(t))$ . Now the Tate-Nakayama pairing reads as

\displaystyle\langle g_{*}(z,t),\hat{f}_{*}(\phi,s)\rangle=\langle\lambda,\hat{f}(s)\rangle\prod_{w}\langle g_{*}(\mu(w)),\phi(w)\rangle^{-1}.

(2.20)

Since $(\lambda,\mu_{w})\in Z_{0}(W_{K/F},X\xrightarrow{f_{*}}Y)_{0}$ and $(\phi,s)\in Z^{1}(W_{K/F},\hat{V}\xrightarrow{\hat{g}}\hat{U})$ , we have

\displaystyle f_{*}(\lambda)=\sum_{w}(w^{-1}\mu(w)-\mu(w)),

(2.21)

and

\displaystyle\hat{g}(\phi(w))=s^{-1}w(s),

(2.22)

for each $w\in W_{K/F}$ . Now we can plug (2.21) and (2.22) into (2.20):

	$\displaystyle\langle g_{}(z,t),\hat{f}_{}(\phi,s)\rangle$	$\displaystyle=\langle f_{*}(\lambda),s\rangle\prod_{w}\langle\mu(w),\hat{g}(\phi(w))\rangle^{-1}$
		$\displaystyle=\langle\sum_{w}(w^{-1}\mu(w)-\mu(w)),s\rangle\prod_{w}\langle\mu(w),s^{-1}w(s)\rangle^{-1}$
		$\displaystyle=\prod_{w}\langle w^{-1}\mu(w),s\rangle\cdot\prod_{w}\langle\mu(w),s^{-1}\rangle\cdot\prod_{w}\langle\mu(w),sw(s)^{-1}\rangle$
		$\displaystyle=\prod_{w}\langle\mu(w),w(s)\rangle\cdot\prod_{w}\langle\mu(w),s^{-1}\rangle\cdot\prod_{w}\langle\mu(w),sw(s)^{-1}\rangle$
		$\displaystyle=1.$

∎

2.4 Global Tate-Nakayama Duality for hypercohomology

In this section, we turn to the global Tate-Nakayama duality for hypercohomology. As its local analogue, this combines the global Langlands correspondence and the global Tate-Nakayama duality. Since the constructions are carried out in the way as the local case, details are omitted.

Let $F$ be a number field, and $C_{F}=F^{\times}\backslash\mathbb{A}_{F}^{\times}$ be its idele class group. Idele class groups will play the same role as multiplicative groups have played in the local setting. Let $W_{F}$ be the global Weil group of $F$ . Let $T$ and $U$ be tori defined over $F$ with cocharacter groups $X$ and $Y$ , respectively, and $f:T\to U$ be an $F$ -morphism. We fix a finite Galois extension $K$ of $F$ over which both $T$ and $U$ split.

First, we introduce some Galois hypercohomology groups:

	$\displaystyle H^{i}(F,T\xrightarrow{f}U)$	$\displaystyle:=H^{i}(F,T(\bar{F})\xrightarrow{f}U(\bar{F})),$
	$\displaystyle H^{i}(\mathbb{A},T\xrightarrow{f}U)$	$\displaystyle:=H^{i}(F,T(\bar{\mathbb{A}})\xrightarrow{f}U(\bar{\mathbb{A}})),$
	$\displaystyle H^{i}(\mathbb{A}/F,T\xrightarrow{f}U)$	$\displaystyle:=H^{i}(F,T(\bar{\mathbb{A}})/T(\bar{F})\xrightarrow{f}U(\bar{\mathbb{A}})/U(\bar{F})).$

It is the last group that will appear in the duality and it can be topologized in the same manner as Step 3 of Section 2.2 (see also Kottwitz-Shelstad [kottwitz1999foundations] for details). We note an elementary long exact sequence involving all the three groups above:

\displaystyle\cdots\to H^{i}(F,T\xrightarrow{f}U)\to H^{i}(\mathbb{A},T\xrightarrow{f}U)\to H^{i}(\mathbb{A}/F,T\xrightarrow{f}U)\to\cdots.

(2.23)

On the dual side, we can define the continuous hypercohomology $H^{1}(W_{F},\hat{U}\xrightarrow{\hat{f}}\hat{T})_{\mathrm{red}}$ and its reduced version $H^{1}(W_{F},\hat{U}\xrightarrow{\hat{f}}\hat{T})_{\mathrm{red}}$ as in the local setting. Now we state the global Tate-Nakayama duality for hypercohomology:

Theorem 2.10.

There is a natural functorial isomorphism

\displaystyle H^{1}(W_{F},\hat{U}\xrightarrow{\hat{f}}\hat{T})_{\mathrm{red}}\cong\mathrm{Hom}_{\mathrm{cts}}(H^{1}(\mathbb{A}/F,T\xrightarrow{f}U),\mathbb{C}^{\times})

that is compatible with the global Langlands correspondence (or more precisely, the isomorphism (LABEL:glcisom)) and the global Tate-Nakayama pairing (LABEL:TN-pairing-global). Moreover, this isomorphism is compatible with the local Tate-Nakayama duality for hypercohomogy, in the sense that the following diagram commutes:

The construction is the same as in the local case, and so is the compatibility with the GLC and the Tate-Nakayama pairing. Meanwhile, the local-global compatibility can be readily checked from the construction.

3 Disconnected reductive groups

In this chapter, we closely follow [kaletha2022local] and introduce a certain class of disconnected groups, for which we reserve the term “disconnected reductive groups” in the rest of this work. Due to our emphasis on disconnected tori, we will present some rank- $1$ examples.

3.1 Convention

Let $F$ be a field of characteristic $0$ with absolute Galois group $\Gamma=\mathrm{Gal}(\bar{F}/F)$ . In this work, we say an affine algebraic group $\tilde{G}$ is a disconnected reductive group, if $\tilde{G}$ satisfies the following conditions:

•

There is an $\bar{F}$ -isomorphism

$\displaystyle\tilde{G}\to G\rtimes A,$

where $G$ is a connected reductive group and $A$ is a (nontrivial) finite group.
•

The action of $A$ on $G$ preserves some fixed $\bar{F}$ -pinning of $G$ .

Non-example 3.1.

The normaliser of the diagonal torus in $\mathrm{SL}_{2}$ does not split as a semidirect product (even over $\bar{F}$ ). Indeed, there does not exist any element of order two in the nonidentity component of $\bar{F}$ -points.

3.2 Classification

In this part, we will recall from [kaletha2022local] the classification of disconnected reductive groups. It is a well-known fact that each connected reductive group has a unique split form, and moreover has a unique quasi-split inner form. The same notions can be extended to the disconnected setting, although there turns out to be an additional type, “translation form”, in the term of [kaletha2022local]. We start from extending the notions of “quasi-split” and “split” groups:

Definition 3.2.

We call $\tilde{G}$ a split disconnected reductive group, if there is an $F$ -isomorphism

\displaystyle\tilde{G}\to G\rtimes A,

where $G$ is a split connected reductive group, and $A$ is a (nontrivial) constant group scheme acting on $G$ by preserving some $F$ -pinning of it.

Definition 3.3.

We call $\tilde{G}$ a quasi-split disconnected reductive group, if there is an $F$ -isomorphism

\displaystyle\tilde{G}\to G\rtimes A,

where $G$ is a quasi-split connected reductive group, and $A$ is a (not necessarily constant) finite group scheme acting on $G$ (the action is defined over $F$ ) and preserving some $F$ -pinning of it.

We note that split disconnected reductive groups can be classified by the root datum of $G$ and the action of $A$ on the root datum. It can be immediately seen from the definitions that every disconnected reductive group is a unique form of a split disconnected reductive group. Now, to obtain a classification of all disconnected reductive groups, it suffices to classify all the forms of a given split disconnected reductive group $G\rtimes A$ . Equivalently, it suffices to understand the Galois cohomology ${H}^{1}(\Gamma,\mathrm{Aut}_{\bar{F}}(G\rtimes A))$ . This leads us to a close examination on the automorphism group $\mathrm{Aut}_{\bar{F}}(G\rtimes A)$ first. From now on, we omit the subscript $\bar{F}$ for brevity.

We highlight three subgroups of $\mathrm{Aut}(G\rtimes A)$ :

•

$G/{Z(G)}^{A}$ , the group of inner automorphisms.
•

$Z^{1}(A,Z(G))$ , the group of translation automorphisms. Each $1$ -cocycle $z\in Z^{1}(A,Z(G))$ induces an automorphism $(g,a)\mapsto(z(a)g,a)$ .
•

$\mathrm{Aut}_{\mathrm{pin}}(G\rtimes A)$ , the group of pinned automorphisms over $\bar{F}$ , as which we refer to automorphisms preserving the pinning of $G$ and the subgroup $1\rtimes A$ .

We note that $G/{Z(G)}^{A}$ intersects nontrivially with $Z^{1}(A,Z(G))$ . The intersection is $Z(G)/{Z(G)}^{A}$ (as a subgroup of $G/{Z(G)}^{A}$ ) or $B^{1}(A,Z(G))$ (as a subgroup of $Z^{1}(A,Z(G))$ ). Now, one can characterise the structure of $\mathrm{Aut}(G\rtimes A)$ (see [kaletha2022local, Section 3.1]):

Fact 3.4.

There is a semidirect product decomposition:

\displaystyle\mathrm{Aut}(G\rtimes A)=(G/{Z(G)}^{A}\cdot Z^{1}(A,Z(G)))\rtimes\mathrm{Aut}_{\mathrm{pin}}(G\rtimes A)).

In view of this result, we resume discussing the classification of disconnected reductive groups. Let $G\rtimes A$ be a split disconnected reductive group. One observes that twisting the rational structure of $G\rtimes A$ by a cocycle in $Z^{1}(\Gamma,\mathrm{Aut}_{\mathrm{pin}}(G\rtimes A))$ yields a quasi-split disconnected reductive group $\tilde{G}$ . It is also clear that each quasi-split disconnected reductive group arises in this way.

Then, one can twist the rational structure on the quasi-split group $\tilde{G}$ by a $1$ -cocycle $z\in Z^{1}(\Gamma,G/Z(G)^{A})$ to obtain $\tilde{G}_{z}$ , an inner form of the quasi-split group. Furthermore, twisting $\tilde{G}_{z}$ by some $z^{\prime}\in Z^{1}(\Gamma,Z^{1}(A,Z(G)))$ yields a translation form $(\tilde{G}_{z})_{z^{\prime}}$ of the inner form $\tilde{G}_{z}$ . Now Fact 3.4 implies that each disconnected reductive group can be obtained in this manner. To put it in a concise way, each disconnected reductive group is a translation form of an inner form of some quasi-split disconnected reductive group.

3.3 Examples of some simple disconnected tori

In this work, we focus on are disconnected tori. As the name suggests, a disconneted torus is a group that has a torus as its identity component and satisfies the Convention 3.1.

In order to provide the reader with a glimpse into disconnected tori and their rational points, we present instances of the most elementary (nontrivial) case, when the identity component is $\mathbb{G}_{m}$ and the component group is $\mathbb{Z}/2\mathbb{Z}$ . In this case, there are two split forms: the direct product $\mathbb{G}_{m}\times\mathbb{Z}/2\mathbb{Z}$ , and the semi-direct product $\mathbb{G}_{m}\rtimes\mathbb{Z}/2\mathbb{Z}$ with $\mathbb{Z}/2\mathbb{Z}$ acting by inverting. We classify their forms and also exhibit the $F$ -rational points of all the forms.

3.3.1 Forms of $\mathbb{G}_{m}\times\mathbb{Z}/2\mathbb{Z}$

Let $\tilde{T}=\mathbb{G}_{m}\times\mathbb{Z}/2\mathbb{Z}$ . We write $\mathbb{Z}/2\mathbb{Z}=\{1,-1\}$ . The automorphism group of $\mathbb{G}_{m}\times\mathbb{Z}/2\mathbb{Z}$ can be worked out as

\displaystyle\mathrm{Aut}(\tilde{T})=Z^{1}(\mathbb{Z}/2\mathbb{Z},\bar{F}^{\times})\rtimes\mathrm{Aut}_{\mathrm{pin}}(\mathbb{G}_{m}\times\mathbb{Z}/2\mathbb{Z})\cong\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}.

We write the nontrivial element in the first (resp. second) $\mathbb{Z}/2\mathbb{Z}$ as $\mu$ (resp. $\omega$ ). As an automorphism of $\tilde{T}$ , $\mu$ fixes the identity component pointwise and sends any $(x,-1)$ from the non-identity component to $(-x,-1)$ . And $\omega$ is the automorphism sending $(x,\pm 1)$ to $(x^{-1},\pm 1)$ . And clearly, the Galois action on $\mathrm{Aut}(\tilde{T})$ is trivial. Thus, the forms of $\mathbb{G}_{m}\times\mathbb{Z}/2\mathbb{Z}$ are classified by the Galois cohomology

\displaystyle H^{1}(\Gamma,\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z})=Z^{1}(\Gamma,\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z})=\mathrm{Hom}_{\mathrm{cts}}(\Gamma,\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}).

Any nontrivial cocycle $z\in Z^{1}(\Gamma,\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z})$ falls into one of the four categories to be described below, and we denote by $\tilde{T}_{z}$ the group obtained from twisting $\tilde{T}$ by $z$ .

•

Case 1: $z$ factors through a quadratic extension $E/F$ and maps onto the first $\mathbb{Z}/2\mathbb{Z}$ -factor: $\Gamma\twoheadrightarrow\mathrm{Gal}(E/F)\xrightarrow{\sim}\{1,\mu\}$ . Let $\mathrm{Gal}(E/F)=\{1,\sigma\}$ . Then after passing to $E$ , the Galois action twisted by $z$ (which we denote by adding a subscript $z$ ) is given by $\sigma_{z}(x,1)=(\sigma(x),1)$ and $\sigma_{z}(x,-1)=(-\sigma(x),-1)$ . Then the group of rational points is

$\displaystyle\tilde{T}_{z}(F)=\{(x,1)|x\in F^{\times}\}\cup\{(x,-1)|x\in E^{\times},\sigma(x)=-x\}.$
•

Case 2: $z$ factors through a quadratic extension $E/F$ and maps onto the second $\mathbb{Z}/2\mathbb{Z}$ -factor: $\Gamma\twoheadrightarrow\mathrm{Gal}(E/F)\xrightarrow{\sim}\{1,\omega\}$ . Let $\mathrm{Gal}(E/F)=\{1,\sigma\}$ . Then after passing to $E$ , the Galois action twisted by $z$ is given by $\sigma_{z}(x,\pm 1)=(\sigma(x)^{-1},\pm 1)$ . Thus we have

$\displaystyle\tilde{T}_{z}(F)=\{(x,\pm 1)|x\in E^{\times},\mathrm{Nm}_{E/F}(x)=1\}.$

•

Case 3: $z$ factors through a quadratic extension $E/F$ and maps into $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ diagonally: $\Gamma\twoheadrightarrow\mathrm{Gal}(E/F)\xrightarrow{\sim}\{1,\mu\omega\}$ . Let $\mathrm{Gal}(E/F)=\{1,\sigma\}$ . Then after passing to $E$ , the Galois action twisted by $z$ is given by $\sigma_{z}(x,1)=(\sigma(x)^{-1},1)$ and $\sigma_{z}(x,-1)=(-\sigma(x)^{-1},-1)$ . So we have

\displaystyle\tilde{T}_{z}(F)=\{(x,1)|x\in E^{\times},\mathrm{Nm}_{E/F}(x)=1\}\cup\{(x,-1)|x\in E^{\times},x+\sigma(x)^{-1}=0\}.

•

Case 4: $z$ factors through a biquadratic extension $K/F$ and maps isomorphically to $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ : $\Gamma\twoheadrightarrow\mathrm{Gal}(K/F)\xrightarrow{\sim}\langle\mu\rangle\times\langle\omega\rangle$ . We write $\mathrm{Gal}(K/F)=\langle\sigma\rangle\times\langle\tau\rangle$ and assume $\sigma\mapsto\mu$ and $\tau\mapsto\omega$ . After passing to $K$ , the twisted Galois action thus obtained is $\sigma_{z}(x,1)=(\sigma(x),1)$ , $\sigma_{z}(x,-1)=(-\sigma(x),-1)$ , and $\tau_{z}(x,\pm 1)=(\tau(x)^{-1},\pm 1)$ . Then we have

	$\displaystyle\tilde{T}_{z}(K^{\sigma})$	$\displaystyle=\{(x,1)\|x\in{K^{\sigma}}^{\times}\}\cup\{(x,-1)\|x\in K^{\times},\sigma(x)=-x\},$
	$\displaystyle\tilde{T}_{z}(K^{\tau})$	$\displaystyle=\{(x,\pm 1)\|x\in K^{\times},\mathrm{Nm}_{K/K^{\tau}}(x)=1\},$
	$\displaystyle\tilde{T}_{z}(F)$	$\displaystyle=\{(x,1)\|x\in{K^{\sigma}}^{\times},\mathrm{Nm}_{K^{\sigma}/F}(x)=1\}$
		$\displaystyle\cup\{(x,-1)\|x\in K^{\times},\sigma(x)=-x,\mathrm{Nm}_{K/K^{\tau}}(x)=1\}.$

3.3.2 Forms of $\mathbb{G}_{m}\rtimes\mathbb{Z}/2\mathbb{Z}$

Let $\tilde{T}=\mathbb{G}_{m}\rtimes\mathbb{Z}/2\mathbb{Z}$ . The semidirect product is given by the inverting action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{G}_{m}$ . An elementary calculation suggests that the inner automorphisms and translation automorphisms coincide over $\bar{F}$ . For convenience, we consider them as translation automorphisms, and one can check

\displaystyle\mathrm{Aut}(\tilde{T})=Z^{1}(\mathbb{Z}/2\mathbb{Z},\bar{F}^{\times})\rtimes\mathrm{Aut}_{\mathrm{pin}}(\mathbb{G}_{m}\times\mathbb{Z}/2\mathbb{Z})\cong\bar{F}^{\times}\rtimes\mathbb{Z}/2\mathbb{Z},

in which $\mathbb{Z}/2\mathbb{Z}$ acts on $\bar{F}^{\times}$ by inverting. For $y\in\bar{F}^{\times}$ , we denote the automorphism that fixes $\mathbb{G}_{m}$ and sends $(x,-1)\mapsto(xy,-1)$ by $\mu_{y}$ . And we denote the nontrivial element in $\mathbb{Z}/2\mathbb{Z}$ by $\omega$ . Again, $\omega$ is the automorphism sending $(x,\pm 1)\mapsto(x^{-1},\pm 1)$ .

One can further check that the Galois group $\Gamma$ acts on $\mathrm{Aut}(\tilde{T})=\bar{F}^{\times}\rtimes\mathbb{Z}/2\mathbb{Z}$ by $\sigma(y,\pm 1)=({\sigma(y)},\pm 1)$ . The forms of $\mathbb{G}_{m}\rtimes\mathbb{Z}/2\mathbb{Z}$ are classified by the Galois cohomology

\displaystyle H^{1}(\Gamma,\mathrm{Aut}(\tilde{T}))=H^{1}(\Gamma,\bar{F}^{\times}\rtimes\mathbb{Z}/2\mathbb{Z}).

To compute the Galois cohomology $H^{1}(\Gamma,\bar{F}^{\times}\rtimes\mathbb{Z}/2\mathbb{Z})$ , we consider the following long exact sequence

\displaystyle\cdots\to\mathbb{Z}/2\mathbb{Z}\to H^{1}(\Gamma,\bar{F}^{\times})\to H^{1}(\Gamma,\bar{F}^{\times}\rtimes\mathbb{Z}/2\mathbb{Z})\to H^{1}(\Gamma,\mathbb{Z}/2\mathbb{Z})\to 0,

where the surjectivity of the last map follows from the existence of a splitting due to the semidirect product. In view of Hilbert 90, we have $H^{1}(\Gamma,\bar{F}^{\times})=0$ , and hence we conclude that the fibre over the trivial element in $H^{1}(F,\mathbb{Z}/2\mathbb{Z})$ is a singleton, which is nothing but the split form. Now, it suffices to understand the fibre $\mathcal{F}_{E}$ of any nontrivial element $[E]\in H^{1}(F,\mathbb{Z}/2\mathbb{Z})$ (which corresponds to a quadratic extension $E/F$ ). To this end, we consider the cocycle $z^{[E]}:\Gamma\twoheadrightarrow\mathrm{Gal}(E/F)=\{1,\sigma\}\hookrightarrow\bar{F}^{\times}\rtimes\mathbb{Z}/2\mathbb{Z}$ sending $\sigma$ to $\omega$ . Clearly, we have $z^{[E]}\in\mathcal{F}_{E}$ .

It remains to investigate whether the fibre $\mathcal{F}_{E}$ contains any other element than $[z^{[E]}]$ . From now on, we fix the quadratic extension $E/F$ . If we twist the Galois action on the short exact sequence by $z^{[E]}$ (abbreviated as $z$ below) and take the long exact sequence, then we obtain

\displaystyle\cdots\to(\bar{F}^{\times}\rtimes\mathbb{Z}/2\mathbb{Z})_{z}^{\Gamma}\to\mathbb{Z}/2\mathbb{Z}\to H^{1}(\Gamma,\bar{F}^{\times}_{z})\to H^{1}(\Gamma,(\bar{F}^{\times}\rtimes\mathbb{Z}/2\mathbb{Z})_{z})\to H^{1}(\Gamma,\mathbb{Z}/2\mathbb{Z}).

We note that there are identifications

	$\displaystyle\mathcal{F}_{E}$	$\displaystyle\cong\mathrm{ker}\left(H^{1}(\Gamma,(\bar{F}^{\times}\rtimes\mathbb{Z}/2\mathbb{Z})_{z})\to H^{1}(\Gamma,\mathbb{Z}/2\mathbb{Z})\right)$
		$\displaystyle\cong H^{1}(\Gamma,\bar{F}_{z}^{\times})/(\mathbb{Z}/2\mathbb{Z}).$

Remark 3.5.

More generally speaking, given a short exact sequence of (not necessarily abelian) $G$ -modules $0\to A\to B\to C\to 0$ , the kernel of the map $H^{1}(G,B)\to H^{1}(G,C)$ (in the associated long exact sequence) is in 1-1 correspondence to the orbit space $H^{1}(G,A)/C^{G}$ , where the action of $C^{G}$ on $H^{1}(G,A)$ is given by: $c\cdot z(\sigma):=bz(\sigma)\sigma(b)^{-1}$ ( $b$ is any lift of $c$ ). When the $G$ -modules are abelian, this recovers the usual quotient. See [serre1979galois, Chapter I] for more details on nonabelian cohomology.

Claim 3.6.

The action of $\mathbb{Z}/2\mathbb{Z}$ on $H^{1}(\Gamma,\bar{F}_{z}^{\times})$ is trivial.

Proof.

Due to the exactness at $\mathbb{Z}/2\mathbb{Z}$ , it suffices to show the surjectivity of the projection $(\bar{F}^{\times}\rtimes\mathbb{Z}/2\mathbb{Z})_{z}^{\Gamma}\to\mathbb{Z}/2\mathbb{Z}$ . It is clear that $(1,-1)$ lies in $(\bar{F}^{\times}\rtimes\mathbb{Z}/2\mathbb{Z})_{z}^{\Gamma}$ according to the definition of $z$ . So the surjectivity follows. ∎

Therefore, $\mathcal{F}_{E}$ is in bijection with $H^{1}(\Gamma,\bar{F}_{z}^{\times})$ . It is clear from the construction that $\bar{F}_{z}^{\times}$ (with the Galois action twisted by $z$ ) coincides with $\bar{F}$ -points of the norm torus $\mathrm{Res}_{E/F}^{1}\mathbb{G}_{m}$ determined by the quadratic extension $E/F$ . We have $H^{1}(\Gamma,\bar{F}_{z}^{\times})\cong H^{1}(E/F,E^{\times}_{z})\cong F^{\times}/\mathrm{Nm}E^{\times}$ . Explicitly, the isomorphism is given by sending $z\in Z^{1}(E/F,E_{z}^{\times})$ to $z(\sigma)$ , where $\sigma\in\mathrm{Gal}(E/F)$ is the nontrivial element.

Based on discussion above, each coset $y\in F^{\times}/\mathrm{Nm}E^{\times}$ represents a class in $H^{1}(\Gamma,\bar{F}_{z}^{\times})$ , and we can define $[z^{\prime}]\in H^{1}(\Gamma,(\bar{F}^{\times}\rtimes\mathbb{Z}/2\mathbb{Z})_{z})$ by setting $z^{\prime}(\sigma)=(y,1)$ after passing to the quotient $\mathrm{Gal}(E/F)$ . After composing $z^{\prime}$ with $z$ , we are able to obtain a $1$ -cocycle in the original (untwisted) sense. Let $z^{\prime\prime}\in Z^{1}(\Gamma,\bar{F}^{\times}\rtimes\mathbb{Z}/2\mathbb{Z})$ defined by (after passing to $E$ ) $z^{\prime\prime}(\sigma)=z^{\prime}(\sigma)z(\sigma)=(y,\omega)$ . We conclude that each form in the fibre $\mathcal{F}_{E}$ arises as a $\tilde{T}_{z^{\prime\prime}}$ thus obtained. The $F$ -rational points can be easily computed:

•

After passing to $E$ , the Galois action twisted by $z^{\prime\prime}$ is given by $\sigma_{z^{\prime\prime}}(x,1)=(\sigma(x)^{-1},1)$ and $\sigma_{z^{\prime\prime}}(x,-1)=(y\sigma(x)^{-1},-1)$ . The $F$ -points are given by

\displaystyle\tilde{T}_{z^{\prime\prime}}(F)=\{(x,1)|x\in E^{\times},\mathrm{Nm}_{E/F}(x)=1\}\cup\{(x,-1)|x\in E^{\times},\mathrm{Nm}_{E/F}(x)=y\}.

We note that, whenever $y\notin\mathrm{Nm}_{E/F}E^{\times}$ , there are no rational points on the non-identity component.

3.4 Rational points on inner forms of quasi-split disconnected groups

In this work, we focus on inner forms of quasi-split groups and do not treat translation forms (that do not fall into the former category). Let $\tilde{G}=G\rtimes A$ be a quasi-split disconnected reductive group. Although the semidirect product is defined over $F$ and $A$ is not necessarily constant as a finite group scheme, we will still abbreviate $A(\bar{F})$ as $A$ .

Let $\bar{z}\in Z^{1}(\Gamma,G/Z(G)^{A})$ . We obtain $\tilde{G}_{\bar{z}}$ by twisting the rational structure of $\tilde{G}$ via $\bar{z}$ . After this twisting process, there is still a short exact sequence of $\Gamma$ -modules:

\displaystyle 1\to G_{\bar{z}}(\bar{F})\to\tilde{G}_{\bar{z}}(\bar{F})\to A\to 1,

where the twisted Galois action on $\tilde{G}_{\bar{z}}(\bar{F})$ is given by

\displaystyle\sigma_{\bar{z}}(g,a)=\bar{z}(\sigma)(\sigma(g),\sigma(a))\bar{z}(\sigma)^{-1}=\left(\bar{z}(\sigma)\sigma(g)\sigma(a)[\bar{z}(\sigma)^{-1}],\sigma(a)\right),

for $\sigma\in\Gamma$ , while the Galois action on $A$ is unchanged.

After taking $\Gamma$ -fixed points, we have

\displaystyle 1\to G_{\bar{z}}(F)\to\tilde{G}_{\bar{z}}(F)\to A(F).

The last projection is not always surjective. In fact, one can see that $(g,a)\in\tilde{G}_{\bar{z}}(F)$ if and only if $a\in A(F)$ and

\displaystyle\bar{z}(\sigma)\sigma(g)a(\bar{z}(\sigma)^{-1})=g

(3.1)

for any $\sigma\in\Gamma$ . We define $A(F)^{[\bar{z}]}$ as the subgroup of $A(F)$ , comprising elements $a$ for which there exists some $g\in G(\bar{F})$ such that (3.1) is satisfied for any $\sigma\in\Gamma$ . Then there is a short exact sequence

\displaystyle 1\to G_{\bar{z}}(F)\to\tilde{G}_{\bar{z}}(F)\to A(F)^{[\bar{z}]}\to 1.

4 The LLC for disconnected tori

Let $F$ be a local field of characteristic zero with absolute Galois group $\Gamma$ . The goal of this chapter is to state the LLC for disconnected tori in terms of pure inner forms.

4.1 Pure inner forms

We start with a quasi-split disconnected torus $\tilde{T}=T\rtimes A$ defined over $F$ . To be precise, $T$ is a (not necessarily split) torus over $F$ , $A$ is a (not necessarily constant) finite group scheme defined over $F$ acting on $T$ , and the action of $A$ on $T$ is also defined over $F$ .

Let $z\in Z^{1}(\Gamma,T)$ . Under the natural map $Z^{1}(\Gamma,T)\to Z^{1}(\Gamma,T/T^{A})$ , $z$ is brought to a $1$ -cocycle $\bar{z}$ taking values in the group of inner automorphisms $T/T^{A}$ . We twist the rational structure of $\tilde{T}$ by $z$ (or more precisely, by $\bar{z}$ ) and obtain an inner form $\tilde{T}_{z}$ , which we call a pure inner form.

According to the discussion made in Section 3.4, any element $(t,a)\in\tilde{T}_{z}(F)$ satisfies

\displaystyle z(\sigma)\sigma(t)a(z(\sigma)^{-1})=t.

(4.1)

If we rewrite (4.1) as $t^{-1}z(\sigma)\sigma(t)=a(z(\sigma))$ , then we see that, for any given $a\in A(F)$ , there exists some $(t,a)\in\tilde{T}_{z}(F)$ if and only if $a\cdot z$ is cohomologous to $z$ . We denote by $A(F)^{[z]}$ the stabliser of the cohomology class $[z]$ in $A ($

	$\displaystyle\tilde{T}_{z}(K^{\sigma})$	$\displaystyle=\{(x,1)\|x\in{K^{\sigma}}^{\times}\}\cup\{(x,-1)\|x\in K^{\times},\sigma(x)=-x\},$
	$\displaystyle\tilde{T}_{z}(K^{\tau})$	$\displaystyle=\{(x,\pm 1)\|x\in K^{\times},\mathrm{Nm}_{K/K^{\tau}}(x)=1\},$
	$\displaystyle\tilde{T}_{z}(F)$	$\displaystyle=\{(x,1)\|x\in{K^{\sigma}}^{\times},\mathrm{Nm}_{K^{\sigma}/F}(x)=1\}$
		$\displaystyle\cup\{(x,-1)\|x\in K^{\times},\sigma(x)=-x,\mathrm{Nm}_{K/K^{\tau}}(x)=1\}.$

On the multiplicity formula for discrete automorphic representations of disconnected tori

Abstract

1 Introduction

1.1 Main results

1.2 Structure of the paper

Acknowledgements.

2 The Tate-Nakayama Duality for hypercohomology

2.1 Convention

2.2 Local Tate-Nakayama duality for hypercohomology

2.2.1 Step 1

Lemma 2.1.

Proof.

Fact 2.2.

Proposition 2.3.

Proof.

Proposition 2.4.

Proof.

2.2.2 Step 2

Remark 2.5.

Proposition 2.6.

Proof (Sketch)..

Proposition 2.7.

Proof.

2.2.3 Step 3

Proposition 2.8.

2.3 A cohomological lemma

Theorem 2.9.

Proof.

2.4 Global Tate-Nakayama Duality for hypercohomology

Theorem 2.10.

3 Disconnected reductive groups

3.1 Convention

Non-example 3.1.

3.2 Classification

Definition 3.2.

Definition 3.3.

Fact 3.4.

3.3 Examples of some simple disconnected tori

3.3.1 Forms of 𝔾m×ℤ/2ℤ\mathbb{G}_{m}\times\mathbb{Z}/2\mathbb{Z}

3.3.2 Forms of 𝔾m⋊ℤ/2ℤ\mathbb{G}_{m}\rtimes\mathbb{Z}/2\mathbb{Z}

Remark 3.5.

Claim 3.6.

Proof.

3.4 Rational points on inner forms of quasi-split disconnected groups

4 The LLC for disconnected tori

4.1 Pure inner forms

3.3.1 Forms of $\mathbb{G}_{m}\times\mathbb{Z}/2\mathbb{Z}$

3.3.2 Forms of $\mathbb{G}_{m}\rtimes\mathbb{Z}/2\mathbb{Z}$