March 27, 2025
We determine the finite group \(\mathcal{S}\) parametrizing a packet in the local Langlands correspondence for a Brylinski-Deligne covering group of an algebraic torus, under some assumption on ramification. Especially, this work generalizes Weissman’s result on covering groups of tori that split over an unramified extension of the base field.
This paper aims to determine the finite group \(\mathcal{S}=\mathcal{S}_{\widetilde{T}}\) whose dual parametrizes a packet in the local Langlands correspondence for a covering group \(\widetilde{T}\to T\) of an algebraic torus. As well as reductive algebraic groups, their covering groups have been studied with arithmetic interest. For instance, Weil studied the Segal-Shale-Weil representations of the non-trivial double cover \(\operatorname{Mp}_{2r}\to\operatorname{Sp}_{2r}\), in order to prove an identity now called the Siegel-Weil formula [1]. Another notable example is an Eisenstein series on a double cover of \(\operatorname{GL}_r\), which gives a Rankin-Selberg integral for the symmetric square L-function of a cuspidal representation of \(\operatorname{GL}_r\) [2]. Such examples motivate us to pursue the Langlands program for covering groups.
In the familiar case of a reductive algebraic group \(G\) over a non-archimedean local field \(F\), we know the notions of the complex dual \(\widehat G\) of the group \(G\) and a Langlands parameter \(\phi\colon\mathrm W_F\times\operatorname{SL}_2(\mathbb{C})\to \vphantom{G}^{\mathrm L}G=\widehat G\rtimes\mathrm W_F\). It has been conjectured that the centralizer \(S_\phi=\operatorname{Cent}_{\widehat G}(\operatorname{Im}\phi)\) indexes the representations in the L-packet \(\Pi_\phi\), through its finite quotient \(\mathcal{S}_\phi=\left.S_\phi\middle/ S_\phi^0\operatorname Z(\widehat G)^{\mathrm W_F}\right.\). Here, \(S_\phi^0\) is the connected component of the identity in the complex reductive group \(S_\phi\), and \(\operatorname Z(\widehat G)^{\mathrm W_F}\) is the subgroup consisting of fixed points in the center \(\operatorname Z(\widehat G)\subset\widehat G\) under the action of the Weil group \(\mathrm W_F\) of the base field \(F\). In this paper, we study an analogue of the group \(\mathcal{S}_\phi\) for covering groups.
Though a local Langlands correspondence for a general covering group has not been satisfactorily formulated, one for a covering \(\widetilde{T}\to T\) of a torus is proposed by Weissman [3]. According to his work, a certain finite abelian group \(\mathcal{S}=\mathcal{S}_{\widetilde{T}}\) parametrizes the representations in a packet (see 3), as \(\mathcal{S}_\phi\) does in the non-cover case. He also explicitly determined the group \(\mathcal{S}_{\widetilde{T}}\) for a cover of an unramified torus \(T\), i.e. a torus splitting over an unramified extension of the base field.
To describe the group \(\mathcal{S}_{\widetilde{T}}\), Weissman introduced certain subgroups \(Y^\#\) and \(Y^{\Gamma\#}\) of the cocharacter lattice \(Y\) of \(T\). In his formulation of the local Langlands correspondence, the cover \(\mu_n\to\widetilde{T}\to T\) is assumed to be a certain functorial one defined by Brylinski and Deligne [4], which induces the groups \(Y^\#\) and \(Y^{\Gamma\#}\) as follows (see 2). First, such a cover \(\widetilde{T}\) is known to be associated with a bilinear form \(B_{\widetilde{T}}\colon Y\times Y\to\mathbb{Z}\). Then for a subgroup \(Y'\subset Y\), the subgroup \(Y'^\#\) of \(Y\) is defined by \[Y'^\#=\{y\in Y\mid B_{\widetilde{T}}(y,Y')\subset n\mathbb{Z}\}.\] In the present case, this subgroup \(Y'\) is \(Y\) itself or the subgroup \(Y^\Gamma\) of fixed points under the action of the absolute Galois group \(\Gamma=\Gamma_F=\operatorname{Gal}(\overline{F}/F)\) of \(F\).
For a subgroup \(Y'\) of \(Y\), let \(\iota\) denote the inclusion map \(Y'\hookrightarrow Y\) and the induced ones, e.g. \(\iota\colon(Y'\otimes\overline{\mathbf{f}}^\times)^\Gamma\to (Y\otimes\overline{\mathbf{f}}^\times)^\Gamma\), where \(\overline{\mathbf{f}}^\times\) is the multiplicative group of an algebraic closure of the residue field \(\mathbf{f}\) of the valuation ring in \(F\). This notation is useful to state the following theorem by Weissman.
Theorem 1 (Weissman [3]). Let \(F\) be a non-archimedean local field of characteristic zero, \(\mathbf{f}\) the residue field of the valuation ring in \(F\), and \(\mu_n\subset \mathbf{f}^\times\) the cyclic subgroup of order \(n\). Suppose that an algebraic torus \(T\) defined over \(F\) splits over an unramified extension \(L/F\), and that \(\mu_n\to\widetilde{T}\to T\) is a Brylinski-Deligne covering group.
Then the group \(\mathcal{S}=\mathcal{S}_{\widetilde{T}}\) parametrizing a packet of representations of \(\widetilde{T}\) is written as the quotient \[\mathcal{S}_{\widetilde{T}} =\left.\iota((Y^{\Gamma\#}\otimes_{\mathbb{Z}} \overline{\mathbf{f}}^\times)^\Gamma)\middle/ \iota((Y^{\#}\otimes_{\mathbb{Z}} \overline{\mathbf{f}}^\times)^\Gamma)\right..\] of subgroups of \((Y\otimes_{\mathbb{Z}}\overline{\mathbf{f}}^\times)^\Gamma\).
In contrast, our main theorem
admits ramification. We just assume that the ramification index \(e\) of a fixed splitting field \(L/F\) is relatively prime to the degree \(n\) of the cover.
Precisely:
Main Theorem 1 (44). Let \(F\) be a non-archimedean local field of characteristic zero, \(\mathbf{f}\) the residue field of the valuation ring of \(F\), and \(\mu_n\subset \mathbf{f}^\times\) the cyclic subgroup of order \(n\). Suppose that an algebraic torus \(T\) defined over \(F\) splits over a finite Galois extension \(L/F\) whose ramification index \(e\) is relatively prime to \(n\). Let \(\mu_n\to\widetilde{T}\to T\) be a Brylinski-Deligne covering group.
Then the group \(\mathcal{S}=\mathcal{S}_{\widetilde{T}}\) parametrizing a packet of representations of \(\widetilde{T}\) is written as the quotient \[\mathcal{S}_{\widetilde{T}} =\left.\iota((Y^{\Gamma\#}\otimes_{\mathbb{Z}} \overline{\mathbf{f}}^\times)^\Gamma)\middle/ \iota((Y^{\#}\otimes_{\mathbb{Z}} \overline{\mathbf{f}}^\times)^\Gamma)\right..\] of subgroups of \((Y\otimes_{\mathbb{Z}}\overline{\mathbf{f}}^\times)^\Gamma\).
As already mentioned, the main theorem generalizes Weissman’s work on covering groups. Moreover, this theorem partially realizes his hope [3] to parametrize a packet in the correspondence for a cover of a ramified torus.
The next two sections are devoted to formulating our subjects, namely a Brylinski-Deligne covering group \(\widetilde{T}\to T\) of a torus and its local Langlands correspondence. Especially, we define the finite group \(\mathcal{S}_{\widetilde{T}}\) as a quotient of the image \(\mathrm Z^\dagger\subset T\) of the center \(\operatorname Z(\widetilde{T})\subset\widetilde{T}\) in 3. To prove the main theorem, we describe the subgroup \(\mathrm Z^\dagger\subset T\) by applying the Galois cohomology theory in 4. In 5, we overcome a critical part by reducing it to a certain orthogonality in the local Tate duality. We then prove the main theorem in the final section.
The author would like to thank his advisor Professor Tamotsu Ikeda for fruitful discussions, and thank Miyu Suzuki for providing me with valuable information. This work was supported by JSPS KAKENHI Grant Number JP23KJ1298.
Let \(\mathbb{Z}_{>0}\), \(\mathbb{Z}\), \(\mathbb{Q}\), and \(\mathbb{C}\) respectively denote the set of positive rational integers, the ring of rational integers, the field of rational numbers, and the field of complex numbers. The symbol \(\otimes\) always denotes the tensor product over \(\mathbb{Z}\). For a map \(f\colon A\to B\) and a subset \(A'\subset A\), let \(f|_{A'}\) denote the restriction, as usual. For a group \(G\), let \(\operatorname Z(G)\subset G\) denote the center.
If a group \(G\) acts on an abelian group \(A\), then \(A^G=\operatorname H^0(G,A)\) is the subgroup consisting of fixed points. For a ring \(R\), let \(R^\times\) denote the group of units.
We write \(F\) for a non-archimedean local field of characteristic zero, \(\mathcal{O}=\mathcal{O}_F\) for its valuation ring, and \(\mathbf{f}\) for the residue field. We fix an algebraic closure \(\overline{F}/F\), and write \(\overline{\mathcal{O}}\subset\overline{F}\) for its valuation ring and \(\overline{\mathbf{f}}\) for the residue field of \(\overline{\mathcal{O}}\). Note that \(\overline{\mathbf{f}}\) is an algebraic closure of the field \(\mathbf{f}\). For a field \(k\), let \(\Gamma_k=\operatorname{Gal}(\overline{k}/k)\) denote its absolute Galois group. In the case of the local field \(F\), we often omit the subscript and simply write \(\Gamma=\Gamma_F\).
In this paper, we treat an algebraic torus \(\mathbb{T}\) defined over the local field \(F\), and fix a finite Galois extension \(L/F\) where \(\mathbb{T}\) splits over \(L\). Let \(Y=\operatorname{Hom}(\mathbb{G}_{\mathrm m},\mathbb{T})\) be the cocharacter lattice, \(T=\mathbb{T}(F)=(Y\otimes_{\mathbb{Z}}\overline{F}^\times)^\Gamma\) the group of rational points, and \(e\in\mathbb{Z}_{>0}\) the ramification index of the extension \(L/F\). We write \(\operatorname{ord}\colon F^\times\to\mathbb{Z}\) for the map of additive valuation, and extend it to \(\overline{F}^\times\to\mathbb{Q}\). We continue to write \(\operatorname{ord}\) for the induced map \[T=(Y\otimes\overline{F}^\times)^\Gamma\to(Y\otimes\mathbb{Q})^\Gamma =Y^\Gamma\otimes\mathbb{Q}.\] For a subgroup \(Y'\subset Y\), let \(\iota\) denote the inclusion map \(Y'\hookrightarrow Y\) and the induced ones. For instance, \(\iota\colon(Y'\otimes\overline{F}^\times)^\Gamma\to (Y\otimes\overline{F}^\times)^\Gamma\) is a morphism between tori.
In the literature on Langlands correspondence for general covering groups [5], [6], the authors often focus on certain covering groups defined by Brylinski and Deligne [4]. In this section, we review the definition and the classification theorem for Brylinski-Deligne covering groups of an algebraic torus \(\mathbb{T}\) defined over a local field \(F\).
Roughly speaking, a Brylinski-Deligne covering group is a central extension by Milnor’s group \(\operatorname K_2\) in algebraic K-theory, or its variation. The groups \(\operatorname{K}_2\) are generalized by Quillen [7] to be defined over schemes, and form a presheaf on the big Zariski site of the base field \(F\). Let \(\mathbb{K}_2\) be the Zariski sheaf associated with the presheaf \(\operatorname K_2\).
To define a Brylinski-Deligne covering group, let \(\mu_n\subset F^\times\) denote the cyclic subgroup of order \(n\).
Definition 2 (Brylinski-Deligne [4]). Let \(\mathbb{T}\to\operatorname{Spec}F\) be an algebraic torus. According to the context, a Brylinski-Deligne covering group of \(\mathbb{T}\) or \(T=\mathbb{T}(F)\) means one of the following:
a central extension \(1\to\mathbb{K}_2\to\widetilde{\mathbb{T}}\to\mathbb{T}\to 1\) as a sheaf of groups on the big Zariski site of \(\operatorname{Spec}F\),
the section \(1\to\operatorname K_2(F)\to\widetilde{\mathbb{T}}(F)\to{T} \to 1\) of [e95BD95sh] on \(\operatorname{Spec}F\), or
the push-out \(1\to\mu_n\to\widetilde{T}\to{T}\to 1\) of [e95BD95sect] via the Hilbert symbol \(\operatorname K_2(F)\to\mu_n\).
A Brylinski-Deligne covering group defined in 2 [e95BD95topl] is indeed a topological covering group [4]. Further, Brylinski and Deligne generally defined their covering groups for any algebraic group.
Example 3 ([5]). For a split torus \(T\), a 2-cocycle of a Brylinski-Deligne cover \(\mu_n\to\widetilde{T}\to{T}\) is written as a product of \(n\)-th Hilbert symbols \((\,,)_n\colon F^\times\times F^\times\to\mu_n\). For example:
Let \(T=F^\times\) be the one-dimensional split torus, i.e. just the multiplicative group. Then for an integer \(a\), the 2-cocycle \((\,,)_n^a\colon F^\times\times F^\times\to\mu_n\) determines a Brylinski-Deligne cover \(\mu_n\to\widetilde{T}\to{T}\).
Let \(T=F^\times\times F^\times\) be the two-dimensional split torus. Then for four integers \((a_{ij})_{i,j\in\{1,2\}}\), the 2-cocycle \[(F^\times \times F^\times)\times(F^\times \times F^\times)\to \mu_n,\;((s_1,s_2),(t_1,t_2))\mapsto\prod_{i,j\in\{1,2\}} (s_i,t_j)_n^{a_{ij}}\] gives a Brylinski-Deligne cover \(\mu_n\to\widetilde{T}\to{T}\). Especially if \(n\geq 2\) and \((a_{ij})_{i,j}= \begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}\), then the covering group \(\widetilde{T}\) is non-abelian.
For any torus \(T\), one has a finite Galois extension \(L/F\) of the base field such that \(T\) splits over \(L\). Then the following classification theorem gives a bilinear form \(Y\times Y\to\mathbb{Z}\) on the cocharacter lattice, which is a key ingredient to describe our main theorem.
Theorem 4 (Classification Theorem of covers [4]). The following two Picard categories are equivalent:
the category of Brylinski-Deligne covers \(\mathbb{K}_2\to\widetilde{\mathbb{T}}\to\mathbb{T}\), and
the category of the following data \((Q,\mathcal{E})\):
\(Q\colon Y\to\mathbb{Z}\) is a Galois-invariant quadratic form on the cocharacter lattice, which gives a bilinear form \(B\colon Y\times Y\to\mathbb{Z}\) by \[(y,y')\mapsto Q(y+y')-Q(y)-Q(y').\]
\(1\to L^\times\to\mathcal{E}\stackrel{\overline{(\;)}}{\to}Y \to 1\) is a Galois-equivariant central extension such that for any \(\epsilon,\zeta\in\mathcal{E}\), \[\epsilon\zeta\epsilon^{-1}\zeta^{-1}=(-1)^{B(\overline{\epsilon},\overline{\zeta})}\] in \(L^\times\).
For any data \((Q,\mathcal{E})\) and \((Q',\mathcal{E}')\), \[\operatorname{Hom}((Q,\mathcal{E}),(Q',\mathcal{E}'))= \begin{cases} \emptyset &\text{if }Q\neq Q'\\ \{\text{morphisms }\mathcal{E}\to\mathcal{E}' \text{ of extensions}\} &\text{if }Q=Q'. \end{cases}\]
Brylinski and Deligne generally proved this type of classification theorem for covering groups of reductive algebraic groups [4].
Definition 5. Let \(\mathbb{K}_2\to\widetilde{\mathbb{T}}\to\mathbb{T}\) be a Brylinski-Deligne cover, and \(Q_{\widetilde{\mathbb{T}}}\colon Y\to\mathbb{Z}\) the Galois-invariant quadratic form corresponding to \(\widetilde{\mathbb{T}}\) by 4. Then we write \(B_{\widetilde{\mathbb{T}}}\colon Y\times Y\to\mathbb{Z}\) for the associated bilinear form \((y,y')\mapsto Q(y+y')-Q(y)-Q(y')\). If \(\mu_n\to\widetilde{T}\to T\) is the cover induced from \(\widetilde{\mathbb{T}}\) (2), then we write \(B=B_{\widetilde{T}}=B_{\widetilde{\mathbb{T}}}\) by abuse of notation.
In this section, we introduce a finite abelian group \(\mathcal{S}=\mathcal{S}_{\widetilde{T}}\), which is the main object of this work (12). Then we shall explain how \(\mathcal{S}_{\widetilde{T}}\) parametrizes a packet in the local Langlands correspondence for a Brylinski-Deligne covering group \(\mu_n\to\widetilde{T}\to T\) of a torus. Our formulation is essentially due to Weissman [3].
The local Langlands correspondence for a covering group \(\mu_n\to\widetilde{T}\to T\) of a torus aims to parametrize certain representations of \(\widetilde{T}\), namely genuine irreducible representations:
Definition 6. We throughout fix an embedding \(j\colon\mu_n\to\mathbb{C}^\times\) of the cyclic group of order \(n\). Generally, let \(\mu_n\to\widetilde{G}\to G\) be an \(n\)-fold covering group of a topological group.
A representation of the covering group \(\widetilde{G}\) is said to be genuine if \(\mu_n\subset\widetilde{G}\) acts as multiplication by scalers in \(\mathbb{C}\) via the embedding \(j\). That is, for any \(\tilde{g}\in\widetilde{G}\) and any point \(v\) in the representation space, one has \[\tilde{g}\cdot v=j(\tilde{g})v.\]
A genuine character \(\chi\colon\widetilde{G}\to\mathbb{C}^\times\) is a character that is genuine as a representation, i.e. \(\chi|_{\mu_n}=j\).
Example 7 (central character). Let \(\mu_n\to\widetilde{T}\to T\) be a covering group of a torus, and \(\rho\) an irreducible genuine representation of \(\widetilde{T}\). Let \(\mathrm Z^\dagger\subset T\) denote the image of the center \(\operatorname Z(\widetilde{T})\subset \widetilde{T}\), and note that \[\mu_n\to\operatorname Z(\widetilde{T})\to\mathrm Z^\dagger\] is a covering group. Then the central character \(\rho|_{\operatorname Z(\widetilde{T})}\colon \operatorname Z(\widetilde{T})\to\mathbb{C}^\times\) is a genuine character.
Remark 8. Since \(\mu_n\to\widetilde{T}\to T\) is a central extension, the commutator map \(\widetilde{T}\times\widetilde{T}\to\widetilde{T}, (\tilde{s},\tilde{t})\mapsto \tilde{s}\tilde{t}\tilde{s}^{-1}\tilde{t}^{-1}\) factors through a map \[\operatorname{comm}_{\widetilde{T}}\colon T\times T\to\mu_n.\] Then the image \(\mathrm Z^\dagger\subset T\) of the center \(\operatorname Z(\widetilde{T})\subset T\) is written as the annihilator \(\mathrm Z^\dagger=\{t\in T\mid \operatorname{comm}_{\widetilde{T}}(t,T)=\{1\}\text{ in }\mu_n\}\) of \(T\).
Fact 9 (an analogue of the Stone-von Neumann Theorem [3]). Let \(\mu_n\to\widetilde{T}\to T\) be a Brylinski-Deligne covering of a torus. Then the correspondence from genuine irreducible representations of the covering group \(\widetilde{T}\) to their central characters (7) is bijective onto the set of genuine characters \(\operatorname Z(\widetilde{T})\to\mathbb{C}^\times\).
Thus, to establish the local Langlands correspondence for the Brylinski-Deligne covering group \(\widetilde{T}\to T\), it suffices to parametrize the genuine characters on the center \(\operatorname Z(\widetilde{T})\). To estimate \(\operatorname Z(\widetilde{T})\subset\widetilde{T}\), we choose the following isogeny \(T^\#\stackrel\iota\to T\) of tori, which approximates the image \(\mathrm Z^\dagger\subset T\) of \(\operatorname Z(\widetilde{T})\).
Definition 10. Let \(Y=\operatorname{Hom}(\mathbb{G}_{\mathrm m},\mathbb{T})\) be the cocharacter group of the torus \(T=\mathbb{T}(F)\). Let \(B_{\widetilde{T}}\colon Y\times Y\to\mathbb{Z}\) be the bilinear form attached to the covering \(\mu_n\to\widetilde{T}\to T\), precisely to the Brylinski-Deligne cover \(\mathbb{K}_2\to\widetilde{\mathbb{T}}\to\mathbb{T}\) defining \(\widetilde{T}\) (5).
We define the subgroup \(Y^\#\subset Y\) by \[Y^\#=\{y\in Y\mid B_{\widetilde{T}}(y,Y)\subset n\mathbb{Z}\}.\]
We write \(T^\#=(Y^\#\otimes\overline{F}^\times)^\Gamma\) for the torus defined by \(Y^\#\). Then the inclusion \(Y^\#\hookrightarrow Y\) induces an isogeny \(\iota\colon T^\#\to T\).
Fact 11 (Weissman [8]). In 10, one has \(\iota(T^\#)\subset\mathrm{Z}^\dagger\), and this index is finite.
Definition 12. For the Brylinski-Deligne covering group \(\widetilde{T}\to T\), we define the finite abelian group \(\mathcal{S}=\mathcal{S}_{\widetilde{T}}\) as the quotient \[\mathcal{S}_{\widetilde{T}}=\mathrm{Z}^\dagger/\iota(T^\#).\]
In Weissman’s work [3], this quotient \(\mathrm{Z}^\dagger/\iota(T^\#)\) is called the “packet group”.
Fact 13 (Langlands correspondence for the torus \(T^\#\) [9]). Let \(\operatorname{W}_F\) be the Weil group of the local field \(F\), and \(\widehat{T^\#}=\operatorname{Hom}(Y^\#,\mathbb{C}^\times)\) the Langlands dual of the torus \(T^\#\). Then one has an isomorphism \[\operatorname{LLC}_{T^\#}\colon \operatorname{Hom}(T^\#,\mathbb{C}^\times)\cong \operatorname{H}^1(\operatorname{W}_F,\widehat{T^\#}).\]
Lemma 14. Let \(\mu_n\to\widetilde{T}\to T\) be a Brylinski-Deligne covering group of a torus. Then we have a diagram \[\begin{array}{*9c} && 1\\ && \uparrow\\ && \mathcal{S}_{\widetilde{T}}\\ && \uparrow\\ 1 &\gets& \mathrm Z^\dagger &\gets& \operatorname Z(\widetilde{T}) &\gets& \mu_n &\gets& 1 \\ && \uparrow \\ 1 &\gets& \iota(T^\#) &\stackrel\iota\twoheadleftarrow& T^\# \\ && \uparrow\\ && 1 \end{array}\] of exact sequences.
To apply the functor \(\operatorname{Hom}(\quad,\mathbb{C}^\times)\) to the diagram in 14, we note the following fact.
Fact 15 (topological injectivity of \(\mathbb{C}^\times\) [10]). The multiplicative group \(\mathbb{C}^\times\) is an injective object in the category of locally compact abelian groups. That is, for any locally compact abelian group \(A\) and its closed subgroup \(C\subset A\), any continuous character \(C\to\mathbb{C}^\times\) extends to a continuous character \(A\to\mathbb{C}^\times\).
Corollary 16. The objects introduced above fit in the diagram \[\begin{array}{*9c} 1 &&to0pt{\hss \{\text{genuine irreducible representations of }\widetilde{T}\} \hss} \\ \downarrow && \llap{\scriptstyle (\;)|_{\operatorname Z(\widetilde{T})}} \downarrow \rlap{\scriptstyle 1:1} \\ \operatorname{Hom}(\mathcal{S}_{\widetilde{T}},\mathbb{C}^\times) && \{\text{genuine characters}\} &\to& \{j\} \\ \downarrow && \cap && \cap \\ \llap{1\to\,} \operatorname{Hom}(\mathrm{Z}^\dagger,\mathbb{C}^\times) &\to& \operatorname{Hom}(\operatorname{Z}(\widetilde{T}), \mathbb{C}^\times) &\to& \operatorname{Hom}(\mu_n,\mathbb{C}^\times) \rlap{\;\to1} \\\downarrow \rlap{\scriptstyle(\;)|_{\,\iota(T^\#)}} \\ \operatorname{Hom}(\iota(T^\#),\mathbb{C}^\times) &\stackrel{\iota^*}\hookrightarrow& \operatorname{Hom}(T^\#,\mathbb{C}^\times) \\ \downarrow && \rotatebox[origin=c]{-90}{\cong} \rlap{\,\scriptstyle\operatorname{LLC}_{T^\#}} \\ 1 && \operatorname{H}^1(\operatorname{W}_F,\widehat{T^\#}), \end{array}\] where the middle horizontal sequence and the left vertical one are exact.
Remark 17.
By definition, the group \(\operatorname{Hom}(\mathrm{Z}^\dagger,\mathbb{C}^\times)\) acts simply transitive on the set of genuine characters \(\operatorname Z(\widetilde{T})\to\mathbb{C}^\times\). Thus, fixing a genuine character \(\chi\) on \(\operatorname Z(\widetilde{T})\) as a base point defines a bijection \[b_\chi\colon \{\text{genuine characters on }\operatorname Z(\widetilde{T})\} \to \operatorname{Hom}(\mathrm{Z}^\dagger,\mathbb{C}^\times).\]
By 16, the composed map \[\operatorname{LLC}_{T^\#}\circ\iota^*\circ(\,)|_{\iota(T^\#)} \colon\operatorname{Hom}(\mathrm{Z}^\dagger,\mathbb{C}^\times) \to\operatorname{H}^1(\operatorname{W}_F,\widehat{T^\#})\] has kernel \(\operatorname{Hom}(\mathcal{S}_{\widetilde{T}},\mathbb{C}^\times)\), which is a finite group.
Definition 18. Fix a genuine character \(\chi\colon\operatorname Z(\widetilde{T})\to\mathbb{C}\). Then the local Langlands correspondence for the Brylinski-Deligne cover \(\widetilde{T}\to T\) is the composition \[\phantom{\text{genuine}} \begin{array}[b]{*9c}to0pt{\hss \{\text{genuine irreducible representations of } \widetilde{T}\} \hss}\\ \llap{\scriptstyle b_\chi\circ (\,)_{\operatorname Z(\widetilde{T})}} \downarrow\rlap{\scriptstyle 1:1}\\ \operatorname{Hom}(\mathrm{Z}^\dagger,\mathbb{C}^\times) \end{array} \xrightarrow{\operatorname{LLC}_{T^\#}\circ\iota^*\circ (\,)|_{\iota(T^\#)}} \operatorname{H}^1(\operatorname{W}_F,\widehat{T^\#}).\] As remarked in 17, this is a finite-to-one correspondence. Note that some packets, i.e., fibers, may be empty.
Proposition 19 (role of \(\mathcal{S}_{\widetilde{T}}\)). In the local Langlands correspondence defined in 18, each non-empty packet \(\Pi\) is just an orbit under the group \(\operatorname{Hom}(\mathcal{S}_{\widetilde{T}},\mathbb{C}^\times)\). Thus, Choosing a base point in the packet \(\Pi\) defines a one-to-one correspondence between \(\Pi\) and \(\operatorname{Hom}(\mathcal{S}_{\widetilde{T}},\mathbb{C}^\times)\).
Proof. This holds by definition. ◻
The correspondence between a packet \(\Pi\) and the group \(\operatorname{Hom}(\mathcal{S}_{\widetilde{T}},\mathbb{C}^\times)\) stated in 19 is an analogue to the usual formulation to parametrize a packet in the local Langlands correspondence for a reductive algebraic group.
To determine the finite group \(\mathcal{S}_{\widetilde{T}}=\mathrm Z^\dagger/\iota(T^\#)\) parametrizing a packet, we prepare exact sequences, and describe the subgroup \(\mathrm{Z}^\dagger\subset T\) via a non-degenerate bilinear form.
For \(k\in\mathbb{Z}\) and a finite \(\Gamma\)-module \(M\), let \(M(k)\) denote the \(k\)-th Tate twist of \(M\). For instance, \((\mathbb{Z}/m\mathbb{Z})(1)=\mu_m\subset\overline{F}^\times\) is the cyclic subgroup of order \(m\), and \((\mathbb{Z}/m\mathbb{Z})(2)=\mu_m\otimes\mu_m\), for \(m\in\mathbb{Z}_{>0}\).
Lemma 20. Let \(A\) be a \(\Gamma\)-module, and \(Y'\subset Y\) a \(\Gamma\)-submodule of the cocharacter lattice. Assume that for some \(m\in\mathbb{Z}_{>0}\), we have the inclusion \(mY\subset Y'\) and exact sequence \[0\to(\mathbb{Z}/m\mathbb{Z})(1)\to A\stackrel{m}{\to}A\to 0.\] Then they induce a short exact sequence \[0\to(Y/Y')(1)\to Y'\otimes A\to Y\otimes A\to 0.\]
Proof. The sequence \(0\to Y'\to Y\to Y/Y'\to 0\) induces an exact sequence \[0\to \operatorname{Tor}_1(Y/Y',A)\to Y'\otimes A\to Y\otimes A \to 0.\] It suffices to give an isomorphism \(\operatorname{Tor}_1(Y/Y',A)\cong(Y/Y')(1)\). Indeed, the sequence \(0\to(\mathbb{Z}/m\mathbb{Z})(1)\to A\stackrel{m}{\to} A\to 0\) gives an exact sequence \[\begin{matrix} \operatorname{Tor}_{1}(Y/Y',A)\stackrel{0}{\to} \operatorname{Tor}_{1}(Y/Y',A)&\to(Y/Y')\otimes(\mathbb{Z}/m\mathbb{Z})(1)\to&(Y/Y')\otimes A\\ &\rotatebox[origin=c]{90}{=} &\rotatebox[origin=c]{90}{=}\\ &(Y/Y')(1) &0. \end{matrix}\] ◻
Example 21. Let \(A=\overline{\mathbf{f}}^\times\) be an algebraic closure of the residue field , and \(Y'=Y^\#=\{y\in Y\mid B_{\widetilde{T}}(y,Y)\in n\mathbb{Z}\}\) as defined in 10. Suppose the inclusion \(\mu_n\subset\mathbf{f}^\times\), so that the assumption in 20 holds for \(m=n\). Then we have a short exact sequence \[0\to(Y/Y^\#)(1)\to Y^\#\otimes \overline{\mathbf{f}}^\times\to Y\otimes \overline{\mathbf{f}}^\times\to 0.\]
Corollary 22. Let \(Y'\subset Y\) be a \(\Gamma\)-submodule of the same rank. Then the tensor products with the valuation sequence \(1\to\overline{\mathcal{O}}^\times\to\overline{F}^\times \stackrel{\operatorname{ord}}{\to}\mathbb{Q}\to 0\) gives the following diagrams of exact sequences:
\[\begin{array}{ccccc} &0&&0&\\ &\rotatebox[origin=c]{-90}{\to}&&\rotatebox[origin=c]{-90}{\to} &\\ 0\to(Y/Y')(1)\to&Y'\otimes \overline{\mathcal{O}}^\times& \to&Y\otimes\overline{\mathcal{O}}^\times&\to 0\\ \rotatebox[origin=c]{90}{=}&\rotatebox[origin=c]{-90}{\to}&&\rotatebox[origin=c]{-90}{\to} &\\ 0\to(Y/Y')(1)\to&Y'\otimes\overline{F}^\times&\to& Y\otimes\overline{F}^\times&\to 0\\ &\rotatebox[origin=c]{-90}{\to}\rlap{\,\scriptstyle\operatorname{ord}}&&\rotatebox[origin=c]{-90}{\to}\rlap{\,\scriptstyle\operatorname{ord}} &\\ &Y'\otimes\mathbb{Q}&\stackrel{\sim}{\to}& Y\otimes\mathbb{Q}\\ &\rotatebox[origin=c]{-90}{\to}&&\rotatebox[origin=c]{-90}{\to} &\\ &0&&0\rlap.& \end{array}\]
\[\begin{matrix} &0 & &0 \\ &\rotatebox[origin=c]{-90}{\to} & &\rotatebox[origin=c]{-90}{\to} \\ 0\to(Y/Y')(1)^\Gamma\to&(Y'\otimes\overline{\mathcal{O}}^\times)^\Gamma &\stackrel{\iota}{\to}&(Y\otimes\overline{\mathcal{O}}^\times)^\Gamma &\stackrel{\partial_{Y/Y'}}{\to} \operatorname{H}^1(\Gamma,(Y/Y')(1))\\ \rotatebox[origin=c]{90}{=}&\rotatebox[origin=c]{-90}{\to} & &\rotatebox[origin=c]{-90}{\to} &\rotatebox[origin=c]{90}{=} \\ 0\to(Y/Y')(1)^\Gamma\to&(Y'\otimes\overline{F}^\times)^\Gamma &\stackrel{\iota}{\to}&(Y\otimes\overline{F}^\times)^\Gamma &\stackrel{\partial_{Y/Y'}}{\to}\operatorname{H}^1(\Gamma,(Y/Y')(1))\\ &\rotatebox[origin=c]{-90}{\to}\rlap{\,\scriptstyle\operatorname{ord}}& &\rotatebox[origin=c]{-90}{\to}\rlap{\,\scriptstyle\operatorname{ord}} \\ &Y'^{\Gamma}\otimes\mathbb{Q} &\stackrel{\sim}{\to} &Y^\Gamma\otimes\mathbb{Q}. & \end{matrix}\]
Proof. [eOF] By assumption, we have some \(m\in\mathbb{Z}_{>0}\) such that \(mY\subset Y'\). Then the pairs \((\overline{\mathcal{O}}^\times,Y')\) and \((\overline{F}^\times,Y')\) respectively satisfy the assumption in 20. [eGl] Just note that \((Y'\otimes\mathbb{Q})^\Gamma=Y'^\Gamma\otimes\mathbb{Q}\). ◻
We write \(\partial=\partial_{Y/Y'}\) for the connecting homomorphisms.
Remark 23. If \(T'=(Y'\otimes\overline{F}^\times)^\Gamma\) is a split torus over \(F\), i.e. \(\Gamma\) acts trivially on \(Y'\), then the connecting homomorphism \(\partial_{Y/Y'}\colon T=(Y\otimes\overline{F}^\times)^\Gamma\to \operatorname{H}^1(\Gamma,(Y/Y')(1))\) is surjective.
Proof. Hilbert’s theorem 90 shows \(\operatorname{H}^1(\Gamma,Y'\otimes\overline{F}^\times) =0\). ◻
Let \(V_{Y'}=\operatorname{ord}((Y'\otimes\overline{F}^\times)^\Gamma)\) be the image of \(\operatorname{ord}\) in the diagram of 22 [eGl].
Corollary 24.
The images of \(\partial\) and \(\operatorname{ord}\) in 22 [eGl] gives the commutative diagram \[\begin{matrix} & & &0 & &0 & &0 &\\ & & &\rotatebox[origin=c]{-90}{\to} & &\rotatebox[origin=c]{-90}{\to} & &\rotatebox[origin=c]{-90}{\to} &\\ 0\to&(Y/Y')(1)^\Gamma&\to&(Y'\otimes\overline{\mathcal{O}}^\times)^\Gamma&\stackrel{\iota}{\to}&(Y\otimes\overline{\mathcal{O}}^\times)^\Gamma&\stackrel\partial\to&\partial((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma)&\to 0\\ &\rotatebox[origin=c]{-90}{=} & &\rotatebox[origin=c]{-90}{\to} & &\rotatebox[origin=c]{-90}{\to} & &\rotatebox[origin=c]{-90}{\hookrightarrow} &\\ 0\to&(Y/Y')(1)^\Gamma&\to&(Y'\otimes\overline{F}^\times)^\Gamma &\stackrel{\iota}{\to}&(Y\otimes\overline{F}^\times)^\Gamma &\stackrel\partial\to&\partial((Y\otimes\overline{F}^\times)^\Gamma) &\to 0\\ & & &\rotatebox[origin=c]{-90}{\to} & &\rotatebox[origin=c]{-90}{\to}\rlap{\,\scriptstyle\operatorname{ord}} & &\rotatebox[origin=c]{-90}{\to} &\\ &0 &\to&V_{Y'} &\hookrightarrow &V_Y &\to&V_Y/V_{Y'} &\to 0\\ & & &\rotatebox[origin=c]{-90}{\to} & &\rotatebox[origin=c]{-90}{\to} & &\rotatebox[origin=c]{-90}{\to} &\\ & & &0 & &0 & &0 & \end{matrix}\] of exact sequences.
In the diagram [eCdi], we have \[\iota((Y'\otimes\overline{F}^\times)^\Gamma)\cap (Y\otimes\overline{\mathcal{O}}^\times)^\Gamma =\iota((Y'\otimes\overline{\mathcal{O}}^\times)^\Gamma).\]
Proof. [eCdi] This follows from the snake lemma. [eePbi] The diagram displays that the map \[\left.(Y\otimes\overline{\mathcal{O}}^\times)^\Gamma\middle/ \iota((Y'\otimes\overline{\mathcal{O}}^\times)^\Gamma)\right.\to \left.(Y\otimes\overline{F}^\times)^\Gamma\middle/ \iota((Y'\otimes\overline{F}^\times)^\Gamma)\right.\] is injective. ◻
For the subgroup \(Y'=nY\) of \(Y\), the connecting homomorphism \[\partial=\partial_{Y/nY}\colon T=(Y\otimes\overline{F}^\times)^\Gamma\to \operatorname{H}^1(\Gamma,(Y/nY)(1))\] is applied to express commutators of lifts of elements in \(T\) to \(\widetilde{T}\).
Fact 25 ([4]). The composite map \[T\times T\stackrel{\partial\times\partial}{\to} \operatorname{H}^1(\Gamma,(Y/nY)(1))^{\oplus2} \stackrel{\cup}{\to}\operatorname{H}^2(\Gamma, (Y/nY)(1)^{\otimes2})\stackrel{B_{\widetilde{T}}}{\to} \operatorname{H}^2(\Gamma,\mathbb{Z}/n(2))=\mu_n\] gives the commutator of lifts of two elements in \(T\) to \(\widetilde{T}\). That is, for any \(s,t\in T\) and their lift \(\tilde{s},\tilde{t}\in\widetilde{T}\), we have \[B_{\widetilde{T}}(\partial s\cup\partial t)= \operatorname{comm}_{\widetilde{T}}(s,t)= \tilde{s}\tilde{t}\tilde{s}^{-1}\tilde{t}^{-1}.\]
Corollary 26. We have \(\mathrm{Z}^\dagger=\{t\in T\mid \forall t'\in T,B_{\widetilde{T}}(\partial t\cup\partial t')=0\}\).
In the rest of this section, we slightly generalize the description of the commutator map \(\operatorname{comm}_{\widetilde{T}}\) in 25, so that we can apply other useful connecting homomorphisms.
For \(i=1,2\), let \(T_i\) be an algebraic torus with cocharacter lattice \(Y_i\).
Definition 27. Let \(B\colon Y_1\times Y_2\to\mathbb{Z}\) be a \(\Gamma\)-invariant bilinear form. Then we define the \(\Gamma\)-submodule \(Y_2^\#\subset Y_1\) as the annihilator \[Y_2^\#=\{y\in Y_1\mid B(y,Y_2)\subset n\mathbb{Z}\}\] of \(Y_2\) for the bilinear form \(Y_1\times Y_2\to\mathbb{Z}/n\). We similarly define the \(\Gamma\)-submodule \(Y_1^\#\subset Y_2\) by \(Y_1^\#=\{y\in Y_2\mid B(Y_1,y)\subset n\mathbb{Z}\}\).
For \(B=B_{\widetilde{T}}\colon Y\times Y\to\mathbb{Z}\), 27 agrees with the definition of the submodule \(Y^\#\) in 10. In general, the modules \(Y_2^\#\) and \(Y_1^\#\) are characterized by the following properties:
Proposition 28. Let \(B\colon Y_1\times Y_2\to\mathbb{Z}\) be a bilinear form. For \(i=1,2\), let \(Y_i'\subset Y_i\) be a submodule.
The form \(B\) induces a bilinear form \(Y_1/Y_1'\times Y_2/Y_2'\to\mathbb{Z}/n\) if and only if \(Y_1'\subset Y_2^\#\) and \(Y_2'\subset Y_1^\#\).
The induced bilinear form \(Y_1/Y_2^\#\times Y_2/Y_1^\#\to\mathbb{Z}/n\) for the subgroups \(Y_2^\#\) and \(Y_1^\#\) is non-degenerate.
Proof. They hold by definition. ◻
Corollary 29. Let \(B\colon Y_1\times Y_2\to\mathbb{Z}\) be a \(\Gamma\)-invariant bilinear form. Then the bilinear form \(B\colon Y_1/Y_2^\#\times Y_2/Y_1^\#\to\mathbb{Z}/n\) induces a non-degenerate bilinear form \[\begin{array}{*9c} \operatorname{H}^1(\Gamma,(Y_1/Y_2^\#)(1))\times \operatorname{H}^1(\Gamma,(Y_2/Y_1^\#)(1)) &\to&\operatorname{H}^2(\Gamma,\mathbb{Z}/n(2))\\ \rotatebox[origin=c]{90}{\in} && \rotatebox[origin=c]{90}{\in}\\ (h_1,h_2) &\mapsto& B(h_1\cup h_2). \end{array}\]
Proof. By definition, the bilinear form \(B\colon(Y_1/Y_2^\#)(1)\times(Y_2/Y_1^\#)(1)\to\mathbb{Z}/n(2)\cong \mathbb{Z}/n(1)\) is non-degenerate, i.e., gives a pairing of dual Galois modules. By local Tate duality [11], the cup product \(\operatorname{H}^1(\Gamma,(Y_1/Y_2^\#)(1))\times \operatorname{H}^1(\Gamma,(Y_2/Y_1^\#)(1))\to \operatorname{H}^2(\Gamma,\mathbb{Z}/n(2))\cong\mathbb{Z}/n\) is a pairing of groups, i.e., a non-degenerate form. ◻
If two submodules \(Y'_1\subset Y^\#_2\) and \(Y'_2\subset Y^\#_1\) satisfy the equalities \(\operatorname{rank}Y'_i=\operatorname{rank}Y_i\) for \(i=1,2\), then the inclusion \(Y_i'\subset Y_i\) induces the connecting homomorphism \(\partial_{Y_i/Y_i'}\colon T_i=(Y_i\otimes\overline{F}^\times)^\Gamma \to\operatorname H^1(\Gamma,(Y_i/Y_i')(1))\). Thus, we may compose the homomorphisms \(\partial_{Y_1/Y_1'}\) and \(\partial_{Y_2/Y_2'}\) with the cup product to obtain a map \[\operatorname{comm}_B\colon T_1\times T_2\stackrel{\partial\times\partial}\to \operatorname H^1(\Gamma,(Y_1/Y_1')(1))\times \operatorname H^1(\Gamma,(Y_2/Y_2')(1))\stackrel B\to \operatorname H^2(\Gamma,(\mathbb{Z}/n)(2)).\]
Lemma 30. For \(i=1,2,3,4\), let \(T_i\) be an algebraic torus with cocharacter lattice \(Y_i\), and \(Y_i'\subset Y_i\) a \(\Gamma\)-submodule of the same rank as \(Y_i\). For \(i=1,3\), we assume the inclusions \(Y_i'\subset Y_{i+1}^\#\) and \(Y_{i+1}'\subset Y_{i}^\#\), so that a \(\Gamma\)-invariant bilinear form \(B_{i,i+1}\colon Y_i\times Y_{i+1}\to\mathbb{Z}\) induces a bilinear form \(Y_i/Y_i'\times Y_{i+1}/Y'_{i+1}\to\mathbb{Z}/n\). For \(i=1,2\), let \(f_i\colon Y_i\to Y_{i+2}\) be a \(\Gamma\)-equivariant homomorphism such that \(f_i(Y_i')\subset Y_{i+2}'\) and that the diagram \[\begin{array}{*9c} Y_1/Y_1'\times Y_2/Y_2' &\stackrel{B_{1,2}}\to& \mathbb{Z}/n\\ \downarrow\rlap{\scriptstyle f_1\times f_2} && \rotatebox[origin=c]{90}=\\ Y_3/Y_3'\times Y_4/Y_4' &\stackrel{B_{3,4}}\to& \mathbb{Z}/n \end{array}\] commutes. Then these maps induce the following commutative diagram: \[\begin{array}{*9c} T_1\times T_2 &\stackrel{\partial\times\partial}\to& \operatorname H^1(\Gamma,(Y_1/Y_1')(1))\times \operatorname H^1(\Gamma,(Y_2/Y_2')(1)) &\stackrel{B_{1,2}}\to& \operatorname H^2(\Gamma,(\mathbb{Z}/n)(2))\\ \downarrow\rlap{\scriptstyle f_1\times f_2} && \downarrow && \rotatebox[origin=c]{90}=\\ T_3\times T_4 &\stackrel{\partial\times\partial}\to& \operatorname H^1(\Gamma,(Y_3/Y_3')(1))\times \operatorname H^1(\Gamma,(Y_4/Y_4')(1)) &\stackrel{B_{3,4}}\to& \operatorname H^2(\Gamma,(\mathbb{Z}/n)(2)). \end{array}\]
Proof. This follows from the functoriality of cup products. ◻
Corollary 31. Let \(T_i\), \(Y_i\supset Y_i'\) and \(B_{i,i+1}\) be as above. For \(i=1,2\), let \(f_i\colon Y_i\to Y_{i+2}\) be any \(\Gamma\)-equivariant homomorphism making the diagram \[\begin{array}{*9c} Y_1\times Y_2 &\stackrel{B_{1,2}}\to& \mathbb{Z}\\ \downarrow\rlap{\scriptstyle f_1\times f_2} && \rotatebox[origin=c]{90}=\\ Y_3\times Y_4 &\stackrel{B_{3,4}}\to& \mathbb{Z} \end{array}\] commute. Then the following diagram commutes: \[\begin{array}{*9c} T_1\times T_2 &\stackrel{\partial\times\partial}\to& \operatorname H^1(\Gamma,(Y_1/Y_1')(1))\times \operatorname H^1(\Gamma,(Y_2/Y_2')(1)) &\stackrel{B_{1,2}}\to& \operatorname H^2(\Gamma,(\mathbb{Z}/n)(2))\\ \downarrow\rlap{\scriptstyle f_1\times f_2} && && \rotatebox[origin=c]{90}=\\ T_3\times T_4 &\stackrel{\partial\times\partial}\to& \operatorname H^1(\Gamma,(Y_3/Y_3')(1))\times \operatorname H^1(\Gamma,(Y_4/Y_4')(1)) &\stackrel{B_{3,4}}\to& \operatorname H^2(\Gamma,(\mathbb{Z}/n)(2)). \end{array}\] Especially, the composite map \(\operatorname{comm}_{B_{1,2}}=B_{1,2}\circ(\partial\times\partial)\) is independent of the choice of subgroups \(Y_1'\) and \(Y_2'\).
Proof.
First, suppose that \(Y_3'=nY_3\) and that \(Y_4'=nY_4\). Since \(f_i(nY_i)\subset nY_{i+2}\) for \(i=1,2\), we have the following commutative diagram: \[\begin{array}{*9c} T_1\times T_2 & \to& \operatorname H^1(\Gamma,(Y_1/Y_1')(1))\times \operatorname H^1(\Gamma,(Y_2/Y_2')(1)) &\stackrel{B_{1,2}}\to& \operatorname H^2(\Gamma,(\mathbb{Z}/n)(2))\\ \rotatebox[origin=c]{90}=&& \downarrow && \rotatebox[origin=c]{90}=\\ T_1\times T_2 & \to& \operatorname H^1(\Gamma,(Y_1/Y_2^\#)(1))\times \operatorname H^1(\Gamma,(Y_2/Y_1^\#)(1)) &\stackrel{B_{1,2}}\to& \operatorname H^2(\Gamma,(\mathbb{Z}/n)(2))\\ \rotatebox[origin=c]{90}=&& \uparrow && \rotatebox[origin=c]{90}=\\ T_1\times T_2 & \to& \operatorname H^1(\Gamma,(Y_1/nY_1)(1))\times \operatorname H^1(\Gamma,(Y_2/nY_2)(1)) &\stackrel{B_{1,2}}\to& \operatorname H^2(\Gamma,(\mathbb{Z}/n)(2))\\ \downarrow\rlap{\scriptstyle f_1\times f_2} && \downarrow && \rotatebox[origin=c]{90}=\\ T_3\times T_4 & \to& \operatorname H^1(\Gamma,(Y_3/nY_3)(1))\times \operatorname H^1(\Gamma,(Y_4/nY_4)(1)) &\stackrel{B_{3,4}}\to& \operatorname H^2(\Gamma,(\mathbb{Z}/n)(2)). \end{array}\]
We may reduce the general case to [eBud] by the diagram \[\begin{array}{*9c} T_1\times T_2 &\to& \operatorname H^1(\Gamma,(Y_1/Y_1')(1))\times \operatorname H^1(\Gamma,(Y_2/Y_2')(1)) &\stackrel{B_{1,2}}\to& \operatorname H^2(\Gamma,(\mathbb{Z}/n)(2))\\ \downarrow\rlap{\scriptstyle f_1\times f_2} && && \rotatebox[origin=c]{90}=\\ T_3\times T_4 &\to& \operatorname H^1(\Gamma,(Y_3/nY_3)(1))\times \operatorname H^1(\Gamma,(Y_4/nY_4)(1)) &\stackrel{B_{3,4}}\to& \operatorname H^2(\Gamma,(\mathbb{Z}/n)(2))\\ \uparrow\rlap{\scriptstyle\operatorname{id}\times \operatorname{id}} && && \rotatebox[origin=c]{90}=\\ T_3\times T_4 &\to& \operatorname H^1(\Gamma,(Y_3/Y_3')(1))\times \operatorname H^1(\Gamma,(Y_4/Y_4')(1)) &\stackrel{B_{3,4}}\to& \operatorname H^2(\Gamma,(\mathbb{Z}/n)(2)). \end{array}\]
◻
Corollary 32. For \(i=1,2\), let \(Y_i\subset Y\) be a \(\Gamma\)-submodule such that the associated homomorphism \(T_i\to T\) is injective. Then the restriction \(B_{\widetilde{T}}|_{Y_1\times Y_2}\colon Y_1\times Y_2\to\mathbb{Z}\) gives the equality \[\operatorname{comm}_{B_{\widetilde{T}}|_{Y_1\times Y_2}}= \operatorname{comm}_{\widetilde{T}}|_{T_1\times T_2}\] as bilinear forms \(T_1\times T_2\to\operatorname H^2(\Gamma,(\mathbb{Z}/n)(2))\).
Note that \(\operatorname{comm}_{B_{\widetilde{T}}}= \operatorname{comm}_{\widetilde{T}}\) for \(B=B_{\widetilde{T}}\).
Definition 33 (annihilators).
Generally, let \(A_1\) and \(A_2\) be finite \(\Gamma\)-modules, and \(B\colon A_1\times A_2\to(\mathbb{Z}/n)(2)\) a \(\Gamma\)-equivariant bilinear form. For a subgroup \(J_1\subset\operatorname{H}^1(\Gamma,A_1)\), we define the subgroup \(J_1^\perp=J_1^{\perp_B}\) of \(\operatorname{H}^1(\Gamma,A_2)\) as the annihilator for the bilinear form \[B(\;\cup\;)\colon \operatorname{H}^1(\Gamma,A_1)\times \operatorname{H}^1(\Gamma,A_2) \to\operatorname{H}^2(\Gamma,\mathbb{Z}/n(2)).\] That is: \[J_1^\perp=\{h\in\operatorname{H}^1(\Gamma,A_2)\mid \forall j_1\in J_1, B(j_1\cup h)=0\}.\] Similarly, for a subgroup \(J_2\subset\operatorname{H}^1(\Gamma,A_2)\), we define the subgroup \(J_2^\perp=J_2^{\perp_B}\) of \(\operatorname{H}^1(\Gamma,A_1)\) by \[J_2^\perp=\{h\in\operatorname{H}^1(\Gamma,A_1)\mid \forall j_2\in J_2, B(h\cup j_2)=0\}.\]
For subgroups \(T_1,T_2\subset T\), we define the subgroup \(\operatorname Z^\dagger_{T_1}(T_2) \subset T_1\) as the annihilator of \(T_2\) for the commutator map \(\operatorname{comm}_{\widetilde{T}}\colon T\times T\to\mu_n\). That is: \[\operatorname Z^\dagger_{T_1}(T_2)= \{t\in T_1\mid \operatorname{comm}_{\widetilde{T}}(t,T_2)= \{1\}\text{ in }\mu_n\}.\] Note that for \(T_2=T\), we have \(\operatorname{Z}^\dagger_{T_1}(T)=\mathrm{Z}^\dagger\cap T_1\).
Remark 34. In 33 [eZdg], suppose that for \(i=1,2\), \(T_i\) is a subtorus with cocharacter lattice \(Y_i\subset Y\). Then the restriction \(B_{\widetilde{T}}|_{Y_1\times Y_2}\colon Y_1\times Y_2\to\mathbb{Z}\) also describes the annihilator as \[\operatorname Z^\dagger_{T_1}(T_2)=\{t\in T_1\mid \operatorname{comm}_{B_{\widetilde{T}}|Y_1\times Y_2}(t,T_2)= \{1\}\text{ in }\mu_n\}\] by 32. Take \(\Gamma\)-submodules \(Y'_1\subset Y_2^\#\) and \(Y'_2\subset Y_1^\#\) such that \(\operatorname{rank}Y_i'=\operatorname{rank}Y_i\) for \(i=1,2\). Then, we may write \[\begin{align} \operatorname Z^\dagger_{T_1}(T_2) &=\{t_1\in T_1\mid \forall t_2\in T_2, B(\partial_{Y_1/Y_1'}(t_1)\cup\partial_{Y_2/Y_2'}(t_2))=0\}\\ &=\partial_{Y_1/Y_1'}^{-1}(\partial_{Y_2/Y_2'}(T_2)^\perp), \end{align}\] where the annihilator \(\partial_{Y_2/Y_2'}(T_2)^\perp\) is defined for the bilinear form \(\operatorname H^1(\Gamma,(Y_1/Y_1')(1))\times \operatorname H^1(\Gamma,(Y_2/Y_2')(1)) \stackrel{B}\to\operatorname H^2(\Gamma,(\mathbb{Z}/n)(2))\).
Example 35. For the connecting homomorphism \(\partial=\partial_{Y/Y^\#}\colon T=(Y\otimes\overline{F}^\times)^\Gamma\to \operatorname H^1(\Gamma,(Y/Y^\#)(1))\), we shall show in 5 that the annihilator of the subgroup \(\partial((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma) \subset\operatorname H^1(\Gamma,(Y/Y^\#)(1))\) is itself: \[\partial((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma)^\perp =\partial((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma).\] Then, by 34, we have \[\begin{align} \operatorname Z^\dagger_T ((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma)= \partial^{-1}\partial ((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma) &=(Y\otimes\overline{\mathcal{O}}^\times)^\Gamma+ \operatorname{Ker}\partial_{Y/Y^\#}\\ &=(Y\otimes\overline{\mathcal{O}}^\times)^\Gamma+\iota(T^\#). \end{align}\]
The following is another example of the annihilator \(\operatorname Z^\dagger_{T_1}(T_2)\) that can be described explicitly:
Proposition 36. For \(i=1,2\), let \(T_i\subset T\) be a subtorus with cocharacter lattice \(Y_i\subset Y\). Recall that the bilinear form \(B_{\widetilde{T}}|_{Y_1\times Y_2}\colon Y_1\times Y_2\to\mathbb{Z}\) defines the subgroups \(Y_2^\#\subset Y_1\) and \(Y_1^\#\subset Y_2\), which induce isogenies \(T_2^\#\stackrel\iota\to T_1\) and \(T_1^\#\stackrel\iota\to T_2\) of tori (27). Assume that the torus \(T_1^\#\) splits over \(F\), i.e. \(\Gamma\) acts trivially on the submodule \(Y_1^\#\subset Y_2\). Then, \[\operatorname Z^\dagger_{T_1}(T_2)= \operatorname{Ker}\partial_{Y_1/Y_2^\#}=\iota(T^\#_2).\]
Proof. The last equality follows from the long exact sequence where the connecting homomorphism \(\partial_{Y_1/Y_2^\#}\) appears. Thus, it suffices to see the first equality. Since \(\operatorname Z^\dagger_{T_1}(T_2) =\partial_{Y_1/Y_2^\#}^{-1}(\partial_{Y_2/Y_1^\#}(T_2)^\perp)\) by 34, it suffices to see that \(\partial_{Y_2/Y_1^\#}(T_2)^\perp=0\). By assumption, the connecting homomorphism \(\partial_{Y_2/Y_1^\#}\colon T_2= (Y_2\otimes\overline{F}^\times)^\Gamma\to \operatorname H^1(\Gamma,(Y_2/Y_1^\#)(1))\) is surjective (23). Since the bilinear form \[\operatorname{H}^1(\Gamma,(Y_1/Y_2^\#)(1))\times \operatorname{H}^1(\Gamma,(Y_2/Y_1^\#)(1)) \stackrel B\to\operatorname{H}^2(\Gamma,\mathbb{Z}/n(2))\] is non-degenerate by 29, we have \[\partial_{Y_2/Y_1^\#}(T_2)^\perp= \operatorname H^1(\Gamma,(Y_2/Y_1^\#)(1))^\perp=0.\] ◻
Example 37. The maximal split subtorus \(T^\Gamma\subset T\) is expressed as \(T^\Gamma=Y^\Gamma\otimes F^\times\). Note that the exact sequence \(0\to Y^\Gamma\hookrightarrow Y\to Y/Y^\Gamma\to 0\) splits, since the quotient \(Y/Y^\Gamma\) is a free abelian group. Hence the sequence \[0\to \begin{array}[t]{c} (Y^\Gamma\otimes\overline{F}^\times)^\Gamma\\ \rotatebox[origin=c]{90}=\\ T^\Gamma \end{array} \to \begin{array}[t]{c} (Y\otimes\overline{F}^\times)^\Gamma\\ \rotatebox[origin=c]{90}=\\ T \end{array} \to ((Y/Y^\Gamma)\otimes\overline{F}^\times)^\Gamma\to 0\] is split exact. Since the action of \(\Gamma\) on \(Y^\Gamma\) is trivial, the pair \((T,T^\Gamma)\) satisfies the assumption of 36. Therefore \[\operatorname Z^\dagger_{T}(T^\Gamma)=\iota(T^{\Gamma\#}),\] where \(T^{\Gamma\#}\stackrel\iota\to T\) is the isogeny corresponding to the inclusion \(Y^{\Gamma\#}\hookrightarrow Y\) of cocharacter lattices.
In this section, we prove the self-orthogonality of the subgroup \(\partial_{Y/Y^\#}((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma) \subset\operatorname H^1(\Gamma,(Y/Y^\#)(1))\) mentioned in 35, by reducing it to the orthogonality theorem for unramified cohomology. Let \(I=I_F\subset\Gamma\) be the inertia group of the extension \(\overline{F}/F\), and \(\Gamma_{\mathbf{f}}= \operatorname{Gal}(\overline{\mathbf{f}}/\mathbf{f})=\Gamma/I\) the absolute Galois group of \(\mathbf{f}\). We fix a finite Galois extension \(L/F\) of the base field where the torus \(T=(Y\otimes\overline{F}^\times)^\Gamma\) splits over \(L\), and write \(e\in\mathbb{Z}_{>0}\) for its ramification index.
From now, we suppose that \(\mu_n\subset\mathbf{f}^\times\), so that we can employ the exact sequence \(0\to (Y/Y^\#)(1)\to Y^\#\otimes\overline{\mathbf{f}}^\times\to Y\otimes\overline{\mathbf{f}}^\times\to 0\) in 21.
Lemma 38. Assume that the degree \(n\) of the cover is prime to the ramification index \(e\) of the fixed splitting field \(L/F\).
The inclusion \(Y^\#\hookrightarrow Y\) induces a short exact sequence \[0\to(Y/Y^\#)^I(1)\to(Y^\#\otimes\overline{\mathbf{f}}^\times)^I \to(Y\otimes\overline{\mathbf{f}}^\times)^I\to 0.\]
The exact sequence in [gSfi] induces an isomorphism \[\iota\colon\operatorname H^1(\Gamma_{\mathbf{f}}, (Y^\#\otimes\overline{\mathbf{f}}^\times)^I)\to \operatorname H^1(\Gamma_{\mathbf{f}}, (Y\otimes\overline{\mathbf{f}}^\times)^I).\]
Proof. [gSfi] Let \(I_L\subset\operatorname{Gal}(\overline{F}/L)\) be the inertia group for \(L\). Note that \(I_L\) acts trivially on \(Y\otimes\overline{\mathbf{f}}^\times\) and \(Y^\#\otimes\overline{\mathbf{f}}^\times\), and that \(\#(I/I_L)=e\). Then the sequence in 21 gives the exact sequence \[(Y^\#\otimes\overline{\mathbf{f}}^\times)^{I/I_L}\to (Y\otimes\overline{\mathbf{f}}^\times)^{I/I_L}\to \operatorname H^1(I/I_L,(Y/Y^\#)(1)).\] The last term of this sequence vanishes, since \((Y/Y^\#)(1)\) is a \(\mathbb{Z}/n\)-module and the integers \(n\) and \(e\) are relatively prime by assumption [12].
[efI] Since the cohomological dimension of the finite field \(\mathbf{f}\) is one, the two maps \[\operatorname H^1(\Gamma_{\mathbf{f}}, (Y\otimes\overline{\mathbf{f}}^\times)^I)\stackrel n\to \operatorname H^1(\Gamma_{\mathbf{f}}, (Y^\#\otimes\overline{\mathbf{f}}^\times)^I)\stackrel\iota\to \operatorname H^1(\Gamma_{\mathbf{f}}, (Y\otimes\overline{\mathbf{f}}^\times)^I)\] are surjective. Since \((Y\otimes\overline{\mathbf{f}}^\times)^I\) is an affine algebraic group defined over \(\mathbf{f}\), we know that the group \(\operatorname H^1(\Gamma_{\mathbf{f}}, (Y\otimes\overline{\mathbf{f}}^\times)^I)\) has finite order [11]. Hence the composite \(\operatorname H^1(\Gamma_{\mathbf{f}}, (Y\otimes\overline{\mathbf{f}}^\times)^I)\stackrel n\to \operatorname H^1(\Gamma_{\mathbf{f}}, (Y\otimes\overline{\mathbf{f}}^\times)^I)\) is an isomorphism, and so is the map \(\iota\). ◻
Corollary 39. The connecting homomorphism \[\partial\colon ((Y\otimes\overline{\mathbf{f}}^\times)^I)^{\Gamma_{\mathbf{f}}}\to \operatorname H^1(\Gamma_{\mathbf{f}},(Y/Y^\#)^I(1))\] induced from the short exact sequence in 38 [gSfi] is surjective.
Proof. In the exact sequence \[((Y\otimes\overline{\mathbf{f}}^\times)^I)^{\Gamma_{\mathbf{f}}} \stackrel\partial\to \operatorname H^1(\Gamma_{\mathbf{f}},(Y/Y^\#)^I(1))\to \operatorname H^1(\Gamma_{\mathbf{f}}, (Y^\#\otimes\overline{\mathbf{f}}^\times)^I)\stackrel\iota\to \operatorname H^1(\Gamma_{\mathbf{f}}, (Y\otimes\overline{\mathbf{f}}^\times)^I),\] the middle map is zero, since the last map \(\iota\) is an isomorphism. ◻
For a \(\Gamma\)-module \(A\), let \(\inf\colon\operatorname H^1(\Gamma_{\mathbf{f}},A^I)\to \operatorname H^1(\Gamma,A)\) be the inflation map on the first cohomology group. This \(\inf\) is injective, and fits into the exact sequence in low dimensions given by the Lyndon-Hochschild-Serre spectral sequence. Thus, we often regard \(\operatorname H^1(\Gamma_{\mathbf{f}},A^I)\subset \operatorname H^1(\Gamma,A)\).
Lemma 40. Let \(A\times A'\stackrel B\to\mu_n\) be a non-degenerate \(\Gamma\)-equivariant pairing between two finite \(\Gamma\)-modules. Then, in the non-degenerate pairing \(\beta\colon\operatorname H^1(\Gamma,A)\times \operatorname H^1(\Gamma,A')\stackrel{B(\,\cup\,)}\to \operatorname H^2(\Gamma,\mu_n)=\mathbb{Z}/n\) due to the local Tate duality, we have \[\inf(\operatorname{H}^1(\Gamma_{\mathbf{f}},A^I))^\perp= \inf(\operatorname{H}^1(\Gamma_{\mathbf{f}},A'^I)).\]
Proof. The restriction \(\operatorname H^1(\Gamma_{\mathbf{f}},A^I)\times \operatorname H^1(\Gamma_{\mathbf{f}},A'^I)\to \operatorname H^2(\Gamma,\mu_n)\) of \(\beta\) factors through \(\operatorname H^2(\Gamma_{\mathbf{f}},\mu_n)=0\). That is, \[\beta\left(\operatorname H^1(\Gamma_{\mathbf{f}},A^I),\, \operatorname H^1(\Gamma_{\mathbf{f}},A'^I)\right)=0.\] It remains to show that \(\#\operatorname H^1(\Gamma,A)= \#\operatorname H^1(\Gamma_{\mathbf{f}},A^I)\cdot \#\operatorname H^1(\Gamma_{\mathbf{f}},A'^I)\). Indeed, we see the equalities \[\begin{align} \#\operatorname H^1(\Gamma,A) &=\#\operatorname H^0(\Gamma,A)\cdot\#\operatorname H^2(\Gamma,A) \\ &=\#\operatorname H^0(\Gamma,A)\cdot\#\operatorname H^0(\Gamma,A') \\ &=\#\operatorname H^0(\Gamma_{\mathbf{f}},A^I)\cdot \#\operatorname H^0(\Gamma_{\mathbf{f}},A'^I) \\ &=\#\operatorname H^1(\Gamma_{\mathbf{f}},A^I)\cdot \#\operatorname H^1(\Gamma_{\mathbf{f}},A'^I) \end{align}\] as follows:
For the first equality, note that \(A\) is a finite \(\mathbb{Z}/n\)-module by definition. Since \(n\) is prime to the residual characteristic \(p\) by assumption, the Euler-Poincaré characteristic is \(\frac{\#\operatorname H^0(\Gamma,A)\cdot \#\operatorname H^2(\Gamma,A)}{\#\operatorname H^1(\Gamma,A)}=1\). The second equality follows from the local Tate duality between \(\operatorname H^2\) and \(\operatorname H^0\). The third one holds by definition. For an explicit proof of the last equality, we may consult Harari’s textbook [13]. ◻
Note that this lemma slightly generalizes the well-known orthogonality theorem for unramified cohomology [11].
Corollary 41. Assume that the degree \(n\) of the cover is prime to the ramification index \(e\) of the fixed splitting field \(L/F\).
For the connecting homomorphism \(\partial=\partial_{Y/Y^\#}\colon T=(Y\otimes\overline{F}^\times)^\Gamma\to \operatorname H^1(\Gamma,(Y/Y^\#)(1))\), the annihilator of the subgroup \(\partial((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma) \subset\operatorname H^1(\Gamma,(Y/Y^\#)(1))\) is itself: \[\partial((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma) ^\perp =\partial((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma).\]
As subgroups of \(T=(Y\otimes\overline{F}^\times)^\Gamma\), we have an equality \[\operatorname Z^\dagger_T ((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma)= (Y\otimes\overline{\mathcal{O}}^\times)^\Gamma+\iota(T^\#).\]
Proof. [eOdd] By assumption, 38 [gSfi] ensures that the horizontal sequences in the commutative diagram \[\begin{array}{*9c} 0&\to& (Y/Y^\#)(1) &\to& Y^\#\otimes\overline{\mathcal{O}}^\times &\to& Y\otimes\overline{\mathcal{O}}^\times &\to& 0\\ & & \rotatebox[origin=c]{90}= & & \downarrow & & \rotatebox[origin=c]{-90}{\twoheadrightarrow}\rlap{\;\scriptstyle\text{split}} & &\\ 0&\to& (Y/Y^\#)(1) &\to& Y^\#\otimes\overline{\mathbf{f}}^\times &\to& Y\otimes\overline{\mathbf{f}}^\times &\to& 0\\ & & \uparrow & & \uparrow & & \uparrow & &\\ 0&\to& (Y/Y^\#)^I(1)&\to& (Y^{\#}\otimes\overline{\mathbf{f}}^\times)^{I}&\to& (Y\otimes\overline{\mathbf{f}}^\times)^I&\to& 0 \end{array}\] are exact. The connecting homomorphisms \(\partial\) on the Galois cohomology groups form the commutative diagram \[\begin{array}{*9c} (Y\otimes\overline{\mathcal{O}}^\times)^\Gamma &\stackrel\partial\to&\operatorname H^1(\Gamma,(Y/Y^\#)(1))\\ \rotatebox[origin=c]{-90}{\twoheadrightarrow} & &\rotatebox[origin=c]{90}=\\ (Y\otimes\overline{\mathbf{f}}^\times)^\Gamma &\stackrel\partial\to&\operatorname H^1(\Gamma,(Y/Y^\#)(1))\\ \rotatebox[origin=c]{90}= & &\rotatebox[origin=c]{90}{\hookrightarrow}\rlap{\scriptstyle\inf}\\ ((Y\otimes\overline{\mathbf{f}}^\times)^I)^{\Gamma_{\mathbf{f}}}&\stackrel\partial\twoheadrightarrow&\operatorname H^1(\Gamma_{\mathbf{f}},(Y/Y^\#)^I(1))\rlap. \end{array}\] In the left column, the downward map \((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma\to (Y\otimes\overline{\mathbf{f}}^\times)^\Gamma\) is surjective, since \(Y\otimes\overline{\mathcal{O}}^\times\to Y\otimes\overline{\mathbf{f}}^\times\) is a split surjection. The bottom \(\partial\) is also surjective by 39. Recall that \(\inf\) in the right column is injective. Then \(\partial((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma)= \partial((Y\otimes\overline{\mathbf{f}}^\times)^\Gamma)= \inf\left(\operatorname{H}^1(\Gamma_{\mathbf{f}}, (Y/Y^\#)^I(1))\right)\). By 40, \[\partial((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma)= \inf\left(\operatorname{H}^1(\Gamma_{\mathbf{f}}, (Y/Y^\#)^I(1))\right)^\perp= \partial((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma)^\perp.\] [eZo] By 34, we have \[\begin{align} \operatorname Z^\dagger_T ((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma)= \partial^{-1}\partial ((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma) &=(Y\otimes\overline{\mathcal{O}}^\times)^\Gamma+ \operatorname{Ker}\partial_{Y/Y^\#}\\ &=(Y\otimes\overline{\mathcal{O}}^\times)^\Gamma+\iota(T^\#). \end{align}\] ◻
41 allows us to describe the intersection \(\mathrm Z^\dagger\cap(Y\otimes\overline{\mathcal{O}}^\times)^\Gamma\) by the maximal split subtorus \(T^\Gamma=Y^\Gamma\otimes F^\times\) of \(T\).
Proposition 42. As in 41, assume that the degree \(n\) of the cover is prime to the ramification index \(e\) of the fixed splitting field \(L/F\). Then the following equalities hold. \[\begin{gather} \setcounter{equation}{0} \tag{1} T=\iota(T^\#)+T^\Gamma+ (Y\otimes\overline{\mathcal{O}}^\times)^\Gamma\\ \tag{2} \mathrm Z^\dagger\cap(Y\otimes\overline{\mathcal{O}}^\times)^\Gamma =\operatorname Z^\dagger_T(T^\Gamma)\cap (Y\otimes\overline{\mathcal{O}}^\times)^\Gamma. \end{gather}\]
Proof. 1 As usual, we normalize the valuation map \(\operatorname{ord}\colon\overline{F}^\times\to\mathbb{Q}\) so that \(\operatorname{ord}(\overline{F}^\times)=\mathbb{Z}\). Then, for the induced map \(\operatorname{ord}\colon T=(Y\otimes\overline{F}^\times)^\Gamma\to (Y\otimes\mathbb{Q})^\Gamma=Y^\Gamma\otimes\mathbb{Q}\), we have \(\operatorname{ord}(T^\Gamma)=Y^\Gamma\) and \(\operatorname{ord}(T)\subset\frac{1}{e}Y^\Gamma\). Since \(nY\subset Y^\#\) by definition, we see the inclusion \(n(Y\otimes\overline{F})^\Gamma\subset \iota((Y^\#\otimes\overline{F})^\Gamma)\) in \(T=(Y\otimes\overline{F})^\Gamma\), and hence \(n\cdot\operatorname{ord}(T)\subset\operatorname{ord}(\iota(T^\#))\). Since the integers \(n\) and \(e\) are relatively prime by assumption, \[\operatorname{ord}(T)= n\operatorname{ord}(T)+e\operatorname{ord}(T)\subset \operatorname{ord}(\iota(T^\#))+Y^\Gamma= \operatorname{ord}(\iota(T^\#)+T^\Gamma).\] Since the map \(\operatorname{ord}\colon T\to Y^\Gamma\otimes\mathbb{Q}\) has kernel \((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma\), the claimed equality holds.
Our main theorem aims to determine the finite abelian group \(\mathcal{S}_{\widetilde{T}}=\mathrm Z^\dagger/\iota(T^\#)\) introduced in 12. Recall that its dual group \(\operatorname{Hom}(\mathcal{S}_{\widetilde{T}},\mathbb{C}^\times)\) parametrizes a packet in the local Langlands correspondence for a Brylinski-Deligne covering \(\mu_n\to\widetilde{T}\to T\) of a torus over a non-archimedean local field \(F\) of characteristic zero. Here, \(\mu_n\subset \mathbf{f}^\times\) denotes the cyclic subgroup of order \(n\).
In this section, we prove the main theorem by reducing it to a comparison between the image \(\mathrm{Z}^\dagger\subset T\) of the center \(\operatorname Z(\widetilde{T})\subset\widetilde{T}\) and the image of the isogeny \(T^\#\stackrel\iota\to T\) of tori. We analyze these two subgroups of \(T\) via the exact sequence \(0\to(Y\otimes\overline{\mathcal{O}}^\times)^\Gamma\to T \stackrel{\operatorname{ord}}\to Y^\Gamma\otimes\mathbb{Q}\) appeared in 22. In addition to the cocharacter lattices \(Y^\#\subset Y\) of \(T^\#\) and \(T\), we employ the subgroup \[Y^{\Gamma\#}=(Y^\Gamma)^\#= \{y\in Y\mid B_{\widetilde{T}}(y,Y^\Gamma)\subset n\mathbb{Z}\}\] of \(Y\), where \(B_{\widetilde{T}}\colon Y\times Y\to\mathbb{Z}\) is the bilinear form attached to the Brylinski-Deligne covering \(\widetilde{T}\to T\). Then the inclusion \(Y^{\Gamma\#}\hookrightarrow Y\) again induces an isogeny \(T^{\Gamma\#}\stackrel\iota\to T\) of tori.
Lemma 43. Let \(L/F\) be a finite Galois extension of the base field where the torus \(T\) splits over \(L\), and \(e\in\mathbb{Z}_{>0}\) the ramification index of \(L/F\). Assume that the ramification index \(e\) and the degree \(n\) of the cover are relatively prime.
For the map \(\operatorname{ord}\colon T=(Y\otimes\overline{F}^\times)^\Gamma \to(Y\otimes\mathbb{Q})^\Gamma=Y^\Gamma\otimes\mathbb{Q}\), we have an equality \(\operatorname{ord}({\mathrm{Z}^\dagger})= \operatorname{ord}({\iota(T^\#)})\) of the images.
Let \(T^{\Gamma\#}\stackrel\iota\to T\) and \(T^{\#}\stackrel\iota\to T\) respectively denote the isogenies of tori. Then as subgroups of \(T=(Y\otimes\overline{F}^\times)^\Gamma\), the following equalities hold: \[\begin{align} \mathrm{Z}^\dagger\cap(Y\otimes \overline{\mathcal{O}}^\times)^\Gamma&= \iota((Y^{\Gamma\#}\otimes \overline{\mathcal{O}}^\times)^\Gamma)\\ \iota(T^\#) \cap(Y\otimes\overline{\mathcal{O}}^\times)^\Gamma&= \iota((Y^{\#}\otimes\overline{\mathcal{O}}^\times)^\Gamma). \end{align}\]
Proof. By assumption, we may apply the results in 5.
[aOd] Since we know that \(\iota(T^\#)\subset\mathrm Z^\dagger\), it suffices to see the inclusion \[\mathrm Z^\dagger\subset \iota(T^\#)+\operatorname{Ker}(\operatorname{ord} )=\iota(T^\#)+(Y\otimes\overline{\mathcal{O}}^\times)^\Gamma.\] Indeed, \(\mathrm Z^\dagger\subset\operatorname Z^\dagger_T ((Y\otimes\overline{\mathcal{O}}^\times)^\Gamma)= \iota(T^\#)+(Y\otimes\overline{\mathcal{O}}^\times)^\Gamma\) by 41.
[aZd] By 42 2 , \(\mathrm Z^\dagger\cap(Y\otimes\overline{\mathcal{O}}^\times)^\Gamma =\operatorname Z^\dagger_T(T^\Gamma)\cap (Y\otimes\overline{\mathcal{O}}^\times)^\Gamma\). Since \(\operatorname Z^\dagger_T(T^\Gamma)=\iota(T^{\Gamma\#})\) as in 37, we have \[\mathrm Z^\dagger\cap(Y\otimes\overline{\mathcal{O}}^\times)^\Gamma =\iota(T^{\Gamma\#})\cap (Y\otimes\overline{\mathcal{O}}^\times)^\Gamma =\iota((Y^{\Gamma\#}\otimes\overline{F}^\times)^\Gamma)\cap (Y\otimes\overline{\mathcal{O}}^\times)^\Gamma.\] Thus 24 [eePbi] for \(Y'=Y^{\Gamma\#}\) ensures the first identity in the statement. The second identity is the special case of 24 [eePbi] for \(Y'=Y^\#\). ◻
Theorem 44. Let \(F\) be a non-archimedean local field of characteristic zero, \(\mathbf{f}\) the residue field of the valuation ring of \(F\), and \(\mu_n\subset \mathbf{f}^\times\) the cyclic subgroup of order \(n\). Suppose that an algebraic torus \(T=\mathbb{T}(F)\) defined over \(F\) splits over a finite Galois extension \(L/F\) whose ramification index \(e\) is relatively prime to \(n\). Let \(\mu_n\to\widetilde{T}\to T\) be a Brylinski-Deligne covering group.
Then the group \(\mathcal{S}=\mathcal{S}_{\widetilde{T}}\) parametrizing a packet of representations of \(\widetilde{T}\) is \[\begin{align} \mathcal{S}_{\widetilde{T}} &=\left.\iota((Y^{\Gamma\#}\otimes_{\mathbb{Z}} \overline{\mathcal{O}}^\times)^\Gamma)\middle/ \iota((Y^{\#}\otimes_{\mathbb{Z}} \overline{\mathcal{O}}^\times)^\Gamma)\right..\\ &=\left.\iota((Y^{\Gamma\#}\otimes_{\mathbb{Z}} \overline{\mathbf{f}}^\times)^\Gamma)\middle/ \iota((Y^{\#}\otimes_{\mathbb{Z}} \overline{\mathbf{f}}^\times)^\Gamma)\right.. \end{align}\]
Proof.
By definition, \(\mathcal{S}_{\widetilde{T}}=\mathrm Z^\dagger/\iota(T^\#)\), where \(T^\#={{(Y^\#\otimes_{\mathbb{Z}}\overline{F}^\times)^\Gamma}}\). The three equalities comparing these two subgroups \(\mathrm Z^\dagger\) and \(\iota(T^\#)\) of \(T\) (43) fit in the diagram \[\begin{array}[t]{*9c} \llap{0\to\;} (Y\otimes\overline{\mathcal{O}}^\times)^\Gamma&\to& (Y\otimes\overline{F}^\times)^\Gamma& \stackrel{\operatorname{ord}}\to& (Y\otimes\overline{\mathbb{Q}}^\times)^\Gamma\\ \cup&&\cup&&\cup\\ \iota((Y^{\Gamma\#}\otimes \overline{\mathcal{O}}^\times)^\Gamma) &\to& {\mathrm{Z}^\dagger}&\to& \operatorname{ord}(\mathrm{Z}^\dagger)\\ \cup&&\cup&&\rotatebox[origin=c]{90}=\\ \iota((Y^{\#}\otimes\overline{\mathcal{O}}^\times)^\Gamma) &\to&{\iota(T^\#)} &\to&\operatorname{ord}({\iota(T^\#)}). \end{array}\] Hence, this gives an isomorphism \[\left.\iota((Y^{\Gamma\#}\otimes \overline{\mathcal{O}}^\times)^\Gamma)\middle/ \iota((Y^{\#}\otimes \overline{\mathcal{O}}^\times)^\Gamma)\right. \stackrel\sim\to\mathrm{Z}^\dagger/\iota(T^\#)= \mathcal{S}_{\widetilde{T}}.\]
Let \(\overline{\mathfrak p}\subset\overline{\mathcal{O}}\) be the maximal ideal of the valuation ring in the algebraic closure \(\overline{F}\). Since \(\mu_n\subset\mathbf{f}^\times\) by assumption, the inclusions \(nY\subset Y^\#\subset Y^{\Gamma\#}\subset Y\) induces isomorphisms \[Y^\#\otimes(1+\overline{\mathfrak p})\stackrel\sim\to Y^{\Gamma\#}\otimes(1+\overline{\mathfrak p})\stackrel\sim\to Y\otimes(1+\overline{\mathfrak p}).\] Then the split exact sequence \(1\to1+\overline{\mathfrak p}\to \overline{\mathcal{O}}^\times \to\overline{\mathbf{f}}^\times\to1\) gives a commutative diagram \[\begin{matrix} 0 & &0 & &0 \\ \downarrow & &\downarrow & &\downarrow \\ (Y^\#\otimes(1+\overline{\mathfrak p}))^\Gamma &\stackrel{\sim}{\to} &(Y^{\Gamma\#}\otimes(1+\overline{\mathfrak p}))^\Gamma &\stackrel{\sim}{\to} &(Y\otimes(1+\overline{\mathfrak p}))^\Gamma \\ \downarrow & &\downarrow & &\downarrow \\ (Y^\#\otimes\overline{\mathcal{O}}^\times)^\Gamma &\to &(Y^{\Gamma\#}\otimes\overline{\mathcal{O}}^\times)^\Gamma &\to &(Y\otimes\overline{\mathcal{O}}^\times)^\Gamma\\ \downarrow & &\downarrow & &\downarrow \\ (Y^\#\otimes\overline{\mathbf{f}}^\times)^\Gamma &\to &(Y^{\Gamma\#}\otimes\overline{\mathbf{f}}^\times)^\Gamma &\to &(Y\otimes\overline{\mathbf{f}}^\times)^\Gamma \\ \downarrow & &\downarrow & &\downarrow \\ 0 & &0 & &0\rlap{,} \\ \end{matrix}\] where the top horizontal maps are isomorphisms, and the vertical sequences are split exact. Thus the map \[\mathcal{S}_{\widetilde{T}}= \iota((Y^{\Gamma\#}\otimes\overline{\mathcal{O}}^\times)^\Gamma) /\iota((Y^\#\otimes\overline{\mathcal{O}}^\times)^\Gamma)\to \iota((Y^{\Gamma\#}\otimes\overline{\mathbf{f}}^\times)^\Gamma) /\iota((Y^\#\otimes\overline{\mathbf{f}}^\times)^\Gamma)\] is an isomorphism.
◻
The trivial example of the main theorem is a Brylinski-Deligne cover \(\widetilde{T}\to T\) of a split torus. Then \(\mathcal{S}_{\widetilde{T}}=1\), since \(Y^\Gamma=Y\). Another elementary example is a cover \(\widetilde{T}\to T\) of an unramified torus, i.e., a torus splitting over an unramified extension \(L/F\) of the base field. Then 44 regains Weissman’s result [3] on covers of an unramified torus.
Thus, our main theorem is a generalization of Weissman’s work on covering groups. Moreover, this theorem partially realizes his hope [3] to parametrize a packet in the local Langlands correspondence for a cover of a ramified torus.