# Jacquet–Langlands: elliptic classes and division-algebra representations

*Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI. Public domain (CC0).*

An irreducible representation of a division algebra's multiplicative group can be finite-dimensional and still correspond to an infinite-dimensional representation of a general linear group. The bridge compares character values on the conjugacy classes that the two groups share. In degree two, the constant character on the division algebra matches the Steinberg character, whose value on an elliptic element is minus one.

We state the general local Jacquet–Langlands theorem. We prove the matching of regular elliptic classes for a division algebra, finite-dimensionality of its smooth irreducible representations, and the degree-two character calculation. We then compose the theorem with the local Langlands correspondence and specify the cohomological realization with its duals and norm twist. The basic references are [Jacquet–Langlands 1970, §15], [Badulescu 2002, Theorem 1.1] and [Arthur 2005, §25].

## 1. Inner forms and the common characteristic polynomial

Throughout, \(F\) is a nonarchimedean local field, of either characteristic, with residue field \(\mathbb F_q\). Let \(D\) be a central division algebra of dimension \(d^2\) over \(F\), and put
\[
A=M_m(D),\qquad n=md,\qquad G'=A^\times=\mathrm{GL}_m(D),\qquad G=\mathrm{GL}_n(F).
\tag{1.1}
\]
After extending scalars to a separable closure, \(A\) becomes a matrix algebra of degree \(n\). Thus \(G'\) is an inner form of \(G\). Here *degree* means the square root of the algebra's dimension.

For \(x\in A\), the **reduced characteristic polynomial** is
\[
P_x(T)=\det(T-x\mid A\otimes_F\overline F\simeq M_n(\overline F)).
\tag{1.2}
\]
The polynomial is independent of the splitting isomorphism and has coefficients in \(F\). Its constant term is \((-1)^n\operatorname{Nrd}(x)\), where \(\operatorname{Nrd}\) is the reduced norm. An element is regular semisimple when this polynomial is separable of degree \(n\). Elements \(g\in G\) and \(g'\in G'\) **match** when they are regular semisimple and have the same polynomial. In particular
\[
\det g=\operatorname{Nrd}(g').
\tag{1.3}
\]

A regular semisimple element of \(G\) is elliptic when its centralizer modulo the center is compact. In \(\mathrm{GL}_n(F)\), these are exactly the elements whose characteristic polynomial is irreducible. Indeed its centralizer is the multiplicative group of the étale algebra generated by the element. For a field extension \(E/F\), the quotient \(E^\times/F^\times\) is compact: its valuation quotient is finite and its unit quotient is compact. A product of two or more fields leaves an unbounded relative valuation direction after dividing by diagonal \(F^\times\).

For the next two sections take \(m=1,d=n\). We use the local Brauer invariant input
\[
\operatorname{Br}(F)\simeq\mathbb Q/\mathbb Z,\qquad
\operatorname{inv}_E(D\otimes_F E)=[E:F]\operatorname{inv}_F(D),
\tag{1.4}
\]
with \(\operatorname{inv}_F(D)\) of exact order \(n\). These are the local class-field and central-simple-algebra theorems [Milne, Chapter III, Theorem 2.1 and equation (29); Chapter IV, Proposition 4.3 and Remark 4.4(b)]. We supply the embedding and conjugacy arguments below.

## 2. Matching elliptic classes, with proof

**Theorem 2.1.** Reduced characteristic polynomials give a bijection between regular semisimple conjugacy classes in \(D^\times\) and regular elliptic conjugacy classes in \(\mathrm{GL}_n(F)\). Both are parametrized by monic separable irreducible polynomials of degree \(n\) with nonzero constant term.

**Proof.** Let \(x\in D^\times\). Its minimal polynomial is irreducible: a proper factorization would give two nonzero elements of the division algebra with product zero. Thus \(F[x]\) is a field. If \(x\) is regular semisimple, its split matrix has \(n\) distinct eigenvalues. The minimal polynomial has all these roots, since a polynomial annihilates a diagonalizable matrix precisely when it vanishes on every eigenvalue. It has degree \(n\) and equals \(P_x\). Its companion matrix is regular elliptic in \(G\).

Conversely let \(P\) be as in the statement and set \(E=F[T]/(P)\). Its degree is \(n\), so (1.4) makes \(D\otimes_F E\) split:
\[
D\otimes_F E\simeq M_n(E).
\]
Let \(W=E^n\) be a simple module for this algebra. Its \(F\)-dimension is \(n^2\). As a left \(D\)-module, it is a vector space over the division ring \(D\); this dimension makes its rank one. Choose a left \(D\)-linear identification \(W\simeq D\). Scalar multiplication by \(E\) commutes with \(D\), and gives an injection
\[
E\longrightarrow\operatorname{End}_D(D)=D^{\mathrm{op}}.
\]
Endomorphisms here are right multiplications. Since \(E\) is commutative, their multiplication also defines an \(F\)-algebra embedding of \(E\) into \(D\). The image \(x\) of \(T\bmod P\) has minimal polynomial \(P\). Its reduced characteristic polynomial has degree \(n\), and is divisible by its minimal polynomial, so it too is \(P\). Its nonzero constant term makes \(x\) invertible.

Uniqueness on the matrix side follows from rational canonical form: a matrix with irreducible characteristic polynomial \(P\) is conjugate to the companion matrix. On the division side, suppose \(x,y\) have polynomial \(P\). Consider
\[
U=\{c\in D:cx=yc\}.
\]
This is the solution space of linear equations over \(F\). Over \(\overline F\), both \(x,y\) diagonalize with the same distinct eigenvalues; they are conjugate and their intertwiner space has dimension \(n\). The rank of a linear system is unchanged by scalar extension, so \(\dim_F U=n\). Choose \(0\ne c\in U\). It is invertible in \(D\), giving \(y=cxc^{-1}\). This proves the bijection. ∎

The separability condition matters in positive characteristic: a purely inseparable maximal subfield does not supply a regular semisimple element. The theorem concerns the regular elliptic locus.

**Example 2.2 (an unramified quadratic class).** Take \(F=\mathbb Q_3\), \(E=F(i)\), \(i^2=-1\). This is unramified because \(T^2+1\) is irreducible modulo three. The quaternion division algebra has presentation
\[
D=E\oplus E\Pi,\qquad \Pi^2=3,\qquad \Pi a\Pi^{-1}=\overline a.
\]
The element \(x=1+i\) has reduced trace 2 and norm 2, hence polynomial \(T^2-2T+2\). It matches
\[
g=\begin{pmatrix}0&-2\\1&2\end{pmatrix}.
\tag{2.1}
\]
Both centralizers are \(E^\times\), compact modulo \(F^\times\). The roots \(1+i,1-i\) give the same conjugacy class; conjugation by \(\Pi\) interchanges them. The determinant of (2.1) equals the reduced norm, including its sign.

In general, for \(a\in E^\times\setminus F^\times\) in an unramified quadratic extension, use \(T^2-\operatorname{Tr}(a)T+N(a)\) and its companion matrix. The class depends on this polynomial, rather than on the embedding of \(E\).

## 3. Smooth irreducible representations of \(D^\times\)

The division algebra has a surjective valuation \(v_D:D^\times\to\mathbb Z\) restricting to \(n v_F\). Put
\[
\mathcal O_D=\{x:v_D(x)\ge0\},\qquad
\mathfrak P_D=\{x:v_D(x)>0\},\qquad U_r=1+\mathfrak P_D^r\quad(r\ge1).
\]
The groups \(U_r\) are compact open and normal in \(D^\times\), since conjugation preserves \(v_D\). They form a neighborhood basis at the identity. This valuation and the cyclic presentation below are the algebraic input [Badulescu 2002, §2.1], valid in both characteristics.

**Proposition 3.1.** Reduction gives
\[
\mathcal O_D/\mathfrak P_D\simeq\mathbb F_{q^n},\qquad
\mathcal O_D^\times/U_1\simeq\mathbb F_{q^n}^\times.
\tag{3.1}
\]
Moreover \(D^\times/F^\times\) is compact.

**Proof.** There is an unramified maximal subfield \(L/F\) of degree \(n\) and a uniformizer \(\Pi\) with
\[
D=\bigoplus_{j=0}^{n-1}\Pi^j L,\qquad \Pi^n=\varpi_F,\qquad
\Pi^{-1}a\Pi=\sigma(a),
\tag{3.2}
\]
where \(\sigma\) generates \(\operatorname{Gal}(L/F)\). In a nonzero sum, nonzero terms have valuations \(j+n v_L(a_j)\). These are distinct modulo \(n\), so the minimum is unique and cannot cancel. The sum is integral exactly when all \(a_j\) are integral. Modulo \(\mathfrak P_D\), only \(a_0\) remains, modulo \(\varpi_F\mathcal O_L\). This proves the residue field assertion. Reduction of units is surjective with kernel \(U_1\), proving the group assertion.

Every element of \(D^\times\) is \(\Pi^j u\), \(u\in\mathcal O_D^\times\). Modulo central powers of \(\varpi_F=\Pi^n\), only \(j=0,\ldots,n-1\) are needed. The quotient \(D^\times/F^\times\) is therefore covered by images of \(n\) compact sets \(\Pi^j\mathcal O_D^\times\). ∎

**Theorem 3.2.** Every irreducible smooth complex representation \(V\) of \(D^\times\) is finite-dimensional and is trivial on an open normal subgroup. No admissibility hypothesis is needed.

**Proof.** Choose \(0\ne v\in V\). Some \(U_r\) fixes \(v\), by smoothness. Normality makes it fix every translate \(gv\). These span \(V\), by irreducibility, so \(U_r\) acts trivially on \(V\).

Let \(z=\varpi_F\) in the center, and make \(V\) a module over \(R=\mathbb C[t,t^{-1}]\) by letting \(t\) act as \(z\). The quotient
\[
D^\times/\langle z\rangle U_r
\]
is finite: the valuations need only \(n\) representatives and \(\mathcal O_D^\times/U_r\) is finite. Choose coset representatives \(g_1,\ldots,g_s\). Their translates \(g_jv\) generate \(V\) over \(R\), since every group translate differs from one of them by a central power of \(z\) and an element of \(U_r\).

The ring \(R\) is a principal ideal domain. The elementary structure theorem for its finitely generated modules applies. The torsion submodule \(V_{\mathrm{tor}}\) is group-stable, because \(t\) is central. If nonzero, it equals \(V\) by irreducibility. A finitely generated torsion \(R\)-module has finite complex dimension: each cyclic summand is \(R/(f)\) for a nonzero Laurent polynomial \(f\).

If \(V_{\mathrm{tor}}=0\), then \(V\simeq R^a\) for \(a>0\). Multiplication by \(t-1\) gives a nonzero proper group-stable submodule. It is nonzero because \(R^a\) is torsion-free, and proper because the quotient is \(\mathbb C^a\). This contradicts irreducibility. Thus the torsion case holds and \(V\) is finite-dimensional. The already obtained \(U_r\) is the open normal subgroup in its kernel. ∎

Finite-dimensionality gives a central character \(\omega_V\): a central operator has a nonzero eigenspace, which is group-stable and hence all of \(V\). Central units have finite image, since their intersection with \(U_r\) is open. An unramified twist through \(\operatorname{Nrd}\) can make \(\omega_V(\varpi_F)\) have absolute value one. Indeed \(\operatorname{Nrd}(\varpi_F)=\varpi_F^n\), so choose an appropriate \(n\)-th root for the twisting character's uniformizer value. Average an inner product over the compact quotient \(D^\times/F^\times\); the integrand descends because the center now acts unitarily. The twist becomes unitary and its matrix coefficients are square-integrable on that quotient. Every smooth irreducible representation of \(D^\times\) is therefore **essentially square-integrable modulo the center**.

The open normal kernel need not have finite index. A scalar value at a central uniformizer can have infinite image.

## 4. The character theorem and the degree-two sign

Let \(\mathcal E_2(H)\) denote the essentially square-integrable irreducible representations of \(H\). Write \(\Theta_\pi\) for the locally constant character function on its regular semisimple locus, representing its trace distribution.

**Theorem 4.1 (local Jacquet–Langlands, stated).** For the groups in (1.1), there is a bijection
\[
\operatorname{JL}_{D,m}:\mathcal E_2(G')\longrightarrow\mathcal E_2(G)
\]
characterized by
\[
\Theta_{\operatorname{JL}_{D,m}(\tau)}(g)
=(-1)^{n-m}\Theta_\tau(g')
\tag{4.1}
\]
on matching regular semisimple elements. In particular it holds on matching elliptic elements. JL preserves central characters and commutes with contragredients and with corresponding twists by \(\chi\circ\operatorname{Nrd}\) and \(\chi\circ\det\).

The characteristic-zero theorem is due to Deligne–Kazhdan–Vignéras and Rogawski; [Arthur 2005, Theorem 25.1] gives its tempered transfer formulation and original locators. The positive-characteristic theorem is [Badulescu 2002, Theorem 1.1], whose \(r\) is our \(m\). For quaternion algebras, [Jacquet–Langlands 1970, Theorem 15.1] identifies the image as special and supercuspidal representations.

The twist property follows from characterization: (1.3) multiplies both values by \(\chi(\det g)\). For the dual, replace \(g,g'\) by their inverses and use \(\Theta_{\pi^\vee}(g)=\Theta_\pi(g^{-1})\). Central-character equality follows by multiplying a matching pair by a central scalar. A square-integrable character is not identically zero on its elliptic locus, by elliptic character orthogonality [Jacquet–Langlands 1970, §15; Badulescu 2002, Theorem 5.3]. A pair with nonzero value then identifies the central scalars.

When \(m=1\), Theorem 3.2 and compactness show that the domain is **all** smooth irreducible representations of \(D^\times\).

**Proposition 4.2 (the sign check).** For the quaternion division algebra,
\[
\operatorname{JL}(1_{D^\times})=\mathrm{St}_2,\qquad
\operatorname{JL}(\chi\circ\operatorname{Nrd})
=\mathrm{St}_2\otimes(\chi\circ\det).
\tag{4.2}
\]

**Proof.** Use the earlier proved rank-two exact sequence
\[
0\longrightarrow1_G\longrightarrow C^\infty(\mathbb P^1(F))
\longrightarrow\mathrm{St}_2\longrightarrow0,
\tag{4.3}
\]
whose middle term is unnormalized induction of the trivial Borel character. An elliptic \(g\) fixes no \(F\)-line and hence no point of \(\mathbb P^1(F)\). We check that the middle character is zero there.

The space \(\mathbb P^1(F)\) is compact and totally disconnected. Choose a finite clopen partition fine enough that \(gC\cap C=\varnothing\) for each part \(C\): choose neighborhoods disjoint from their translates, take a finite cover, and refine it to a partition. Choose a small compact open group \(U\) preserving every part. Its orbits lie in these parts, and there are finitely many of them on \(\mathbb P^1(F)\). Their indicators give a basis of the \(U\)-invariants. The averaging operator \(e_U\pi(g)e_U\) has zero diagonal in this basis, because no orbit meets its translate. Its trace is zero. Shrink \(U\) also so the regular character is constant on \(UgU\). The normalized averaging measure has total mass one; the trace-distribution identity gives \(\Theta_{C^\infty(\mathbb P^1)}(g)=0\).

Characters add in (4.3), so \(\Theta_{\mathrm{St}_2}(g)=0-1=-1\). The trivial character on \(D^\times\) is one, and the sign in (4.1) is \((-1)^{2-1}=-1\). Uniqueness in Theorem 4.1 identifies the first transfer. Twisting and (1.3) identify the second. ∎

At the pair in Example 2.2 the values are \(1,-1\); after twisting they are \(\chi(2),-\chi(2)\). This checks the reduced norm as well as the sign.

The supercuspidal support theorem [Deligne 1976, the theorem on p. 156] places the character's support over elements compact in the adjoint group. Elliptic elements satisfy this condition. The support theorem supplies context for the comparison; it neither determines the values nor proves (4.1).

## 5. Parameters for division-algebra representations

For \(D\) of degree \(n\), define
\[
\operatorname{rec}_D(\tau)=\operatorname{rec}_F(\operatorname{JL}(\tau)).
\tag{5.1}
\]
The representation classification and monodromy-block lesson identify the essentially square-integrable parameters exactly as
\[
\sigma\otimes S_k,\qquad \sigma\text{ irreducible},\qquad k\dim\sigma=n.
\tag{5.2}
\]
These are the indecomposable Frobenius-semisimple Weil–Deligne pairs. Here \(S_k=\|\cdot\|^{-(k-1)/2}\mathrm{Sp}(k)\) is our centered block. Thus (5.1) is a bijection from irreducible smooth representations of \(D^\times\) to (5.2). It does not equate the dimension of \(\tau\) with its parameter's dimension \(n\).

For \(n=2\), \(\chi\circ\operatorname{Nrd}\) has parameter \(\widehat\chi S_2\), with nonzero monodromy. Any other irreducible quaternion representation has supercuspidal transfer and an irreducible two-dimensional Weil parameter with zero monodromy. Indeed a special transfer has a unique preimage already given by (4.2). A principal series or a determinant character of \(G\) is outside this square-integrable image.

If \(\chi\) is unramified and \(a=\chi(\varpi_F)\), the parameter of the quaternion character has Frobenius eigenvalues \(a q^{1/2},a q^{-1/2}\), with monodromy from the first line to the second. Hence
\[
L(s,\operatorname{rec}_D(\chi\circ\operatorname{Nrd}))
=(1-aq^{-s-1/2})^{-1},\qquad a(\operatorname{rec}_D(\chi\circ\operatorname{Nrd}))=1.
\]
For a conductor-zero additive character its epsilon factor is \(-a q^{1/2-s}\), by the proved special-factor formula. These are the factors of its JL transfer. Defining parameter factors by (5.1) is immediate; comparison with division-algebra zeta integrals requires the separate normalization theorem [Deligne 1973b, §3.2.8(K)].

## 6. What the Lubin–Tate tower realizes

Here take a finite extension \(F/\mathbb Q_p\), choose \(\ell\ne p\), and identify complex and algebraically closed \(\ell\)-adic coefficients. Let \(D_n\) have invariant \(1/n\). A one-dimensional formal \(\mathcal O_F\)-module of height \(n\) over an algebraic closure of the residue field has rational endomorphism algebra \(D_n\). Its deformation ring is a power-series ring in \(n-1\) variables over the integers of the completed maximal unramified extension. Drinfeld level structures give the Lubin–Tate tower. Its cohomology has commuting linear-group, division-algebra and Weil actions. These are geometric inputs [Wedhorn 2000, (4.4.1) and §§5.2–5.5].

The cohomology and actions must be fixed before writing a tensor product. In [Wedhorn 2000, (4.4.1) and (5.5.4)], \(\Psi^i(\rho)\) is obtained from the level-limit vanishing-cycle spaces using the \(\mathcal O_{D_n}^\times\)-multiplicity functor and action extension specified there. Put
\[
[\Psi(\rho)]=(-1)^{n-1}\sum_{i=0}^{n-1}(-1)^i[\Psi^i(\rho)].
\]
The **construction theorem** of Harris and Taylor [Wedhorn 2000, (4.4.1)] states that for a supercuspidal \(\pi\) of \(G\) there is a true \(n\)-dimensional Weil representation \(r_\ell(\pi)\) such that
\[
[\Psi(\operatorname{JL}^{-1}(\pi)^\vee)]
=[\pi\otimes r_\ell(\pi)],\qquad
\operatorname{rec}_F(\alpha)
=r_\ell(\alpha^\vee\otimes\nu^{(1-n)/2}).
\tag{6.1}
\]
Here \(\nu(g)=|\det g|_F\); the second equality is the Harris–Taylor definition on supercuspidals. Harris and Taylor prove that this is the factor-normalized correspondence used here; see [Wedhorn 2000, (4.4.1)] and the outline of their proof in [Carayol 2000, §§5–8].

Apply (6.1) to \(\pi^\vee\) and use the twist property. In our geometric Weil norm convention this gives
\[
[\Psi(\operatorname{JL}^{-1}(\pi))]
=[\pi^\vee\otimes\operatorname{rec}_F(\pi)
\otimes\|\cdot\|^{(1-n)/2}].
\tag{6.2}
\]
Extracting the division-algebra type, then removing this norm twist, realizes the pair \(\pi^\vee,\operatorname{rec}_F(\pi)\). Choosing a linear-group multiplicity functor instead transfers a dual to the division-algebra side. Different Hom conventions explain the common formulations with \(\operatorname{JL}^{-1}(\pi)^\vee\).

Formula (6.1) is an equality of **virtual representations** in a Grothendieck group. [Carayol 2000, §5.5] distinguishes it from the conjectured individual-degree isomorphism. A stronger middle-degree statement uses compactly supported cohomology of the full tower and concentration of its supercuspidal part in degree \(n-1\). [Dat 2010, §1.1, Theorem 1 and its following remarks] states this formulation with its Tate twist, and identifies its characteristic-zero input as Harris–Taylor together with concentration results of Mieda and Strauch. In that source's convention,
\[
\operatorname{Hom}_G(H_c^{n-1},\pi)
\simeq \rho(\pi)\otimes\sigma(\pi)((n-1)/2),
\]
where \(\rho,\sigma\) are its Jacquet–Langlands and Langlands assignments. We use (6.1)–(6.2) for the exact normalization of [Wedhorn 2000, (4.4.1)], retaining its twist and multiplicity convention.

At height one Harris and Taylor give \(\Psi^0(\rho)=\rho^{-1}\boxtimes\widehat\rho\) and higher degrees vanish. The half twist in (6.2) is zero, recovering local reciprocity and checking the dual placement. At height two the construction recovers the rank-two correspondence, including the primitive parameters. The next lesson explains the global geometry that produces all heights.

## 7. Exercises and complete solutions

**Exercise 7.1 (easy).** Prove \(\mathcal O_D^\times/(1+\varpi_D\mathcal O_D)\simeq\mathbb F_{q^n}^\times\).

**Solution.** The valuation-one element \(\varpi_D\) generates the two-sided ideal \(\mathfrak P_D\). In (3.2), integral elements have integral coefficients, and reduction retains precisely the constant coefficient in \(\mathcal O_L/\varpi_F\mathcal O_L=\mathbb F_{q^n}\). A lift of a nonzero residue has valuation zero and is a unit. Reduction on units has kernel \(1+\mathfrak P_D\), so the first isomorphism theorem applies. This is a group isomorphism although \(\mathcal O_D\) itself need not be commutative.

**Exercise 7.2 (medium).** Match a regular elliptic class from an unramified quadratic extension with the quaternion division algebra.

**Solution.** Let \(E/F\) be unramified quadratic and \(a\in E^\times\setminus F^\times\). Its minimal polynomial is \(T^2-tT+c\), with \(t=\operatorname{Tr}(a),c=N(a)\). The splitting and module argument of Theorem 2.1 embeds \(E\) into \(D\). The embedded \(a\) matches the companion matrix \(\left(\begin{smallmatrix}0&-c\\1&t\end{smallmatrix}\right)\). Both centralizers are \(E^\times\), compact modulo \(F^\times\). The intertwiner argument makes the class independent of the embedding. For \(F=\mathbb Q_3,a=1+i\), this is the pair in (2.1), with \(t=c=2\).

**Exercise 7.3 (medium).** Check the character relation between the trivial quaternion representation and Steinberg.

**Solution.** An elliptic matrix fixes no point of \(\mathbb P^1(F)\). The clopen partition and compact averaging argument of Proposition 4.2 gives character zero for the induced representation. Subtract the character one of its constant subrepresentation in (4.3): Steinberg has value \(-1\). The matching division-algebra element has trivial character one, so \(\Theta_{\mathrm{St}_2}(g)=(-1)^{2-1}\Theta_1(g')\). A reduced-norm twist multiplies both sides by \(\chi(\det g)\), by (1.3).

**Exercise 7.4 (hard).** Prove finite-dimensionality without assuming admissibility.

**Solution.** One nonzero vector is fixed by some \(U_r\). Normality makes \(U_r\) fix all its translates and hence the representation. Modulo \(U_r\) and central powers of \(\varpi_F\), the group has finitely many cosets. Their translates generate the representation over \(\mathbb C[t,t^{-1}]\), with \(t\) the central uniformizer action. The torsion submodule is group-stable. If nonzero, irreducibility makes it the whole module; finite generation then gives finite complex dimension. If zero, the module is nonzero finite free over the Laurent polynomial ring; multiplication by \(t-1\) gives a nonzero proper group-stable submodule, a contradiction. Thus the torsion case holds. The kernel includes the open normal \(U_r\). Admissibility follows afterwards.

## What this lesson does not prove

The central-simple-algebra structure and local Brauer invariant are inputs [Milne, Chapter III, Theorem 2.1 and equation (29); Chapter IV, Proposition 4.3 and Remark 4.4(b)]. The valuation and cyclic presentation are [Badulescu 2002, §2.1]. We proved the degree-\(n\) embedding from splitting, conjugacy uniqueness, regular elliptic class matching, the residue-unit formula, compactness modulo the center and smooth finite-dimensionality. The elementary structure theorem for finitely generated modules over a principal ideal domain is an algebra prerequisite.

The general representation correspondence is [Badulescu 2002, Theorem 1.1] in positive characteristic and the Deligne–Kazhdan–Vignéras/Rogawski theorem in characteristic zero, described in [Arthur 2005, §25, Theorem 25.1]. The quaternion image theorem is [Jacquet–Langlands 1970, Theorem 15.1]. Character distributions and elliptic orthogonality are local character-theory inputs from these works. The Steinberg sequence was proved earlier in the course; its elliptic trace and sign were proved here. The supercuspidal support theorem is [Deligne 1976, p. 156].

The essentially square-integrable classification and centered monodromy formulas are earlier course results. Combining them with rec and JL gives (5.1)–(5.2). Comparison with division-algebra zeta integrals requires [Deligne 1973b, §3.2.8(K)].

The deformation, level and cohomology construction and its virtual compatibility are [Wedhorn 2000, (4.4.1) and §5] and [Carayol 2000, §§5.1–5.6]. Its individual-degree refinement uses the additional concentration theorem [Dat 2010, §1.1]. We converted their definition to (6.2). The algebraic matching argument does not prove the general trace-formula or cohomological correspondence.

## References

- [Jacquet–Langlands 1970] Hervé Jacquet and Robert P. Langlands, [*Automorphic Forms on GL(2)*](https://publications.ias.edu/sites/default/files/Automorphic-forms-on-GL2.pdf), Lecture Notes in Mathematics 114, Springer, 1970, §15, Theorem 15.1 and elliptic orthogonality.
- [Badulescu 2002] Alexandru Ioan Badulescu, [“Correspondance de Jacquet–Langlands pour les corps locaux de caractéristique non nulle”](https://www.numdam.org/item/ASENS_2002_4_35_5_695_0/), *Annales scientifiques de l'École Normale Supérieure* 35 (2002), 695–747, Theorem 1.1, §2.1 and Theorem 5.3.
- [Arthur 2005] James Arthur, *An Introduction to the Trace Formula*, in [*Harmonic Analysis, the Trace Formula, and Shimura Varieties*](https://www.claymath.org/wp-content/uploads/2022/03/cmip04c.pdf), Clay Mathematics Proceedings 4, AMS/CMI, 2005, §25, particularly Theorem 25.1.
- [Milne] James S. Milne, [*Class Field Theory*](https://www.jmilne.org/math/CourseNotes/CFT.pdf), Chapter III, Theorem 2.1 and equation (29); Chapter IV, Proposition 4.3 and Remark 4.4.
- [Deligne 1973b] Pierre Deligne, [*Formes modulaires et représentations de GL(2)*](https://publications.ias.edu/sites/default/files/Number21.pdf), in *Modular Functions of One Variable II*, Lecture Notes in Mathematics 349, Springer, 1973, 55–105, §3.2.8.
- [Deligne 1976] Pierre Deligne, [*Le support du caractère d'une représentation supercuspidale*](https://publications.ias.edu/sites/default/files/31_LeSupport.pdf), *Comptes rendus de l'Académie des sciences de Paris* 283 (1976), 155–157, theorem on p. 156.
- [Wedhorn 2000] Torsten Wedhorn, [*The local Langlands correspondence for GL(n) over p-adic fields*](https://arxiv.org/abs/math/0011210v2), lectures at the School on Automorphic Forms on GL(n), ICTP Trieste, 2000, (4.4.1) and §5.
- [Carayol 2000] Henri Carayol, “[Preuve de la conjecture de Langlands locale pour GL_n : travaux de Harris–Taylor et Henniart](https://www.numdam.org/item/SB_1998-1999__41__191_0/),” Séminaire Bourbaki, exposé 857 (1998–1999), *Astérisque* 266 (2000), 191–243, §§5–8.
- [Dat 2010] Jean-François Dat, [*Théorie de Lubin–Tate non abélienne ℓ-entière*](https://arxiv.org/abs/1011.1887), preprint, §1.1, Theorem 1 and the following characteristic-zero and concentration remarks.