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The Jacquet–Langlands correspondence

We have $F/\mathbb{Q}$ a totally real field and we had the space $$ \mathcal{A}_ 0(\GL_2(F) \backslash \GL_2(\mathbb{A}_ F))_ 0 $$ of cuspidal automorphic forms $\varphi \in \mathcal{A}_ 0(\GL_2(F) \backslash \GL_2(\mathbb{A}_ F))$ satisfying

  1. for $v \mid \infty$ we have $\mathfrak{z}_ v$ acts with Harish-Chandra parameters $\lbrace 0, -1 \rbrace$,
  2. $\varphi(gu) = \prod_{v \mid \infty} j^{-2}(u_v) \varphi(g)$ for all $u \in \prod u_v \in \prod_{v \mid \infty} \mathrm{SO}(2)$, where $j^{-2}(\begin{smallmatrix} a & b \br -b & a \end{smallmatrix}) = (a + bi)^2$,
  3. $\varphi(gz) = \lVert z \rVert^{-1} \varphi(g)$ for all $z \in \mathbb{A}_ F^\times$.

We now lose the action at infinity because in the second condition we are picking a vector in each $\pi_\infty$. But we still have an action of $\GL_2(\mathbb{A}_ F^\infty)$ and get a decomposition $$ \GL_2(\mathbb{A}_ F^\infty) = \bigoplus_\pi \pi^\infty $$ where $\pi$ is over cuspidal automorphic representations with Harish-Chandra parameters $\lbrace 0, -1 \rbrace$ at all infinite places and $\chi_\pi = \lVert -\rVert^{-1}$.

The Jacquet–Langlands correspondence

Assume moreover that $F/\mathbb{Q}$ has even degree. Let $D$ be the quaternion algebra with center $F$ ramified at the infinite places of $F$. Then we have $$ D \otimes_{\mathbb{Q}} \mathbb{R} \cong \mathbb{H}^{[F:\mathbb{Q}]}, \quad D \otimes_{\mathbb{Q}} \mathbb{A}^\infty \cong M_{2 \times 2}(\mathbb{A}_ F^\infty). $$ We can then look at the corresponding space $$ \mathcal{A}(D^\times \backslash (D \otimes \mathbb{A}^\infty)^\times, \mathbb{C})_ 0 = \lbrace \varphi \colon D^\times \backslash (D \otimes \mathbb{A}^\infty)^\times / \mathbb{A}_ F^\times \to \mathbb{C} \text{ locally constant} \rbrace $$ where the condition $0$ only has this center condition because $D^\times / F^\times \hookrightarrow (D \otimes \mathbb{A}^\infty)^\times / \mathbb{A}_ F^\times$ is discrete and cocompact.

Theorem 1 (Jacquet–Langlands). There is an isomorphism of $\GL_2(\mathbb{A}_ F^\infty)$-modules $$ \mathcal{A}(D^\times \backslash (D \otimes \mathbb{A}^\infty)^\times) \cong \mathcal{A}_ 0(\GL_2(F) \backslash \GL_2(\mathbb{A}_ F))_ 0 \otimes \lVert \det \rVert^{1/2} \oplus \bigoplus_{\chi \colon \mathbb{A}_ F^\times / \overline{F^\times (F_\infty^\times)^0} \to \lbrace \pm 1 \rbrace} \chi \circ \lVert-\rVert_\mathrm{red}. $$

Upon choosing an isomorphism $i \colon \mathbb{C} \cong \bar{\mathbb{Q}}_ \ell$, we can also write it as $$ \mathcal{A}(D^\times \backslash (D \otimes \mathbb{A}^\infty)^\times, \bar{\mathbb{Q}}_ \ell)_ 0 \cong \bigoplus_\pi i(\pi^\infty \otimes \lVert \det \rVert^{1/2}) \oplus \bigoplus_\chi \chi \circ \lVert - \rVert_\mathrm{red}. $$ In fact, for every irreducible $\pi \subseteq \mathcal{A}(D^\times \backslash (D \otimes \mathbb{A}^\infty)^\times, \bar{\mathbb{Q}}_ \ell)_ 0$ we can define a Galois representation $$ r(\pi) \colon G_F \to \GL_2(\bar{\mathbb{Q}}_ \ell) $$ so that

  • if $\pi = i(\pi^{\prime\infty} \otimes \lVert \det \rVert^{1/2})$ then $r(\pi) = r_{\ell,i}(\pi^\prime)$,
  • if $\pi = \chi \circ \lVert - \rVert_\mathrm{red}$ then $r(\pi) = (\chi \circ \mathrm{Art}^{-1}) \oplus (\chi \circ \mathrm{Art}^{-1}) \epsilon_\ell^{-1}$, where $\mathrm{Art} \colon \mathbb{A}_ F^\times / \overline{F^\times (F_\infty^\times)^0} \cong \Gal(F^\mathrm{ab} / F)$.

These are independent of the choice of $i$ now, and satisfy the following properties.

  1. $\det r(\pi) = \epsilon_\ell^{-1}$
  2. If $v \nmid \ell$ and $\pi_v$ is unramified then $\tr r(\pi)(\mathrm{Frob}_ v)$ is the eigenvalue of $T_v$ on $\pi_v^{\GL_2(\mathcal{O}_ {F,v})}$.
  3. If $v \mid \ell$ and $\pi_v$ is unramified then $r(\pi) \vert_{G_{F_v}}$ is Fontaine–Laffaille with Hodge–Tate weights $\lbrace 0, 1 \rbrace$.
  4. If $v \nmid \ell$ and $\pi_v^{\mathrm{Iw}_ v^1} \neq 0$, where $\mathrm{Iw}_ v^1 = \lbrace (\begin{smallmatrix} a & b \br c & d \end{smallmatrix}) : c \equiv 0, a \equiv d \pmod{v} \rbrace$ then one of the following hold:
    • $\dim \pi_v^{\mathrm{Iw}_ v^1} = 2$ and $r(\pi) \vert_{G_{F_v}}$ factors through $G_{F_v}^\mathrm{ab}$ and for all $a \in F_v^\times \cap \mathcal{O}_ {F,v}$ the matrix $r(\pi) \vert_{G_{F_v}}(\mathrm{Art}_ {F_v} a)$ has the same eigenvalues as $U_a$ has on $\pi_v^{\mathrm{Iw}_ v^1}$,
    • $\dim \pi_v^{\mathrm{Iw}_ v^1} = 1$ and $\pi_v$ is infinite-dimensional, in which case we have $$ r(\pi) \vert_{G_{F_v}} \sim \begin{pmatrix} \psi & \ast \br 0 & \psi \epsilon_\ell^{-1} \end{pmatrix} $$ where $\psi(\mathrm{Art}_ {F_v} a)$ is the eigenvalue $U_a$ on the $1$-dimensional space $\pi_v^{\mathrm{Iw}_ v^1}$ for all $a \in F_v^\times \cap \mathcal{O}_ {F,v}^\times$ and $\psi$ is of finite order,
    • $\dim \pi_v^{\mathrm{Iw}_ v^1} = 1$ and $\pi_v$ is $1$-dimensional, in which case we have $$ r(\pi) \vert_{G_{F_v}} \cong \psi \oplus \psi \epsilon_\ell $$ and $\psi(\mathrm{Art}_ {F_v} a)$ is the eigenvalue of $U_a$ on $\pi_v^{\mathrm{Iw}_ v^1}$ for $a \in F_v^\times \cap \mathcal{O}_ {F,v}^\times$ as before, and $\psi$ is $\epsilon_l^{-1}$ times a finite order character.

Some tame levels

We now fix $L/\mathbb{Q}_ \ell$ a finite extension and write $\mathcal{O} = \mathcal{O}_ L$ and $\lambda \subseteq \mathcal{O}$ and $\mathcal{O}/\lambda = \mathbb{F}$ as before. Let $R$ be a finite set of finite places of $F$, containing no place above $\ell$. For each $v \in R$, choose a subgroup and a character $$ k(v)^\times \supseteq \Delta_v \xrightarrow{\chi_v} \mathcal{O}^\times $$ and we write $$ \Delta = \prod_{v \in R} \Delta_v, \quad \chi = \prod_{v \in R} \chi_v. $$ We also define the congruence subgroup $$ U_\Delta(R) = \biggl\lbrace u \in \GL_2(\widehat{\mathcal{O}_ F}) : u_v = \begin{pmatrix} a_v & b_v \br c_v & d_v \end{pmatrix} \text{ with } c_v \equiv 0, a_v \equiv d_v \pmod{v}, a_v \in \Delta_v \text{ for all } v \in R \biggr\rbrace. $$ For example, if $\Delta = \prod_{v \in R} k(v)^\times$ then $U_\Delta(R) = U_0(R)$ and if $\Delta = \lbrace 1 \rbrace$ then $U_\Delta(R) = U_1(R)$.

For $A$ an $\mathcal{O}$-algebra, we can now define the space of automorphic forms with coefficients $\mathcal{A}$, $$ \mathcal{S}(U_\Delta(R), \chi, A) = \lbrace \varphi \colon D^\times \backslash (D \otimes \mathbb{A}^\infty)^\times / (\mathbb{A}_ F^\infty)^\times \to A : \varphi(gu) = \chi(u) \varphi(g) \text{ for } u \in U_\Delta(R) \rbrace. $$ Using the discussion above, we can identify this as $$ \mathcal{S}(U_\Delta(R), \chi, \bar{\mathbb{Q}}_ \ell)_ 0 = \bigoplus \pi^{U_\Delta(R),\chi} \subseteq \bigoplus \pi = \mathcal{A}(D^\times \backslash (D \otimes \mathbb{A}^\infty)^\times, \bar{\mathbb{Q}}_ \ell)_ 0. $$

On the other hand, we can also understand this space by decomposing into double cosets $$ \mathcal{S}(U_\Delta(R), \chi, A) = \bigoplus_{g \in D^\times \backslash (D \otimes \mathbb{A}^\infty)^\times / (\mathbb{A}_ F^\infty)^\times U_\Delta(R)} A(\chi)^{g^{-1} D^\times g \cap (\mathbb{A}_ F^\infty)^\times U_\Delta(R)}, $$ where the action is by $\chi \colon (\mathbb{A}_ F^\infty)^\times U_\Delta(R) \to \mathcal{O}^\times$. Here, we can understand this subgroup as $$ 0 \to F^\times \to (\mathbb{A}_ F^\infty)^\times U_\Delta(R) \cap g^{-1} D^\times g \twoheadrightarrow \Sigma(g) \subseteq D^\times / F^\times, $$ where the second map is $x \mapsto g x g^{-1}$. Here $\Sigma(g)$ is compact and discrete and hence finite. We can also check that $\ell \nmid \lvert \Sigma(g) \rvert$, and therefore this fixed points $A(\chi)^{g^{-1} D^\times g \cap ((\mathbb{A}_ F^\infty)^\times U_\Delta(R))}$ is either $0$ or all of $A$.

Proposition 2. This space $\mathcal{S}(U_\Delta(R), \chi, A)$ is a finite free $A$-module, and moreover this it is compatible with base change in $A$, i.e., $$ \mathcal{S}(U_\Delta, \chi, A) \otimes_A B = \mathcal{S}(U_\Delta, \chi, B). $$