Last time we discussed how certain automorphy lifting theorems can prove Fermat’s last theorem. These are of the form of, if $\bar{r}$ a mod $\ell$ Galois representation is automorphic, then a suitable $\ell$-adic lift is automorphic.
Space of automorphic forms
Here is a concrete definition of automorphy that we can work with. Let $F/\mathbb{Q}$ be a totally real field with $[F:\mathbb{Q}]$ even (this is not that serious because we can increase $F$). Let $\ell \gt 3$ be a prime unramified in $F$. There is a unique quaternion algebra $D/F$ that is precisely ramified at the infinite places, i.e., $$ D \otimes_F F_v = \begin{cases} \mathbb{H} & v \mid \infty, \br M_{2 \times 2}(F_v) & v \nmid \infty. \end{cases} $$ Then we can look at $D^\times \subseteq (D \otimes \mathbb{A}_ F^\infty)^\times$ that is discrete with compact quotient.
We choose $L/\mathbb{Q}_ \ell$ be a finite extension (for the coefficient side) and let $\mathcal{O} = \mathcal{O}_ L$ with $\mathcal{O} / \lambda = \mathbb{F}$. Let $S$ be a finite set of finite places of $F$ not containing primes above $\ell$, and for $v \in S$ we choose a subgroup $k(v)^\times \supseteq \Delta_v$ as well as a character $\chi_v \colon \Delta_v \to \mathcal{O}^\times$. We also write $$ \Delta = \prod_{v \in S} \Delta_v, \quad \chi = \prod_{v \in S} \chi_v. $$ Consider the level subgroup $$ U_\Delta(S) = \prod_{v \notin S} \mathrm{GL}_ 2(\mathcal{O}_ {F,v}) \times \prod_{v \in S} \biggl\lbrace \begin{pmatrix} a & b \br c & d \end{pmatrix} \in \mathrm{GL}_ 2(\mathcal{O}_ {F,v}) : v(c) \gt 0, a/d \bmod{v} \in \Delta_v \biggr\rbrace \subseteq \mathrm{GL}_ 2(\mathbb{A}_ F^\infty) $$ which is open and compact. This has a character $$ \chi \colon U_\Delta(S) \to \mathcal{O}^\times; \quad \begin{pmatrix} a & b \br c & d \end{pmatrix}_ v \mapsto \chi_v(a/d). $$ Now we can define for $A$ an $\mathcal{O}$-algebra the space of automorphic forms $$ S(U_\Delta(S), A)_ \chi = \lbrace \varphi \colon D^\times \backslash (D \otimes_F \mathbb{A}_ F^\infty)^\times / (\mathbb{A}_ F^\infty)^\times \to A : \varphi(gu) = \chi(u) \varphi(g) \text{ for all } u \in U_\Delta(S) \rbrace. $$ This can be identified with the direct sum $$ S(U_\Delta(S), A) = \bigoplus_{g \in D^\times \backslash (D \otimes \mathbb{A}_ F^\infty)^\times / (\mathbb{A}_ F^\infty)^\times U_\Delta(S)} A(\chi)^{((\mathbb{A}_ F^\infty)^\times U_\Delta(S) \cap g^{-1} D^\times g) / F^\times}. $$ Here, the number of double cosets $g$ is a finite set, and this group we’re taking the fixed points with is a finite group with order prime to $\ell$ because $\ell$ is unramified in $F$. It follows that this space $S(U_\Delta(S), A)_ \chi$ of automorphic forms is finite free over $A$.
As with any other space of automorphic forms, this has an Hecke algebra action. Let $\mathbb{T}(U_\Delta(S), A)_ \chi$ be the $A$-subalgebra of $\operatorname{End}_ A(S(U_\Delta(S), A)_ \chi)$ generated by $T_v$ for $v \notin S$ and $v \nmid \ell$. This is given by $$ (T_v \varphi)(h) = \sum_{\alpha \in \mathcal{O}_ {F,v}/v} \varphi\biggl( h \begin{pmatrix} \pi_v & \alpha \br 0 & 1 \end{pmatrix} \biggr) + \varphi\biggl( h \begin{pmatrix} 1 & 0 \br 0 & \pi_v \end{pmatrix} \biggr).$$ They all commute with each other, so that $\mathbb{T}(U_\Delta(S), A)_ \chi$ is a commutative subalgebra of a finite free $A$-module. This behaves well with base change, so that if $A \to B$ then $$ \mathbb{T}(U_\Delta(S), A) \otimes_A B \twoheadrightarrow \mathbb{T}(U_\Delta(S), B) $$ has nilpotent kernel and an isomorphism if $B$ is $\ell$-torsion free.
Associated Galois representations
We also have a decomposition $$ \mathbb{T}(U_\Delta(S), \mathcal{O})_ \chi \cong \bigoplus_{\mathfrak{m}} \mathbb{T}(U_\Delta(S), \mathcal{O})_ {\chi,\mathfrak{m}}. $$
Theorem 1. If $\mathfrak{m}$ is a maximal ideal of $\mathbb{T}(U_\Delta(S), \mathcal{O})_ \chi$, then there exists a unique $$ \bar{r}_ \mathfrak{m} \colon G_F \to \mathrm{GL}_ 2(k(\mathfrak{m})) $$ a continuous semisimple representation such that
- $\det \bar{r}_ \mathfrak{m} = \bar{\epsilon}_ \ell^{-1}$,
- if $v \notin S$ and $v \nmid \ell$ then $\bar{r}_ \mathfrak{m}$ is unramified at $v$ and $\tr \bar{r}_ \mathfrak{m}(\mathrm{Frob}_ v) = T_v$,
- if $v \mid \ell$ then $\bar{r}_ \mathfrak{m} \vert_{G_{F_v}}$ is Fontaine–Laffaille with Hodge–Tate weights $\lbrace 0, 1 \rbrace$,
- if $v \in S$ then there exist characters $\psi_{v,1}, \psi_{v,2} \colon G_{F_v}^\mathrm{ab} \to \overline{k(\mathfrak{m})}^\times$ tamely ramified with $$ \bar{r}_ \mathfrak{m} \vert_{G_{F_v}}^\mathrm{ss} = \psi_{v,1} \oplus \psi_{v,2}. $$ Note that because these are tamely ramified the composition $$ \mathcal{O}_ {F,v}^\times \xrightarrow{\mathrm{Art}} I_{F_v^\mathrm{ab}/F_v} \xrightarrow{\psi_{v,1}} \overline{k(\mathfrak{m})}^\times $$ factors through $k(v)^\times$. Now this has the additional property that its restriction to $\Delta_v$ recovers $\chi_v$, and then $\psi_{v,2}$ is just going to be its inverse on the inertia.
Definition 2. We say that $\mathfrak{m}$ is non-Eisenstein when $\bar{r}_ \mathfrak{m}$ is irreducible.
We now always assume that $\mathfrak{m}$ non-Eisenstein. In this case, we can lift the representation to $$ r_\mathfrak{m} \colon G_F \to \mathrm{GL}_ 2(\mathbb{T}(U_\Delta(S), \mathcal{O})_ {\chi,\mathfrak{m}}) $$ such that
- $\det r_\mathfrak{m} = \epsilon_\ell^{-1}$,
- if $v \notin S$ and $r \nmid \ell$ then $r_\mathfrak{m}$ is unramified at $v$ and $\tr r_\mathfrak{m}(\mathrm{Frob}_ v) = T_v$,
- if $v \mid \ell$ then $r_\mathfrak{m} \vert_{G_{F_v}}$ is Fontaine–Laffaille with Hodge–Tate numbers $\lbrace 0, 1 \rbrace$,
- if $v \in S$ then $r_\mathfrak{m}$ is tamely ramified at $v$.
Now we can give an ad hoc definition of automorphy. Let $L^\prime/L$ and $\mathbb{F}^\prime / \mathbb{F}$ be finite extensions.
Definition 3. For a representation $$ r \colon G_F \to \mathrm{GL}_ 2(L^\prime) \text{ or } \bar{r} \colon G_F \to \mathrm{GL}_ 2(\mathbb{F}^\prime), $$ with $r$ continuous semisimple, we say that $r$ (respectively $\bar{r}$) is automorphic (of weight $0$ level prime to $\ell$ tamely ramified away from $\ell$, if $v \mid \ell$ and $r$ is ramified at $v$ then $r \vert_{G_{F_v}}^\mathrm{ss} \cong \psi_1 \oplus \psi_2$ for $\psi_i$ tamely ramified) when for some $S, \Delta, \chi$ there exists a $\theta \colon \mathbb{T}(U_\Delta(S), \mathcal{O})_ \chi \to L^\prime$ or $\mathbb{F}^\prime$ such that
- $\theta(T_v) = \tr r(\mathrm{Frob}_ v)$ or $\tr \bar{r}(\mathrm{Frob}_ v)$ for $v \notin S$ and $v \nmid \ell$,
- $r / \bar{r}$ is unramified outside $S \cup \lbrace v \mid \ell \rbrace$.
If $\bar{r}$ is irreducible, this just means that $\bar{r} = \theta \circ r_\mathfrak{m}$ or the same for $r$.
Auxiliary primes
Let us now specialize to the case when $S = Q \amalg R$ where $R$ is supposed to be the primes where $r$ ramifies, and $Q$ is some auxiliary set of primes. For $v \in R$, we will want
- $\lvert k(v) \rvert \equiv 1 \pmod{\ell}$,
- $\bar{r}_ \mathfrak{m} \vert_{G_{F_v}}$ is trivial,
- $\Delta_v = k(v)^\times$ and $\chi_v$ has $\ell$-power order,
and for $v \in Q$ we will want
- $\lvert k(v) \rvert \equiv 1 \pmod{\ell}$,
- $\Delta_v$ is the maximal subgroup of $k(v)^\times$ of order prime to $\ell$,
- $H_v = k(v)^\times / \Delta_v$ has $\ell$-power order,
- $\chi_v = 1$,
- $\bar{r}_ \mathfrak{m}$ is unramified at $v$,
- $\bar{r}_ \mathfrak{m}(\mathrm{Frob}_ v)$ has distinct eigenvalues $\alpha_v, \beta_v$.
For $v \in Q$, if we look at the Frobenius eigenvalue, we can use Hensel’s lemma to lift it to $$ \operatorname{char}_ {r_\mathfrak{m}(\sigma)}(x) = (x - A_{v,\sigma}) (x - B_{v,\sigma}), \quad A_{v,\sigma} \equiv \alpha_v, B_{v,\sigma} \equiv \beta_v \pmod{\mathfrak{m}}. $$ So there exist characters $$ \chi_{\alpha_v}, \chi_{\beta_v} \colon G_{F_v} \to \mathbb{T}(U_\Delta(S), \mathcal{O})_ {\chi,\mathfrak{m}}^\times $$ such that for all $\sigma \in G_{F_v}$ we have $$ \operatorname{char}_ {r\mathfrak{m}(\sigma)}(x) = (x - \chi_{\alpha_v}(\sigma)) (x - \chi_{\beta_v}(\sigma)). $$
For $v \in Q$ and $a \in \mathcal{O}_ {F,v} - \lbrace 0 \rbrace$, we have these Hecke operators $$ (U_a \varphi)(h) = \int_{U_\Delta(S)_ v \begin{pmatrix} a & 0 \br 0 & 1 \end{pmatrix} U_\Delta(S)_ v} \varphi(hg) dg, \quad dg(U_\Delta(S)_ v) = 1. $$ These all commute with each other, $U_{ab} = U_a U_b$ and commute with $T_v$. By a concrete form of local-global compatibility, these $U_a$ acting on $S(U_\Delta(S), \mathcal{O})_ {\chi,\mathfrak{m}}$ also satisfy $$ (x - \chi_{\alpha_v}(\mathrm{Art}_ {F_v}(a))) (x - \chi_{\beta_v}(\mathrm{Art}_ {F_v}(a))). $$ Fix $\sigma_0 \in G_{F_v}^\mathrm{ab}$ above $\mathrm{Frob}_ v$ which is $\mathrm{Art}(a_0)$ so that $a_0 \in \mathcal{O}_ {F,v}$ with $v(a_0) = 1$. Then we define the idempotents $$ e_{\alpha_v} = \frac{U_{a_0} - B_{v,\sigma_0}}{A_{v,\sigma_0} - B_{v,\sigma_0}}, \quad e_{\beta_v} = \frac{U_{a_0} - A_{v,\sigma_0}}{B_{v,\sigma_0} - A_{v,\sigma_0}}. $$ Then we can define $$ S(\chi, Q, \mathcal{O}) = \prod_{v \in Q} e_{\alpha_v} S(U_\Delta(S), \mathcal{O})_ {\chi,\mathfrak{m}} $$ and the Hecke algebra $$ \mathbb{T}(\chi, Q) = \mathbb{T}(U_\Delta(S), \mathcal{O})_ {\chi,\mathfrak{m}} $$ acting on it. Now if we look at the Galois representation $$ r_{\chi,Q}^\mathrm{mod} \colon G_F \to \mathrm{GL}(\mathbb{T}(\chi, Q)) $$ this has the property that for $v \in Q$ we have $$ r_{\chi,Q}^\mathrm{mod} \vert_{G_{F_v}} \cong \chi_{\alpha_v} \oplus \chi_{\beta_v}. $$ Moreover, the $U_a$-action on $S(\chi, Q, \mathcal{O})$ is given by $\chi_{\alpha_v}(\mathrm{Art}(a)) \in \mathbb{T}(\chi, Q)$.
Now let us consider $H_v = k(v)^\times / \Delta_v$ for $v \in Q$ which is cyclic of order the larges power of $\ell$ dividing $\lvert k(v)^\times \rvert$. Then we have an action $$ H_Q = \prod_{v \in Q} H_v \to \mathbb{T}(\chi, Q)^\times; \quad a \mapsto \chi_{\alpha_v}(\mathrm{Art}(a)). $$
Lemma 4. 1. The space $S(\chi, Q, \mathcal{O})$ is a free $\mathcal{O}[H_Q]$-module.
- The trace map $S(\chi, Q, \mathcal{O})_ {H_Q} \to S(\chi, \emptyset, \mathcal{O})$ is an isomorphism.
This is in some sense the only input from automorphic forms that we need.