A Rank-Two Case of Local-Global Compatibility for $l=p$

The goal of this work is to prove a new case of local-global compatibility for automorphic Galois representations over CM fields at $l=p$. Let $F$ be a CM field and let $G_{F}=\operatorname{Gal}(\overline{F}/F)$ denote the absolute Galois group. Let $\pi$ be a regular algebraic cuspidal automorphic representation of $\operatorname{GL}_{n}(\mathbb{A}_{F})$. Fix a prime $l$ and an isomorphism $\iota:\overline{\mathbb{Q}}_{l}\cong\mathbb{C}$. Conjecturally there exists a continuous semisimple Galois representation $r_{\iota}(\pi):G_{F}\to\operatorname{GL}_{n}(\overline{\mathbb{Q}}_{l})$ that is potentially semistable (which is known to be equivalent to de Rham) above $l$, such that we have an isomorphism of Weil–Deligne representations

for any finite place $v$ in $F$. Here on the automorphic side $\pi_{v}$ is the local component of $\pi=\otimes^{\prime}_{v}\pi_{v}$ at $v$, and $\operatorname{rec}_{F_{v}}$ is the local Langlands correspondence for $F_{v}$ normalized as in [HT01]; on the Galois side $G_{F_{v}}=\operatorname{Gal}(\overline{F}_{v}/F_{v})$ is the decomposition group at $v$, and $\operatorname{WD}$ is a functor taking a continuous semisimple Galois representation to a Weil–Deligne representation by Grothendieck’s $l$-adic monodromy theorem when $v\nmid l$ and by a construction of Fontaine [Fon94] when $v\mid l$ and $r_{\iota}(\pi)|_{G_{F_{v}}}$ is potentially semistable. We may then study the above isomorphism for all finite places not dividing $l$ and for all places above $l$ separately. The former is called the $l\neq p$ case and the latter is called the $l=p$ case.

From now on we will focus on the $l=p$ case. If $\pi$ is polarizable, the $l=p$ local-global compatibility has been established by work of many people [BLGGT12, BLGGT14a, Car14], etc. For non-polarizable $\pi$, the representations $r_{\iota}(\pi)$ have been constructed by [HLTT16] and [Sch15] using different methods. Recently, A’Campo proved that under certain conditions the representation $r_{\iota}(\pi)$ is de Rham with specified Hodge–Tate weights [A’C24, Theorem 4.3.3], and Hevesi proved that under the same conditions the $l=p$ local-global compatibility holds up to semisimplification [Hev23, Theorem 1.1]. This means that to prove the $l=p$ local-global compatibility, we only need to compare the monodromy operators of the two Weil–Deligne representations. It turns out when $\pi$ is of rank $2$ and weight $0$, we can solve the problem most of the time.

Method of proof. By [Hev23, Theorem 1.1], it suffices to show that when $\pi_{v}$ is special (i.e. a twist of Steinberg), the Galois representation $r_{\iota}(\pi)|_{G_{F_{v}}}$ has nontrivial monodromy. Assume otherwise. Combined with [A’C24, Theorem 4.3.3], after a base change we may assume that $r_{\iota}(\pi)|_{G_{F_{v}}}$ is crystalline. Next we apply a potential automorphy theorem to find an automorphic representation $\pi_{1}$ that is unramified above $l$ such that there is a congruence between mod $l$ Galois representations $\overline{r}_{\iota}(\pi)\cong\overline{r}_{\iota}(\pi_{1})$ when restricted to the absolute Galois group of some finite extension of $F$. Then we are able to apply an automorphy lifting theorem to find an automorphic representation $\pi_{2}$ that is unramified above $l$ such that $r_{\iota}(\pi)\cong r_{\iota}(\pi_{2})$ when restricted to the absolute Galois group of some field extension of $F$. From there we deduce a contradiction with genericity.

The idea to approach the local-global compatibility conjecture using automorphy lifting theorems is inspired by Luu [Luu15]. This method has been used in [AN20] and [Yan21] to prove cases of the $l\neq p$ local-global compatibility, where the isomorphism is known up to semisimplification by the main result of [Var24]. However, in the $l=p$ case there is extra subtlety coming from the construction of the $\operatorname{WD}$ functor. We need to use the fact that this construction is independent of field extensions in some sense. See the next chapter for a review with more details.

Our method is confined to rank two and weight $0$. The proof of our potential automorphy theorem [AN20, Theorem 3.9] uses a moduli space of Hilbert–Blumenthal abelian varieties which realizes a fixed mod $l$ Galois representation, as in Taylor’s original work on the subject [Tay02]. This method only works for $\operatorname{GL}_{2}$. On the other hand, our automorphy lifting theorem [ACC⁺23, Theorem 6.1.1] (of Fontaine–Laffaille version) does not allow a change of weight, and our potential automorphy theorem produces an automorphic representation of weight $0$ as an input of the automorphy lifting theorem. Thus we are also confined to weight $0$.

When dealing with the $l\neq p$ local-global compatibility, [AN20] also has the restriction to rank two and weight $0$. By using the ordinary automorphy lifting theorem [ACC⁺23, Theorem 6.1.2], [Yan21] removes the weight $0$ condition at the cost of imposing an ordinarity assumption on the automorphic representation at the prime $l$ (the residue characteristic of the coefficients). However, this can not be translated to the $l=p$ local-global compatibility problem, since the ordinary automorphy lifting theorem does not guarantee that the lift $\pi_{2}$ is unramified above $l$, which is essential for our proof.

To go beyond rank two, we need to upgrade our potential automorphy theorems. When $\pi$ is polarizable such theorems are known [HSBT10, BLGHT11, BLGGT14b] by using Dwork family of motives. In the non-polarizable case, Qian proved potential automorphy theorems under general settings [Qia23, Theorem 1.1] using Dwork families. Recently in [Mat23], based on [BLGGT14b, Qia23], Matsumoto proved many versions of potential automorphy theorems, and through a careful investigation of the Jordan canonical form of the Weil–Deligne representations, he generalized the $l\neq p$ local-global compatibility results [AN20, Yan21] to $\operatorname{GL}_{n}$. If this method can be translated to the $l=p$ case, then we believe analogous versions of our main theorems can be proved in higher rank.

Notations. Throughout $F$ will be a CM field. If $v$ is a place of $F$, $F_{v}$ is the completion of $F$ at $v$. Let $\mathbb{A}_{F}$ be the ring of adeles of $F$. Fix $\overline{F}$ an algebraic closure of $F$. We write $G_{F}=\operatorname{Gal}(\overline{F}/F)$ for the absolute Galois group, and similarly $G_{F_{v}}=\operatorname{Gal}(\overline{F}_{v}/F_{v})$.

For a non-archimedean local field $K$, let $I_{K}$ be the inertia subgroup and let $W_{K}$ be the Weil group.

We write $\epsilon_{l}$ for the $l$-adic cyclotomic character. We normalize our Hodge–Tate weights such that $\epsilon_{l}$ has Hodge–Tate weight $-1$. Let $B_{\text{dR}}$, $B_{\text{st}}$, $B_{\text{cris}}$ be Fontaine’s rings of de Rham, semistable, and crystalline periods, respectively. We take $K$ and $E$ to be finite extensions of $\mathbb{Q}_{l}$, with $E$ sufficiently large to contain the images of all embeddings $\tau:K\hookrightarrow\overline{\mathbb{Q}}_{l}$. Let $K_{0}$ be the maximal unramified extension of $\mathbb{Q}_{l}$ inside $K$. If $\rho:G_{K}\to\operatorname{GL}(V)\cong\operatorname{GL}_{d}(E)$ is a de Rham representation, we write $D_{\text{dR}}(\rho)=D_{\text{dR}}(V)=(B_{\text{dR}}\otimes_{\mathbb{Q}_{l}}V)^{G_{K}}$, which is a $K\otimes_{\mathbb{Q}_{l}}E$-module. We define $D_{\text{st}}$ and $D_{\text{cris}}$ analogously but they are both $K_{0}\otimes_{\mathbb{Q}_{l}}E$-modules. We write $\operatorname{HT}_{\tau}(\rho)$ for the multiset of $\tau$-labelled Hodge–Tate weights of $\rho$, which are the degrees $i$ at which $\operatorname{gr}^{i}(D_{\text{dR}}(V)\otimes_{K_{0}\otimes E,\tau\otimes 1}E)\neq 0$, counted with multiplicities equal to the $E$-dimensions of the corresponding graded pieces.

Let $\pi$ be a regular algebraic cuspidal automorphic representation of $\operatorname{GL}_{2}(\mathbb{A}_{F})$. We say that $\pi$ has weight $0$ if it has the same infinitesimal character as the trivial (algebraic) representation of $\operatorname{Res}_{F/\mathbb{Q}}\operatorname{GL}_{2}$. For any finite place $v$ in $F$, $\pi_{v}$ is the local component of $\pi=\otimes^{\prime}_{v}\pi_{v}$ at $v$, and $\operatorname{rec}_{F_{v}}$ is the local Langlands correspondence for $F_{v}$ of [HT01].

For a Weil–Deligne representation $(r,N)$, its Frobenius semisimplification is $(r,N)^{\operatorname{F-ss}}=(r^{\operatorname{ss}},N)$, and its semisimplification is $(r,N)^{\operatorname{ss}}=(r^{\operatorname{ss}},0)$.

Acknowledgements. I would like to thank Patrick Allen for bringing my attention to this project and answering my questions in the early stage with great patience. I would like to thank Liang Xiao and Yiwen Ding for their helpful suggestions on an early draft of this paper. I would like to thank Shanxiao Huang and Yiqin He for helpful discussions on $p$-adic Hodge theory and the $\mathcal{L}$-invariants.

We first recall Fontaine’s construction of the $\operatorname{WD}$ functor following [Fon94, All16]. Let $K$ and $E$ be finite extensions of $\mathbb{Q}_{l}$ with $E$ sufficiently large.

For fixed finite extension $L/K$, we write $G_{L/K}=\operatorname{Gal}(L/K)$ and make the following definition.

We first associate a Weil–Deligne representation $(r_{D},N_{D})$ to a $(\varphi,N,G_{L/K})$-module $D$. Let $N_{D}=N$. Extend the action of $\operatorname{Gal}(L/K)$ to $W_{K}$ by letting $I_{L}$ act trivially. For $w\in W_{K}$, let $\alpha(w)\in\mathbb{Z}$ be such that the image of $w$ in $W_{K}/I_{K}$ is $\sigma^{-\alpha(w)}$, where $\sigma$ is the image of the absolute arithmetic Frobenius. We then define an $L_{0}\otimes_{\mathbb{Q}_{l}}E$-linear action $r_{D}$ of $W_{K}$ on $D$ by $r_{D}(w)=w\varphi^{\alpha(w)}$. Since $L_{0}\otimes_{\mathbb{Q}_{l}}E\cong\prod_{\tau:L_{0}\hookrightarrow E}E$, we have a decomposition $D=\prod_{\tau:L_{0}\hookrightarrow E}D_{\tau}$ where $D_{\tau}=D\otimes_{L_{0}\otimes_{\mathbb{Q}_{l}}E,\tau\otimes 1}E$, and an induced Weil–Deligne representation $(r_{\tau},N_{\tau})$ of dimension $[L_{0}:\mathbb{Q}_{l}]$ over $E$. By [BM02, 2.2.1], the isomorphism class of $(r_{\tau},N_{\tau})$ is independent of $\tau$, so we may denote any element in the isomorphism class by $\operatorname{WD}(D)$.

Now let $\rho:G_{K}\to\operatorname{GL}_{d}(E)$ be a continuous potentially semistable representation. Choose $L/K$ such that $\rho|_{G_{L}}$ is semistable. There is a standard way to associate it with a $(\varphi,N,G_{L/K})$-module

Still by [BM02, §2.2.1], the isomorphism class of $\operatorname{WD}(D_{\text{st},L}(\rho))$ is independent of the choice of $L$. Hence we may set $\operatorname{WD}(\rho)=\operatorname{WD}(D_{\text{st},L}(\rho))$.

Next we explain some important results used in our proof. The first is an automorphy lifting theorem [AN20, Theorem 2.1], which is essentially a special case of [ACC⁺23, Theorem 6.1.1].

We give the definitions of decomposed generic and enormous image in the technical condition 3. Take a continuous mod $l$ representation $\overline{\rho}:G_{F}\to\operatorname{GL}_{2}(\overline{\mathbb{F}}_{l})$.

Next is a potential automorphy theorem [AN20, Theorem 3.9].

Finally we state two remarkable recent results. The first is [A’C24, Theorem 4.3.3]. The original theorem is proved for $\pi$ of rank $n$ and arbitrary weight, but for simplicity we only state the special case when $\pi$ is of rank $2$ and weight $0$.

The second is the up-to-monodromy $l=p$ local-global compatibility [Hev23, Theorem 1.1, Remark 1.2].

We also modify the statement for simplicity. The original theorem proves that under certain partial order, the monodromy of the Galois side is “less than” the monodromy of the automorphic side ([Hev23, Remark 1.2]). In particular, if the Galois side has nontrivial monodromy, so does the automorphic side. It remains to prove the converse.

Now we are ready to prove our main theorem.

In [Maz94, §9], Mazur defined a collection of invariants attached to the isomorphism classes of certain filtered $(\varphi,N)$-modules called the two-dimensional monodromy modules. We first recall the definitions. Let $D$ be a filtered $(\varphi,N)$-module, which is a $K_{0}$-vector space with actions of $\varphi$ and $N$ and an exhaustive and separated decreasing filtration $\{\operatorname{Fil}^{i}\}_{i}$ on $D_{K}:=D\otimes_{K_{0}}K$, satisfying the following:

1. (1)

   $\varphi$ is $K_{0}$-semilinear;

2. (2)

   $N$ is $K_{0}$-linear;

3. (3)

   $N\varphi=l\varphi N$.

Note that $D$ has a basis of $\varphi$-eigenvectors, and $N$ takes $\varphi$-eigenvectors to $\varphi$-eigenvectors or $0$.

Let $f$ be a classical normalized eigenform of $\Gamma_{0}(N)$ of even weight $k\geq 2$ (and Nebentypus character $\chi$, for some integer $N$), and $\rho_{f}:G_{\mathbb{Q}}\to\operatorname{GL}_{2}(E)$ its associated $l$-adic Galois representation for a suitable $E/\mathbb{Q}_{l}$. If $\rho_{f,l}:=\rho_{f}|_{G_{\mathbb{Q}_{l}}}$ is semistable non-crystalline, it is automatically non-critical, which means that $D_{\text{st}}(\rho_{f,l})$ is a monodromy module, and therefore we may make sense of the Fontaine–Mazur $\mathcal{L}$-invariant $\mathcal{L}(D_{\text{st}}(\rho_{f,l}))$ attached to $f$. Note that this $\mathcal{L}$-invariant only depends on $\rho_{f,l}$, so in particular it only depends on $f$ and $l$.

Now we return to the settings of this paper and slightly generalize the above idea. Let $F$ be a CM field and let $l$ be a rational prime that is inert in $F$. Fix an isomorphism $\iota:\overline{\mathbb{Q}}_{l}\cong\mathbb{C}$. Let $\pi$ be a regular algebraic cuspidal automorphic representation of $\operatorname{GL}_{2}(\mathbb{A}_{F})$ of weight $0$. Assume that $\pi_{v}$ is special for the finite place $v$ in $F$ lying above $l$. The proof of Theorem 3.1 shows that $r_{\iota}(\pi)|_{G_{F_{v}}}$ is (potentially) semistable non-crystalline, and is non-critical since $G_{F_{v}}\cong G_{\mathbb{Q}_{l}}$. Hence we can make sense of the Fontaine–Mazur $\mathcal{L}$-invariant $\mathcal{L}(D_{\text{st}}(r_{\iota}(\pi)|_{G_{F_{v}}}))$ attached to $\pi$. Similar to the classical modular form case, the $\mathcal{L}$-invariant only depends on $r_{\iota}(\pi)|_{G_{F_{v}}}$, so in particular it only depends on $\pi$ and $l$. More generally, if we only assume that $l$ is unramified in $F$, then we need to assume that $r_{\iota}(\pi)|_{G_{F_{v}}}$ is non-critical to make $D_{\text{st}}(r_{\iota}(\pi)|_{G_{F_{v}}})$ a monodromy module.

Once we have the definition, it is a natural question to ask if it coincides with other versions of $\mathcal{L}$-invariants other than the one due to Fontaine–Mazur. In the classical modular form case the question is extensively studied and the answer is yes (for instance see the introduction of [BDI10]), but in general the question is widely open. We hope to see the question being answered with the emergence of generalizations of more versions of $\mathcal{L}$-invariants.