A Local Analogue of the Ghost Conjecture of Bergdall–Pollack

Abstract

We formulate a local analogue of the ghost conjecture of Bergdall and Pollack, which essentially relies purely on the representation theory of ${{\,\textrm{GL}\,}}_2({\mathbb {Q}}_p)$. We further study the combinatorial properties of the ghost series as well as its Newton polygon, in particular, giving a characterization of the vertices of the Newton polygon and proving an integrality result of the slopes. In a forthcoming sequel, we will prove this local ghost conjecture under some mild hypothesis and give arithmetic applications.

Appendices

Appendix A: Recollection of Representation Theory of ${\mathbb {F}}[{{\,\textrm{GL}\,}}_2({\mathbb {F}}_p)]$

In this appendix, we recall some basic facts on (right) ${\mathbb {F}}$-representations of ${{\,\textrm{GL}\,}}_2({\mathbb {F}}_p)$. Our main references are [11] and [48] which work with left representations. Our convention will be the corresponding right representation given by transpose.

A.1 Setup

Throughout this appendix, we fix an odd prime number p. We set

$$\begin{aligned} {\bar{\textrm{G}}}= & {} {{\,\textrm{GL}\,}}_2({\mathbb {F}}_p), \quad {\bar{\textrm{U}}} = {\begin{pmatrix}1&{}{\mathbb {F}}_p\\ 0&{} 1\end{pmatrix}}, \quad \text {and }\ {\bar{\textrm{B}}} = {\begin{pmatrix}{\mathbb {F}}_p^\times &{}{\mathbb {F}}_p\\ 0&{} {\mathbb {F}}_p^\times \end{pmatrix}}.\\ {\bar{\textrm{T}}}= & {} {\begin{pmatrix}{\mathbb {F}}_p^\times &{}0\\ 0&{} {\mathbb {F}}_p^\times \end{pmatrix}}, \quad {\bar{\textrm{U}}}^- = {\begin{pmatrix}1&{}0\\ {\mathbb {F}}_p&{} 1\end{pmatrix}}, \quad \text {and }\ {\bar{\textrm{B}}}^- = {\begin{pmatrix}{\mathbb {F}}_p^\times &{}0\\ {\mathbb {F}}_p&{} {\mathbb {F}}_p^\times \end{pmatrix}}. \end{aligned}$$

The coefficient field will be a finite extension ${\mathbb {F}}$ of ${\mathbb {F}}_p$. Write $\omega : {\mathbb {F}}_p^\times \rightarrow {\mathbb {F}}^\times$ for the character induced by inclusion. For a character $\chi _{a,b}:=\omega ^a \times \omega ^b$ and an ${\mathbb {F}}$-representation V of ${\bar{\textrm{G}}}$, we write $V(\chi )$ for the subspace of V on which ${\bar{\textrm{T}}}$ acts by $\chi$.

Let ${\bar{s}}={\big ({\begin{matrix}0&{}1\\ 1&{} 0\end{matrix}} \big )}$ represent the nontrivial element in the Weyl group of ${\bar{\textrm{G}}}$. If $\chi$ is a character of ${\bar{\textrm{T}}}$, we use $\chi ^s$ to denote the character defined by

$$\begin{aligned} \chi ^s({\bar{t}})=\chi ({\bar{s}}^{-1}{\bar{t}}{\bar{s}}), \quad \text {for all }{\bar{t}}\in {\bar{\textrm{T}}}. \end{aligned}$$

For a pair of non-negative integers (a, b), we use $\sigma _{a,b}$ to denote the right ${\mathbb {F}}$-representation ${{\,\textrm{Sym}\,}}^a{\mathbb {F}}^{\oplus 2} \otimes \textrm{det}^b$ of ${\bar{\textrm{G}}}$ (in particular, b is taken modulo $p-1$). The right action is the transpose of the left action in [11, 48]; explicitly, it is, realized on the space of homogeneous polynomials of degree a

$$\begin{aligned} h(X, Y) \big |_{{\big ({{\begin{matrix}{\bar{\alpha }}&{}{}{\bar{\beta }}\\ {\bar{\gamma }}&{}{} {\bar{\delta }}\end{matrix}}} \big )}}: = h({\bar{\alpha }} X + {\bar{\beta }} Y, {\bar{\gamma }} X + {\bar{\delta }} Y) \quad \text{ for } {\bigg ({\begin{matrix}{\bar{\alpha }}&{}{}{\bar{\beta }}\\ {\bar{\gamma }}&{}{} {\bar{\delta }}\end{matrix}} \bigg )} \in {\bar{\text {G}}}. \end{aligned}$$

When $a \in \{0, \dots , p-1\}$ and $b \in \{0, \dots , p-2\}$, $\sigma _{a,b}$ is irreducible. These representations exhaust all irreducible (right) ${\mathbb {F}}$-representations of ${{\,\textrm{GL}\,}}_2({\mathbb {F}}_p)$, and are called Serre weights. We use $\textrm{Proj}_{a,b}$ to denote the projective envelope of $\sigma _{a,b}$ as a (right) ${\mathbb {F}}[{{\,\textrm{GL}\,}}_2({\mathbb {F}}_p)]$-module.

Lemma A.2

The projective envelope $\textrm{Proj}_{a,b}$ can be described explicitly as follows.

• (1) For any b, $\textrm{Proj}_{0,b}$ is of dimension p over ${\mathbb {F}}$, and admits a socle filtration

  

  where the $\sigma _{0,b}$ on the left represents the socle, the $\sigma _{0,b}$ on the right represents the cosocle, and each line represents a (unique up to isomorphism) nontrivial extension between the two Jordan–Hölder factors.

• (2) When $1 \le a \le p-2$, $\textrm{Proj}_{a,b}$ is of dimensional 2p over ${\mathbb {F}}$, and admits a socle filtration

  

  where the $\sigma _{a,b}$ on the left represents the socle, the $\sigma _{a,b}$ on the right represents the cosocle, and each line represents a (unique up to isomorphism) nontrivial extension between the two factors. (Here, when $a=p-2$, the term $\sigma _{p-3-a, a+b+1}$ is interpreted as 0.)

• (3) For any b, the Serre weight $\sigma _{p-1,b}$ is projective, i.e., $\textrm{Proj}_{p-1,b} = \sigma _{p-1,b}$. In particular, $\dim _{\mathbb {F}}\textrm{Proj}_{p-1,b} = p$.

Moreover, in either case $\textrm{Proj}_{a,b}$ is also the injective envelope of $\sigma _{a,b}$ as an ${\mathbb {F}}$-representation of ${{\,\textrm{GL}\,}}_2({\mathbb {F}}_p)$.

Proof

The dimensions of the projective envelopes follow from [48, Lemma 4.1.7]. The socle filtrations of the projective envelopes in (1) and (2) follow from [14, Lemma 3.4, Lemma 3.5]. Part (3) follows from [48, Corollary 4.2.22]. $\square$

A.3 Induced Representations

Consider the following character $\eta$ of ${\bar{\textrm{B}}}$:

$$\begin{aligned} \eta : {\bar{\text {B}}}\rightarrow {\mathbb {F}}^\times ,\quad {\bigg ({\begin{matrix}{\bar{\alpha }}&{}{}{\bar{\beta }}\\ 0&{}{} {\bar{\delta }}\end{matrix}} \bigg )} \mapsto {\bar{\alpha }}. \end{aligned}$$

For $k \in {\mathbb {Z}}$, we have an induced representation

$$\begin{aligned} {{\,\text {Ind}\,}}_{{\bar{\text {B}}}}^{{\bar{\text {G}}}}(\eta ^k): = \big \{f: {\bar{\text {G}}} \rightarrow {\mathbb {F}}\, \big |\, f({\bar{b}} {\bar{g}}) = \eta ^k({\bar{b}})f({\bar{g}}), \text{ for } \text{ all } {\bar{b}} \in {\bar{\text {B}}}\big \},\end{aligned}$$

which we equip with the right action given by

$$\begin{aligned} f|_{{\bar{h}}}({\bar{g}}): = f({\bar{g}} {\bar{h}}^\textrm{T}), \quad \text {for } {\bar{g}}, {\bar{h}} \in {\bar{\textrm{G}}}. \end{aligned}$$

This is the transpose of the usual left action.

Lemma A.4

• (1) For any integer $1\le k\le p-1$, we have an exact sequence of ${\bar{\textrm{G}}}$-representations

  $$\begin{aligned} 0\rightarrow \sigma _{k,0}\rightarrow {{\,\textrm{Ind}\,}}_{{\bar{\textrm{B}}}}^{{\bar{\textrm{G}}}}\eta ^{k}\rightarrow \sigma _{p-1-k, k}\rightarrow 0.\end{aligned}$$

  (A.4.1)

  The above exact sequence splits if and only if $k=p-1$.

• (2) Taking ${\bar{U}}$-invariants of (A.4.1) gives an exact sequence of ${\bar{\textrm{U}}}$-invariants

  $$\begin{aligned} 0\rightarrow \sigma _{k,0}^{{\bar{\textrm{U}}}}\rightarrow \big ({{\,\textrm{Ind}\,}}_{{\bar{\textrm{B}}}}^{{\bar{\textrm{G}}}}\eta ^{k}\big )^{{\bar{\textrm{U}}}}\rightarrow \sigma _{p-1-k, k}^{{\bar{\textrm{U}}}}\rightarrow 0. \end{aligned}$$

• (3) In the description of $\textrm{Proj}_{a,b}$ of Lemma A.2 (2), the extensions

  

  are twists of induced representations ${{\,\textrm{Ind}\,}}_{{\bar{\textrm{B}}}}^{{\bar{\textrm{G}}}}\eta ^{a} \otimes \textrm{det}^b$ and ${{\,\textrm{Ind}\,}}_{{\bar{\textrm{B}}}}^{{\bar{\textrm{G}}}}\eta ^{p-1-a} \otimes \textrm{det}^{a+b}$, respectively.

Proof

(1) follows from [48, Proposition 3.2.2, Lemma 4.1.3]. (2) follows from [48, Lemma 3.1.11]. (3) follows from [48, Lemma 4.1.1]. $\square$

Lemma A.5

When $1 \le a\le p-2$, in terms of the socle filtration in Lemma A.2 (2), the ${\bar{\textrm{U}}}$-invariant subspace of $\textrm{Proj}_{a,b}$ is a 2-dimensional subspace given by

$$\begin{aligned} (\textrm{Proj}_{a,b})^{{\bar{\textrm{U}}}} \cong \sigma _{a,b}^{{\bar{\textrm{U}}}} \oplus \sigma _{p-1-a, a+b}^{{\bar{\textrm{U}}}}, \end{aligned}$$

where the first term is the $\sigma _{a,b}$ in the socle. In particular, the actions of ${\bar{\textrm{T}}}$ on these two 1-dimensional subspaces are given by the characters

$$\begin{aligned} \omega ^{b} \times \omega ^{a+b} \quad \text{ and } \quad \omega ^{a+b} \times \omega ^{b}, \quad \text {respectively}.\end{aligned}$$

Proof

Since $\textrm{Proj}_{a,b}$ is a free module over ${\mathbb {F}}[{\bar{\textrm{U}}}]$, we know that $\dim (\textrm{Proj}_{a,b})^{{\bar{\textrm{U}}}} = 2$. However, ${{\,\textrm{Ind}\,}}_{{\bar{\textrm{B}}}}^{{\bar{\textrm{G}}}}\eta ^a \otimes \textrm{det}^b$ is a subrepresentation of $\textrm{Proj}_{a,b}$ by Lemma A.4 (3) and has already 2-dimensional ${\bar{\textrm{U}}}$-invariants by Lemma A.4 (2). Therefore, we have

$$\begin{aligned} (\textrm{Proj}_{a,b})^{{\bar{\textrm{U}}}} \cong \big ({{\,\textrm{Ind}\,}}_{{\bar{\textrm{B}}}}^{{\bar{\textrm{G}}}}\eta ^{a}\big )^{{\bar{\textrm{U}}}} \otimes \textrm{det}^b \cong \sigma _{a,b}^{{\bar{\textrm{U}}}} \oplus \sigma _{p-1-a, a+b}^{{\bar{\textrm{U}}}}. \end{aligned}$$

$\square$

Lemma A.6

For any integer $k\ge p+1$, we have an exact sequence of ${\bar{\textrm{G}}}$-represent-ations

$$\begin{aligned} 0\rightarrow \sigma _{k-(p+1),1}\xrightarrow {i} \sigma _{k,0}\xrightarrow {\pi } {{\,\textrm{Ind}\,}}_{{\bar{\textrm{B}}}}^{{\bar{\textrm{G}}}}\eta ^{k}\rightarrow 0. \end{aligned}$$

Proof

This is [52, Proposition 2.4]. For the convenience of the readers, we recall the definition of the two maps. The map i is given by multiplication by the homogeneous polynomial $f_p(X,Y)=X^pY-XY^p$ of degree $p+1$. It is straightforward to check that i is ${\bar{\textrm{G}}}$-equivariant and injective. The map $\pi$ comes from adjunction

$$\begin{aligned} {{\,\textrm{Hom}\,}}_{{\bar{\textrm{G}}}}(\sigma _{k,0}, {{\,\textrm{Ind}\,}}_{{\bar{\textrm{B}}}}^{{\bar{\textrm{G}}}}\eta ^k) \cong {{\,\textrm{Hom}\,}}_{{\bar{\textrm{B}}}^-}(\sigma _{k,0}, \eta ^k). \end{aligned}$$

(Note that both $\sigma _{k,0}$ and the induced representations are right representations, coming from the transpose of the natural left action.) The vector $X^k \in \sigma _{k,0}$ is ${\bar{\textrm{U}}}^-$-invariants and is an eigenvector of ${\bar{\textrm{T}}}$ with eigencharacter $\eta ^k$. $\square$

We conclude the appendix with the following result that is claimed in Remark 2.23(2).

Proposition A.7

Let ${\bar{\rho }} = {\big ({\begin{matrix}\omega _1^{a+b+1}&{}*\ne 0\\ 0&{} \omega _1^b\end{matrix}} \big )}$ with $1\le a \le p-4$ and $0 \le b \le p-2$ be as in Definition 2.22 and let ${\widetilde{\textrm{H}}}$ is a primitive ${\mathcal {O}}\llbracket \textrm{K}_p\rrbracket$-projective augmented module of type ${\bar{\rho }}$. In particular, ${\overline{\textrm{H}}} = {\widetilde{\textrm{H}}} / (\varpi , \textrm{I}_{1+p\textrm{M}_2({\mathbb {Z}}_p)})$ is isomorphic to $\textrm{Proj}_{a,b}$ as a right ${\mathbb {F}}[{{\,\textrm{GL}\,}}_2({\mathbb {F}}_p)]$-module. Then, $\textrm{S}_2^\textrm{Iw}(\omega ^b\times \omega ^{a+b})$ is of rank 1 over ${\mathcal {O}}$, and the $U_p$-action on this space is invertible (over ${\mathcal {O}}$); in particular, the $U_p$-slope on this space is zero.

Proof

By Proposition 4.1, $\textrm{S}_2^\textrm{Iw}(\omega ^b\times \omega ^{a+b})$ is free of rank 1 over ${\mathcal {O}}$.

Consider three characters $\varepsilon =1\times \omega ^a$, $\varepsilon '=\omega ^a\times 1$ and $\psi =\omega ^a\times \omega ^a$. By Corollary 4.4 and Proposition 4.7, we have $d_{p+1-a}^\textrm{Iw}(\psi )=2$ and $d_{p+1-a}^\textrm{ur}(\psi )=0$. We choose a set $\{\tilde{u}_j={\big ({{\begin{matrix}1&{}{}0\\ j&{}{} 1\end{matrix}}} \big )},j=0,\dots , p-1,\tilde{u}_\infty ={\big ({{\begin{matrix}0&{}{}1\\ 1&{}{} 0\end{matrix}}} \big )} \}$ of coset representatives of $\textrm{Iw}_p\setminus \textrm{K}_p$. The formula

$$\begin{aligned} \text {pr}(\varphi )(m)=\sum _{j=0,\dots ,p-1,\infty }\varphi (m\tilde{u}_j^{-1})|_{\tilde{u}_j},\quad \varphi \in \text {S}_{p+1-a}^\text {Iw}(\psi ),\ m\in \tilde{\text {H}} \end{aligned}$$

defines a map $\textrm{pr}:\textrm{S}_{p+1-a}^\textrm{Iw}(\psi )\rightarrow \textrm{S}_{p+1-a}^\textrm{ur}(\psi )$. Here $(\cdot )|_{\tilde{u}_j}$ in the above formula is the right action of $\textrm{K}_p$ defined in (2.20.1). On the other hand, a straightforward computation gives an equality of operators $\textrm{AL}_{p+1-a,\psi }\circ \textrm{pr}-\textrm{AL}_{p+1-a,\psi }=U_p$ on $\textrm{S}_{p+1-a}^\textrm{Iw}(\psi )$. Since $\textrm{S}_{p+1-a}^\textrm{ur}(\psi )=0$, we actually have $\textrm{U}_p=-\textrm{AL}_{p+1-a,\psi }$ on $\textrm{S}_{p+1-a}^\textrm{Iw}(\psi )$. Therefore, by Proposition 3.12, the $U_p$-slopes on $\textrm{S}_{p+1-a}^\textrm{Iw}(\psi )$ are $\frac{p-1-a}{2}$ with multiplicity 2. By Proposition 3.10, we have $v_p(c_1^{(\varepsilon ')}(w_{p+1-a}))\ge \frac{p-1-a}{2}>1$. Since $c_1^{(\varepsilon ')}(w)\in {\mathcal {O}}\llbracket w\rrbracket$ and $v_p(w_{p+1-a})\ge 1$, $v_p(w_2)\ge 1$, we have $v_p(c_1^{(\varepsilon ')}(w_2))\ge 1$. Since $d_2^\textrm{Iw}(\varepsilon ')=1$, by Proposition 3.10 and Proposition 3.12, we see that the $U_p$-slope on $\textrm{S}_2^\textrm{Iw}(\varepsilon ')$ is exactly 1. Now, by Proposition 3.12, the $U_p$-slope on $\textrm{S}_2^\textrm{Iw}(\varepsilon )$ is 0.

$\square$

Appendix B: Errata for [44]

We include some errata for [44] here.

B.1 [44, Proposition 3.4]

This is just a bad convention, the action $||_{\delta _p}^{[\cdot ]}$ is a right action; however, [44, Proposition 3.4] uses the column vector convention. Ideally, one should swap the indices of $P_{m,n}(\delta _p)$. This does not create an mathematical error.

B.2 [44, §5.4]

In [44, (5.4.1)], we defined a closed subspace

$$\begin{aligned} {{\,\text {Ind}\,}}_{B^\text {op}({\mathbb {Z}}_p)}^{\text {Iw}_q}([\cdot ]')^{\text {mod}} = \widehat{\bigoplus \limits _{n \ge 0}}\ T^n \Lambda ^{>1/p}\cdot \left( {\begin{array}{c}z\\ n\end{array}}\right) \end{aligned}$$

of the space of continuous functions ${\mathcal {C}}^0({\mathbb {Z}}_p; \Lambda ^{>1/p})$. Unfortunately, as stated, this subspace is not stable under the ${\textbf{M}}_1$-action. If one insists on having an ${\textbf{M}}_1$-action, one has to replace the completed direct sum above with direct product, then the rest of the estimate stays valid, except that the space $S_{\textrm{int}}^{D, \dagger , 1}$ is not a Banach module over $\Lambda ^{>1/p}$ in the usual sense; it is the $\Lambda ^{>1/p}$-linear dual of a Banach module over $\Lambda ^{>1/p}$. The compactness of $U_p$ needs be interpreted otherwise.