A note on depth preservation

TL;DR

This paper demonstrates that for certain wildly ramified tori, the depth characteristic is not maintained under the local Langlands correspondence, highlighting a specific limitation in the current understanding.

Contribution

It reveals that depth preservation fails for wildly ramified tori in the local Langlands correspondence, providing new insights into ramification behavior.

Findings

Depth is not preserved for wildly ramified tori.

The failure of depth preservation is specific to certain ramification conditions.

This challenges assumptions about the general behavior of the local Langlands correspondence.

Abstract

We show that for a wildly ramified torus, depth is not preserved in general under local Langlands correspondence for tori.

Equations

$$G_u=G_i\text{ where }i\text{ is the least integer }\ge u.$$

$$(G/H{)}^v=G^{v}H/H.$$

$$G^v=\underset{F/K\text{ finite}}{\underleftarrow{\lim }}\mathrm{Gal}(F/K{)}^v.$$

$$(G/I{)}^r\cap (H/I)$$

$${\mathrm{dep}}_G(M)=\inf \{r\ge 0\mid M^{G^s}\ne 0\forall s>r\}.$$

$${\mathrm{dep}}_G(\phi )=\inf \{r\ge 0\mid G^s\subset \ker (\phi )\forall s>r\}.$$

$${\mathrm{Res}}_{G^r}({\mathrm{Ind}}_H^{G}N)=\underset{g\in G^r\backslash G\text{/}/H}{⨁}{\mathrm{Ind}}_{G^r\cap H}^{G^r}{N}^g.$$

$$({\mathrm{Ind}}_H^{G}N{)}^{G^r}\ne 0$$

$$\mathrm{Sh}:{\mathrm{H}}^1(G,{\mathrm{Ind}}_H^{G}N)\to {\mathrm{H}}^1(H,N)$$

$$\gamma \mapsto (h\mapsto \gamma (h)(1))$$

$$({\mathrm{Ind}}_B^{A}M{)}^C\cong {\mathrm{Ind}}_{B/B\cap C}^{A/C}M{}^{C\cap B}.$$

$${\mathrm{H}}^1(G/G^r,({\mathrm{Ind}}_H^{G}N{)}^{G^r})$$

$$\mathrm{Hom}(T(K),{\mathbb{C}}^{\times })$$

$$\cong$$

$${\lambda }_T:\mathrm{Hom}(T(K),{\mathbb{C}}^{\times })\to {\mathrm{H}}^1(W_K,\hat{T})$$

$$M_r^{\times }\cong (W_M^r{)}^{\mathrm{ab}}.$$

$$\inf \{r\ge 0\mid T(K{)}_s\subset \ker (\chi )\text{ for }s>r\}.$$

$$T(K{)}_r$$

$$=$$

$$\mathrm{Hom}(T(K)/T(K{)}_r,{\mathbb{C}}^{\times })\cong {\mathrm{H}}^1(W_K/W_K^{{\phi }_{K^{\prime }/K}(er)},{\hat{T}}^{W_K^{{\phi }_{K^{\prime }/K}(er)}}).$$

$$\mathrm{Hom}(T(K)/T(K{)}_r,{\mathbb{C}}^{\times })$$

$$\cong$$

$${\phi }_{K^{\prime }/K}(e\cdot {\mathrm{dep}}_T(\chi ))={\mathrm{dep}}_{W_K}({\lambda }_T(\chi )).$$

$${\mathrm{dep}}_T(\chi )={\mathrm{dep}}_{W_K}({\lambda }_T(\chi )).$$

$${\mathrm{dep}}_T(\chi )={\mathrm{dep}}_{T^{\prime }}({\chi }^{\prime })=\sup \{{\mathrm{dep}}_{T_i^{\prime }}({\chi }_i^{\prime })\mid 1\le i\le n\}.$$

$${\mathrm{dep}}_{W_K}({\lambda }_{T^{\prime }}({\chi }^{\prime }))$$

$${\mathrm{dep}}_{W_K}({\lambda }_T(\chi ))\ge {\mathrm{dep}}_T(\chi ).$$

$${\phi }_{L/K}(r)$$

$$G(m{)}^s:=\{a\in G(m)\mid a\equiv 1\text{ mod }{p}^s\}.$$

$$G_0=G$$

$$\text{if }1\le u\le p-1$$

Full text

1. Introduction

Let $K$ be a non-archimedean local field and let $W_{K}$ denote its Weil group. Local class field theory (LCFT) tells us that there is a canonical isomorphism $K^{\times}\cong W_{K}^{\mathrm{ab}}$ and this isomorphism respects the numbering on the filtration subgroups $\{K_{r}^{\times}\}_{r\geq 0}$ of $K^{\times}$ and the upper numbering on the filtration subgroups $\{W_{K}^{r}\}_{r\geq 0}$ of $W_{K}^{\mathrm{}}$. Local Langlands Correspondence (LLC) stipulates a vast generalization of the LCFT isomorphism. In LLC, irreducible representations $\mathrm{Irr}(G(K))$ of the $K$-points of a reductive $K$-group $G$ are expected to be parametrized by arithmetic objects called Langlands parameters $\Phi(G)$ in a certain natural way. For each $\pi\in\mathrm{Irr}(G(K))$, Moy-Prasad theory associates an invariant called depth $\mathrm{dep}(\pi)$. Also for each $\phi\in\Phi(G)$, one defines the notion of depth $\mathrm{dep}(\phi)$. It is the smallest number such that $\phi$ is trivial on $W_{K}^{r}$ for all $r>\mathrm{dep}(\phi)$. If $\phi$ associates to $\pi$ under LLC, then one expects that quite fairly $\mathrm{dep}(\pi)=\mathrm{dep}(\phi)$. This is known in many cases (see the introduction in [ABPS2016] for a survey). However, counter examples have been constructed for inner forms of $\mathrm{SL}_{n}(K)$ [ABPS2016] and in the case of $\mathrm{SL}_{2}(K)$ when $K$ has characteristic $2$ [AMPS2017].

Now let $T=\mathrm{R}_{K^{\prime}/K}\mathbb{G}_{\mathrm{m}}$ where $K^{\prime}$ is a finite separable extension of $K$ and $R_{K^{\prime}/K}$ denotes the Weil restriction and let $\lambda_{T}:\chi\in\mathrm{Irr}(T(K))\mapsto\lambda_{T}(\chi)\in\Phi(T)$ under LLC. In this note, we show that $\varphi_{K^{\prime}/K}(e\cdot\mathrm{dep}(\chi))=\mathrm{dep}(\lambda_{T}(\chi))$ where $\varphi_{K^{\prime}/K}$ is the Hasse-Herbrand function and $e$ is the ramification index of $K^{\prime}/K$. Thus for all postitive depth characters $\chi$, $\mathrm{dep}(\lambda_{T}(\chi))>\mathrm{dep}(\chi)$. When $T$ is a tamely induced wildly ramified torus (see Sec. 7.1), we show that $T(K)$ admits characters for which depth is not preserved under LLC. In Section 8, we compute Hasse-Herbrand function for a certain wildly ramified extension of a cyclotomic field to illustrate the failure of depth preservation.

The proofs in Section 7 follow closely the proofs in [Yu] and [MM2015].

2. Review of ramification groups

Let $K$ be a non-archimedean local field and let $L$ be a finite Galois extension of $K$. Write $\mathcal{O}_{L}$, $\mathfrak{p}_{L}$ for the ring of integers of $L$ and the maximal ideal of $\mathcal{O}_{L}$. For $i\geq-1$, define $G_{i}$ to be the set of all $s\in G:=\mathrm{Gal}(L/K)$ such that $s$ operates trivially on $\mathcal{O}_{L}/\mathfrak{p}_{L}^{i+1}$. Then $G_{-1}=G$. The groups $G_{i}$ are called ramification groups. They form a decreasing filtration of normal subgroups. Extend the definition of $G_{u}$ for all real numbers $u\geq-1$ by setting

[TABLE]

This numbering of ramification groups is called lower numbering. Lower numbering behaves well with respect to intersections, i.e., if $H$ is a subgroup of $G$, then $G_{u}\cap H=H_{u}$.

Upper numbering of ramification groups

Define $\varphi_{L/K}:[-1,\infty)\rightarrow\mathbb{R}$ to be the map $r\mapsto\int_{0}^{r}\frac{1}{(G_{0}:G_{t})}dt$ where $(G_{0}:G_{u}):=(G_{u}:G_{0})^{-1}$ for $u\in[-1,0)$. The function $\varphi_{L/K}$ is called the Hasse-Herbrand function. It has the basic properties [Serre]:

• (a)

$\varphi_{L/K}$ is continuous, piecewise linear, increasing and concave.

• (b)

$\varphi_{L/K}(0)=0$.

• (c)

$\varphi_{L/K}$ is a homeomorphism of $[-1,\infty)$ onto itself.

• (d)

If $H$ is a normal subgroup of $G$, then $\varphi_{L/K}=\varphi_{L^{H}/K}\circ\varphi_{L/L^{H}}$.

If an extension $M/K$ is not Galois, define $\varphi_{M/K}=\varphi_{M^{\prime}/K}\circ\varphi_{M/M^{\prime}}^{-1}$, where $M^{\prime}$ is a Galois extension of $K$ containing $M$. The inverse $\varphi_{M/K}^{-1}$ is denoted $\psi_{M/K}$.

Define an *upper numbering *on ramification groups by setting $G^{v}=G_{u}$ if $v=\varphi_{L/K}(u)$. Upper numbering behaves well with respect to quotients, i.e., if $H$ is a normal subgroup of $G$, then

[TABLE]

For an infinite Galois extension $\Omega$ of $K$, define the ramification groups on $G=\mathrm{Gal}(\Omega/K)$ by:

[TABLE]

Now let $L/K$ be Galois extension of local fields and let $F$ be a finite extension of $K$ contained in $L$. Write $G=\mathrm{Gal}(L/K)$ and $H=\mathrm{Gal}(L/F)$.

Lemma 1**.**

For all $r\geq 0$, $G^{r}\cap H=H^{\psi_{F/K}(r)}$.

Proof.

Let $E$ be a fintie Galois extension of $K$ in $L$ containing $F$. Write $I=\mathrm{Gal}(L/E)$. Then

[TABLE]

The lemma now follows by taking inverse limit over $E$. ∎

3. Notion of depth

Let $G=\mathrm{Gal}(L/K)$ where $L$ and $K$ are local fields and let $M$ be a $G$-module. Define the depth of $M$ to be:

[TABLE]

Define the depth of a co-cycle $\varphi\in\mathrm{H}^{1}(G,M)$ to be:

[TABLE]

4. Depth change under induction

Let $G=\mathrm{Gal}(L/K)$ and $H=\mathrm{Gal}(L/F)$ where $K\subseteq F\subseteq L$ and $F/K$ finite Galois. Let $N$ be an $H$-module.

Proposition 2**.**

$\mathrm{dep}_{H}(N)=\psi_{F/K}(\mathrm{dep}_{G}(\mathrm{Ind}_{H}^{G}N)).$**

Proof.

By Mackey theory,

[TABLE]

Here $\mathrm{Res}$ denotes the restriction functor and $N^{g}$ denotes the $g$-twisted module $N$. By Lemma 1, $G^{r}\cap H=H^{\psi_{F/K}(r)}$. Thus

[TABLE]

∎

5. Depth change under Shapiro’s isomorphism

Again $G=\mathrm{Gal}(L/K)$ and $H=\mathrm{Gal}(L/F)$ where $K\subseteq F\subseteq L$ and $F/K$ is any finite extension and let $N$ be an $H$-module.

Shapiro’s lemma states that the map

[TABLE]

defined by

[TABLE]

is an isomorphism. We wish to relate the depth of co-cycles under this isomorphism. We first observe the following:

Lemma 3**.**

Let A be a group, $B$ and $C$ subgroups of $A$ with $C$ being normal in $A$. Let $M$ be a $B$-module. Then there is a canonical isomorphism of $A/C$-modules:

[TABLE]

Proof.

The map $f\in(\mathrm{Ind}_{B}^{A}M)^{C}\mapsto\tilde{f}:=(gC\mapsto f(g))\in(\mathrm{Ind}_{B/B\cap C}^{A/C}M{}^{C\cap B})$ is easily verified to be the required isomorphism. ∎

Lemma 4**.**

For $r\geq 0$, Shapiro’s lemma induces an isomorphism $\mathrm{H^{1}}(G/G^{r},(\mathrm{Ind}_{H}^{G}N)^{G^{r}})\cong\mathrm{H^{1}}(H/H^{\psi_{F/K}(r)},N^{H^{\psi_{F/K}(r)}}).$

Proof.

We have

[TABLE]

The first isomorphism follows from Lemma 3, second from Lemma 1 and the last from Shapiro’s lemma. ∎

Write $\mathrm{H^{1}}(G,\mathrm{Ind}_{H}^{G}N)_{\mathrm{adm}}=\bigcup_{r\geq 0}\mathrm{H^{1}}(G/G^{r},(\mathrm{Ind}_{H}^{G}N)^{G^{r}}).$

Corollary 5**.**

If $\lambda\in\mathrm{H^{1}}(G,\mathrm{Ind}_{H}^{G}N)_{\mathrm{adm}}$ , then $\mathrm{dep}_{G}(\lambda)=\varphi_{F/K}(\mathrm{dep}_{H}(\mathrm{Sh}(\lambda))$.

Proof.

Let $\mathrm{dep}_{G}(\lambda)=r$. Then $G^{s}\subset\mathrm{ker}(\lambda)$ if $s>r$. By Lemma 4, this implies $H^{\psi_{F/K}(s)}\subset\mathrm{ker}(\mathrm{Sh}(\lambda))$ if $s>r$. Therefore $\mathrm{dep}_{H}(\mathrm{Sh}(\lambda))\leq\psi_{F/K}(\mathrm{dep}_{G}(\lambda))$. The argument is reversible showing that $\mathrm{dep}_{H}(\mathrm{Sh}(\lambda))\geq\psi_{F/K}(\mathrm{dep}_{G}(\lambda))$. Therefore $\mathrm{dep}_{H}(\mathrm{Sh}(\lambda))=\psi_{F/K}(\mathrm{dep}_{G}(\lambda))$.

∎

6. Langlands correspondence for tori

We review here the statement of local Langlands correspondence for tori as stated and proved in [Yu].

6.1. Special case

Let $T=\mathrm{R}_{K^{\prime}/K}\mathbb{G}_{m}$ where $K^{\prime}$ is a finite separable extension of $K$ and $R_{K^{\prime}/K}$ denotes the Weil restriction. Then $T(K)=K^{\prime\times}$ and the group of characters $X^{*}(T)$ is canonically a free $\mathbb{Z}$-module with basis $W_{K}/W_{K^{\prime}}$ where $W_{K}$ (resp. $W_{K^{\prime}}$) denotes the Weil group of $K$ (resp. $K^{\prime}$). From this, it follows that the complex dual $\hat{T}$ of $T$ is canonically isomorphic to $\mathrm{Ind}_{W_{K^{\prime}}}^{W_{K}}\mathbb{C}^{\times}$. We get,

[TABLE]

The isomorphism 6.1 follows by class field theory and the isomorphism 6.1 by Shapiro’s lemma.

6.2. The LLC for tori in general

Theorem**.**

*[L97]*There is a unique family of homomorphisms

[TABLE]

with the following properties:

1. (1)

$\lambda_{T}$* is additive functorial in $T$, i.e., it is a morphism between two additive functors from the category of tori over $K$ to the category of abelian groups;* 2. (2)

For $T=R_{K^{\prime}/K}\mathbb{G}_{m}$, where $K^{\prime}/K$ is a finite separable extension, $\lambda_{T}$ is the isomorphism described in Section 6.1.

7. Depth change for tori under LLC

We keep the notations as in Section 6. Let $M$ be a local field. Recall that $M^{\times}$ admits a filtration $\{M_{r}^{\times}\}_{r\geq 0}$ where $M_{0}^{\times}$ is the units of the ring of integers and for $r>0$, $M_{r}^{\times}:=\{x\in M\mid\mathrm{ord}_{M}(x-1)\geq r\}$. Here $\mathrm{ord}_{M}$ is the valuation of $M$ normalised so that $\mathrm{ord}_{M}(M^{\times})=\mathbb{Z}$. Under local class field theory isomorphism

[TABLE]

We recall that $T(K)$ carries a Moy-Prasad filtration $\{T(K)_{r}\}_{r\geq 0}$ [MP96]. The depth $\mathrm{dep}_{T}(\chi)$ of a character $\chi:T(K)\rightarrow\mathbb{C}^{\times}$ is defined to be

[TABLE]

The group $T(K)_{0}$ is called the Iwahori subgroup of $T(K)$. It is a subgroup of finite index in the maximal compact subgroup of $T(K)$. When $T=\mathrm{R}_{K^{\prime}/K}\mathbb{G}_{\mathrm{m}}$, then for $r>0$,

[TABLE]

Here $\mathrm{ord}_{K}$ is the valuation on $K^{\prime}$ normalised so that $\mathrm{ord}_{K}(K^{\times})=\mathbb{Z}$ and $e$ is the ramification index of $K^{\prime}/K$. The equality 7.1 follows from [Yu03, Sec. 4.2] and the equality 7 follows from the fact that $\mathrm{ord}_{K^{\prime}}(\alpha)=e\cdot\mathrm{ord}_{K}(\alpha)$ for all $\alpha\in K^{\times}.$

Theorem 6**.**

Let $T=\mathrm{R}_{K^{\prime}/K}\mathbb{G}_{\mathrm{m}}$, where $K^{\prime}/K$ is a finite separable extension of local fields of ramification index $e$. Then for $r\geq 0$, the local Langlands correspondence for tori induces an isomorphism:

[TABLE]

Proof.

The case $r=0$ is a special case of [MM2015, Theorem 7]. For $r>0$, this follows by

[TABLE]

Here, the isomorphism 7 follows from Lemma 4. ∎

Corollary 7**.**

For $T$ as in Theorem 6

[TABLE]

Proof.

This follows from an argument analogous to the argument in the proof of the Corollary 5. ∎

Remark 8*.*

The slope of the map $r\mapsto\varphi_{K^{\prime}/K}(er)$ at a differentiable point $r$ is $\frac{e}{(G_{0}:G_{r})}\geq 1$. Thus, when $K^{\prime}/K$ is a wildly ramified extension, $\varphi_{K^{\prime}/K}(er)>r$ and consequently $\mathrm{dep}_{T}(\chi)<\mathrm{dep}_{W_{K}}(\lambda_{T}(\chi))$.

When $K^{\prime}/K$ is a tamely ramified extension, $\varphi_{K^{\prime}/K}(r)=\frac{r}{e}$. Therefore in this case, Corollary 7 simplifies to,

[TABLE]

This is a special case of Depth-preservation Theorem of Yu for tamely ramified tori [Yu, Sec. 7.10].

7.1. Case of a tamely induced tori

Recall that a $K$-torus is called induced if it is of the form $\Pi_{i=1}^{k}R_{L_{i}/K}\mathbb{G}_{\mathrm{m}}$, where $L_{i}$ are finite separable extensions of $K$. A $K$-torus $\mathcal{T}$ is called tamely induced if $\mathcal{T}\otimes_{K}K_{t}$ is an induced torus for some tamely ramified extension $K_{t}$ of $K$. In this section, we compare depths under LLC for such tori following the proof in [Yu, Sec. 7.10].

Let $T$ be a tamely induced $K$-torus. Then there exists an induced torus $T^{\prime}=\prod_{i=1}^{n}\mathrm{R}_{K_{i}^{\prime}/K}\mathbb{G}_{\mathrm{m}}$ such that $T^{\prime}\twoheadrightarrow T$ and $C_{0}:=\mathrm{ker}(T^{\prime}\rightarrow T)$ is connected. Further $T^{\prime}(K)_{r}\twoheadrightarrow T(K)_{r}$ $\forall r>0$ (see proof in [Yu03, Lemma 4.7.4]). Let $\chi\in\mathrm{Hom}(T(K),\mathbb{C}^{\times})$ and let $\chi^{\prime}$ denote its lift to $T^{\prime}(K)$. Then

[TABLE]

Here $T_{i}^{\prime}$ denotes $\mathrm{R}_{K_{i}^{\prime}/K}\mathbb{G}_{\mathrm{m}}$ and $\chi_{i}^{\prime}=\chi^{\prime}|_{T_{i}^{\prime}}$. By functoriality, $\lambda_{T^{\prime}}(\chi^{\prime})$ is the image of $\lambda_{T}(\chi)$ under $\mathrm{H}^{1}(W_{K},\hat{T})\rightarrow\mathrm{H}^{1}(W_{K},\hat{T^{\prime}})$ and therefore $\mathrm{dep}_{W_{K}}(\lambda_{T}(\chi))=\mathrm{dep}_{W_{K}}(\lambda_{T^{\prime}}(\chi^{\prime}))$. But

[TABLE]

Here $e_{i}$ denotes the ramification index of $K_{i}^{\prime}/K$. Thus

[TABLE]

Now assume $T$ is wildly ramified. We will now produce a character of $T(K)$ for which the inequality (7.6) is strict. We can assume without loss of generality that $T_{0}:=\mathrm{R}_{K_{1}^{\prime}/K}\mathbb{G}_{\mathrm{m}}$ is wildly ramified. Let $\chi_{0}^{\prime}$ be a positive depth character of $T_{0}(K)$ which is trivial on $C_{0}\cap T_{0}(K)$. Extend $\chi_{0}^{\prime}$ trivially to a character $\chi^{\prime}$ of $T^{\prime}(K)$. Then since $\mathbb{C}^{\times}$ is divisible, the character $\chi^{\prime}$ lifts to a character $\chi$ of $T(K)$. By Remark 8, $\mathrm{dep}_{T_{0}}(\chi_{0}^{\prime})<\mathrm{dep}_{W_{K}}(\lambda_{T_{0}}(\chi_{0}^{\prime}))$. Since $\mathrm{dep}_{T}(\chi)=\mathrm{dep}_{T_{0}}(\chi_{0}^{\prime})$ and $\mathrm{dep}_{W_{K}}(\lambda_{T}(\chi))=\mathrm{dep}_{W_{K}}(\lambda_{T_{0}}(\chi_{0}^{\prime}))$, it follows that the inequality (7.6) is strict for this choice of $\chi$.

8. An Example

Let $K=\mathbb{Q}_{p}$, $L=K(\zeta_{p^{n}})$, where $\zeta_{p^{n}}$ denotes a primitive $p^{n}$th root of unity, $n\geq 1$. Then $L/K$ is a totally ramified extension of degree $(p-1)p^{n-1}.$ Consider the intermediate extension $F=K(\zeta_{p})$ of $K$ of degree $p-1$ over $K$. Then $L/F$ is a wildly ramified extension. Write $G=\mathrm{Gal}(L/K)$ and $H=\mathrm{Gal}(L/F)$.

Lemma 9**.**

For $1\leq r$ and $\varphi_{L/F}(r)=(p-1)\varphi_{L/K}(r)$.

Proof.

We first note that since we considering abelian extensions, the jumps in filtration occur at integer values. We have for $r\geq 1$,

[TABLE]

The last equality holds because $G_{t}=H_{t}$ for $t\geq 1$ and $(G_{0}:H_{0})=p-1$. Thus

[TABLE]

∎

Write $m=p^{n}$ and let $G(m)=(\mathbb{Z}/m\mathbb{Z})^{\times}$. By [Serre, Chap IV, Prop. 17], $G=G(m)$. Define

[TABLE]

Then $G(m)^{s}=\mathrm{Gal}(L/K(\zeta_{p^{s}}))$. The ramification groups $G_{u}$ of $G$ are [Serre, Chap IV, Prop. 18]:

[TABLE]

We now calculate $\varphi_{L/F}.$

Proposition 10**.**

The Hasse-Herbrand function of the wildly ramified extension $L/F$ is given by

[TABLE]

Proof.

We consider various cases:

• •

Case $0<r\leq 1$

[TABLE]

• •

Case $1<r\leq p-1$

[TABLE]

Therefore, $\varphi_{L/F}(r)=r$.

• •

Case $p^{k}-1<r\leq p^{k+1}-1$ with $1\leq k<n-1$

[TABLE]

Therefore, $\varphi_{L/F}(r)=k(p-1)+\frac{r-p^{k}+1}{p^{k}}$.

• •

Case $r>p^{n-1}-1$

[TABLE]

Therefore, $\varphi_{L/F}(r)=(n-1)(p-1)+\frac{r-p^{n-1}+1}{p^{n-1}}.$

∎

Now write $T=\mathrm{R}_{L/F}\mathbb{G}_{\mathrm{m}}$ and let $\lambda_{T}$ be as denoted in Sec. 6. It then immediately follows from Prop. 10:

Lemma 11**.**

$\varphi_{L/K}(p^{n-1}r)>r$* $\forall r>0.$ Consequently, for all positive depth $\chi\in\mathrm{Hom}(T(K),\mathbb{C}^{\times})$, $\mathrm{dep}_{T}(\chi)<\mathrm{dep}_{W_{K}}(\lambda_{T}(\chi))$.*

9. Acknowledgement

The authors would like to thank Anne-Marie Aubert for carefully going over this article and suggesting several improvements.