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Local Langlands correspondence and ramification for Carayol representations

Published online by Cambridge University Press:  06 September 2019

Colin J. Bushnell
Affiliation:
King’s College London, Department of Mathematics, Strand, London WC2R 2LS, UK email colin.bushnell@kcl.ac.uk
Guy Henniart
Affiliation:
Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France email Guy.Henniart@math.u-psud.fr
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Abstract

Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with Weil group ${\mathcal{W}}_{F}$. Let $\unicode[STIX]{x1D70E}$ be an irreducible smooth complex representation of ${\mathcal{W}}_{F}$, realized as the Langlands parameter of an irreducible cuspidal representation $\unicode[STIX]{x1D70B}$ of a general linear group over $F$. In an earlier paper we showed that the ramification structure of $\unicode[STIX]{x1D70E}$ is determined by the fine structure of the endo-class $\unicode[STIX]{x1D6E9}$ of the simple character contained in $\unicode[STIX]{x1D70B}$, in the sense of Bushnell and Kutzko. The connection is made via the Herbrand function $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ of $\unicode[STIX]{x1D6E9}$. In this paper we concentrate on the fundamental Carayol case in which $\unicode[STIX]{x1D70E}$ is totally wildly ramified with Swan exponent not divisible by $p$. We show that, for such $\unicode[STIX]{x1D70E}$, the associated Herbrand function satisfies a certain functional equation, and that this property essentially characterizes this class of representations. We calculate $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ explicitly, in terms of a classical Herbrand function arising naturally from the classification of simple characters. We describe exactly the class of functions arising as Herbrand functions $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6EF}}$, as $\unicode[STIX]{x1D6EF}$ varies over the set of totally wild endo-classes of Carayol type. In a separate argument, we derive a complete description of the restriction of $\unicode[STIX]{x1D70E}$ to any ramification subgroup and hence a detailed interpretation of the Herbrand function. This gives concrete information concerning the Langlands correspondence.

Type
Research Article
Copyright
© The Authors 2019 

Let $F$ be a non-Archimedean, locally compact field with residual characteristic $p$ . Let $\mathscr{W}_{F}$ be the Weil group of a separable closure $\bar{F}/F$ . For a real variable $x\geqslant 0$ , let $\mathscr{R}_{F}(x)=\mathscr{W}_{F}^{x}$ be the corresponding ramification subgroup of $\mathscr{W}_{F}$ and $\mathscr{R}_{F}^{+}(x)$ the closure of $\bigcup _{y>x}\mathscr{R}_{F}(y)$ . We use the conventions of [Reference SerreSer68] here, so that $\mathscr{R}_{F}(0)$ is the inertia group $\mathscr{I}_{F}$ and $\mathscr{R}_{F}^{+}(0)$ is the wild inertia group $\mathscr{P}_{F}$ in $\mathscr{W}_{F}$ . If $\mathscr{G}$ is any of this list of locally profinite groups, $\widehat{\mathscr{G}}$ will denote the set of equivalence classes of irreducible, smooth, complex representations of $\mathscr{G}$ . We shall be concerned with the ramification structure of certain $\unicode[STIX]{x1D70E}\in \widehat{\mathscr{W}}_{F}$ , that is, the structure of the restricted representations $\unicode[STIX]{x1D70E}\,|\,\mathscr{R}_{F}(x)$ and $\unicode[STIX]{x1D70E}\,|\,\mathscr{R}_{F}^{+}(x)$ , for $x>0$ .

On the other side, let $\mathscr{A}_{n}^{0}(F)$ denote the set of equivalence classes of irreducible, cuspidal, complex representations of the general linear group $\operatorname{GL}_{n}(F)$ , $n\geqslant 1$ , and set $\widehat{\operatorname{GL}}_{F}=\bigcup _{n\geqslant 1}\mathscr{A}_{n}^{0}(F)$ . For $\unicode[STIX]{x1D70B}\in \widehat{\operatorname{GL}}_{F}$ , write $\text{gr}(\unicode[STIX]{x1D70B})=n$ to indicate $\unicode[STIX]{x1D70B}\in \mathscr{A}_{n}^{0}(F)$ . Such a representation $\unicode[STIX]{x1D70B}$ contains a simple character $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70B}}$ in $\operatorname{GL}_{n}(F)$ [Reference Bushnell and KutzkoBK93] and, up to conjugation, only one [Reference Bushnell and HenniartBH13]. The endo-class $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B}}$ of $\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D70B}}$ is therefore uniquely determined by $\unicode[STIX]{x1D70B}$ . Let $\mathbf{\mathscr{E}}(F)$ denote the set of endo-classes of simple characters over $F$ . (For the notion of endo-class, see [Reference Bushnell and HenniartBH96] or the summary in any of [Reference Bushnell, Diamond, Kassei and KimBus14, Reference Bushnell and HenniartBH03, Reference Bushnell and HenniartBH13].)

Denote by $\unicode[STIX]{x1D70B}\mapsto ^{\,\text{L}}\unicode[STIX]{x1D70B}$ the Langlands correspondence $\widehat{\operatorname{GL}}_{F}\rightarrow \widehat{\mathscr{W}}_{F}$ [Reference Harris and TaylorHT01, Reference HenniartHen00, Reference Laumon, Rapoport and StuhlerLRS93, Reference ScholzeSch13]. Writing $\unicode[STIX]{x1D70E}=^{\,\text{L}}\unicode[STIX]{x1D70B}$ , the fine structure of the endo-class $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B}}$ and the ramification structure of $\unicode[STIX]{x1D70E}$ determine each other [Reference Bushnell and HenniartBH17, 6.5 Corollary]. The relationship is expressed via a certain Herbrand function $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B}}}$ attached to the endo-class $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B}}$ . In this paper we consider a particularly interesting class of representations, comprising what we call Carayol representations. We compute the associated Herbrand functions. We list the functions which arise as Herbrand functions. We interpret the results in terms of the ramification structure of the associated Galois representations, from which we extract information about the Langlands correspondence.

We review the background from [Reference Bushnell and HenniartBH17] with as little formality as possible. If $\unicode[STIX]{x1D70B}\in \widehat{\operatorname{GL}}_{F}$ and $\unicode[STIX]{x1D70E}=^{\,\text{L}}\unicode[STIX]{x1D70B}\in \widehat{\mathscr{W}}_{F}$ , the endo-class $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B}}$ determines the restriction $\unicode[STIX]{x1D70E}\,|\,\mathscr{P}_{F}$ . More precisely, $\unicode[STIX]{x1D70E}$ defines an element $[\unicode[STIX]{x1D70E}]_{0}^{+}$ of the orbit space $\mathscr{W}_{F}\backslash \widehat{\mathscr{P}}_{F}$ , namely the orbit of irreducible components of $\unicode[STIX]{x1D70E}\,|\,\mathscr{P}_{F}$ . The Langlands correspondence induces a canonical bijection ([Reference Bushnell and HenniartBH03, 8.2 Theorem], [Reference Bushnell and HenniartBH14b, 6.1])

(A) $$\begin{eqnarray}\begin{array}{@{}rll@{}}\mathbf{\mathscr{E}}(F)\ & \longrightarrow \ & \mathscr{W}_{F}\backslash \widehat{\mathscr{P}}_{F},\\ \unicode[STIX]{x1D6E9}\ & \longmapsto \ & ^{\,\text{L}}\unicode[STIX]{x1D6E9}\end{array}\end{eqnarray}$$

by

$$\begin{eqnarray}[\hspace{0.0pt}^{\,\text{L}}\unicode[STIX]{x1D70B}]_{0}^{+}=^{\,\text{L}}\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B}},\quad \unicode[STIX]{x1D70B}\in \widehat{\operatorname{GL}}_{F}.\end{eqnarray}$$

Results developed in [Reference Bushnell and HenniartBH96, Reference Bushnell and HenniartBH99, Reference Bushnell and HenniartBH03, Reference Bushnell and HenniartBH05a, Reference Bushnell and HenniartBH05b, Reference Bushnell and HenniartBH10] and particularly [Reference Bushnell and HenniartBH14b] show that the map (A) is central to understanding of the Langlands correspondence.

The starting point of [Reference Bushnell and HenniartBH17] is that each of the sets $\mathbf{\mathscr{E}}(F)$ , $\mathscr{W}_{F}\backslash \widehat{\mathscr{P}}_{F}$ carries a canonical ultrametric. That on $\mathbf{\mathscr{E}}(F)$ , denoted by $\mathbb{A}$ , is built on the fact that simple characters are characters of compact groups carrying canonical filtrations, and those filtrations provide a medium via which the characters may be compared. The ultrametric $\mathbb{A}$ relates to Swan exponents of pairs of representations, as defined from the local constants of [Reference Jacquet, Piatetski-Shapiro and ShalikaJPS83, Reference ShahidiSha84]. Let $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}(F)$ and choose $\unicode[STIX]{x1D70B}\in \widehat{\operatorname{GL}}_{F}$ such that $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B}}=\unicode[STIX]{x1D6E9}$ . There is a unique continuous function $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6E9}}(x)$ , $x\geqslant 0$ , such that

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6E9}}(\mathbb{A}(\unicode[STIX]{x1D6E9},\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70C}}))=\frac{\text{sw}(\check{\unicode[STIX]{x1D70B}}\times \unicode[STIX]{x1D70C})}{\text{gr}(\unicode[STIX]{x1D70B})\,\text{gr}(\unicode[STIX]{x1D70C})},\end{eqnarray}$$

for any $\unicode[STIX]{x1D70C}\in \widehat{\operatorname{GL}}_{F}$ . The function $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6E9}}$ is piecewise linear, strictly increasing and convex. It is given by an explicit formula [Reference Bushnell and HenniartBH17, (4.4.1)] derived from the conductor formula of [Reference Bushnell, Henniart and KutzkoBHK98, 6.5 Theorem].

We call $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6E9}}$ the structure function of $\unicode[STIX]{x1D6E9}$ .

The ultrametric on $\mathscr{W}_{F}\backslash \widehat{\mathscr{P}}_{F}$ , denoted by $\unicode[STIX]{x1D6E5}$ , is defined by comparing representations via the canonical filtration of $\mathscr{P}_{F}$ by ramification groups: for $\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\in \widehat{\mathscr{W}}_{F}$ ,

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}([\unicode[STIX]{x1D70E}]_{0}^{+},[\unicode[STIX]{x1D70F}]_{0}^{+})=\inf \{x>0:\operatorname{Hom}_{\mathscr{ R}_{F}(x)}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F})\neq 0\}.\end{eqnarray}$$

The ultrametric $\unicode[STIX]{x1D6E5}$ likewise relates to Swan exponents of tensor products of pairs of representations of $\mathscr{W}_{F}$ [Reference HeiermannHei96]. For $\unicode[STIX]{x1D70E}\in \widehat{\mathscr{W}}_{F}$ , there is a unique continuous function $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D70E}}(x)$ , $x\geqslant 0$ , such that

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D70E}}(\unicode[STIX]{x1D6E5}([\unicode[STIX]{x1D70E}]_{0}^{+},[\unicode[STIX]{x1D70F}]_{0}^{+}))=\frac{\text{sw}(\check{\unicode[STIX]{x1D70E}}\otimes \unicode[STIX]{x1D70F})}{\dim \unicode[STIX]{x1D70E}\cdot \dim \unicode[STIX]{x1D70F}},\end{eqnarray}$$

for all $\unicode[STIX]{x1D70F}\in \widehat{\mathscr{W}}_{F}$ . The function $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D70E}}$ is piecewise linear, strictly increasing and convex. It is given by a formula derived from the ramification structure of $\unicode[STIX]{x1D70E}$ , reproduced in (2.2.2) below. If $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D70E}}$ is smooth at $x$ , its derivative satisfies

$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D70E}}^{\prime }(x)=\dim \operatorname{End}_{\mathscr{ R}_{F}(x)}(\unicode[STIX]{x1D70E})/(\dim \unicode[STIX]{x1D70E})^{2}.\end{eqnarray}$$

We call $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D70E}}$ the decomposition function of $\unicode[STIX]{x1D70E}$ : it depends only on the orbit $[\unicode[STIX]{x1D70E}]_{0}^{+}$ .

If $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}(F)$ , set $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}=\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6E9}}^{-1}\circ \unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D70E}}$ , for any $\unicode[STIX]{x1D70E}\in \widehat{\mathscr{W}}_{F}$ such that $[\unicode[STIX]{x1D70E}]_{0}^{+}=^{\,\text{L}}\unicode[STIX]{x1D6E9}$ . The Langlands correspondence respects Swan exponents of pairs and $\dim (\text{}^{\,\text{L}}\unicode[STIX]{x1D70B})=\text{gr}(\unicode[STIX]{x1D70B})$ , $\unicode[STIX]{x1D70B}\in \widehat{\operatorname{GL}}_{F}$ , so

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}(\unicode[STIX]{x1D6E5}(\text{}^{\,\text{L}}\unicode[STIX]{x1D6E9},^{\,\text{L}}\unicode[STIX]{x1D6EF}))=\mathbb{A}(\text{}\unicode[STIX]{x1D6E9},\unicode[STIX]{x1D6EF}),\quad \unicode[STIX]{x1D6EF}\in \mathbf{\mathscr{E}}(F).\end{eqnarray}$$

The function $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ is called the Herbrand function of $\unicode[STIX]{x1D6E9}$ . It is continuous, strictly increasing and piecewise linear.

If we take the view that $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}(F)$ has been given, in terms of the standard classification from [Reference Bushnell and KutzkoBK93], it is a simple matter to write down the function $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6E9}}$ . The Interpolation Theorem [Reference Bushnell and HenniartBH17, 7.5] shows, in principle, how to compute $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$  directly from  $\unicode[STIX]{x1D6E9}$ , without reference to $^{\,\text{L}}\unicode[STIX]{x1D6E9}$ . It yields the decomposition function $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D70E}}$ and therefore a numerical account of the ramification structure of $\unicode[STIX]{x1D70E}$ , just in terms of $\unicode[STIX]{x1D6E9}$ . The Interpolation Theorem is not easy to apply directly, but it is the foundation of much of what we do here.

We specify the classes of representation on which we focus.

Let $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}(F)$ . Assuming, as we invariably do, that $\unicode[STIX]{x1D6E9}$ is non-trivial, it is the endo-class of a simple character $\unicode[STIX]{x1D703}\in \mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FD})$ attached to a simple stratum $[\mathfrak{a},m,0,\unicode[STIX]{x1D6FD}]$ in some matrix algebra $\text{M}_{n}(F)$ (following the conventions of [Reference Bushnell and KutzkoBK93]). In particular, $\unicode[STIX]{x1D6FD}\in \operatorname{GL}_{n}(F)$ and the algebra $F[\unicode[STIX]{x1D6FD}]$ is a field: one says that $F[\unicode[STIX]{x1D6FD}]$ is a parameter field for $\unicode[STIX]{x1D6E9}$ . The positive integers $\deg \unicode[STIX]{x1D6E9}=[F[\unicode[STIX]{x1D6FD}]\,:\,F]$ and $e_{\unicode[STIX]{x1D6E9}}=e(F[\unicode[STIX]{x1D6FD}]|F)$ are invariants of $\unicode[STIX]{x1D6E9}$ . The slope $\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}$ of $\unicode[STIX]{x1D6E9}$ , defined by $\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}=m/e_{\mathfrak{a}}$ , where $e_{\mathfrak{a}}$ is the period of the hereditary $\mathfrak{o}_{F}$ -order $\mathfrak{a}$ , is likewise an invariant of $\unicode[STIX]{x1D6E9}$ . If $\unicode[STIX]{x1D70B}\in \widehat{\operatorname{GL}}_{F}$ satisfies $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D70B}}=\unicode[STIX]{x1D6E9}$ , then $\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}=\text{sw}(\unicode[STIX]{x1D70B})/\text{gr}(\unicode[STIX]{x1D70B})$ . However, neither $\unicode[STIX]{x1D703}$ nor $\unicode[STIX]{x1D6E9}$ determines the parameter field $F[\unicode[STIX]{x1D6FD}]$ : see the later parts of § 6.

Say that $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}(F)$ is totally wild if $\deg \unicode[STIX]{x1D6E9}=e_{\unicode[STIX]{x1D6E9}}=p^{r}$ , for an integer $r\geqslant 0$ . If $\unicode[STIX]{x1D6E9}$ is totally wild, say that it is of Carayol type if $\deg \unicode[STIX]{x1D6E9}>1$ and the integer $e_{\unicode[STIX]{x1D6E9}}\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}$ is not divisible by $p$ . Let $\mathbf{\mathscr{E}}^{\text{C}}(F)$ denote the set of endo-classes $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}(F)$ that are totally wild of Carayol type. We aim to calculate $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ for all $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}^{\text{C}}(F)$ .

We concentrate on this case for two reasons. First, [Reference Bushnell and HenniartBH17, 7.1 Proposition] reduces the problem of calculating Herbrand functions to the totally wild case. Second, we have to work with simple characters. The definition of simple character in [Reference Bushnell and KutzkoBK93] is rigidly hierarchical in nature and proofs are almost always inductive along this hierarchy. The first inductive step concerns the case where the element $\unicode[STIX]{x1D6FD}$ (as above) is minimal over $F$ [Reference Bushnell and KutzkoBK93, (1.4.14)]. For totally wild endo-classes, this is the Carayol case.

On the other side, say that $\unicode[STIX]{x1D70E}\in \widehat{\mathscr{W}}_{F}$ is totally wild if the restriction $\unicode[STIX]{x1D70E}\,|\,\mathscr{P}_{F}$ is irreducible. In particular, $\dim \unicode[STIX]{x1D70E}=p^{r}$ , for some $r\geqslant 0$ . Denote by $\widehat{\mathscr{W}}_{F}^{\text{wr}}$ the set of totally wild elements of $\widehat{\mathscr{W}}_{F}$ . An endo-class $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}(F)$ is then totally wild if and only if there exists $\unicode[STIX]{x1D70E}\in \widehat{\mathscr{W}}_{F}^{\text{wr}}$ such that $[\unicode[STIX]{x1D70E}]_{0}^{+}=^{\,\text{L}}\unicode[STIX]{x1D6E9}$ (cf. [Reference Bushnell and HenniartBH14b, §6]). Say that $\unicode[STIX]{x1D70E}\in \widehat{\mathscr{W}}_{F}^{\text{wr}}$ is of Carayol type if $\dim \unicode[STIX]{x1D70E}\neq 1$ and $p$ does not divide $\text{sw}(\unicode[STIX]{x1D70E})$ . Thus $\unicode[STIX]{x1D70E}\in \widehat{\mathscr{W}}_{F}^{\text{wr}}$ is of Carayol type if and only if $[\unicode[STIX]{x1D70E}]_{0}^{+}=^{\,\text{L}}\unicode[STIX]{x1D6E9}$ , for some $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}^{\text{C}}(F)$ . We shall see that these representations $\unicode[STIX]{x1D70E}$ exhibit a family of quite singular properties, reflecting the special nature of the endo-classes $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}^{\text{C}}(F)$ .

We review our main results. They are organized into three principal theorems, that complement and support each other, followed by a substantial application.

For any $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}(F)$ , the Herbrand function $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}(x)$ satisfies $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}(0)=0$ and $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}(x)=x$ for $x\geqslant \unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}$ [Reference Bushnell and HenniartBH17, 6.2 Proposition]. The derivative $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}^{\prime }(x)$ has only finitely many discontinuities in the interesting region $0<x<\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}$ : we call them the jumps of $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ . When $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}^{\text{C}}(F)$ , the function $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}(x)$ is convex in the region $0\leqslant x\leqslant \unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}$ . The reasons for this are simple (§ 2.4), but the property is very useful.

Theorem 1. Let $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}^{\text{C}}(F)$ . The graph $y=\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}(x)$ , $0\leqslant x\leqslant \unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}$ , is symmetric with respect to the line $x+y=\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}$ . That is,

(B) $$\begin{eqnarray}\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}-x=\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}(\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}-\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}(x)),\quad 0\leqslant x\leqslant \unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}.\end{eqnarray}$$

Theorem 1 has a satisfying converse. The group of characters of $U_{F}^{1}$ acts on the set $\mathbf{\mathscr{E}}(F)$ following the natural twisting action of characters of $F^{\times }$ or $\mathscr{W}_{F}$ on $\widehat{\operatorname{GL}}_{F}$ or $\widehat{\mathscr{W}}_{F}$ . We denote this action by $(\unicode[STIX]{x1D712},\unicode[STIX]{x1D6E9})\mapsto \unicode[STIX]{x1D712}\unicode[STIX]{x1D6E9}$ . It has the property $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D712}\unicode[STIX]{x1D6E9}}=\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ [Reference Bushnell and HenniartBH17, 7.4 Proposition]. We obtain the following corollary.

Corollary. Let $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}(F)$ be totally wild of degree $p^{r}$ , for some $r\geqslant 1$ , and suppose that $\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}\leqslant \unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D712}\unicode[STIX]{x1D6E9}}$ for all characters $\unicode[STIX]{x1D712}$ of $U_{F}^{1}$ . The function $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ then has the symmetry property (B) if and only if $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}^{\text{C}}(F)$ .

Theorem 1, together with some preliminary calculations, suggests the definition of a family of elementary functions. Let $r\geqslant 1$ and let $E/F$ be a totally ramified field extension of degree $p^{r}$ . Let $m$ be a positive integer not divisible by $p$ and set $\unicode[STIX]{x1D70D}=m/p^{r}$ . Let $\unicode[STIX]{x1D713}_{E/F}$ be the classical Herbrand function of $E/F$ [Reference DeligneDel84, Reference SerreSer68]. Define $c$ by the equation $c+p^{-r}\unicode[STIX]{x1D713}_{E/F}(c)=\unicode[STIX]{x1D70D}$ . There is then a unique function $^{2\,\!}\unicode[STIX]{x1D6F9}_{(E/F,\unicode[STIX]{x1D70D})}(x)$ , defined for $0\leqslant x\leqslant \unicode[STIX]{x1D70D}$ , such that the graph $y=^{2\,\!}\unicode[STIX]{x1D6F9}_{(E/F,\unicode[STIX]{x1D70D})}(x)$ is symmetric with respect to the line $x+y=\unicode[STIX]{x1D70D}$ and $^{2\,\!}\unicode[STIX]{x1D6F9}_{(E/F,\unicode[STIX]{x1D70D})}(x)=p^{-r}\unicode[STIX]{x1D713}_{E/F}(x)$ , for $0\leqslant x\leqslant c$ . Functions of this form will be called bi-Herbrand functions.

Our strategy is to identify $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ , $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}^{\text{C}}(F)$ , as a specific bi-Herbrand function. Let $\deg \unicode[STIX]{x1D6E9}=p^{r}$ . There is a simple stratum $[\mathfrak{a},m,0,\unicode[STIX]{x1D6FC}]$ in $\text{M}_{p^{r}}(F)$ such that $\unicode[STIX]{x1D6E9}$ is the endo-class of some $\unicode[STIX]{x1D703}\in \mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FC})$ . Thus $F[\unicode[STIX]{x1D6FC}]/F$ is totally ramified of degree $p^{r}$ and $p$ does not divide $m=-\unicode[STIX]{x1D710}_{F[\unicode[STIX]{x1D6FC}]}(\unicode[STIX]{x1D6FC})$ . In this notation, $\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}=m/p^{r}$ . If $\Vert \mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FC})\Vert$ denotes the set of endo-classes of elements of $\mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FC})$ , then $\Vert \mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FC})\Vert \subset \mathbf{\mathscr{E}}^{\text{C}}(F)$ .

The set $\Vert \mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FC})\Vert$ is not well adapted to our purposes, because the function $\unicode[STIX]{x1D6E9}\mapsto \unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ is not constant there. Indeed, it may vary widely: see 7.2 Theorem 1. To overcome this problem, we specify a non-empty subset $\mathscr{C}^{\star }(\mathfrak{a},\unicode[STIX]{x1D6FC})$ of $\mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FC})$ , using an explicit formula given in 7.1 below: we say that $\unicode[STIX]{x1D703}$ conforms to $\unicode[STIX]{x1D6FC}$ to indicate $\unicode[STIX]{x1D703}\in \mathscr{C}^{\star }(\mathfrak{a},\unicode[STIX]{x1D6FC})$ . Let $\Vert \mathscr{C}^{\star }(\mathfrak{a},\unicode[STIX]{x1D6FC})\Vert$ denote the set of endo-classes of characters $\unicode[STIX]{x1D703}\in \mathscr{C}^{\star }(\mathfrak{a},\unicode[STIX]{x1D6FC})$ .

Theorem 2. Let $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}^{\text{C}}(F)$ have degree $p^{r}$ and $\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}=m/p^{r}$ . There is a simple stratum $[\mathfrak{a},m,0,\unicode[STIX]{x1D6FC}]$ in $\text{M}_{p^{r}}(F)$ such that $\unicode[STIX]{x1D6E9}\in \Vert \mathscr{C}^{\star }(\mathfrak{a},\unicode[STIX]{x1D6FC})\Vert$ . For any such stratum,

(C) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}(x)=^{2\,\!}\unicode[STIX]{x1D6F9}_{(F[\unicode[STIX]{x1D6FC}]/F,\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}})}(x),\quad 0\leqslant x\leqslant \unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}.\end{eqnarray}$$

Theorem 2 has the following consequence.

Corollary. Let $E/F$ be a totally ramified field extension of degree $p^{r}$ , $r\geqslant 1$ , and let $m$ be a positive integer not divisible by $p$ . There exists $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}^{\text{C}}(F)$ , with parameter field $E/F$ , such that

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}(x)=^{2\,\!}\unicode[STIX]{x1D6F9}_{(E/F,m/p^{r})}(x),\quad 0\leqslant x\leqslant m/p^{r}=\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}.\end{eqnarray}$$

The corollary is an effective tool for constructing representations of $\mathscr{W}_{F}$ with specified ramification properties. An application of the technique is given in 9.7.

Our third result looks at the problem from the Galois side. Let $\unicode[STIX]{x1D70E}\in \widehat{\mathscr{W}}_{F}^{\text{wr}}$ be of Carayol type and dimension $p^{r}$ . Define $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}^{\text{C}}(F)$ by $[\unicode[STIX]{x1D70E}]_{0}^{+}=^{\,\text{L}}\unicode[STIX]{x1D6E9}$ . As $r\geqslant 1$ , the function $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ has at least one jump [Reference Bushnell and HenniartBH17, 7.7]. If $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ has exactly one jump, we say that $\unicode[STIX]{x1D70E}$ is H-singular. In § 8, we analyse the structure of such representations in some detail: they belong to a rather special class of ‘Heisenberg representations’ (as one says).

Without restriction on the number of jumps, define a number $c_{\unicode[STIX]{x1D6E9}}$ by the equation

$$\begin{eqnarray}c_{\unicode[STIX]{x1D6E9}}+\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}(c_{\unicode[STIX]{x1D6E9}})=\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}},\quad \unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}^{\text{C}}(F).\end{eqnarray}$$

By the symmetry of Theorem 1, $c_{\unicode[STIX]{x1D6E9}}$ is a jump of $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ if and only if $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ has an odd number of jumps and, in that case, $c_{\unicode[STIX]{x1D6E9}}$ is the middle one.

Theorem 3. Let $\unicode[STIX]{x1D70E}\in \widehat{\mathscr{W}}_{F}^{\text{wr}}$ be of Carayol type and dimension $p^{r}$ . Let $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}^{\text{C}}(F)$ satisfy $^{\,\text{L}}\unicode[STIX]{x1D6E9}=[\unicode[STIX]{x1D70E}]_{0}^{+}$ .

  1. (1) The restriction $\unicode[STIX]{x1D70E}\,|\,\mathscr{R}_{F}^{+}(c_{\unicode[STIX]{x1D6E9}})$ is a direct sum of characters.

  2. (2) Let $\unicode[STIX]{x1D709}$ be a character of $\mathscr{R}_{F}^{+}(c_{\unicode[STIX]{x1D6E9}})$ occurring in $\unicode[STIX]{x1D70E}$ , let $\mathscr{W}_{L_{\unicode[STIX]{x1D709}}}$ be the $\mathscr{W}_{F}$ -stabilizer of $\unicode[STIX]{x1D709}$ , and let $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D709}}$ be the natural representation of $\mathscr{W}_{L_{\unicode[STIX]{x1D709}}}$ on the $\unicode[STIX]{x1D709}$ -isotypic subspace of $\unicode[STIX]{x1D70E}\,|\,\mathscr{R}_{F}^{+}(c_{\unicode[STIX]{x1D6E9}})$ . The field extension $L_{\unicode[STIX]{x1D709}}/F$ is totally ramified of degree dividing $p^{r}$ and $\unicode[STIX]{x1D70E}=\operatorname{Ind}_{L_{\unicode[STIX]{x1D709}}/F}\,\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D709}}$ . Moreover,

    (D) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}(x)=p^{-r}\unicode[STIX]{x1D713}_{L_{\unicode[STIX]{x1D709}}/F}(x),\quad 0\leqslant x\leqslant c_{\unicode[STIX]{x1D6E9}}.\end{eqnarray}$$
  3. (3) If $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ has an odd number of jumps, then $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D709}}$ is irreducible, totally wild, H-singular, of Carayol type and of dimension $p^{r}\big/[L_{\unicode[STIX]{x1D709}}\,:\,F]\neq 1$ .

  4. (4) If $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ has an even number of jumps, then $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D709}}$ is a character and $[L_{\unicode[STIX]{x1D709}}\,:\,F]=p^{r}$ .

By symmetry, relation (D) determines $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ completely. Any two choices of the character $\unicode[STIX]{x1D709}$ are $\mathscr{W}_{F}$ -conjugate, so the same applies to the field $L_{\unicode[STIX]{x1D709}}$ . The field extension $L_{\unicode[STIX]{x1D709}}/F$ is not usually Galois but, after a suitable tamely ramified base field extension, it has a canonical presentation as a tower of elementary abelian extensions faithfully reflecting the ramification structure of $\unicode[STIX]{x1D70E}$ .

The canonical presentation of $\unicode[STIX]{x1D70E}$ as an induced representation,

$$\begin{eqnarray}\unicode[STIX]{x1D70E}=\operatorname{Ind}_{L_{\unicode[STIX]{x1D709}}/F}\,\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D709}}=\operatorname{Ind}_{\mathscr{W}_{L_{\unicode[STIX]{x1D709}}}}^{\mathscr{ W}_{F}}\,\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D709}},\end{eqnarray}$$

is derived from arithmetic considerations. It can claim to be more natural than anything provided by a purely group-theoretic approach.

The restrictions $\unicode[STIX]{x1D70E}\,|\,\mathscr{R}_{F}(x)$ , $\unicode[STIX]{x1D70E}\,|\,\mathscr{R}_{F}^{+}(x)$ follow a clear pattern, underlying the symmetry property of Theorem 1. To give the flavour, suppose there are at least two jumps. Let $j$ be the least and $\bar{\jmath }$ the greatest. The restriction $\unicode[STIX]{x1D70E}\,|\,\mathscr{R}_{F}(j)$ is irreducible, while $\unicode[STIX]{x1D70E}\,|\,\mathscr{R}_{F}^{+}(\bar{\jmath })$ is a multiple of a character. The restriction $\unicode[STIX]{x1D70E}\,|\,\mathscr{R}_{F}^{+}(j)$ is a multiplicity-free direct sum of irreducible representations while $\unicode[STIX]{x1D70E}\,|\,\mathscr{R}_{F}(\bar{\jmath })$ is a direct sum of characters, its isotypic components being the restrictions of the irreducible components of $\unicode[STIX]{x1D70E}\,|\,\mathscr{R}_{F}^{+}(j)$ . The pattern repeats for the second and penultimate jump, and so on.

We now have two expressions, (C) and (D), for the Herbrand function $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ of $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}^{\text{C}}(F)$ . Together they show how to read the algebraic structure of the decompositions $\unicode[STIX]{x1D70E}\,|\,\mathscr{R}_{F}(x)$ , $x>0$ , directly from the presentation $\unicode[STIX]{x1D6E9}\in \Vert \mathscr{C}^{\star }(\mathfrak{a},\unicode[STIX]{x1D6FC})\Vert$ . Our final tranche of results treats this in some detail.

In the same context, the number $c_{\unicode[STIX]{x1D6E9}}$ (as in part 6 above) and the function $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ , as $\unicode[STIX]{x1D6E9}$ ranges over $\Vert \mathscr{C}^{\star }(\mathfrak{a},\unicode[STIX]{x1D6FC})\Vert$ , depend only on $\unicode[STIX]{x1D6FC}$ . We therefore denote them by $c_{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6FC}}$ , respectively. Let $j_{\infty }(\unicode[STIX]{x1D6FC})=j_{\infty }(F[\unicode[STIX]{x1D6FC}]|F)$ be the largest jump of the classical Herbrand function $\unicode[STIX]{x1D713}_{F[\unicode[STIX]{x1D6FC}]/F}$ . The definition of $^{2\,\!}\unicode[STIX]{x1D6F9}_{(F[\unicode[STIX]{x1D6FC}]/F,\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}})}$ and Theorem 2 show that $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6FC}}$ has an even number of jumps if and only if $j_{\infty }(\unicode[STIX]{x1D6FC})<c_{\unicode[STIX]{x1D6FC}}$ .

Let $\mathscr{G}^{\star }(\unicode[STIX]{x1D6FC})$ be the set of $\unicode[STIX]{x1D70E}\in \widehat{\mathscr{W}}_{F}^{\text{wr}}$ such that $[\unicode[STIX]{x1D70E}]_{0}^{+}\in ^{\,\text{L}}\Vert \mathscr{C}^{\star }(\mathfrak{a},\unicode[STIX]{x1D6FC})\Vert$ .

Theorem 4A. If $\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\in \mathscr{G}^{\star }(\unicode[STIX]{x1D6FC})$ , the representations $\unicode[STIX]{x1D70E}\,|\,\mathscr{R}_{F}^{+}(c_{\unicode[STIX]{x1D6FC}})$ , $\unicode[STIX]{x1D70F}\,|\,\mathscr{R}_{F}^{+}(c_{\unicode[STIX]{x1D6FC}})$ are equivalent. In particular, any character $\unicode[STIX]{x1D709}$ of $\mathscr{R}_{F}^{+}(c_{\unicode[STIX]{x1D6FC}})$ occurring in $\unicode[STIX]{x1D70E}\,|\,\mathscr{R}_{F}^{+}(c_{\unicode[STIX]{x1D6FC}})$ also occurs in $\unicode[STIX]{x1D70F}\,|\,\mathscr{R}_{F}^{+}(c_{\unicode[STIX]{x1D6FC}})$ .

All representations $\unicode[STIX]{x1D70E}\in \mathscr{G}^{\star }(\unicode[STIX]{x1D6FC})$ therefore give rise to the same conjugacy class of field extensions $L_{\unicode[STIX]{x1D709}}/F$ and the associated representations $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D709}}$ all have the same dimension $p^{r}/[L_{\unicode[STIX]{x1D709}}\,:\,F]$ .

To go further, there is a second field extension to be taken into account. If $\unicode[STIX]{x1D70C}\in \widehat{\mathscr{W}}_{F}$ has dimension $n$ , let $\bar{\unicode[STIX]{x1D70C}}:\mathscr{W}_{F}\rightarrow \text{PGL}_{n}(\mathbb{C})$ be the associated projective representation. The kernel of $\bar{\unicode[STIX]{x1D70C}}$ is of the form $\mathscr{W}_{E}$ , where $E/F$ is finite and Galois. One calls $E/F$ the centric field of $\unicode[STIX]{x1D70C}$ . Returning to the main topic, let $\widetilde{L}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D709}}/L_{\unicode[STIX]{x1D709}}$ be the centric field of the H-singular representation $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D709}}\in \widehat{\mathscr{W}}_{L_{\unicode[STIX]{x1D709}}}^{\text{wr}}$ . The extension $\widetilde{L}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D709}}/L_{\unicode[STIX]{x1D709}}$ is Galois. It is non-trivial if and only if $\dim \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D709}}>1$ , that is, $^{2\,\!}\unicode[STIX]{x1D6F9}_{(F[\unicode[STIX]{x1D6FC}]/F,\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}})}$ has an odd number of jumps.

Let $w_{\unicode[STIX]{x1D6FC}}=w_{F[\unicode[STIX]{x1D6FC}]/F}$ be the wild exponent (1.6.1) of the field extension $F[\unicode[STIX]{x1D6FC}]/F$ . We consider two cases. Say that $\unicode[STIX]{x1D6FC}$ is $\star$ -exceptional if $j_{\infty }(\unicode[STIX]{x1D6FC})=c_{\unicode[STIX]{x1D6FC}}$ and the integer $l_{\unicode[STIX]{x1D6FC}}=m-w_{\unicode[STIX]{x1D6FC}}$ is even and positive. Otherwise, say that $\unicode[STIX]{x1D6FC}$ is $\star$ -ordinary. (This terminology is suggested by the usage of [Reference KutzkoKut84], but is not equivalent to it.)

For our next result, we fix a character $\unicode[STIX]{x1D709}$ of $\mathscr{R}_{F}^{+}(c_{\unicode[STIX]{x1D6FC}})$ occurring in $\unicode[STIX]{x1D70E}\in \mathscr{G}^{\star }(\unicode[STIX]{x1D6FC})$ and abbreviate $L=L_{\unicode[STIX]{x1D709}}$ , $\widetilde{L}_{\unicode[STIX]{x1D70E}}=\widetilde{L}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D709}}$ . Let $T_{\unicode[STIX]{x1D70E}}/F$ be the maximal tame sub-extension of $\widetilde{L}_{\unicode[STIX]{x1D70E}}/L$ . Let $d_{\unicode[STIX]{x1D70E}}$ be the number of characters $\unicode[STIX]{x1D712}$ of $\mathscr{W}_{L}$ such that $\unicode[STIX]{x1D719}\otimes \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D709}}\cong \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D709}}$ .

Theorem 4B.

  1. (1) If $\unicode[STIX]{x1D6FC}$ is $\star$ -ordinary, then $\widetilde{L}_{\unicode[STIX]{x1D70E}}=\widetilde{L}_{\unicode[STIX]{x1D70F}}$ , for all $\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\in \mathscr{G}^{\star }(\unicode[STIX]{x1D6FC})$ .

  2. (2) If $\unicode[STIX]{x1D6FC}$ is $\star$ -exceptional, then $T_{\unicode[STIX]{x1D70E}}=T_{\unicode[STIX]{x1D70F}}$ and $d_{\unicode[STIX]{x1D70E}}=d_{\unicode[STIX]{x1D70F}}$ , for all $\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\in \mathscr{G}^{\star }(\unicode[STIX]{x1D6FC})$ . There are at most $d_{\unicode[STIX]{x1D70E}}$ Galois extensions of the form $\widetilde{L}_{\unicode[STIX]{x1D70F}}/L$ , $\unicode[STIX]{x1D70F}\in \mathscr{G}^{\star }(\unicode[STIX]{x1D6FC})$ .

The bound in part (2) is achieved when $[T_{\unicode[STIX]{x1D70E}}\,:\,L]$ is not divisible by $p$ . (In general, we do not know what happens here, but $p$ can divide $[T_{\unicode[STIX]{x1D70E}}\,:\,L]$ : see 9.6 Example.) In part (1), the set $\mathscr{G}^{\star }(\unicode[STIX]{x1D6FC})$ bears a canonical structure as principal homogeneous space over an easily described group of characters of $L^{\times }$ .

We give an overview of our methods and the layout of the paper.

Section 1 is a free-standing account of the classical Herbrand functions $\unicode[STIX]{x1D713}_{E/F}$ , $\unicode[STIX]{x1D711}_{E/F}$ of a finite field extension $E/F$ . For Galois extensions $E/F$ , much of what we need can be deduced from the standard account in [Reference SerreSer68]. We develop the same level of detail for non-Galois extensions, starting from Deligne’s notes [Reference DeligneDel84].

The development proper starts with § 2. We introduce the main players and fix the basic notation. We take a simple stratum $[\mathfrak{a},m,0,\unicode[STIX]{x1D6FC}]$ in the matrix algebra $\text{M}_{p^{r}}(F)$ , $r\geqslant 1$ , as in part 4 above, and a simple character $\unicode[STIX]{x1D703}\in \mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FC})$ of endo-class $\unicode[STIX]{x1D6E9}$ . Thus $\unicode[STIX]{x1D6E9}\in \mathbf{\mathscr{E}}^{\text{C}}(F)$ and $\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}=m/p^{r}$ . The Interpolation Theorem of [Reference Bushnell and HenniartBH17] readily yields $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}(x)=p^{-r}\unicode[STIX]{x1D713}_{F[\unicode[STIX]{x1D6FC}]/F}(x)$ in the range $0\leqslant x\leqslant \unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}/2$ . In the region $\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}/2<\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}(x)\leqslant \unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}$ it interprets the value $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}(x)$ in terms of intertwining properties of certain simple strata.

Section 3 is devoted to the proof of Theorem 1. The argument is couched almost entirely in terms of Galois representations. Take $\unicode[STIX]{x1D70E}\in \widehat{\mathscr{W}}_{F}^{\text{wr}}$ of dimension greater than $1$ . After a tame base field extension, [Reference Bushnell and HenniartBH17, 8.3 Theorem] gives a sufficiently canonical presentation $\unicode[STIX]{x1D70E}=\operatorname{Ind}_{K/F}\,\unicode[STIX]{x1D70F}$ , where $K/F$ is cyclic of degree $p$ . After an elementary change of variables, the jumps of $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D70F}}$ are among those of $\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D70E}}$ but one or two of them are ‘flattened’, in an obvious sense. One of these is invariably the first. If $\unicode[STIX]{x1D70E}$ is of Carayol type, the other is the last: this follows from an application of the conductor formula of [Reference Bushnell, Henniart and KutzkoBHK98, 6.5 Theorem], which also gives a relation between the first and last jumps. One may then assume that $\unicode[STIX]{x1D70F}$ has the symmetry property and proceed by induction on dimension.

Section 4 makes a transition back to the GL side. The combination of convexity and symmetry imposes significant restrictions on the piecewise linear graph $y=\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}(x)$ in the relevant region $0\leqslant x\leqslant \unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}$ . We abstract these properties in the definition of the bi-Herbrand function $^{2\,\!}\unicode[STIX]{x1D6F9}_{(E/F,\unicode[STIX]{x1D70D})}$ . Much of the section is devoted to listing elementary, but useful, geometric properties of the graphs of $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ and $^{2\,\!}\unicode[STIX]{x1D6F9}_{(E/F,\unicode[STIX]{x1D70D})}$ . Our strategy is to identify $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ as a bi-Herbrand function. In many cases, one can do that immediately; see 4.6 Example. This simple case also has a role in the more complicated arguments that follow.

Sections 5 and 6 are highly technical in nature, preparing the way for the arguments of § 7. In § 5, we use the Interpolation Theorem to identify, via some delicate intertwining and conjugacy arguments, a subset of $\Vert \mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FC})\Vert$ on which the Herbrand function $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ takes the expected value $^{2\,\!}\unicode[STIX]{x1D6F9}_{(F[\unicode[STIX]{x1D6FC}]/F,\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}})}$ . The specification of this set, which we temporarily call $\mathscr{L}_{\unicode[STIX]{x1D6FC}}$ , is quite subtle. There is nothing canonical or natural about $\mathscr{L}_{\unicode[STIX]{x1D6FC}}$ , but it is a vital computational device.

The set $\mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FC})$ does not determine $\unicode[STIX]{x1D6FC}$ , although it does determine $\mathfrak{a}$ and the integer $m$ . Let $\text{P}(\mathfrak{a},\unicode[STIX]{x1D6FC})$ be the set of $\unicode[STIX]{x1D6FD}\in \operatorname{GL}_{p^{r}}(F)$ for which $[\mathfrak{a},m,0,\unicode[STIX]{x1D6FD}]$ is a simple stratum satisfying $\mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FD})=\mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FC})$ . In § 6 we examine various ways in which one can construct elements $\unicode[STIX]{x1D6FD}$ of $\text{P}(\mathfrak{a},\unicode[STIX]{x1D6FC})$ while keeping track of the relation between the sets $\mathscr{L}_{\unicode[STIX]{x1D6FC}}$ and $\mathscr{L}_{\unicode[STIX]{x1D6FD}}$ .

In § 7 we first define the subset $\mathscr{C}^{\star }(\mathfrak{a},\unicode[STIX]{x1D6FC})$ of simple characters $\unicode[STIX]{x1D703}\in \mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FC})$ that conform to $\unicode[STIX]{x1D6FC}$ . We show that, if $\unicode[STIX]{x1D703}^{\prime }\in \mathscr{C}(\mathfrak{a},\unicode[STIX]{x1D6FC})$ , there exists $\unicode[STIX]{x1D6FC}^{\prime }\in \text{P}(\mathfrak{a},\unicode[STIX]{x1D6FC}^{\prime })$ to which $\unicode[STIX]{x1D703}^{\prime }$ conforms. The calculations in §§5 and 6 give a first result (7.2 Theorem 1) from which Theorem 2 follows.

With § 8 we return to the Galois side. We first recast the general theory of representations of, loosely speaking, Heisenberg type and so identify the class of representations with Herbrand function having a single jump. This is in preparation for § 9, where we prove Theorem 3. That result is given in two tranches. In the first, (9.2), we assume that $\unicode[STIX]{x1D70E}$ is ‘absolutely wild’, in the sense that its centric field extension is totally wildly ramified. The argument there develops the method of § 3.

The general case is presented separately as 9.5 Corollary. The transition to the general case is, we found, surprising in both its simplicity and its exactness. It marks a change in direction in the paper. Until the end of § 7 we rely on the fact that, when using the Interpolation Theorem to compute the Herbrand function, one can impose an arbitrary finite, tamely ramified, base field extension while losing no control: the method is illustrated in the proof of 2.6 Proposition and then used repeatedly until the end of the proof of 9.2 Theorem. From 9.5 Corollary on, we have to take account of the tame structures destroyed by such a process. Theorems 4A and 4B follow in § 10, where we combine and compare the main results of §§7 and 9.

Some parts of Theorems 4 are foreshadowed, often in more detail, in the classical literature of dimension $p$ [Reference HenniartHen84, Reference KutzkoKut80, Reference KutzkoKut84, Reference MœglinMœ90]. There is a device from [Reference MœglinMœ90] that allows us to remove the distinction between ordinary and exceptional elements $\unicode[STIX]{x1D6FC}$ , provided  $p\neq 2$ . We summarize this in 10.6, and then briefly review the historical context.

Background and notation

General notations are quite familiar: $\mathfrak{o}_{F}$ is the discrete valuation ring in $F$ , $\mathfrak{p}_{F}$ is the maximal ideal of $\mathfrak{o}_{F}$ and $\unicode[STIX]{x1D710}_{F}$ is the normalized additive valuation. For $k\geqslant 1$ , $U_{F}^{k}$ is the congruence unit group $1+\mathfrak{p}_{F}^{k}$ . Similarly, if $\mathfrak{a}$ is a hereditary $\mathfrak{o}_{F}$ -order in some matrix algebra, then $U_{\mathfrak{a}}^{k}=1+\mathfrak{p}^{k}$ , where $\mathfrak{p}$ is the Jacobson radical $\operatorname{rad}\mathfrak{a}$ of $\mathfrak{a}$ . For real $x$ , $x\mapsto [x]$ is the greatest integer function.

If $E/F$ is a finite field extension, then $\unicode[STIX]{x1D713}_{E/F}$ , $\unicode[STIX]{x1D711}_{E/F}$ are the classical Herbrand functions discussed in § 1. If $E/F$ is Galois and $\unicode[STIX]{x1D6E4}=\operatorname{Gal}(E/F)$ , then $\unicode[STIX]{x1D6E4}_{a}$ , $\unicode[STIX]{x1D6E4}^{a}$ , $a\geqslant 0$ , are the ramification subgroups of $\unicode[STIX]{x1D6E4}$ in the lower, upper numbering conventions of [Reference SerreSer68]. The symbols $\mathscr{W}_{F}$ , $\widehat{\mathscr{W}}_{F}$ , $\mathscr{P}_{F}$ , $\widehat{\mathscr{P}}_{F}$ , $\widehat{\operatorname{GL}}_{F}$ , $\mathbf{\mathscr{E}}(F)$ , $^{\,\text{L}}\unicode[STIX]{x1D6E9}$ , $[\unicode[STIX]{x1D70E}]_{0}^{+}$ , $\mathscr{R}_{F}(x)$ , $\mathscr{R}_{F}^{+}(x)$ all retain the meaning given them in the introduction. Notation concerned with simple characters is all taken from [Reference Bushnell and KutzkoBK93, Reference Bushnell and HenniartBH96]. For the special cases considered here, full definitions are given in 2.12.3. The broader summary in [Reference Bushnell, Diamond, Kassei and KimBus14] may be found helpful. Certain special notations recur sporadically. Their definitions may be found as follows: $\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D6E9}}$  (2.1), $\unicode[STIX]{x1D70D}_{\unicode[STIX]{x1D70E}}$  (2.2), $\widehat{\mathscr{W}}_{F}^{\text{wr}}$  (3.2), $\widehat{\mathscr{W}}_{F}^{\text{awr}}$  (3.2), $\mathbf{\mathscr{E}}^{\text{C}}(F)$  (2.3), $j_{\infty }(E|F)$  (1.5), $w_{E/F}$  (1.6), $\mathscr{C}^{\star }$  (7.1).

1 Classical Herbrand functions

Let $E/F$ be a finite, separable field extension. As we go through the paper, we rely on properties of the classical Herbrand function $\unicode[STIX]{x1D713}_{E/F}$ and its inverse $\unicode[STIX]{x1D711}_{E/F}$ . For Galois extensions $E/F$ , many of these are to be found in [Reference SerreSer68]. In the general case, we develop them from the outline in [Reference DeligneDel84]. Beyond that, we need estimates of the jumps of $\unicode[STIX]{x1D713}_{E/F}$ , that is, the discontinuities of the derivative $\unicode[STIX]{x1D713}_{E/F}^{\prime }(x)$ , $x>0$ . With only minor changes, the formalism applies equally well to inseparable extensions $E/F$ : we indicate how this is done in 1.7.

We conclude the section with what seems to be a novel result on the structure of a broad class of totally ramified extensions. We do not need this until near the end of the paper but it fits well in the present context. The reader may wish to skip that, or even the entire section, referring back to it as needed.

1.1

Let $E/F$ be a finite Galois extension. The Herbrand function $\unicode[STIX]{x1D713}_{E/F}(x)$ is defined, for $x\geqslant -1$ , in [Reference SerreSer68, IV § 3] but we shall always assume $x\geqslant 0$ . If $K/F$ is a Galois extension contained in $E$ , the fundamental transitivity property $\unicode[STIX]{x1D713}_{E/F}=\unicode[STIX]{x1D713}_{E/K}\,\circ \,\unicode[STIX]{x1D713}_{K/F}$ holds. If the finite separable extension $E/F$ is not Galois, we follow [Reference DeligneDel84]. Let $E^{\prime }/F$ be a finite Galois extension containing $E$ . The function $\unicode[STIX]{x1D713}_{E^{\prime }/E}$ is positive and strictly increasing, so we may set

(1.1.1) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{E/F}=\unicode[STIX]{x1D713}_{E^{\prime }/E}^{-1}\circ \unicode[STIX]{x1D713}_{E^{\prime }/F}.\end{eqnarray}$$

Because of the transitivity property for Galois extensions, this definition does not depend on the choice of $E^{\prime }/F$ . The relation

(1.1.2) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{E/F}=\unicode[STIX]{x1D713}_{E/K}\circ \unicode[STIX]{x1D713}_{K/F}\end{eqnarray}$$

then holds for any tower $F\subset K\subset E$ of finite separable extensions. In all cases, $\unicode[STIX]{x1D711}_{E/F}$ shall be the inverse function for $\unicode[STIX]{x1D713}_{E/F}$ ,

(1.1.3) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{E/F}\circ \unicode[STIX]{x1D713}_{E/F}(x)=x=\unicode[STIX]{x1D713}_{E/F}\circ \unicode[STIX]{x1D711}_{E/F}(x),\quad x\geqslant 0.\end{eqnarray}$$

Lemma.

  1. (1) If $K/F$ is finite and tamely ramified, then $\unicode[STIX]{x1D713}_{K/F}(x)=ex$ , where $e=e(K|F)$ .

  2. (2) If $E/F$ is finite separable and $K/F$ is finite and tamely ramified, with $e(K|F)=e$ , then $\unicode[STIX]{x1D713}_{EK/K}(x)=e(EK|E)\,\unicode[STIX]{x1D713}_{E/F}(x/e)$ . If $E/F$ is totally wildly ramified, then $\unicode[STIX]{x1D713}_{EK/K}(x)=e\,\unicode[STIX]{x1D713}_{E/F}(x/e)$ .

Proof. Part (1) follows immediately from the definitions here and in [Reference SerreSer68]. By (1.1.2) and part (1), $\unicode[STIX]{x1D713}_{EK/F}(x)=\unicode[STIX]{x1D713}_{EK/K}\circ \unicode[STIX]{x1D713}_{K/F}(x)=\unicode[STIX]{x1D713}_{EK/K}(ex)$ . On the other hand, $\unicode[STIX]{x1D713}_{EK/F}(x)=\unicode[STIX]{x1D713}_{EK/E}\circ \unicode[STIX]{x1D713}_{E/F}(x)=e(EK|E)\unicode[STIX]{x1D713}_{E/F}(x)$ , whence part (2) follows.◻

The lemma reduces most questions to the totally wildly ramified case.

1.2

We list some properties of the graph $y=\unicode[STIX]{x1D713}_{E/F}(x)$ , $x\geqslant 0$ .

Proposition 1. Let $E/F$ be a finite separable extension and write $e=e(E|F)=e_{0}p^{r}$ , where $e_{0}$ is an integer not divisible by $p$ .

  1. (1) The function $\unicode[STIX]{x1D713}_{E/F}$ is continuous, piecewise linear, strictly increasing and convex.

  2. (2) If $x$ is sufficiently large, then $\unicode[STIX]{x1D713}_{E/F}^{\prime }(x)=e$ .

  3. (3) There exists $\unicode[STIX]{x1D716}>0$ such that $\unicode[STIX]{x1D713}_{E/F}(x)=e_{0}x$ , for $0\leqslant x<\unicode[STIX]{x1D716}$ .

  4. (4) The derivative $\unicode[STIX]{x1D713}_{E/F}^{\prime }$ is continuous except at a finite number of points.

Proof. All assertions are standard when $E/F$ is Galois, and (2)–(4) then follow from (1.1.2) in general. In (1), the first two properties are clear while, by (3), $\unicode[STIX]{x1D713}_{E/F}^{\prime }(x)=e_{0}\geqslant 1$ for $x$ positive and sufficiently small. It is enough, therefore, to show that $\unicode[STIX]{x1D713}_{E/F}$ is convex. By 1.1 Lemma (2), we need only prove that $\unicode[STIX]{x1D713}_{EK/K}$ is convex for some finite tame extension $K/F$ . We choose $K/F$ to be the maximal tame sub-extension of the normal closure $E^{\prime }/F$ of $E/F$ . This reduces us to the case in which $E^{\prime }/F$ is totally wildly ramified. If $E=F$ , there is nothing to prove, so assume otherwise. The proper subgroup $\operatorname{Gal}(E^{\prime }/E)$ of the finite $p$ -group $\operatorname{Gal}(E^{\prime }/F)$ is contained in a normal subgroup of index $p$ . That is, there is a Galois sub-extension $F^{\prime }/F$ of $E/F$ of degree $p$ . In the relation $\unicode[STIX]{x1D713}_{E/F}=\unicode[STIX]{x1D713}_{E/F^{\prime }}\circ \unicode[STIX]{x1D713}_{F^{\prime }/F}$ , the function $\unicode[STIX]{x1D713}_{F^{\prime }/F}$ is convex since $F^{\prime }/F$ is Galois. By induction on degree, $\unicode[STIX]{x1D713}_{E/F^{\prime }}$ is convex, whence so is $\unicode[STIX]{x1D713}_{E/F}$ .◻

This technique of the proof of the proposition will be used again, so we make a formal definition.

Definition. Let $E/F$ be a finite separable extension, with normal closure $E^{\prime }/F$ . Say that $E/F$ is absolutely wildly ramified if $E^{\prime }/F$ is totally wildly ramified.

In the notation of the definition, let $K/F$ be the maximal tame sub-extension of $E^{\prime }/F$ . The extension $EK/K$ is then absolutely wildly ramified. From the proof of Proposition 1, we extract a useful property.

Gloss. If $E/F$ is absolutely wildly ramified, there exists a Galois extension $F^{\prime }/F$ , of degree $p$ , such that $F^{\prime }\subset E$ .

We give a second application.

Proposition 2. Let $E/F$ be finite, separable and totally wildly ramified. If $\unicode[STIX]{x1D713}_{E/F}$ is smooth at $x$ , then the value $\unicode[STIX]{x1D713}_{E/F}^{\prime }(x)$ is a non-negative power of $p$ .

Proof. The result is standard when $E/F$ is Galois. Otherwise, let $K/F$ be finite and tamely ramified. Part (2) of 1.1 Lemma implies that the result holds for $E/F$ if and only if it holds for $EK/K$ . It is therefore enough to treat the case of $E/F$ absolutely wild. As in the Gloss, let $F^{\prime }/F$ be a sub-extension of $E/F$ that is Galois of degree $p$ . The extension $F^{\prime }/F$ has the desired property since it is Galois. By induction on the degree, we may assume that it holds equally for $E/F^{\prime }$ . The proposition then follows from the transitivity relation $\unicode[STIX]{x1D713}_{E/F}=\unicode[STIX]{x1D713}_{E/F^{\prime }}\circ \unicode[STIX]{x1D713}_{F^{\prime }/F}$ .◻

1.3

As in the Galois case, the function $\unicode[STIX]{x1D713}_{E/F}$ reflects properties of the norm map $\text{N}_{E/F}:E^{\times }\rightarrow F^{\times }$ .

Proposition. Let $E/F$ be a finite separable extension. Let $\unicode[STIX]{x1D712}$ be a character of $F^{\times }$ such that $\text{sw}(\unicode[STIX]{x1D712})=k\geqslant 1$ . The character $\unicode[STIX]{x1D712}\circ \text{N}_{E/F}$ of $E^{\times }$ then has the following properties:

  1. (1) $\text{sw}(\unicode[STIX]{x1D712}\circ \text{N}_{E/F})\leqslant \unicode[STIX]{x1D713}_{E/F}(k)$ ;

  2. (2) if $\unicode[STIX]{x1D713}_{E/F}^{\prime }$ is continuous at $k$ , then $\text{sw}(\unicode[STIX]{x1D712}\circ \text{N}_{E/F})=\unicode[STIX]{x1D713}_{E/F}(k)$ .

Proof. The result is standard when $E/F$ is Galois [Reference SerreSer68, V Proposition 9].

Suppose next that $E/F$ is tamely ramified and set $e=e(E|F)$ . Thus $\unicode[STIX]{x1D713}_{E/F}(x)=ex$ , $x\geqslant 0$ . If $\unicode[STIX]{x1D712}$ is a character of $F^{\times }$ with $\text{sw}(\unicode[STIX]{x1D712})=k\geqslant 1$ , then $\text{sw}(\unicode[STIX]{x1D712}\circ \text{N}_{E/F})=ek$ and there is nothing to prove.

Transitivity now reduces us to the case where $E/F$ is totally wildly ramified. Also, if $K/F$ is a finite tame extension, the result holds for $E/F$ if and only if it holds for $EK/K$ . We may therefore assume that $E/F$ is absolutely wildly ramified. Let $F^{\prime }$ be a field, $F\subset F^{\prime }\subset E$ , such that $F^{\prime }/F$ is Galois of degree $p$ (as in 1.2 Gloss). The result holds for the extension $F^{\prime }/F$ and so, in general, by induction on $[E\,:\,F]$ .◻

Definition. A jump of $\unicode[STIX]{x1D713}_{E/F}$ is a point $x>0$ at which the derivative $\unicode[STIX]{x1D713}_{E/F}^{\prime }$ is not continuous. Let $J_{E/F}$ denote the set of jumps of $\unicode[STIX]{x1D713}_{E/F}$ .

The set $J_{E/F}$ is finite by 1.2 Proposition 1(4).

Corollary. Let $E/F$ be totally wildly ramified, and let $K/F$ be a finite tame extension, with $e=e(K|F)$ . If $\unicode[STIX]{x1D712}$ is a character of $K^{\times }$ with $\text{sw}(\unicode[STIX]{x1D712})=k\geqslant 1$ , such that $e^{-1}k\notin J_{E/F}$ , then

$$\begin{eqnarray}\text{sw}(\unicode[STIX]{x1D712}\circ \text{N}_{EK/K})=\unicode[STIX]{x1D713}_{EK/K}(k)=e\,\unicode[STIX]{x1D713}_{E/F}(e^{-1}k).\end{eqnarray}$$

Proof. The second equality is 1.1 Lemma, whence $J_{EK/K}=eJ_{E/F}$ . The result now follows from the proposition.◻

1.4

Another familiar property extends to the general case.

Proposition. Let $E/F$ be a finite separable extension. If $\unicode[STIX]{x1D716}>0$ , then

$$\begin{eqnarray}\begin{array}{@{}rcl@{}}\mathscr{R}_{F}(\unicode[STIX]{x1D716})\cap \mathscr{W}_{E}\ & =\ & \mathscr{R}_{E}(\unicode[STIX]{x1D713}_{E/F}(\unicode[STIX]{x1D716})),\\ \mathscr{R}_{F}^{+}(\unicode[STIX]{x1D716})\cap \mathscr{W}_{E}\ & =\ & \mathscr{R}_{E}^{+}(\unicode[STIX]{x1D713}_{E/F}(\unicode[STIX]{x1D716})).\end{array}\end{eqnarray}$$

Proof. If $E/F$ is Galois, the result follows from [Reference SerreSer68, IV Proposition 14]. The case of $E/F$ tame readily follows. If $K/F$ is a finite tame extension, the result therefore holds for $E/F$ if and only if it holds for $EK/K$ (cf. 1.1 Lemma). Thus we need only treat the case where $E/F$ is absolutely wildly ramified. There is a Galois sub-extension $F^{\prime }/F$ of $E/F$ of degree $p$ . If $F^{\prime }=E$ , there is nothing to do, so we assume otherwise. We have

$$\begin{eqnarray}\begin{array}{@{}rcl@{}}\mathscr{R}_{F}(\unicode[STIX]{x1D716})\cap \mathscr{W}_{E}\ & =\ & \mathscr{R}_{F}(\unicode[STIX]{x1D716})\cap \mathscr{W}_{F^{\prime }}\cap \mathscr{W}_{E}\\ \ & =\ & \mathscr{R}_{F^{\prime }}(\unicode[STIX]{x1D713}_{F^{\prime }/F}(\unicode[STIX]{x1D716}))\cap \mathscr{W}_{E}\\ \ & =\ & \mathscr{R}_{E}(\unicode[STIX]{x1D713}_{E/F^{\prime }}(\unicode[STIX]{x1D713}_{F^{\prime }/F}(\unicode[STIX]{x1D716})))\\ \ & =\ & \mathscr{R}_{E}(\unicode[STIX]{x1D713}_{E/F}(\unicode[STIX]{x1D716})),\end{array}\end{eqnarray}$$

by induction on $[E\,:\,F]$ . The second assertion follows.◻

For a sharper result of this kind, see 1.9 Corollary 2 below.

1.5

Let $j_{\infty }(E|F)$ be the largest element of $J_{E/F}$ .

Proposition. Let $E/F$ be separable and totally wildly ramified. If $\bar{E}/F$ is the normal closure of $E/F$ , then $j_{\infty }(\bar{E}|F)=j_{\infty }(E|F)$ .

Proof. Let $K/F$ be a finite tame extension. The result then holds for $E/F$ if and only if it holds for $EK/K$ . We may therefore assume that $E/F$ is absolutely wildly ramified.

The relation $\unicode[STIX]{x1D713}_{\bar{E}/F}=\unicode[STIX]{x1D713}_{\bar{E}/E}\circ \unicode[STIX]{x1D713}_{E/F}$ implies that

$$\begin{eqnarray}J_{\bar{E}/F}=J_{E/F}\cup \unicode[STIX]{x1D713}_{E/F}^{-1}(J_{\bar{E}/E}).\end{eqnarray}$$

We have to show that $j_{\infty }(E|F)$ is the largest element of this set. Set $\unicode[STIX]{x1D6E4}=\operatorname{Gal}(\bar{E}/F)$ and $\unicode[STIX]{x1D6E5}=\operatorname{Gal}(\bar{E}/E)$ . The definition of $\unicode[STIX]{x1D6E4}_{x}$ [Reference SerreSer68, IV § 1] gives $\unicode[STIX]{x1D6E5}_{x}=\unicode[STIX]{x1D6E4}_{x}\cap \unicode[STIX]{x1D6E5}$ , for all $x\geqslant 0$ . Let $k_{\infty }$ be the largest jump of $\unicode[STIX]{x1D6E4}$ in this numbering. Thus $\unicode[STIX]{x1D6E4}_{k_{\infty }}\neq \{1\}=\unicode[STIX]{x1D6E4}_{k_{\infty }+\unicode[STIX]{x1D700}}$ , for all $\unicode[STIX]{x1D700}>0$ . As $\bar{E}/F$ is the least Galois extension containing $E$ , so $\bigcap _{\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6E5}\unicode[STIX]{x1D6FE}^{-1}=1$ . That is, $\unicode[STIX]{x1D6E5}$ has no non-trivial subgroup normal in $\unicode[STIX]{x1D6E4}$ . Since $\bar{E}/F$ is totally wildly ramified, $\unicode[STIX]{x1D6E4}_{k_{\infty }}$ is central in $\unicode[STIX]{x1D6E4}$ , so $\unicode[STIX]{x1D6E5}_{k_{\infty }}=\unicode[STIX]{x1D6E4}_{k_{\infty }}\cap \unicode[STIX]{x1D6E5}$ is normal in $\unicode[STIX]{x1D6E4}$ , whence $\unicode[STIX]{x1D6E5}_{k_{\infty }}=1$ . The largest jump of $\unicode[STIX]{x1D6E5}$ is therefore strictly less than $k_{\infty }$ . Translating back, the largest jump $j_{\infty }(\bar{E}|E)$ of $\unicode[STIX]{x1D713}_{\bar{E}/E}$ is strictly less than $\unicode[STIX]{x1D713}_{E/F}(j_{\infty }(E|F))$ .◻

1.6

Let $E/F$ be a finite separable extension. Denote by $d_{E/F}$ the differental exponent of $E/F$ : thus $\mathfrak{p}_{E}^{d_{E/F}}$ is the different of $E/F$ . Define the wild exponent $w_{E/F}$ of $E/F$ by

(1.6.1) $$\begin{eqnarray}w_{E/F}=d_{E/F}+1-e(E|F).\end{eqnarray}$$

We record, for use throughout the paper, some basic facts involving the wild exponent.

Lemma. Let $E/F$ be finite, with $E\subset \bar{F}$ .

  1. (1) If $F\subset K\subset E$ , then

    $$\begin{eqnarray}w_{E/F}=e(E|K)w_{K/F}+w_{E/K}.\end{eqnarray}$$
  2. (2) If $\unicode[STIX]{x1D70F}$ is an irreducible representation of $\mathscr{W}_{E}$ , then

    $$\begin{eqnarray}\text{sw}(\operatorname{Ind}_{E/F}\,\unicode[STIX]{x1D70F})=(\text{sw}(\unicode[STIX]{x1D70F})+w_{E/F}\,\dim \unicode[STIX]{x1D70F})\,f(E|F).\end{eqnarray}$$
    In particular,
    $$\begin{eqnarray}w_{E/F}=\text{sw}(\operatorname{Ind}_{E/F}\,1_{E})/f(E|F),\end{eqnarray}$$
    where $1_{E}$ is the trivial character of $\mathscr{W}_{E}$ .

Proof. Assertion (1) follows from the multiplicativity property of the different and a short calculation. Part (2) follows from the corresponding properties of the Artin exponent [Reference SerreSer68, ch. VI § 2]. ◻

The main business of the subsection concerns estimates relating the wild exponent $w_{E/F}$ to the largest jump $j_{\infty }(E|F)$ of $\unicode[STIX]{x1D713}_{E/F}$ .

Proposition. If $E/F$ is separable and totally wildly ramified of degree $p^{r}$ , then

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{E/F}(x)=p^{r}x-w_{E/F},\quad x\geqslant j_{\infty }(E|F).\end{eqnarray}$$

Proof. Let $K/F$ be tamely ramified with $e=e(K|F)$ . Thus $w_{EK/K}=e\,w_{E/F}$ by the lemma. The result therefore holds for $E/F$ if and only if it holds for $EK/K$ . Taking $K/F$ to be the maximal tame sub-extension of the normal closure of $E/F$ , we reduce to the case where $E/F$ is absolutely wildly ramified. Part (2) of 1.2 Proposition 1 implies that there is a constant $c_{E/F}$ such that $\unicode[STIX]{x1D713}_{E/F}(x)=p^{r}x-c_{E/F}$ , for $x\geqslant j_{\infty }(E|F)$ . We show that $c_{E/F}=w_{E/F}$ .

Let $F^{\prime }/F$ be a sub-extension of $E/F$ that is Galois of degree $p$ . In this case, $j_{\infty }(F^{\prime }|F)$ is the only jump of $\unicode[STIX]{x1D713}_{F^{\prime }/F}$ , and it equals $w_{F^{\prime }/F}/(p-1)$ [Reference SerreSer68, V § 3]. The proposition thus holds for $F^{\prime }/F$ . If $E/F$ is Galois, we may assume inductively that $c_{E/F^{\prime }}=w_{E/F^{\prime }}$ . So, taking $x$ sufficiently large, we get

$$\begin{eqnarray}\displaystyle p^{r}x-c_{E/F} & = & \displaystyle \unicode[STIX]{x1D713}_{E/F^{\prime }}(\unicode[STIX]{x1D713}_{F^{\prime }/F}(x))=\unicode[STIX]{x1D713}_{E/F^{\prime }}(px-w_{F^{\prime }/F})\nonumber\\ \displaystyle & = & \displaystyle p^{r}x-p^{r-1}w_{F^{\prime }/F}-w_{E/F^{\prime }}=p^{r}x-w_{E/F},\nonumber\end{eqnarray}$$

by the lemma. Thus $c_{E/F}=w_{E/F}$ when $E/F$ is Galois.

Suppose that $E/F$ is not Galois. The normal closure $E^{\prime }/F$ of $E/F$ is totally wildly ramified by hypothesis. So, with $p^{s}=[E^{\prime }\,:\,F]$ and $x$ sufficiently large, we get

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{E^{\prime }/F}(x) & = & \displaystyle p^{s}x-w_{E^{\prime }/F}=\unicode[STIX]{x1D713}_{E^{\prime }/E}(\unicode[STIX]{x1D713}_{E/F}(x))\nonumber\\ \displaystyle & = & \displaystyle p^{s-r}(p^{r}x-c_{E/F})-w_{E^{\prime }/E}.\nonumber\end{eqnarray}$$

Thus $w_{E^{\prime }/F}=e(E^{\prime }|E)c_{E/F}-w_{E^{\prime }/E}$ , and the lemma implies $c_{E/F}=w_{E/F}$ .◻

Corollary. Let $E/F$ be totally wildly ramified of degree $p^{r}$ . If $j_{\infty }=j_{\infty }(E|F)$ is the largest jump of $\unicode[STIX]{x1D713}_{E/F}$ , then

$$\begin{eqnarray}(p^{r}-1)j_{\infty }\geqslant w_{E/F}\geqslant p^{r-1}(p-1)j_{\infty }\geqslant p^{r}j_{\infty }/2.\end{eqnarray}$$

Moreover, $w_{E/F}=(p^{r}-1)j_{\infty }$ if and only if $j_{\infty }$ is the only jump of $\unicode[STIX]{x1D713}_{E/F}$ .

Proof. Since $\unicode[STIX]{x1D713}_{E/F}(x)\geqslant x$ for all $x\geqslant 0$ , the first inequality follows directly from the proposition, likewise the final remark.

Observe that $\unicode[STIX]{x1D713}_{E/F}^{\prime }(x)\leqslant p^{r-1}$ , for all points $0<x<j_{\infty }$ at which the derivative is defined (1.2 Proposition 2). The function $\unicode[STIX]{x1D717}(x)=\unicode[STIX]{x1D713}_{E/F}(x)-p^{r-1}x$ is therefore decreasing on the interval $0<x<j_{\infty }$ . Thus $\unicode[STIX]{x1D717}(j_{\infty })\leqslant 0$ , or $p^{r}j_{\infty }-w_{E/F}\leqslant p^{r-1}j_{\infty }$ , as required.◻

1.7

If $E/F$ is a finite, purely inseparable extension, we set $\unicode[STIX]{x1D713}_{E/F}(x)=x$ , $x\geqslant 0$ . If $E/F$ is a finite extension, define

(1.7.1) $$\begin{eqnarray}\unicode[STIX]{x1D713}_{E/F}=\unicode[STIX]{x1D713}_{E/E_{0}}\circ \unicode[STIX]{x1D713}_{E_{0}/F}=\unicode[STIX]{x1D713}_{E_{0}/F},\end{eqnarray}$$

where $E_{0}/F$ is the maximal separable sub-extension of $E/F$ . Assuming $E\neq E_{0}$ , the derivative of $\unicode[STIX]{x1D713}_{E/F}$ satisfies $\unicode[STIX]{x1D713}_{E/F}^{\prime }(x)<[E\,:\,F]$ for all $x$ . We therefore set $j_{\infty }(E|F)=\infty$ when $E/F$ is not separable. With these definitions, all the results of 1.11.31.5 and 1.6 remain valid.

1.8

We anticipate a phenomenon arising later on, in §§5 and 6.

Let $E/F$ be totally ramified of degree $p^{r}$ , $r\geqslant 1$ . Thus $E=F[\unicode[STIX]{x1D6FC}]$ , where $\unicode[STIX]{x1D6FC}$ is a root of an Eisenstein polynomial $f(X)=X^{p^{r}}+a_{1}X^{p^{r}-1}+\cdots +a_{p^{r}-1}X+a_{p^{r}}\in \mathfrak{o}_{F}[X]$ , and one has $d_{E/F}=\unicode[STIX]{x1D710}_{E}(f^{\prime }(\unicode[STIX]{x1D6FC}))$ .

Set $a_{0}=1$ . If $E/F$ is inseparable, the coefficient $a_{j}$ is zero unless $j\equiv 0\hspace{0.6em}({\rm mod}\hspace{0.2em}p)$ . Each term $(p^{r}-j)a_{j}\unicode[STIX]{x1D6FC}^{j-1}$ in $f^{\prime }(\unicode[STIX]{x1D6FC})$ vanishes, giving $d_{E/F}=w_{E/F}=\infty$ .

Proposition. Suppose $E/F$ is separable and totally ramified of degree $p^{r}$ . There is an integer $k$ such that $0\leqslant k\leqslant p^{r}-1$ , and

$$\begin{eqnarray}d_{E/F}=\min _{0\leqslant j\leqslant p^{r}-1}\unicode[STIX]{x1D710}_{E}((p^{r}-j)a_{j}\unicode[STIX]{x1D6FC}^{j-1})\equiv k-1\hspace{0.6em}({\rm mod}\hspace{0.2em}p^{r}).\end{eqnarray}$$

In particular, $w_{E/F}\equiv k\hspace{0.6em}({\rm mod}\hspace{0.2em}p)$ . If $F$ has characteristic $p$ , then $k\not \equiv 0\hspace{0.6em}({\rm mod}\hspace{0.2em}p)$ .

Proof. For $0\leqslant j\leqslant p^{r}-1$ , the term $(p^{r}-j)a_{j}\unicode[STIX]{x1D6FC}^{j-1}$ is either zero or

$$\begin{eqnarray}\unicode[STIX]{x1D710}_{E}((p^{r}-j)a_{j}\unicode[STIX]{x1D6FC}^{j-1})\equiv j-1\hspace{0.6em}({\rm mod}\hspace{0.2em}p^{r}).\end{eqnarray}$$

This gives the expression for $d_{E/F}$ . If $F$ has characteristic $p$ , any term with $j\equiv 0\hspace{0.6em}({\rm mod}\hspace{0.2em}p)$ has valuation $\infty$ and the second assertion follows.◻

If $F$ has characteristic zero, an Eisenstein polynomial $f(X)=X^{p}-a$ gives a field extension $E/F$ of degree $p$ such that $w_{E/F}\equiv 0\hspace{0.6em}({\rm mod}\hspace{0.2em}p)$ .

1.9

We prove a simple, but under-appreciated, result concerning absolutely wildly ramified extensions $E/F$ (1.2 Definition). It reappears naturally in the analysis of representations in § 9.

Let $E/F$ be a finite separable extension. As before, let $J_{E/F}$ be the set of jumps of the piecewise linear function $\unicode[STIX]{x1D713}_{E/F}$ . For $x>0$ , define

$$\begin{eqnarray}w_{x}(E|F)=\lim _{\unicode[STIX]{x1D716}\rightarrow 0}\unicode[STIX]{x1D713}_{E/F}^{\prime }(x+\unicode[STIX]{x1D716})/\unicode[STIX]{x1D713}_{E/F}^{\prime }(x-\unicode[STIX]{x1D716}).\end{eqnarray}$$

By 1.2 Proposition 2, $w_{x}(E|F)$ is a non-negative power of $p$ while $w_{x}(E|F)>1$ if and only if $x\in J_{E/F}$ .

If $E/F$ is a finite Galois extension with $\operatorname{Gal}(E/F)=\unicode[STIX]{x1D6E4}$ , we use the notation $\unicode[STIX]{x1D6E4}^{y+}=\bigcup _{z>y}\unicode[STIX]{x1D6E4}^{z}$ , and similarly for the lower numbering.

Proposition. Let $E/F$ be separable and absolutely wildly ramified. Let $a$ be the least element of $J_{E/F}$ .

  1. (1) The number $a$ is an integer and there exists a character $\unicode[STIX]{x1D712}$ of $F^{\times }$ such that $\text{sw}(\unicode[STIX]{x1D712})=a$ and $\unicode[STIX]{x1D712}\circ \text{N}_{E/F}=1$ .

  2. (2) Let $D=D_{(1)}(E|F)$ be the group of characters $\unicode[STIX]{x1D712}$ of $F^{\times }$ such that $\text{sw}(\unicode[STIX]{x1D712})\leqslant a$ and $\unicode[STIX]{x1D712}\circ \text{N}_{E/F}=1$ . All non-trivial elements of $D$ have Swan exponent $a$ , and $D$ is elementary abelian of order $w_{a}(E|F)$ .

  3. (3) If $E_{1}/F$ is class field to the group $D$ , then $F\subset E_{1}\subset E$ , $\unicode[STIX]{x1D713}_{E_{1}/F}(a)=a$ and

    $$\begin{eqnarray}J_{E/E_{1}}=\unicode[STIX]{x1D713}_{E_{1}/F}(J_{E/F})\smallsetminus \{a\}.\end{eqnarray}$$

Proof. We proceed by induction on $[E\,:\,F]$ . If