# Semi-stable representations as limits of crystalline representations

Anand Chitrao  School of Mathematics, Tata Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400005, India  Eknath Ghate  School of Mathematics,
Tata Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400005, India  Seidai Yasuda  Department of
Mathematics, Hokkaido University,
Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan

###### Abstract

We construct an explicit sequence of crystalline representations converging to
a given irreducible two-dimensional semi-stable representation of
$\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$. The convergence
takes place in the blow-up space of two-dimensional trianguline
representations studied by Colmez and Chenevier. The process of blow-up is
described in detail in the rigid analytic setting and may be of independent
interest.

Our convergence result can be used to compute the reductions of any
irreducible two-dimensional semi-stable representation in terms of the
reductions of certain crystalline representations of exceptional weight. In
particular, this provides an alternative approach to computing the reductions
of irreducible two-dimensional semi-stable representations that circumvents
the somewhat technical machinery of integral $p$-adic Hodge theory. For
instance, using the zig-zag conjecture made in [Gha21] on the reductions of
crystalline representations of exceptional weights, we recover completely the
work of Breuil-Mézard and Guerberoff-Park on the reductions of irreducible
semi-stable representations of weights at most $p+1$, at least on the inertia
subgroup. In the cases where the zig-zag conjecture is known, we are further
able to obtain some new information about the reductions for small odd
weights.

Finally, we use the above ideas to explain some apparent violations to local
constancy in the weight of the reductions of crystalline representations of
small weight which were noticed in [Gha21].

††footnotetext: Keywords: Galois representations, $(\varphi,\Gamma)$-modules,
$\mathcal{L}$-invariants, rigid geometry, blow-ups††footnotetext: 2020
Mathematics Subject Classification: 11F80, 14G22

##  1 Introduction

Let $p$ be an odd prime. Let $E$ be a finite extension of $\mathbb{Q}_{p}$.
Let $D_{\mathrm{st}}$ be Fontaine’s functor inducing an equivalence of
categories between semi-stable representations of the Galois group
$\mathrm{Gal}(\overline{{\mathbb{Q}}}_{p}/\mathbb{Q}_{p})$ over $E$ and
admissible filtered $(\varphi,N)$-modules over $E$. We introduce two kinds of
representations using this functor.

For every integer $k\geq 2$ and $a_{p}\in E$ of positive valuation, there is
an irreducible two-dimensional crystalline representation $V_{k,a_{p}}$ over
$E$ of the Galois group
$\mathrm{Gal}(\overline{{\mathbb{Q}}}_{p}/\mathbb{Q}_{p})$ with Hodge-Tate
weights $(0,k-1)$ and $D_{\mathrm{cris}}(V_{k,a_{p}}^{*})=D_{k,a_{p}}$, where
$D_{k,a_{p}}=Ee_{1}\oplus Ee_{2}$ is the filtered $\varphi$-module defined by:

$\displaystyle\begin{cases}\varphi(e_{1})=p^{k-1}e_{2}\\\
\varphi(e_{2})=-e_{1}+a_{p}e_{2},\end{cases}$ and
$\displaystyle\mathrm{Fil}^{i}{D_{k,a_{p}}}=\begin{cases}D_{k,a_{p}},&\text{
if }i\leq 0\\\ Ee_{1},&\text{ if }1\leq i\leq k-1\\\ 0,&\text{ if }i\geq
k.\end{cases}$

Similarly, for every integer $k\geq 2$ and $\mathcal{L}\in\mathbb{P}^{1}(E)$
(called the $\mathcal{L}$-invariant), there is a two-dimensional semi-stable
representation $V_{k,\mathcal{L}}$ over $E$ of the Galois group
$\mathrm{Gal}(\overline{{\mathbb{Q}}}_{p}/\mathbb{Q}_{p})$ with Hodge-Tate
weights $(0,k-1)$ and
$D_{\mathrm{st}}(V_{k,\mathcal{L}}^{*})=D_{k,\mathcal{L}}$, where
$D_{k,\mathcal{L}}=Ee_{1}\oplus Ee_{2}$ is the filtered $(\varphi,N)$-module
defined by:

$\displaystyle\begin{cases}\varphi(e_{1})=p^{k/2}e_{1}\\\
\varphi(e_{2})=p^{(k-2)/2}e_{2},\end{cases}$ and
$\displaystyle\mathrm{Fil}^{i}{D_{k,\mathcal{L}}}=\begin{cases}D_{k,\mathcal{L}},&\text{
if }i\leq 0\\\ E(e_{1}+\mathcal{L}e_{2}),&\text{ if }1\leq i\leq k-1\text{ and
}\mathcal{L}\neq\infty\\\ E(e_{1}+e_{2}),&\text{ if }1\leq i\leq k-1\text{ and
}\mathcal{L}=\infty\\\ 0,&\text{ if }i\geq k,\end{cases}$

and

$\displaystyle\begin{cases}N(e_{1})=e_{2}\\\ N(e_{2})=0,\end{cases}\text{ if
}\mathcal{L}\neq\infty,\quad\text{ and }\quad N=0\>\text{ if
}\mathcal{L}=\infty.$

If $k\geq 3$, then the semi-stable representation $V_{k,\mathcal{L}}$ is
irreducible and when $\mathcal{L}=\infty$ is isomorphic to the representation
$V_{k,a_{p}}$ with $a_{p}=p^{k/2}+p^{(k-2)/2}$.

This paper studies several relationships between the crystalline
representations $V_{k,a_{p}}$ and the semi-stable representations
$V_{k,\mathcal{L}}$ for $k\geq 3$.

###  1.1 Limits of crystalline representations

Colmez [Col08] and Chenevier [Che13] have constructed a moduli space of non-
split trianguline $(\varphi,\Gamma)$-modules of rank $2$ over the Robba ring.
The first goal of this paper is to construct for $k\geq 3$ an explicit
sequence of crystalline representations converging in this space to the (dual
of the) semi-stable representation $V_{k,\mathcal{L}}$ for a prescribed
$\mathcal{L}$-invariant $\mathcal{L}$. We will use this in conjunction with a
local constancy result in [Che13] to study the reductions of semi-stable
representations.

In order to state our result, let us recall the definition of the rigid-
analytic space constructed by Colmez and Chenevier. Let $\mathcal{T}$ be the
parameter space for characters of $\mathbb{Q}_{p}^{*}$. For each $i\geq 0$,
let $F_{i}$ and $F_{i}^{\prime}$ be the closed analytic subvarieties of
$\mathcal{T}\times\mathcal{T}$ such that for every finite extension $E$ of
$\mathbb{Q}_{p}$, we have

$\begin{array}[]{rl}F_{i}(E)\\!&=\\{(\delta_{1},\delta_{2})\in\mathcal{T}(E)\times\mathcal{T}(E)\text{
}|\text{ }\delta_{1}\delta_{2}^{-1}=x^{i}\chi\\},\\\
F_{i}^{\prime}(E)\\!&=\\{(\delta_{1},\delta_{2})\in\mathcal{T}(E)\times\mathcal{T}(E)\text{
}|\text{ }\delta_{1}\delta_{2}^{-1}=x^{-i}\\}.\end{array}$

For $i\geq 0$, we call the characters $x^{i}\chi$ and $x^{-i}$ exceptional.
Let $F=\cup_{i\geq 0}\>F_{i}$ and $F^{\prime}=\cup_{i\geq 0}\>F_{i}^{\prime}$.
The Colmez-Chenevier space $\widetilde{\mathcal{T}}_{2}$ is the blow-up of
$(\mathcal{T}\times\mathcal{T})\setminus F^{\prime}$ along $F$ in the category
of rigid-analytic spaces.

Our first main theorem is the following:

######  Theorem 1.1.

Let $k\geq 3$, $r=k-2$ and $\mathcal{L}\in\mathbb{P}^{1}(E)$. For $n\geq 1$,
let

$(k_{n},a_{n})=\begin{cases}(k+p^{n}(p-1),\>p^{r/2}(1+\mathcal{L}p^{n}(p-1)/2)),&\text{if
$\mathcal{L}\not=\infty$}\\\ (k+p^{n^{2}}(p-1),\>p^{r/2}(1+p^{n})),&\text{if
$\mathcal{L}=\infty$}.\end{cases}$ (1)

Then the sequence of crystalline representations $V^{*}_{k_{n},a_{n}}$
converges in $\widetilde{\mathcal{T}}_{2}$ to the semi-stable representation
$V^{*}_{k,\mathcal{L}}$.

The weights $k_{n}$ appearing in the theorem are for sufficiently large $n$
exceptional in the sense that they are two more than twice the slope of
$a_{n}$ modulo $(p-1)$. This will be of key importance in what follows.

Remark. We remark that the techniques used to prove the theorem can also be
used to prove that the limit of a sequence of irreducible two-dimensional
crystalline representations of
$\mathrm{Gal}(\overline{{\mathbb{Q}}}_{p}/\mathbb{Q}_{p})$ with Hodge-Tate
weights belonging to an interval $[a,b]$ is also irreducible crystalline with
Hodge-Tate weights in $[a,b]$, at least if the difference of the Hodge-Tate
weights of the representations in the sequence is at least $2$ infinitely
often. This gives another (geometric) proof of a special case of a general
result of Berger [Ber04, Théorème 1] (see Section 9).

###  1.2 Reductions of semi-stable representations

In [Che13, Proposition 3.9], Chenevier proved that the representations in the
space $\widetilde{\mathcal{T}}_{2}$ lie in families. Furthermore (see, e.g.,
the discussion on [Che13, p. 1513]), the semi-simplification of the reduction
of representations living in connected families are isomorphic. This means
that if the points in $\widetilde{\mathcal{T}}_{2}$ corresponding to two
representations are close, then the semi-simplifications of the reductions of
the two representations are the same. Moreover, the dual of the reduction of a
lattice in a $p$-adic representation is the same as the reduction of the dual
lattice in the dual representation. Using these facts, along with Theorem 1.1,
we see that if one knows the reductions of the crystalline representations
appearing in Theorem 1.1, then one can compute the reduction of any two-
dimensional semi-stable representation.

This is of some importance because the reductions of these two classes of
representations are nowadays largely studied by completely different methods:
the crystalline case uses the compatibility of reduction between the $p$-adic
and mod $p$ Local Langlands correspondences, or computes the reduction of the
corresponding Wach module, whereas the reductions in the semi-stable case are
determined by studying the reductions of the corresponding strongly divisible
modules. In our experience, the former methods, while quite intricate, are not
as complicated as the latter method. However, in view of the remarks above,
the techniques used in the crystalline case may be brought to bear on the
study of the reductions of semi-stable representations.

Let us illustrate this with some examples. The reductions of semi-stable
representations have been computed completely for even weights in the range
$[2,p]$ by Breuil and Mézard [BM02], and for odd weights in the same range by
Guerberoff and Park [GP19] at least on inertia. In [Gha21], the second author
made a conjecture called the zig-zag conjecture describing the reductions of
crystalline representations of exceptional weights and half-integral slopes in
terms of an alternating sequence of reducible and irreducible mod $p$
representations. This conjecture has been verified for some small slopes (cf.
[BG13] for slope $1/2$, [BGR18] for slope $1$ and [GR19] for slope $3/2$).
This “crystalline” conjecture is intimately connected to the “semi-stable”
results in [BM02, Theorem 1.2] and [GP19, Theorem 5.0.5]. More precisely, the
zig-zag conjecture and Theorem 1.1 can be used to completely recover the
description of the reductions of semi-stable representations in these theorems
when $k\neq 2$, at least on the inertia subgroup (see Theorem 1.2 below).
Moreover, in the odd weight cases for which the zig-zag conjecture has been
proved, we obtain new information about the reductions of semi-stable
representations on the full Galois group
$\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$.

To elaborate further, let us set up some notation. For an integer $k$ in the
interval $[3,p+1]$, define $v_{-}$ and $v_{+}$ to be the largest and smallest
integers, respectively, such that $v_{-}<(k-2)/2<v_{+}$. For $l\geq 1$, let

$H_{l}=\sum_{i=1}^{l}\frac{1}{i}$

be the $l$-th partial harmonic sum and set $H_{0}=0$. For convenience, write
$H_{-}=H_{v_{-}}$ and $H_{+}=H_{v_{+}}$. For any $\mathcal{L}$ in a finite
extension of $\mathbb{Q}_{p}$, let

$\nu=v_{p}(\mathcal{L}-H_{-}-H_{+})$

be the $p$-adic valuation (normalized so that $v_{p}(p)=1$) of the
$\mathcal{L}$-invariant shifted by the partial harmonic sums $H_{-}$ and
$H_{+}$. Let $\omega$ and $\omega_{2}$ denote the mod $p$ fundamental
characters of levels $1$ and $2$, respectively, on the inertia group
$I_{\mathbb{Q}_{p}}$. The character $\omega$ may be thought of as a character
of $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$; we choose an
extension of $\omega_{2}$ to
$\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p^{2}})$ such that for any
integer $c$ with $p+1\nmid c$, the representation
$\mathrm{ind}(\omega_{2}^{c})$ obtained by inducing $\omega_{2}^{c}$ from
$\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p^{2}})$ to
$\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ has determinant
$\omega^{c}$. Let $\mu_{\lambda}$ be the unramified character of
$\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ mapping a geometric
Frobenius element at $p$ to $\lambda\in\overline{\mathbb{F}}_{p}^{*}$ or
$\overline{\mathbb{Q}}_{p}^{*}$. Later we also think of $\mu_{\lambda}$ as a
character of $\mathbb{Q}_{p}^{*}$ taking $p$ to $\lambda$ and
$\mathbb{Z}_{p}^{*}$ to $1$.

Our second main theorem is the following:

######  Theorem 1.2.

If the zig-zag conjecture is true, then for any weight $k$ satisfying $3\leq
k\leq p+1$, we have the following $(k-1)$-fold description of the (semi-
simplification of the) reduction of the semi-stable representation
$V_{k,\mathcal{L}}$ for
$\mathcal{L}\in{\mathbb{P}}^{1}(\overline{\mathbb{Q}}_{p})$ on the inertia
subgroup:

$\overline{V}_{k,\mathcal{L}}|_{I_{\mathbb{Q}_{p}}}\sim\begin{cases}\omega_{2}^{k-1}\oplus\omega_{2}^{p(k-1)},&\text{if
$\nu<1-(k-2)/2$}\\\\[2.0pt] \omega^{k-2}\oplus\omega,&\text{if
$\nu=1-(k-2)/2$}\\\\[2.0pt]
\omega_{2}^{k-2+p}\oplus\omega_{2}^{p(k-2)+1},&\text{if
$1-(k-2)/2<\nu<2-(k-2)/2$}\\\\[2.0pt] \omega^{k-3}\oplus\omega^{2},&\text{if
$\nu=2-(k-2)/2$}\\\\[2.0pt] \qquad\vdots&\qquad\qquad\qquad\vdots\\\\[2.0pt]
\omega^{(k-1)/2}\oplus\omega^{(k-1)/2},&\text{if $\nu\geq 1/2$, and $k$ is
odd,}\\\\[2.0pt] \qquad\text{or}&\\\\[2.0pt]
\omega^{k/2}\oplus\omega^{(k-2)/2},&\text{if $\nu=0$}\\\\[-5.0pt]
&\qquad\qquad\text{and k is even.}\\\\[-5.0pt]
\omega_{2}^{k/2+p(k/2-1)}\oplus\omega_{2}^{k/2-1+pk/2},&\text{if
$\nu>0$},\end{cases}$

Remark.  The theorem holds when $\mathcal{L}=\infty$ even without assuming the
zig-zag conjecture by the work of Fontaine and Edixhoven [Edi92], since, by
convention, $v_{p}(\infty)=-\infty$.

Theorem 1.2 above indeed matches with the results in [BM02] and [GP19] for
$\mathcal{L}\neq\infty$:

  * •

Even $k\in[3,p+1]$: In [BM02, Theorem 1.2], Breuil and Mézard have computed
the reduction of $V_{k,\mathcal{L}}$ in terms of the valuations $v_{p}(a)$ and
$v_{p}(\mathcal{L})$, where

$a=(-1)^{k/2}\left(-1+\frac{k}{2}\left(\frac{k}{2}-1\right)(-\mathcal{L}+2H_{k/2-1})\right).$

Since $k\in[3,p+1]$, we see that
$v_{p}\left(\frac{k}{2}\left(\frac{k}{2}-1\right)\right)=0$, and therefore

$\displaystyle v_{p}(a)$ $\displaystyle=$ $\displaystyle
v_{p}\left(\frac{-1}{(k/2)(k/2-1)}+(-\mathcal{L}+2H_{k/2-1})\right)$
$\displaystyle=$ $\displaystyle
v_{p}\left(\frac{1}{k/2}-\frac{1}{k/2-1}-\mathcal{L}+2\sum_{i=1}^{k/2-1}\frac{1}{i}\right)$
$\displaystyle=$ $\displaystyle
v_{p}\left(-\mathcal{L}+\sum_{i=1}^{k/2}\frac{1}{i}+\sum_{i=1}^{k/2-2}\frac{1}{i}\right)$
$\displaystyle=$ $\displaystyle v_{p}(\mathcal{L}-H_{k/2}-H_{k/2-2})$
$\displaystyle=$ $\displaystyle\nu,$

where we have used $k$ is even in the last equality. Now:

    * –

If $\nu>0$, then Theorem 1.2 yields
$\overline{V}_{k,\mathcal{L}}|_{I_{\mathbb{Q}_{p}}}\sim\omega_{2}^{k/2+p(k/2-1)}\oplus\omega_{2}^{k/2-1+pk/2}$,
which agrees with [BM02, Theorem 1.2 (ii)].

    * –

If $\nu=0$, then Theorem 1.2 yields
$\overline{V}_{k,\mathcal{L}}|_{I_{\mathbb{Q}_{p}}}\sim\omega^{k/2}\oplus\omega^{(k-2)/2}$,
which agrees with [BM02, Theorem 1.2 (i)].

    * –

So assume $\nu<0$. Then $\nu=v_{p}(\mathcal{L})$. If $\nu<2-k/2$, then Theorem
1.2 yields
$\overline{V}_{k,\mathcal{L}}|_{I_{\mathbb{Q}_{p}}}\sim\omega_{2}^{k-1}\oplus\omega_{2}^{p(k-1)}$,
which agrees with [BM02, Theorem 1.2 (iii)]. Now if $2-k/2\leq\nu<0$ and
$\nu\in\mathbb{Z}$, then [BM02, Theorem 1.2 (iii)] yields
$\overline{V}_{k,\mathcal{L}}|_{I_{\mathbb{Q}_{p}}}\sim\omega^{k/2-\nu}\oplus\omega^{k/2+\nu-1}$,
which is the same as the reduction computed in Theorem 1.2. Finally, if
$2-k/2\leq\nu<0$ and $\nu\not\in\mathbb{Z}$, then [BM02, Theorem 1.2 (iii)]
yields
$\overline{V}_{k,\mathcal{L}}|_{I_{\mathbb{Q}_{p}}}\sim\omega_{2}^{k/2-\lfloor\nu\rfloor+p(k/2+\lfloor\nu\rfloor-1)}\oplus\omega_{2}^{k/2+\lfloor\nu\rfloor-1+p(k/2-\lfloor\nu\rfloor)}$,
which matches with the reduction computed in Theorem 1.2.

Remark.  Since the zig-zag conjecture has already been proved for $p\geq 5$
and slope $1$ in [BGR18, Theorem 1.1], even on
$\mathrm{Gal}(\overline{{\mathbb{Q}}}_{p}/\mathbb{Q}_{p})$, we recover the
following result of Breuil and Mézard when $k=4$ (cf. [BM02, Theorem
1.2]):111The computation
$\overline{a}\overline{\left(\frac{a_{p}}{p}\right)}=\overline{(-1+2(-\mathcal{L}+2))}\overline{\left(\frac{p+p^{2}}{p}\right)}=\overline{-2\left(\mathcal{L}-\frac{3}{2}\right)}$
shows that the reduction agrees with the reduction computed in [BM02, Theorem
1.2]. if $p\geq 5$ and $k=4$, then

$\overline{V}_{k,\mathcal{L}}\sim\begin{cases}\mathrm{ind}(\omega_{2}^{3}),&\text{if
$\nu<0$}\\\\[5.0pt]
\mu_{\lambda}\>\omega^{2}\oplus\mu_{\lambda^{-1}}\>\omega,&\text{if
$\nu=0$}\\\\[5.0pt] \mathrm{ind}(\omega_{2}^{2+p}),&\text{if
$\nu>0$},\end{cases}$

where

$\lambda=\overline{-2\left(\mathcal{L}-\frac{3}{2}\right)}.$

If points close in $\widetilde{\mathcal{T}}_{2}$ have isomorphic reductions
(not just up to semi-simplification), then one can read off more subtle
information about whether the representation $\overline{V}_{k,\mathcal{L}}$ is
peu or très ramifiée (when it is reducible and $\lambda=\pm 1$) from the
corresponding information of a sufficiently close crystalline representation
$V_{k_{n},a_{n}}$, which, in turn, is controlled by the size of
$v_{p}(u_{n}-\varepsilon_{n})$ in the notation of [BGR18, Theorem 1.3].
Indeed, in the middle case of the trichotomy above, we have
$v_{p}(3-2\mathcal{L})=0$, putting us in part (i) of [BM02, Theorem 1.2]. Note
that $\overline{\frac{a_{n}}{p}}=1$ and $\overline{k_{n}-2}=2$. There are now
two cases to consider depending on whether $\varepsilon_{n}=\pm 1$. If
$v_{p}(\mathcal{L}-2)=0$, then a small check shows that $\varepsilon_{n}=1$,
so by [BGR18, Theorem 1.3 (1) (a)], the reduction
$\overline{V}_{k,\mathcal{L}}$ (without semi-simplification) is peu ramifiée.
If $v_{p}(\mathcal{L}-2)>0$, then another small check shows that
$\varepsilon_{n}=-1$ and that
$v_{p}(u_{n}-\varepsilon_{n})=v_{p}(\mathcal{L}-2)$. Thus, by [BGR18, Theorem
1.3 (1) (b)], the reduction $\overline{V}_{k,\mathcal{L}}$ (without semi-
simplification) is peu ramifiée if and only if $v_{p}(\mathcal{L}-2)<1$, at
least if $\mathcal{L}$ lies in an unramified extension of $\mathbb{Q}_{p}$.
Both these conclusions are consistent with the corresponding conclusions in
[BM02, Theorem 1.2 (i)]!

  * •

Odd $k\in[3,p+1]$: In [GP19, Theorem 5.0.5], Guerberoff and Park compute the
reduction of $V_{k,\mathcal{L}}$ in terms of the valuation
$v_{p}(\mathcal{L}-a(k-1))$, where

$a(j)=H_{j/2}+H_{j/2-1},$

for $j\geq 1$. We clearly have $a(k-1)=H_{-}+H_{+}$. Therefore, the regions
used to classify the reductions in Theorem 1.2 match with those used in [GP19,
Theorem 5.0.5]. Now:

    * –

If $\nu<1-(k-2)/2$, then Theorem 1.2 yields
$\overline{V}_{k,\mathcal{L}}|_{I_{\mathbb{Q}_{p}}}\sim\omega_{2}^{k-1}\oplus\omega_{2}^{p(k-1)}$,
which agrees with the reduction computed in [GP19, Theorem 5.0.5 (2)].

    * –

Assume $-1/2-l<\nu<1/2-l$ for some $l\in\\{0,1,\cdots,(k-5)/2\\}$. This region
can be written as $(-l+(k-3)/2)-(k-2)/2<\nu<(-l+(k-1)/2)-(k-2)/2$. Theorem 1.2
yields
$\overline{V}_{k,\mathcal{L}}|_{I_{\mathbb{Q}_{p}}}\sim\omega_{2}^{(k-1)+(p-1)(-l+(k-3)/2)}\oplus\omega_{2}^{p((k-1)+(p-1)(-l+(k-3)/2))}$,
which agrees with the reduction computed in [GP19, Theorem 5.0.5 (1)].

    * –

Assume $\nu=-1/2-l$ for some $l\in\\{0,1,\cdots,(k-5)/2\\}$. This can be
written as $\nu=(-l+(k-3)/2)-(k-2)/2$. Theorem 1.2 yields
$\overline{V}_{k,\mathcal{L}}|_{I_{\mathbb{Q}_{p}}}\sim\omega^{k-1-(-l+(k-3)/2)}\oplus\omega^{(-l+(k-3)/2)}$,
which agrees with the reduction computed in [GP19, Theorem 5.0.5].

    * –

Finally, if $\nu\geq 1/2$, then Theorem 1.2 yields
$\overline{V}_{k,\mathcal{L}}|_{I_{\mathbb{Q}_{p}}}\sim\omega^{(k-1)/2}\oplus\omega^{(k-1)/2}$,
which agrees with the reduction computed in [GP19, Theorem 5.0.5].

For some small weights ($k=3$ and $5$), we may in fact improve on [GP19,
Theorem 5.0.5] by computing the reductions $\overline{V}_{k,\mathcal{L}}$ on
the full Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$
(by giving formulas for $\lambda$). Indeed, since the zig-zag conjecture has
been proved for slopes $1/2$ and $3/2$ in [BG13, Theorem A] and [GR19, Theorem
1.1], respectively, we obtain the following theorems.

######  Theorem 1.3.

Let $p\geq 3$ and $k=3$. We have the following dichotomy for the shape of the
semi-simplification of the reduction of the semi-stable representation
$V_{k,\mathcal{L}}$:

$\overline{V}_{k,\mathcal{L}}\sim\begin{cases}\mathrm{ind}(\omega_{2}^{2}),&\text{if
$\nu<1/2$}\\\ \mu_{\lambda}\omega\oplus\mu_{\lambda^{-1}}\omega,&\text{if
$\nu\geq 1/2$},\end{cases}$

where

$\displaystyle\lambda+\frac{1}{\lambda}$ $\displaystyle=$
$\displaystyle\overline{-p^{-1/2}(\mathcal{L}-1)}.$

######  Theorem 1.4.

Let $p\geq 5$ and $k=5$. We have the following tetrachotomy for the shape of
the semi-simplification of the reduction of the semi-stable representation
$V_{k,\mathcal{L}}$:

$\overline{V}_{k,\mathcal{L}}\sim\begin{cases}\mathrm{ind}(\omega_{2}^{4}),&\text{if
$\nu<-1/2$}\\\\[5.0pt]
\mu_{\lambda_{1}}\omega^{3}\oplus\mu_{\lambda_{1}^{-1}}\omega,&\text{if
$\nu=-1/2$}\\\\[5.0pt] \mathrm{ind}(\omega_{2}^{3+p}),&\text{if
$-1/2<\nu<1/2$}\\\\[5.0pt]
\mu_{\lambda_{2}}\omega^{2}\oplus\mu_{\lambda_{2}^{-1}}\omega^{2},&\text{if
$\nu\geq 1/2$},\end{cases}$

where the constants $\lambda_{i}$ are given by

$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle\overline{-3p^{1/2}\left(\mathcal{L}-\frac{5}{2}\right)},$
$\displaystyle\lambda_{2}+\frac{1}{\lambda_{2}}$ $\displaystyle=$
$\displaystyle\overline{2p^{-1/2}\left(\mathcal{L}-\frac{5}{2}\right)}.$

###  1.3 Rescuing local constancy

In [Ber12], Berger proved the following theorem on the local constancy in the
weight of the semi-simplification of the reduction of crystalline
representations.

######  Theorem 1.5 ([Ber12, Theorem B]).

Let $a_{p}\not=0$ and $k>3v_{p}(a_{p})+\alpha(k-1)+1$, where
$\alpha(j)=\sum_{n\geq 1}\left\lfloor\frac{j}{p^{n-1}(p-1)}\right\rfloor$.
Then there exists $m=m(k,a_{p})$ such that
$\overline{V}_{k^{\prime},a_{p}}=\overline{V}_{k,a_{p}}$, if $k^{\prime}\geq
k$ and $k^{\prime}-k\in p^{m-1}(p-1)\mathbb{Z}$.

This local constancy result does not extend to small weights in the sense that
the bound in the theorem is sharp. This was noticed in [Gha21] using the
following examples.

  * •

Let $k=4$ and $a_{p}=p\geq 5$. Then $4\not>3(1)+1$ and the crystalline
representation $V_{k,a_{p}}$ does not satisfy Berger’s bound on the weight.
Now let $k^{\prime}=4+p^{n}(p-1)$, for large $n$. If the above local constancy
result were to hold for $V_{k,a_{p}}$, then we would have
$\overline{V}_{k^{\prime},a_{p}}\sim\overline{V}_{k,a_{p}}$. However,
$\overline{V}_{k,a_{p}}$ is irreducible as $k$ belongs to the range $[2,p+1]$
treated by Fontaine and Edixhoven [Edi92], but
$\overline{V}_{k^{\prime},a_{p}}\sim\mu_{3}\>\omega^{2}\oplus\mu_{3^{-1}}\>\omega$
is reducible by [BGR18, Theorem 1.1].

  * •

Let $k=5$ and $a_{p}=p^{3/2}$, for $p\geq 7$. Then $5\not>3(1.5)+1$ and the
crystalline representation $V_{k,a_{p}}$ again does not satisfy the bound on
the weight. Now let $k^{\prime}=5+p^{n}(p-1)$, for large $n$. If the above
local constancy result were to hold for $V_{k,a_{p}}$, then we would have
$\overline{V}_{k^{\prime},a_{p}}\sim\overline{V}_{k,a_{p}}$. However,
$\overline{V}_{k,a_{p}}\sim\mathrm{ind}(\omega_{2}^{4})$ since $k$ belongs to
the Fontaine-Edixhoven range, but
$\overline{V}_{k^{\prime},a_{p}}\sim\mathrm{ind}(\omega_{2}^{3+p})$ by [GR19,
Theorem 1.1].

However, Berger’s local constancy result can be generalized to the small
weights above if one uses the correct (non-crystalline) limit provided by the
following corollary to Theorem 1.1.

######  Corollary 1.6.

For $k\geq 3$, the sequence of crystalline representations
$V^{*}_{k+p^{n}(p-1),p^{r/2}}$ converges to the semi-stable representation
$V^{*}_{k,\mathcal{L}}$ for $\mathcal{L}=0$.

Indeed, since

  * •

if $k=4$ and $p\geq 5$, then by [BM02, Theorem 1.2], the reduction of the
semi-stable representation $V_{4,\mathcal{L}}$ for $\mathcal{L}=0$ is
$\mu_{3}\>\omega^{2}\oplus\mu_{3^{-1}}\>\omega$, and,

  * •

if $k=5$ and $p\geq 7$, then by [GP19, Theorem 5.0.5] (or, better, by Theorem
1.4), the reduction of the semi-stable representation $V_{5,\mathcal{L}}$ for
$\mathcal{L}=0$ is $\mathrm{ind}(\omega_{2}^{3+p})$,

we see that there is no contradiction to local constancy in the weight for the
reductions at the small weights above if one interprets local constancy
correctly!222We remark that in this context the case $k=3$ and $a_{p}=p^{1/2}$
is uninteresting, at least if $p\geq 5$. On the one hand, we have
$3>3(0.5)+1$, so that Berger’s local constancy result holds for $p\geq 5$.
Indeed, by [BG13, Theorem A], the reduction of $V_{3+p^{n}(p-1),p^{1/2}}$ for
large $n$ is $\mathrm{ind}(\omega_{2}^{2})$, which is the result one would
obtain by Fontaine-Edixhoven [Edi92] for the reduction of $V_{3,p^{1/2}}$. On
the other hand, by [GP19, Theorem 5.0.5] (more precisely, by Theorem 1.3) the
reduction of the semi-stable representation $V_{3,\mathcal{L}}$ for
$\mathcal{L}=0$ is also $\mathrm{ind}(\omega_{2}^{2})$, so that ‘naïve’ and
‘real’ local constancy are indistinguishable. This phenomenon can be explained
by noting that the reduction of the ‘real’ (semi-stable) limit
$V^{*}_{3,\mathcal{L}}$ for $\mathcal{L}=0$ of the sequence
$V^{*}_{3+p^{n}(p-1),p^{1/2}}$ is the same as the reduction of its ‘fake’
(crystalline) limit $V^{*}_{3,p^{1/2}}$. Indeed, consider a second sequence of
crystalline representations
$V^{*}_{3+p^{n}(p-1),p^{1/2}+p^{2+p^{n}(p-1)-(1/2)}}$. The (duals of the)
terms of the first sequence are close to the (duals of the) corresponding
terms of the second sequence in the sense of [Ber12, Theorem A], so they have
the same reduction. Hence, the ‘real’ limit $V^{*}_{3,\mathcal{L}}$ for
$\mathcal{L}=0$ of the first sequence has the same reduction as the limit of
the second sequence, namely $V^{*}_{3,p^{1/2}+p^{2-(1/2)}}$. The (dual of
this) last representation is in turn close to the (dual of the) ‘fake’ limit
$V^{*}_{3,p^{1/2}}$, again in the sense of [Ber12, Theorem A], at least if
$p\geq 5$, so again they have the same reduction. Therefore, for large $n$,
the duals of the reductions $\overline{V}_{3,\mathcal{L}}$ for $\mathcal{L}=0$
and $\overline{V}_{3,p^{1/2}}$ are isomorphic, hence so are the reductions
themselves.

##  2 Computing the $\mathcal{L}$-invariant

In this section, we explain the techniques involved in proving Theorem 1.1.

Let $E$ be a finite extension of $\mathbb{Q}_{p}$. Let $\mathcal{R}_{E}$ be
the Robba ring over $E$ consisting of bidirectional power series having
coefficients in $E$ that converge on the elements of
$\overline{\mathbb{Q}}_{p}$ with valuation in $]0,M]$, for some $M>0$. For a
more precise description, see Section 4. Let $k\geq 3$ and $r=k-2$, and for
$n\geq 1$, let $(k_{n},a_{n})$ be as in (1). Let $\chi$ be the cyclotomic
character.

The functor $\mathbf{D}_{\mathrm{rig}}$ sets up an equivalence of categories
between the category of $E$-linear representations of
$\mathrm{Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})$ and the category of
$(\varphi,\Gamma)$-modules over $\mathcal{R}_{E}$ of slope $0$. By [Ber12,
Proposition $3.1$], $\mathbf{D}_{\mathrm{rig}}(V_{k_{n},a_{n}}^{*})$ is an
extension of $\mathcal{R}_{E}(\mu_{1/y_{n}}\chi^{1-k_{n}})$ by
$\mathcal{R}_{E}(\mu_{y_{n}})$, where

$y_{n}=\frac{a_{n}+\sqrt{a_{n}^{2}-4p^{k_{n}-1}}}{2}.$ (2)

This allows us to associate to it a point of $\widetilde{\mathcal{T}}_{2}$.
Namely, to (the dual of) $V_{k_{n},a_{n}}$ is associated the unique point of
$\widetilde{\mathcal{T}}_{2}$ lying above the point
$(\mu_{y_{n}},\mu_{1/y_{n}}\chi^{1-k_{n}})$ in
$(\mathcal{T}\times\mathcal{T})\setminus F^{\prime}$. We therefore get a
sequence of points in $\widetilde{\mathcal{T}}_{2}$ corresponding to the
sequence of crystalline representations $V^{*}_{k_{n},a_{n}}$. This sequence
converges to a point in the fiber above the exceptional point
$(\mu_{p^{r/2}},\mu_{1/p^{r/2}}\chi^{1-k})$. Each point in this fiber
corresponds to a semi-stable representation of the form
$V^{*}_{k,\mathcal{L}}$ for some
$\mathcal{L}\in\mathbb{P}^{1}(\overline{\mathbb{Q}}_{p})$. We wish to compute
the $\mathcal{L}$-invariant of the limit point.

The first step to compute the $\mathcal{L}$-invariant is to transfer this
sequence of points from $\widetilde{\mathcal{T}}_{2}$ to
$\widetilde{\mathcal{T}}$, the blow-up of
$\mathcal{T}\setminus\\{x^{-j}\\}_{j\geq 0}$ at $\\{x^{i}\chi\\}_{i\geq 0}$,
by the map of rigid analytic varieties
$\widetilde{\mathcal{T}}_{2}\to\widetilde{\mathcal{T}}$ induced by the map
$\mathcal{T}\times\mathcal{T}\to\mathcal{T}$ which on $E$-valued points sends
$(\delta_{1},\delta_{2})$ to $\delta_{1}\delta_{2}^{-1}$. The second step is
to set up local coordinates $U_{r}$ around the point $\mu_{p^{r}}\chi^{k-1}$
and to compute the blow-up $\widetilde{U}_{r}$ of this coordinate patch at the
exceptional point. The third step is to compute the limit of the transferred
sequence in $\widetilde{U}_{r}$. The last but most important step is to apply
Theorem 6.1 to compute the $\mathcal{L}$-invariant associated to the limit
point.

We now describe the local coordinate patch $U_{r}$ in $\mathcal{T}$ around the
exceptional point and its blow-up $\widetilde{U}_{r}$. Throughout this paper,
let $\zeta_{p-1}$ be a fixed primitive $(p-1)^{\mathrm{th}}$ root of unity.
Associating the tuple $(\delta(p),\delta(\zeta_{p-1}),\delta(1+p)-1)$ to a
character $\delta\in\mathcal{T}(\mathbb{Q}_{p})$ identifies
$\mathcal{T}(\mathbb{Q}_{p})$ with $\mathbb{Q}_{p}^{*}\times\mu_{p-1}\times
p\mathbb{Z}_{p}$. Under this identification, the exceptional character
$\mu_{p^{r}}\chi^{k-1}$ ($=x^{r}\chi$) goes to the tuple
$(p^{r},\zeta_{p-1}^{k-1},(1+p)^{k-1}-1)$. The set
$p^{r}\mathbb{Z}_{p}^{*}\times\\{\zeta_{p-1}^{k-1}\\}\times p\mathbb{Z}_{p}$
is a neighborhood of $\mu_{p^{r}}\chi^{k-1}$ in $\mathcal{T}(\mathbb{Q}_{p})$.
This leads us to consider the following affinoid algebra

$U_{r}=\text{Sp }\mathbb{Q}_{p}\langle
S_{1},S_{2},T_{1},T_{2},T_{3}\rangle/(p^{r}T_{1}-S_{1},1-T_{1}T_{2},pT_{3}-S_{2})$

as a neighbourhood of $\mu_{p^{r}}\chi^{k-1}$ in $\mathcal{T}$ because clearly
$U_{r}(\mathbb{Q}_{p})=p^{r}\mathbb{Z}_{p}^{*}\times p\mathbb{Z}_{p}$. The
variable $S_{1}$ corresponds to the first factor and $S_{2}$ to the second
factor. From now on, by fixing the tame part of the characters under
consideration, we identify $U_{r}(E)$ with the subset
$p^{r}\mathcal{O}_{E}^{*}\times\\{\zeta_{p-1}^{k-1}\\}\times p\mathcal{O}_{E}$
of $\mathcal{T}(E)$, where $\mathcal{O}_{E}$ is the ring of integers of $E$.

The character $\mu_{p^{r}}\chi^{k-1}$ corresponds to the maximal ideal
$m=(S_{1}-p^{r},S_{2}-((1+p)^{k-1}-1))$ of $\mathcal{O}(U_{r})$. The blow-up
$\widetilde{U}_{r}$ of $U_{r}$ at the maximal ideal $m$ has the following
standard description:

$\widetilde{U}_{r}(E)=\\{(s_{1},s_{2},\>\xi_{1}:\xi_{2})\in
U_{r}(E)\times\mathbb{P}^{1}(E)\text{ }|\text{
}(s_{1}-p^{r})\xi_{2}=(s_{2}-((1+p)^{k-1}-1))\xi_{1}\\}.$

For large $n$, the sequence of points of $\widetilde{\mathcal{T}}$ associated
with $V^{*}_{k_{n},a_{n}}$ lies in $\widetilde{U}_{r}$. We prove that the
sequence converges to the point (see Section 7)

$(p^{r},(1+p)^{k-1}-1,\>\mathcal{L}\>\dfrac{p^{r}}{(1+p)^{k-1}\>\mathrm{log}(1+p)}:1)$, | if $\mathcal{L}\not=\infty$
---|---
$(p^{r},(1+p)^{k-1}-1,\>1:0)$, | if $\mathcal{L}=\infty$

in the blow up $\widetilde{U}_{r}$, where log is normalized so that
$\log(p)=0$. The proof of Theorem 1.1 then follows immediately from Theorem
6.1, a formula for the $\mathcal{L}$-invariant of a point in the exceptional
fiber, noting that the fudge factor there cancels with the extra factor
appearing in the third coordinate of the limit point above when
$\mathcal{L}\neq\infty$.

Theorem 6.1 is proved as follows. We first convert a point in the exceptional
fiber to a tangent direction in $U_{r}$ at the point $(p^{r},(1+p)^{k-1}-1)$,
i.e., an element of
$\mathbb{P}(\mathrm{Hom}(m/m^{2}\otimes_{\mathbb{Q}_{p}}E,E))$. This is done
using Proposition 3.5. We then explicitly describe the isomorphism
$\mathbb{P}(\mathrm{Hom}(m/m^{2}\otimes_{\mathbb{Q}_{p}}E,E))\to\mathbb{P}(H^{1}(\mathcal{R}_{E}(x^{r}\chi)))$
stated in [Che13, Theorem 2.33], using some preparatory material on the
cohomology of ‘big’ $(\varphi,\Gamma)$-modules in Section 5. We then represent
the cohomology class corresponding to the above limit point as an explicit
linear combination of the basis elements of
$H^{1}(\mathcal{R}_{E}(x^{r}\chi))$ studied by Benois in [Ben11, Proposition
1.5.4]. The original formula for the $\mathcal{L}$-invariant in the literature
is however in terms of a different basis of
$H^{1}(\mathcal{R}_{E}(x^{r}\chi))$, namely the one constructed by Colmez in
[Col08, Proposition 2.19] (see Section 4). Restating the formula for the
$\mathcal{L}$-invariant in terms of Benois’s basis (Definition 4.6) allows us
to finally compute the $\mathcal{L}$-invariant.

##  3 The blow-up of $U_{r}$

In this section, we provide details about blow-ups in the rigid analytic
setting which may be of independent interest. This is inspired by the work of
Schoutens (cf. [Sch95]), but we carefully work out the details and also do not
work over algebraically closed fields. For background on rigid analytic
geometry, we refer the reader to [BGR84]. We also recall the important
Proposition 3.5, which for each finite extension $E$ of $\mathbb{Q}_{p}$
establishes a bijection between the $E$-valued points of $\widetilde{U}_{r}$
that lie above $(p^{r},(1+p)^{k-1}-1)$ and the tangent directions in $U_{r}$
at the exceptional point.

###  3.1 The blow-up as a rigid analytic variety

The blow-up of $U_{r}$ consists of a rigid analytic variety
$\widetilde{U}_{r}$ and a map $\pi:\widetilde{U}_{r}\to U_{r}$ satisfying the
properties given in [Sch95, Definition 1.2.1]. In this subsection, we will
construct $\widetilde{U}_{r}$.

Recall that $U_{r}=\text{Sp }\mathcal{O}(U_{r})$, where

$\mathcal{O}(U_{r})=\mathbb{Q}_{p}\langle
S_{1},S_{2},T_{1},T_{2},T_{3}\rangle/(p^{r}T_{1}-S_{1},1-T_{1}T_{2},pT_{3}-S_{2}).$

The $\mathbb{Q}_{p}$-valued point $(p^{r},(1+p)^{k-1}-1)$ corresponds to the
maximal ideal $m=(f_{1},f_{2})$ of $\mathcal{O}(U_{r})$, where
$f_{1}=S_{1}-p^{r}$ and $f_{2}=S_{2}-((1+p)^{k-1}-1)$. By the blow-up of
$U_{r}$ at $(p^{r},(1+p)^{k-1}-1)$, we mean the blow-up at the maximal ideal
$m$.

We will construct $\widetilde{U}_{r}$ by a patching argument. Let
$Q_{1},Q_{2},Q_{1}^{\prime}$ and $Q_{2}^{\prime}$ be indeterminates. For
$i=0,1,\cdots$, consider the affinoid algebra

$A_{i}=\mathcal{O}(U_{r})\langle p^{i}Q_{1}\rangle/(f_{2}-Q_{1}f_{1}),$

which describes the subset of $U_{r}$ where $|f_{2}|\leq p^{i}|f_{1}|$.
Sending $p^{i+1}Q_{1}$ to $p\cdot p^{i}Q_{1}$ and $X$ to $p^{i}Q_{1}$ induces
an isomorphism $A_{i+1}\langle X\rangle/(pX-p^{i+1}Q_{1})\cong A_{i}$, showing
that $\text{Sp }A_{i}$ is the affinoid subdomain (hence an open subvariety) of
$\text{Sp }A_{i+1}$ consisting of maximal ideals $\mathfrak{m}$ such that
$|Q_{1}\mod{\mathfrak{m}}|\leq p^{i}$. We therefore get a sequence of
inclusions $\text{Sp }A_{0}\to\text{Sp }A_{1}\to\text{Sp }A_{2}\to\cdots$
associated to the sequence of affinoid algebra homomorphisms $\cdots\to
A_{2}\to A_{1}\to A_{0}$. Using [BGR84, Proposition $9.3.2/1$] we paste
together the $\text{Sp }A_{i}$ for $i\geq 0$ to get a rigid analytic variety
$\widetilde{V}_{1}$. The same proposition also states that $\\{\text{Sp
}A_{i}\\}_{i\geq 0}$ is an admissible cover of $\widetilde{V}_{1}$. Similarly,
for $i\geq 0$, we define

$B_{i}=\mathcal{O}(U_{r})\langle p^{i}Q_{2}\rangle/(f_{1}-Q_{2}f_{2})$

and glue the corresponding affinoid spaces together to get a rigid analytic
variety $\widetilde{V}_{2}$. The blow-up $\widetilde{U}_{r}$ is the rigid
analytic variety obtained by further glueing $\widetilde{V}_{1}$ and
$\widetilde{V}_{2}$ along certain open subvarieties.

We describe these open subvarieties now. For each $i\geq 0$, consider the
affinoid algebra

$A_{i}^{\prime}=\mathcal{O}(U_{r})\langle
p^{i}Q_{1},p^{i}Q_{1}^{\prime}\rangle/(f_{2}-Q_{1}f_{1},1-Q_{1}Q_{1}^{\prime}),$

which describes a Laurent subdomain $\text{Sp }A_{i}^{\prime}$ of $\text{Sp
}A_{i}$ given by the condition $|Q_{1}|\geq p^{-i}$. Since we have an
isomorphism $A_{i+1}^{\prime}\langle
X,Y\rangle/(pX-p^{i+1}Q_{1},pY-p^{i+1}Q_{1}^{\prime})\cong A_{i}^{\prime}$
(sending $X$ to $p^{i}Q_{1}$ and $Y$ to $p^{i}Q_{1}^{\prime}$), we see that
$\text{Sp }A_{i}^{\prime}$ is an affinoid subdomain (and hence an open
subvariety) of $\text{Sp }A_{i+1}^{\prime}$. Using [BGR84, Proposition
$9.3.2/1$], we paste together $\text{Sp }A_{i}^{\prime}$ for $i\geq 0$ to get
a rigid analytic variety $\widetilde{V}_{1}^{\prime}$. Moreover, using [BGR84,
Proposition $9.3.3/1$], we see that the canonical inclusions $\text{Sp
}A_{i}^{\prime}\to\text{Sp }A_{i}$ induce an inclusion
$\widetilde{V}_{1}^{\prime}\xhookrightarrow{}\widetilde{V}_{1}$, identifying
$\widetilde{V}_{1}^{\prime}$ as a subvariety of $\widetilde{V}_{1}$.
Similarly, we construct a subvariety $\widetilde{V}_{2}^{\prime}$ of
$\widetilde{V}_{2}$ by glueing all the $\text{Sp }B_{i}^{\prime}$ together,
where

$B_{i}^{\prime}=\mathcal{O}(U_{r})\langle
p^{i}Q_{2},p^{i}Q_{2}^{\prime}\rangle/(f_{1}-Q_{2}f_{2},1-Q_{2}Q_{2}^{\prime}).$

These spaces are summarized by the following picture:

Figure 1: Exceptional fiber of the blow-up in the rigid setting.

The space $\widetilde{V}_{1}$ corresponds to the surface of the sphere except
the south pole in the figure above whereas $\widetilde{V}_{2}$ corresponds to
the surface of the sphere except the north pole.

We claim that $\widetilde{V}_{i}^{\prime}$ is an open subvariety of
$\widetilde{V}_{i}$, for $i=1,2$. Without loss of generality, assume $i=1$. To
prove this claim, we need to show that $\widetilde{V}_{1}^{\prime}\cap\text{Sp
}A_{i}$ is an admissible open subset of $\text{Sp }A_{i}$ for each $i\geq 0$.
Write $\widetilde{V}_{1}^{\prime}\cap\text{Sp
}A_{i}=\cup_{k=i}^{\infty}\>(\text{Sp }A_{k}^{\prime}\cap\text{Sp }A_{i})$. If
$k\geq i$, then $\text{Sp }A_{k}^{\prime}\cap\text{Sp }A_{i}$ is the set of
maximal ideals $\mathfrak{m}$ of $A_{i}$ satisfying $|Q_{1}\text{ mod
}\mathfrak{m}|\geq p^{-k}$. Therefore $\widetilde{V}_{1}^{\prime}\cap\text{Sp
}A_{i}$ is the set of maximal ideals $\mathfrak{m}$ of $A_{i}$ such that
$|Q_{1}\text{ mod }\mathfrak{m}|>0$. In other words, it is the complement of
the vanishing set of $Q_{1}$ in $\text{Sp }A_{i}$, i.e., it is a Zariski open
subset. Since Zariski open subsets are admissible open, we have proved the
claim.

To glue $\widetilde{V}_{1}$ and $\widetilde{V}_{2}$, we define an isomorphism
$\phi:\widetilde{V}_{1}^{\prime}\to\widetilde{V}_{2}^{\prime}$ as follows.
Recall that for $i\geq 0$,

$\begin{array}[]{l}A_{i}^{\prime}=\mathcal{O}(U_{r})\langle
p^{i}Q_{1},p^{i}Q_{1}^{\prime}\rangle/(f_{2}-Q_{1}f_{1},1-Q_{1}Q_{1}^{\prime}),\\\
B_{i}^{\prime}=\mathcal{O}(U_{r})\langle
p^{i}Q_{2},p^{i}Q_{2}^{\prime}\rangle/(f_{1}-Q_{2}f_{2},1-Q_{2}Q_{2}^{\prime}).\end{array}$

Consider an isomorphism of affinoid algebras $B_{i}^{\prime}\to
A_{i}^{\prime}$ given by

${p^{i}Q_{2}}$${p^{i}Q_{1}^{\prime},}$${p^{i}Q_{2}^{\prime}}$${p^{i}Q_{1}.}$

This isomorphism gives rise to an isomorphism $\phi_{i}:\text{Sp
}A_{i}^{\prime}\to\text{Sp }B_{i}^{\prime}$ of the corresponding affinoid
spaces. For $i,j\geq 0$, these maps clearly restrict to the same map on
$\text{Sp }A_{i}^{\prime}\cap\text{Sp }A_{j}^{\prime}$. Therefore applying
[BGR84, Proposition $9.3.3/1$], we get an isomorphism of rigid analytic
varieties $\phi:\widetilde{V}_{1}^{\prime}\to\widetilde{V}_{2}^{\prime}$.

Define $\widetilde{U}_{r}$ to be the rigid analytic variety obtained by
glueing $\widetilde{V}_{1}$ and $\widetilde{V}_{2}$ along
$\widetilde{V}_{1}^{\prime}$ and $\widetilde{V}_{2}^{\prime}$, respectively,
using the isomorphism $\phi$. This rigid analytic variety is analogous to the
blow-up in the classical algebraic geometry setting.

We see this by computing its $E$-valued points for any finite extension $E$ of
$\mathbb{Q}_{p}$. Since $\widetilde{U}_{r}$ is covered by $\widetilde{V}_{1}$
and $\widetilde{V}_{2}$, we compute $\widetilde{V}_{1}(E)$ and
$\widetilde{V}_{2}(E)$ first. Recall for $i\geq 0$,

$A_{i}=\mathcal{O}(U_{r})\langle p^{i}Q_{1}\rangle/(f_{2}-Q_{1}f_{1}).$

We identify the set of $E$-valued points of $\text{Sp }A_{i}$ with

$\\{(s_{1},s_{2},q_{1})\in U_{r}(E)\times E\text{ }\big{|}\text{
}(1+s_{2}-(1+p)^{k-1})=q_{1}(s_{1}-p^{r}),\>|q_{1}|\leq p^{i}\\}.$

The first condition is a consequence of the relation $f_{2}=Q_{1}f_{1}$. For
$i\geq j$, the glueing map $A_{i}\to A_{j}$ induces the canonical inclusion
map on $E$-valued points $\text{Sp }A_{j}(E)\to\text{Sp }A_{i}(E)$. Therefore,
the set of $E$-valued points of $\widetilde{V}_{1}$ is the union of all the
$\text{Sp }A_{i}(E)$, i.e.,

$\widetilde{V}_{1}(E)=\\{(s_{1},s_{2},q_{1})\in U_{r}(E)\times E\text{
}\big{|}\text{ }(1+s_{2}-(1+p)^{k-1})=q_{1}(s_{1}-p^{r})\\}.$

We similarly have

$\widetilde{V}_{2}(E)=\\{(s_{1},s_{2},q_{2})\in U_{r}(E)\times E\text{
}\big{|}\text{ }(s_{1}-p^{r})=q_{2}(1+s_{2}-(1+p)^{k-1})\\}.$

We can now describe $\widetilde{U}_{r}(E)$. If $P$ is an $E$-valued point of
$\widetilde{V}_{1}\cap\widetilde{V}_{2}$, it is an $E$-valued point of
$\text{Sp }A_{i}\cap\text{Sp }B_{i}=\text{Sp }A_{i}^{\prime}$, for some $i\geq
0$. Being a point of $\text{Sp }A_{i}$, it is of the form
$(s_{1},s_{2},q_{1})$, for some $s_{1},s_{2},q_{1}\in E$. Since the glueing
map between $B_{i}^{\prime}$ and $A_{i}^{\prime}$ takes $S_{1}$ to $S_{1}$,
$S_{2}$ to $S_{2}$ and $Q_{2}$ to the inverse of $Q_{1}$ in $A_{i}^{\prime}$,
we see that $(s_{1},s_{2},q_{1}^{-1})$ represents $P$ as an $E$-valued point
of $\text{Sp }B_{i}$ (the condition $|q_{1}^{-1}|\leq p^{i}$ is satisfied
because $P\in\text{Sp }A_{i}^{\prime}$ implies $|q_{1}|\geq p^{-i}$). We can
therefore identify $(s_{1},s_{2},q_{1})\in\widetilde{V}_{1}(E)$ with the point
$(s_{1},s_{2},1:q_{1})\in U_{r}(E)\times\mathbb{P}^{1}(E)$ and
$(s_{1},s_{2},q_{2})\in\widetilde{V}_{2}(E)$ with the point
$(s_{1},s_{2},q_{2}:1)\in U_{r}(E)\times\mathbb{P}^{1}(E)$, so that a point in
the intersection goes to the same point in $U_{r}(E)\times\mathbb{P}^{1}(E)$
under both of these identifications. As a result of this discussion, we see
that

$\widetilde{U}_{r}(E)=\\{(s_{1},s_{2},\>\xi_{1}:\xi_{2})\in
U_{r}(E)\times\mathbb{P}^{1}(E)\text{ }\big{|}\text{
}(s_{1}-p^{r})\xi_{2}=(1+s_{2}-(1+p)^{k-1})\xi_{1}\\}$

exactly as in the classical algebraic geometry setting.

###  3.2 The blow-up map $\pi:\widetilde{U}_{r}\to U_{r}$

In this subsection, we define a candidate for the blow-up map
$\pi:\widetilde{U}_{r}\to U_{r}$ using [BGR84, Proposition $9.3.3/1$].

To define $\pi$, we first define its restrictions $\pi_{i}$ to
$\widetilde{V}_{i}$, for $i=1,2$. We first define the map
$\pi_{1}:\widetilde{V}_{1}\to U_{r}$. For each $i\geq 0$, consider the map
$\text{Sp }A_{i}\to U_{r}$ associated to the canonical homomorphism

$\mathcal{O}(U_{r})\to\mathcal{O}(U_{r})\langle
p^{i}Q_{1}\rangle/(f_{2}-Q_{1}f_{1})=A_{i}.$

These maps clearly restrict to the same map on $\text{Sp }A_{i}\cap\text{Sp
}A_{j}$, for $i,j\geq 0$. So using [BGR84, Proposition $9.3.3/1$], we glue
these maps together to get a map $\pi_{1}:\widetilde{V}_{1}\to U_{r}$ of rigid
analytic varieties. The other map $\pi_{2}:\widetilde{V}_{2}\to U_{r}$ is
constructed similarly.

Now, to get a map $\pi:\widetilde{U}_{r}\to U_{r}$, we have to check that
$\pi_{1}$ and $\pi_{2}$ agree on the intersection of $\widetilde{V}_{1}$ and
$\widetilde{V}_{2}$ in $\widetilde{U}_{r}$. Since this intersection is equal
to $\widetilde{V}_{1}^{\prime}$ $(\cong\widetilde{V}_{2}^{\prime})$, we have
to check that the following diagram commutes

${\widetilde{V}_{1}^{\prime}}$${U_{r}.}$${\widetilde{V}_{2}^{\prime}}$$\scriptstyle{\pi_{1}|_{\widetilde{V}_{1}^{\prime}}}$$\scriptstyle{\phi}$$\scriptstyle{\pi_{2}|_{\widetilde{V}_{2}^{\prime}}}$

Since $\phi$ is obtained by pasting the maps $\phi_{i}:\text{Sp
}A_{i}^{\prime}\to\text{Sp }B_{i}^{\prime}$, the commutativity of the diagram
above is equivalent to the commutativity of the following diagram, for all
$i\geq 0$

${\text{Sp }A_{i}^{\prime}}$${U_{r}.}$${\text{Sp
}B_{i}^{\prime}}$$\scriptstyle{\pi_{1}|_{\text{Sp
}A_{i}^{\prime}}}$$\scriptstyle{\phi_{i}}$$\scriptstyle{\pi_{2}|_{\text{Sp
}B_{i}^{\prime}}}$

The commutativity of the diagram above can be checked by reversing all the
arrows and noting that the resulting maps are $\mathcal{O}(U_{r})$-algebra
homomorphisms. This shows that $\pi_{1}$ and $\pi_{2}$ agree on the
intersection of $\widetilde{V}_{1}$ and $\widetilde{V}_{2}$ in
$\widetilde{U}_{r}$. Therefore, we glue $\pi_{1}$ and $\pi_{2}$ using [BGR84,
Proposition $9.3.3/1$] to get a map $\pi:\widetilde{U}_{r}\to U_{r}$.

###  3.3 Proof that $\pi:\widetilde{U}_{r}\to U_{r}$ is the blow-up

In this subsection, we prove that the map $\pi:\widetilde{U}_{r}\to U_{r}$
defined in the previous subsection is the blow-up of $U_{r}$ at the maximal
ideal $m=(f_{1},f_{2})$. To prove this, we need to check that $\pi$ satisfies
the two properties stated in [Sch95, Definition 1.2.1], namely the
invertibility of a certain sheaf, and a corresponding universal property.

We first show that the ideal sheaf on $U_{r}$ corresponding to the ideal $m$
of $\mathcal{O}(U_{r})$ extends to an invertible ideal sheaf on
$\widetilde{U}_{r}$ in the sense of [Sch95, Definition $1.1.1$]. Since this is
a local criterion (this means that the criterion can be checked over an
admissible cover), we need to check that for all $i\geq 0$, the maps
$\pi\big{|}_{\text{Sp }A_{i}}:\text{Sp }A_{i}\to U_{r}$ and
$\pi\big{|}_{\text{Sp }B_{i}}:\text{Sp }B_{i}\to U_{r}$ satisfy the same
property. Let us prove this statement for $\pi\big{|}_{\text{Sp
}A_{i}}:\text{Sp }A_{i}\to U_{r}$. In this case, the extended ideal sheaf
corresponds to the ideal $mA_{i}$ on $\text{Sp }A_{i}$. Moreover, [Sch95,
Proposition 1.1.4] states that the invertibility of this sheaf is equivalent
to the invertibility of the ideal $mA_{i,\mathfrak{m}}$ of
$A_{i,\mathfrak{m}}$ for all maximal ideals $\mathfrak{m}$ of $A_{i}$, where
$A_{i,\mathfrak{m}}$ is the localization of $A_{i}$ at $\mathfrak{m}$. To show
this, it is enough to prove that $mA_{i}$ is generated by a regular element of
$A_{i}$ because regular elements go to regular elements under flat base
change.

Since $mA_{i}$ is generated by $f_{1}$, we prove that $f_{1}$ is not a zero
divisor in $A_{i}$ for $i\geq 0$. Note that we only need to prove this
statement for $i=0$ because of the injections $A_{i}\xhookrightarrow{}A_{0}$.

######  Lemma 3.1.

$f_{1}$ is not a zero divisor in $A_{0}=\mathcal{O}(U_{r})\langle
Q_{1}\rangle/(f_{2}-Q_{1}f_{1})$.

###### Proof.

Recall that

$\mathcal{O}(U_{r})=\mathbb{Q}_{p}\langle
S_{1},S_{2},T_{1},T_{2},T_{3}\rangle/(p^{r}T_{1}-S_{1},1-T_{1}T_{2},pT_{3}-S_{2}).$

To prove that $f_{1}$ is not a zero divisor in $A_{0}$, it is enough to prove
that $f_{1}$ is not a zero divisor in $\mathbb{Q}_{p}\langle
S_{1},S_{2},Q_{1}\rangle/(f_{2}-Q_{1}f_{1})$. Indeed, since $\text{Sp }A_{0}$
is an affinoid subdomain (a Laurent subdomain) of $\text{Sp
}\mathbb{Q}_{p}\langle S_{1},S_{2},Q_{1}\rangle/(f_{2}-Q_{1}f_{1})$, the
canonical map $\mathbb{Q}_{p}\langle
S_{1},S_{2},Q_{1}\rangle/(f_{2}-Q_{1}f_{1})\to A_{0}$ is flat by [BGR84,
Corollary $7.3.2/6$]. Therefore, if $f_{1}$ is not a a zero divisor in the
former then, by flatness, it is not a zero divisor in the latter.

So let us now prove that $f_{1}$ is not a zero divisor in
$\mathbb{Q}_{p}\langle S_{1},S_{2},Q_{1}\rangle/(f_{2}-Q_{1}f_{1})$. Assume
that there exists $h\in\mathbb{Q}_{p}\langle S_{1},S_{2},Q_{1}\rangle$ such
that $f_{2}-Q_{1}f_{1}$ divides $f_{1}h$. We know that
$f_{2}-Q_{1}f_{1}=1+S_{2}-(1+p)^{k-1}-Q_{1}(S_{1}-p^{r})$ is
$S_{2}$-distinguished of degree $1$ (cf. [BGR84, Definition 5.2.1/1]).
Applying the Weierstrass division theorem (cf. [BGR84, Theorem 5.2.1/2]) to
$h$ and $f_{2}-Q_{1}f_{1}$, we see that there exist unique power series
$q\in\mathbb{Q}_{p}\langle S_{1},S_{2},Q_{1}\rangle$ and
$r\in\mathbb{Q}_{p}\langle S_{1},Q_{1}\rangle$ (note that $S_{2}$ does not
appear in $r$ because $f_{2}-Q_{1}f_{1}$ is of degree $1$ in $S_{2}$) such
that $h=q(f_{2}-Q_{1}f_{1})+r$. The fact that $f_{2}-Q_{1}f_{1}$ divides
$f_{1}h$ implies that $f_{2}-Q_{1}f_{1}$ divides $f_{1}r$. This is possible
only if $r=0$, i.e., only if $f_{2}-Q_{1}f_{1}$ divides $h$. Therefore $h=0$
in $\mathbb{Q}_{p}\langle S_{1},S_{2},Q_{1}\rangle/(f_{2}-Q_{1}f_{1})$. ∎

Having checked the first property of blow-ups for $\pi$, we now check the
second property. Using [Sch95, Lemma 1.2.4], we note that it is enough to
check it for affinoid spaces. In other words, given any affinoid space
$Y=\text{Sp }R$ and a map of rigid analytic varieties $f:Y\to U_{r}$ such that
the ideal sheaf on $Y$ associated to the ideal $mR$ is invertible, we prove
that there exists a unique map $g:Y\to\widetilde{U}_{r}$ such that the
following diagram commutes

${\widetilde{U}_{r}}$${Y}$${U_{r}.}$$\scriptstyle{\pi}$$\scriptstyle{f}$$\scriptstyle{g}$

For $i=1,2$, define $\mathfrak{a}_{i}=(f^{*}(f_{i})R:mR)$ and let
$Y_{i}=Y\setminus V(\mathfrak{a}_{i})$. We prove two lemmas about the $Y_{i}$.

######  Lemma 3.2.

For $i=1$, $2$, a point $y\in Y$ belongs to $Y_{i}$ if and only if
$m\mathcal{O}_{Y,y}=f^{*}(f_{i})\mathcal{O}_{Y,y}$.

###### Proof.

Without loss of generality, we assume $i=1$. Let $y\in Y$ be arbitrary. Let
$\mathfrak{m}_{y}$ be the maximal ideal of $R$ corresponding to $y$.

($\implies$) Let $y\in Y_{1}$. Therefore
$\mathfrak{a}_{1}\not\subseteq\mathfrak{m}_{y}$. By the definition of
$\mathfrak{a}_{1}$ if $a\in\mathfrak{a}_{1}$, then $amR\subseteq
f^{*}(f_{1})R$. Extending to the localization of $R$ at $\mathfrak{m}_{y}$, we
see that $amR_{\mathfrak{m}_{y}}\subseteq f^{*}(f_{1})R_{\mathfrak{m}_{y}}$.
For $a\in\mathfrak{a}_{1}\setminus\mathfrak{m}_{y}$, we get
$mR_{\mathfrak{m}_{y}}=amR_{\mathfrak{m}_{y}}\subseteq
f^{*}(f_{1})R_{\mathfrak{m}_{y}}$. Therefore
$mR_{\mathfrak{m}_{y}}=f^{*}(f_{1})R_{\mathfrak{m}_{y}}$. Extending these
ideals to $\mathcal{O}_{Y,y}$ by the map
$R_{\mathfrak{m}_{y}}\to\mathcal{O}_{Y,y}$ described in [BGR84, Proposition
7.3.2/3], we get $m\mathcal{O}_{Y,y}=f^{*}(f_{1})\mathcal{O}_{Y,y}$.

($\impliedby$) Assume that $m\mathcal{O}_{Y,y}=f^{*}(f_{1})\mathcal{O}_{Y,y}$.
Extending to the completion $\widehat{\mathcal{O}}_{Y,y}$, we get
$m\widehat{\mathcal{O}}_{Y,y}=f^{*}(f_{1})\widehat{\mathcal{O}}_{Y,y}$. Using
the equality $\widehat{\mathcal{O}}_{Y,y}=\widehat{R}_{\mathfrak{m}_{y}}$
given in [BGR84, Proposition 7.3.2/3], we rewrite this as
$m\widehat{R}_{\mathfrak{m}_{y}}=f^{*}(f_{1})\widehat{R}_{\mathfrak{m}_{y}}$.
Since the canonical inclusion
$R_{\mathfrak{m}_{y}}\to\widehat{R}_{\mathfrak{m}_{y}}$ is faithfully flat, we
get $mR_{\mathfrak{m}_{y}}=f^{*}(f_{1})R_{\mathfrak{m}_{y}}$. This implies
$f^{*}(f_{2})\in f^{*}(f_{1})R_{\mathfrak{m}_{y}}$. Therefore there exists
$a\in R$ and $s\in R\setminus\mathfrak{m}_{y}$ such that
$f^{*}(f_{2})=\frac{a}{s}f^{*}(f_{1})$. By the definition of equality in
localizations, there exists $s^{\prime}\in R\setminus\mathfrak{m}_{y}$ such
that $s^{\prime}(sf^{*}(f_{2})-af^{*}(f_{1}))=0$ in $R$. Therefore
$s^{\prime}sf^{*}(f_{2})\in f^{*}(f_{1})R$. Since $mR$ is generated by
$f^{*}(f_{1})$ and $f^{*}(f_{2})$, we see that
$s^{\prime}s\in(f^{*}(f_{1})R:mR)=\mathfrak{a}_{1}$. In other words,
$\mathfrak{a}_{1}\not\subseteq\mathfrak{m}_{y}$ (because
$s^{\prime}s\not\in\mathfrak{m}_{y}$). Therefore $y\in Y_{1}$. ∎

Using [Sch95, Lemma 1.1.2] and the lemma above one easily checks that
$Y=Y_{1}\cup Y_{2}$. The sets $Y_{1}$ and $Y_{2}$, being Zariski open subsets
of $Y$, are admissible open and form an admissible cover of $Y$.

Before proving the next lemma, we remark that, by [Sch95, Lemma 0.4], we have
$\mathfrak{a}_{i}\mathcal{O}(Y_{i})=\mathcal{O}(Y_{i})$, which implies that
$m\mathcal{O}(Y_{i})=f^{*}(f_{i})\mathcal{O}(Y_{i})$, for $i=1,2$.

######  Lemma 3.3.

Let $i=1$, $2$. Then, $f^{*}(f_{i})$ is not a zero divisor in
$\mathcal{O}(Y^{\prime})$ for any admissible open subset $Y^{\prime}$ of $Y$
contained in $Y_{i}$.

###### Proof.

The proof has three steps. In the first step, we prove that $f^{*}(f_{i})$ is
not a zero divisor in $\mathcal{O}_{Y,y}$ for any $y\in Y^{\prime}$. In the
second step, we prove that the canonical map
$\mathcal{O}(Y^{\prime})\to\prod_{y\in Y^{\prime}}\mathcal{O}_{Y,y}$ is
injective. In the third step, we use these two facts to conclude that
$f^{*}(f_{i})$ is not a zero divisor in $\mathcal{O}(Y^{\prime})$. Let $i=1$
or $2$.

  * •

Using the remark preceding this lemma, we get
$m\mathcal{O}(Y^{\prime})=f^{*}(f_{i})\mathcal{O}(Y^{\prime})$. Extending to
the stalks, we get $m\mathcal{O}_{Y,y}=f^{*}(f_{i})\mathcal{O}_{Y,y}$ for each
$y\in Y^{\prime}$. Since the sheaf associated to the ideal $m\mathcal{O}(Y)$
is invertible, we see that $m\mathcal{O}_{Y,y}$ is generated by a regular
element of $\mathcal{O}_{Y,y}$ for each $y\in Y$. Therefore $f^{*}(f_{i})$ is
not a zero divisor in $\mathcal{O}_{Y,y}$ for each $y\in Y^{\prime}$.

  * •

Consider the map $\mathcal{O}(Y^{\prime})\to\prod_{y\in
Y^{\prime}}\mathcal{O}_{Y,y}$. Let $a$ be an element of
$\mathcal{O}(Y^{\prime})$ that maps to $0$ in $\mathcal{O}_{Y,y}$ for each
$y\in Y^{\prime}$. Choose an admissible cover $\\{U_{j}\\}_{j\in J}$ of
$Y^{\prime}$ by affinoid subdomains of $Y$. For any $j\in J$, the image of $a$
in $\mathcal{O}_{Y,y}$ is $0$ for each $y\in U_{j}$. Using [BGR84, Corollary
$7.3.2/4$], we see that the restriction of $a$ to $\mathcal{O}(U_{j})$ is $0$
for each $j\in J$. Since the cover $\\{U_{j}\\}_{j\in J}$ is admissible, we
see that $a=0$ in $\mathcal{O}(Y^{\prime})$.

  * •

Suppose there exists a $b\in\mathcal{O}(Y^{\prime})$ such that
$f^{*}(f_{i})b=0$. This means that $f^{*}(f_{i})b=0$ in $\mathcal{O}_{Y,y}$
for each $y\in Y^{\prime}$. Using the fact that $f^{*}(f_{i})$ is not a zero
divisor in $\mathcal{O}_{Y,y}$ for $y\in Y^{\prime}$, we see that the image of
$b$ in $\mathcal{O}_{Y,y}$ is $0$. The injectivity of the map
$\mathcal{O}(Y^{\prime})\to\prod_{j\in J}\mathcal{O}_{Y,y}$ implies that $b=0$
in $\mathcal{O}(Y^{\prime})$. Therefore $f^{*}(f_{i})$ is not a zero divisor
in $\mathcal{O}(Y^{\prime})$. ∎

We now construct $g:Y\to\widetilde{U}_{r}$. In order to do this, we use the
above lemmas to define the restrictions of $g$ to the elements of a refinement
of the cover $Y=Y_{1}\cup Y_{2}$ and then apply the usual patching argument.

As $f^{*}(f_{2})\in m\mathcal{O}(Y_{1})=f^{*}(f_{1})\mathcal{O}(Y_{1})$, there
exists a $q_{1}\in\mathcal{O}(Y_{1})$ such that
$f^{*}(f_{2})=q_{1}f^{*}(f_{1})$. Similarly, there exists a
$q_{2}\in\mathcal{O}(Y_{2})$ such that $f^{*}(f_{1})=q_{2}f^{*}(f_{2})$.
Furthermore, $q_{1}$ and $q_{2}$ are unique because of Lemma 3.3.

Since $Y_{1}$ is an admissible open subset of $Y$, there exists an admissible
cover $\\{Y_{1,m}\\}_{m\in I}$ of $Y_{1}$ by affinoid subdomains $Y_{1,m}$ of
$Y$. For each $m\in I$, there is a restriction map
$\mathcal{O}(Y_{1})\to\mathcal{O}(Y_{1,m})$ and so we can think of $q_{1}$ as
an element of $\mathcal{O}(Y_{1,m})$. For each $m\in I$, fix a natural number
$\underline{m}$ such that $|p^{\underline{m}}q_{1}|\leq 1$. Consider the
affinoid algebra map $f^{*}:\mathcal{O}(U_{r})\to\mathcal{O}(Y)$ and compose
it with the restriction map $\mathcal{O}(Y)\to\mathcal{O}(Y_{1,m})$. Extend
this composition to a map $\mathcal{O}(U_{r})\langle
p^{\underline{m}}Q_{1}\rangle\to\mathcal{O}(Y_{1,m})$ by sending
$p^{\underline{m}}Q_{1}$ to $p^{\underline{m}}q_{1}$. This is possible because
of [BGR84, Corollary 1.4.3/2]. Since we have the relation
$f^{*}(f_{1})=q_{1}f^{*}(f_{2})$ in $\mathcal{O}(Y_{1,m})$, we see that this
extension factors through an affinoid algebra map
$A_{\underline{m}}\to\mathcal{O}(Y_{1,m})$. Let $g_{1,m}:Y_{1,m}\to\text{Sp
}A_{\underline{m}}$ be the corresponding map of affinoid spaces. Similarly,
there is an admissible cover $\\{Y_{2,n}\\}_{n\in J}$ of $Y_{2}$ by affinoid
subdomains $Y_{2,n}$ of $Y$ and for each $n\in J$, a fixed natural number
$\underline{n}$ such that $|p^{\underline{n}}q_{2}|\leq 1$ and a map of
affinoid spaces $g_{2,n}:Y_{2,n}\to\text{Sp }B_{\underline{n}}$ associated to
the map of affinoid algebras $B_{\underline{n}}\to\mathcal{O}(Y_{2,n})$
sending $p^{\underline{n}}Q_{2}$ to $p^{\underline{n}}q_{2}$. Since
$g_{1,m}^{*}$ and $g_{2,n}^{*}$ are $\mathcal{O}(U_{r})$-algebra
homomorphisms, we see that $\pi\circ g_{1,m}=f$ on $Y_{1,m}$ and $\pi\circ
g_{2,n}=f$on $Y_{2,n}$, for $m\in I$, $n\in J$.

Note that $\\{Y_{1,m},Y_{2,n}\\}_{m\in I,n\in J}$ is an admissible cover of
$Y$. To obtain $g:Y\to\widetilde{U}_{r}$, we glue all the maps $g_{1,m}$ and
$g_{2,n}$. Without loss of generality, assume
$\underline{m}\geq\underline{m}^{\prime}$. Then, $g_{1,m}$ and
$g_{1,m^{\prime}}$ agree on $Y_{1,m}\cap Y_{1,m^{\prime}}$ because the
following diagram commutes

${A_{\underline{m}^{\prime}}}$${\mathcal{O}(Y_{1,m}\cap
Y_{1,m^{\prime}}),}$${A_{\underline{m}}}$$\scriptstyle{g_{1,m^{\prime}}^{*}}$$\scriptstyle{g_{1,m}^{*}}$

A similar argument shows that $g_{2,n}$ and $g_{2,n^{\prime}}$ agree on
$Y_{2,n}\cap Y_{2,n^{\prime}}$.

We check that the maps $g_{1,m}$ and $g_{2,n}$ agree on $Y_{1,m}\cap Y_{2,n}$.
Let $N\geq\underline{m}$, $\underline{n}$. Think of $g_{1,m}$ as a map from
$Y_{1,m}$ to $\text{Sp }A_{N}$ (by composing with the map $\text{Sp
}A_{\underline{m}}\to\text{Sp }A_{N}$) and $g_{2,n}$ as a map from $Y_{2,n}$
to $\text{Sp }B_{N}$ (by composing with the map $\text{Sp
}B_{\underline{n}}\to\text{Sp }B_{N}$). We claim that $g_{1,m}$ maps
$Y_{1,m}\cap Y_{2,n}$ into $\text{Sp }A_{N}^{\prime}$ and that $g_{2,n}$ maps
$Y_{1,m}\cap Y_{2,n}$ into $\text{Sp }B_{N}^{\prime}$. To show this, we first
prove that $q_{1}$ is the inverse of $q_{2}$ in $\mathcal{O}(Y_{1,m}\cap
Y_{2,n})$. Transfer the relations $f^{*}(f_{2})=q_{1}f^{*}(f_{1})$ and
$f^{*}(f_{1})=q_{2}f^{*}(f_{2})$ from $\mathcal{O}(Y_{1})$ and
$\mathcal{O}(Y_{2})$, respectively, to $\mathcal{O}(Y_{1,m}\cap Y_{2,n})$ by
the restriction maps. Substituting the first relation into the second
relation, we get $f^{*}(f_{1})=q_{2}q_{1}f^{*}(f_{1})$. Using Lemma 3.3, we
cancel $f^{*}(f_{1})$ to see that $q_{1}$ is indeed the inverse of $q_{2}$ in
$\mathcal{O}(Y_{1,m}\cap Y_{2,n})$.

Compose $g_{1,m}^{*}:A_{N}\to\mathcal{O}(Y_{1,m})$ with the restriction map
$\mathcal{O}(Y_{1,m})\to\mathcal{O}(Y_{1,m}\cap Y_{2,n})$. Extend this
composition to a map $A_{N}\langle
p^{N}Q_{1}^{\prime}\rangle\to\mathcal{O}(Y_{1,m}\cap Y_{2,n})$ by sending
$p^{N}Q_{1}^{\prime}$ to $p^{N}q_{2}$. This is possible because
$|p^{N}q_{2}|\leq 1$ since the intersection of two affinoid subdomains of an
affinoid space is again an affinoid space and because under the homomorphism
$\mathcal{O}(Y_{2,n})\to\mathcal{O}(Y_{1,m}\cap Y_{2,n})$ of affinoid
algebras, the image of an element has norm at most the norm of the element
itself. Using the fact that $q_{2}$ is the inverse of $q_{1}$ in
$\mathcal{O}(Y_{1,m}\cap Y_{2,n})$, we may further extend the above map to
$A_{N}^{\prime}$ to obtain the following commutative diagram

${A_{N}=\mathcal{O}(U_{r})\langle
p^{N}Q_{1}\rangle/(f_{2}-Q_{1}f_{1})}$${\mathcal{O}(Y_{1,m})}$${A_{N}^{\prime}=\mathcal{O}(U_{r})\langle
p^{N}Q_{1},p^{N}Q_{1}^{\prime}\rangle/(f_{2}-Q_{1}f_{1},1-Q_{1}Q_{1}^{\prime})}$${\mathcal{O}(Y_{1,m}\cap
Y_{2,n}).}$$\scriptstyle{g_{1,m}^{*}}$

This shows that the affinoid algebra map $A_{N}\to\mathcal{O}(Y_{1,m}\cap
Y_{2,n})$ factors through the map $A_{N}\to A_{N}^{\prime}$. Reversing the
arrows, we see that the restriction of $g_{1,m}$ to $Y_{1,m}\cap Y_{2,n}$
factors through the inclusion $\text{Sp }A_{N}^{\prime}\to\text{Sp }A_{N}$. In
other words, $g_{1,m}$ maps $Y_{1,m}\cap Y_{2,n}$ into $\text{Sp
}A_{N}^{\prime}$. Similarly, we can prove that $g_{2,n}$ maps $Y_{1,m}\cap
Y_{2,n}$ into $\text{Sp }B_{N}^{\prime}$.

Now we can prove that $g_{1,m}$ and $g_{2,n}$ agree on $Y_{1,m}\cap Y_{2,n}$.
Indeed, this statement is equivalent to the commutativity of the following
diagram

${A_{N}^{\prime}}$${\mathcal{O}(Y_{1,m}\cap
Y_{2,n}),}$${B_{N}^{\prime}}$$\scriptstyle{g_{1,m}^{*}}$$\scriptstyle{\phi_{N}^{*}}$$\scriptstyle{g_{2,n}^{*}}$

where the vertical map is the glueing map. This diagram commutes because under
the top two maps $Q_{2}\mapsto Q_{1}^{\prime}\mapsto q_{2}$ and
$Q_{2}^{\prime}\mapsto Q_{1}\mapsto q_{1}$, and these are exactly the images
under the lower map.

By [BGR84, Proposition 9.3.3/1], there exists a map of rigid analytic spaces
$g:Y\to\widetilde{U}_{r}$ such that the following diagram commutes

${\widetilde{U}_{r}}$${Y}$${U_{r}.}$$\scriptstyle{\pi}$$\scriptstyle{f}$$\scriptstyle{g}$

To prove that $\pi:\widetilde{U}_{r}\to U_{r}$ is the blow-up of $U_{r}$ at
the ideal $m=(f_{1},f_{2})$, it remains to check that $g$ is unique making the
above diagram commute. Suppose $g^{\prime}:Y\to\widetilde{U}_{r}$ is another
candidate. We prove that $g=g^{\prime}$. We need:

######  Lemma 3.4.

For $i=1,2$, we have $g^{\prime-1}(\widetilde{V}_{i})=Y_{i}$.

###### Proof.

Without loss of generality, assume that $i=1$.

  * •

We prove that $g^{\prime-1}(\widetilde{V}_{1})\subseteq Y_{1}$. Recall that
$\widetilde{V}_{1}=\cup_{i\geq 0}\>\text{Sp }A_{i}$. Therefore it is enough to
prove that $Y_{1,i}:=g^{\prime-1}(\text{Sp }A_{i})\subseteq Y_{1}$ for each
$i\geq 0$. (The notation $Y_{1,i}$ here and $Y_{2,i}$ below are local to this
proof.) Fix $i\geq 0$. Consider the following commutative diagram

${\text{Sp }A_{i}}$${Y_{1,i}=g^{\prime-1}(\text{Sp
}A_{i})}$${U_{r}.}$$\scriptstyle{\pi}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f}$

We know that $mA_{i}=f_{1}A_{i}$. Extending this ideal from $A_{i}$ to
$\mathcal{O}(Y_{1,i})$ using $g^{\prime*}$, we see that
$m\mathcal{O}(Y_{1,i})=g^{\prime*}(f_{1})\mathcal{O}(Y_{1,i})=f^{*}(f_{1})\mathcal{O}(Y_{1,i})$.
Let $y\in Y_{1,i}$. Noting that $Y_{1,i}$ is an admissible open subset of $Y$,
we have $m\mathcal{O}_{Y,y}=f^{*}(f_{1})\mathcal{O}_{Y,y}$. By Lemma 3.2, we
see that $y\in Y_{1}$.

  * •

We prove that $Y_{1}\subseteq g^{\prime-1}(\widetilde{V}_{1})$. This is
equivalent to proving
$g^{\prime-1}(\widetilde{U}_{r}\setminus\widetilde{V}_{1})\subseteq Y\setminus
Y_{1}$. Recall that $\widetilde{U}_{r}$ is covered by $\widetilde{V}_{1}$ and
$\widetilde{V}_{2}$. Since $\widetilde{V}_{2}=\cup_{i\geq 0}\>\text{Sp
}B_{i}$, we have

$\widetilde{U}_{r}\setminus\widetilde{V}_{1}=\cup_{i\geq 0}\>(\text{Sp
}B_{i}\setminus\widetilde{V}_{1}).$

Thus, it is enough to prove that $Y_{2,i}:=g^{\prime-1}(\text{Sp
}B_{i}\setminus\widetilde{V}_{1})\subseteq Y\setminus Y_{1}$ for each $i\geq
0$. Fix $i\geq 0$. We prove that if $y\in Y_{2,i}$, then $y\not\in Y_{1}$. We
do so by proving that the ideal $m\mathcal{O}_{Y,y}$ of $\mathcal{O}_{Y,y}$ is
not generated by $f^{*}(f_{1})$ (cf. Lemma 3.2). So, let $y\in Y_{2,i}$ map to
$\text{Sp }B_{i}\setminus\widetilde{V}_{1}$ under $g^{\prime}$. Assume towards
a contradiction that $y\in Y_{1}$. By Lemma 3.2, we see that
$m\mathcal{O}_{Y,y}$ is generated by $f^{*}(f_{1})$. By the analog of the
first bullet point, $Y_{2,i}\subseteq Y_{2}$, so we also have
$m\mathcal{O}_{Y,y}=f^{*}(f_{2})\mathcal{O}_{Y,y}$. Therefore there exists a
unit $u\in\mathcal{O}_{Y,y}$ such that $f^{*}(f_{1})=uf^{*}(f_{2})$. Consider
the following commutative diagram

${\text{Sp }B_{i}}$${Y_{2,i}=g^{\prime-1}(\text{Sp
}B_{i}\setminus\widetilde{V}_{1})}$${U_{r}.}$$\scriptstyle{\pi}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f}$

Note that $Y_{2,i}$ is an admissible open subset of $Y$. We obtain the
relation $f^{*}(f_{1})=g^{\prime*}(Q_{2})f^{*}(f_{2})$ in $\mathcal{O}_{Y,y}$
by pushing the relation $f_{1}=Q_{2}f_{2}$ from $B_{i}$ to
$\mathcal{O}(Y_{2,i})$ using $g^{\prime*}$ and then from
$\mathcal{O}(Y_{2,i})$ to $\mathcal{O}_{Y,y}$. Since $f^{*}(f_{2})$ is not a
zero divisor in $\mathcal{O}_{Y,y}$, we see that $g^{\prime*}(Q_{2})=u$, which
is a unit of $\mathcal{O}_{Y,y}$. We prove that this is a contradiction. We
know that $\text{Sp }B_{i}\,\cap\,\widetilde{V}_{1}$ is the set of maximal
ideals $\mathfrak{m}$ of $B_{i}$ satisfying the extra condition
$|Q_{2}\mod{\mathfrak{m}}|>0$. Therefore $\text{Sp
}B_{i}\setminus\widetilde{V}_{1}$ is the set of maximal ideals $\mathfrak{m}$
of $B_{i}$ satisfying $|Q_{2}\mod{\mathfrak{m}}|=0$. Since
$g^{\prime}(y)\in\text{Sp }B_{i}\setminus\widetilde{V}_{1}$, we see that
$Q_{2}$ vanishes at $g^{\prime}(y)$. In other words, if we denote the maximal
ideal of $\mathcal{O}_{\widetilde{U}_{r},\>g^{\prime}(y)}$ by
$\mathfrak{m}_{g^{\prime}(y)}$, then we have $Q_{2}\>\text{mod
}\mathfrak{m}_{g^{\prime}(y)}=0$. If $\mathfrak{n}_{y}$ is the maximal ideal
of $\mathcal{O}_{Y,y}$, then using the following commutative diagram

${\mathcal{O}_{\widetilde{U}_{r},\>g^{\prime}(y)}}$${\mathcal{O}_{Y,y}}$${\mathcal{O}_{\widetilde{U}_{r},\>g^{\prime}(y)}/\mathfrak{m}_{g^{\prime}(y)}}$${\mathcal{O}_{Y,y}/\mathfrak{n}_{y},}$$\scriptstyle{g^{\prime*}}$

we get $g^{\prime*}(Q_{2})\text{ mod }\mathfrak{n}_{y}=0$. This means that
$g^{\prime*}(Q_{2})$ is contained in the maximal ideal of $\mathcal{O}_{Y,y}$.
In other words, $g^{\prime*}(Q_{2})$ is not a unit. This is a contradiction.
Therefore $y\not\in Y_{1}$. ∎

To show that $g=g^{\prime}$, we prove that their restrictions to $Y_{1,m}$ and
$Y_{2,n}$ are equal, for all $m\in I$ and $n\in J$. We prove that $g$ and
$g^{\prime}$ agree on $Y_{1,m}$. Using the definition of $g$, we see that the
restriction of $g$ to $Y_{1,m}$ factors as $Y_{1,m}\to\text{Sp
}A_{\underline{m}}\to\widetilde{U}_{r}$. Since $g^{\prime}$ maps the affinoid
space $Y_{1,m}$ to $\widetilde{U}_{r}$ and $\\{\text{Sp }A_{i},\text{Sp
}B_{j}\\}_{i,j\geq 0}$ is an admissible cover of $\widetilde{U}_{r}$, we
deduce that $g^{\prime}$ maps $Y_{1,m}$ into the union of finitely many
$\text{Sp }A_{i}$ and $\text{Sp }B_{j}$. By Lemma 3.4, we see that
$g^{\prime}:Y_{1,m}\to\widetilde{U}_{r}$ factors as $Y_{1,m}\to\text{Sp
}A_{i}\to\widetilde{U}_{r}$ for some $i\gg 0$. We may assume that
$i\geq\underline{m}$. To check that $g$ and $g^{\prime}$ agree on $Y_{1,m}$,
it is therefore enough to check that the following diagram commutes

${\text{Sp }A_{i}}$${Y_{1,m}}$${\text{Sp
}A_{\underline{m}}.}$$\scriptstyle{g^{\prime}}$$\scriptstyle{g}$

In other words, we have to check that the following diagram commutes

${A_{i}}$${\mathcal{O}(Y_{1,m})}$${A_{\underline{m}}.}$$\scriptstyle{g^{\prime*}}$$\scriptstyle{g^{*}}$

Since both $g^{*}$ and $g^{\prime*}$ are equal to $f^{*}$ on
$\mathcal{O}(U_{r})$, we only need to check that
$g^{*}(Q_{1})=g^{\prime*}(Q_{1})$. Pushing the relation $f_{2}=Q_{1}f_{1}$
from $A_{i}$ to $\mathcal{O}(Y_{1,m})$ under $g^{\prime*}$, we get
$f^{*}(f_{2})=g^{\prime*}(Q_{1})f^{*}(f_{1})$. Similarly, pushing the same
relation from $A_{\underline{m}}$ to $\mathcal{O}(Y_{1,m})$ under $g^{*}$, we
get $f^{*}(f_{2})=g^{*}(Q_{1})f^{*}(f_{1})$. Using Lemma 3.3, we get
$g^{*}(Q_{1})=g^{\prime*}(Q_{1})$. This proves the commutativity of the
diagram above. In other words, $g$ and $g^{\prime}$ agree on $Y_{1,m}$. A
similar argument shows that $g$ and $g^{\prime}$ agree on $Y_{2,n}$. Therefore
$g=g^{\prime}$ on $Y$. This finally proves that $\pi:\widetilde{U}_{r}\to
U_{r}$ is the blow-up of $U_{r}$ at the maximal ideal $(f_{1},f_{2})$.

###  3.4 Points in the exceptional fiber as tangent directions

The following standard fact allows us to realize $E$-valued points of the
fiber over the exceptional point $(p^{r},(1+p)^{r+1}-1)$ as tangent directions
in $U_{r}$ at this point. Recall that the maximal ideal $m$ of
$\mathcal{O}(U_{r})$ corresponding to this exceptional point is the ideal
generated by $f_{1}=S_{1}-p^{r}$ and $f_{2}=S_{2}-((1+p)^{k-1}-1)$.

######  Proposition 3.5.

There is a a bijection

$\pi^{-1}(p^{r},(1+p)^{r+1}-1)(E)\rightarrow\mathbb{P}(\mathrm{Hom}(m/m^{2}\otimes_{\mathbb{Q}_{p}}E,E))$

between the $E$-valued points of the fiber over the point
$(p^{r},(1+p)^{k-1}-1)$ and the elements of the projectivization of the
tangent space $\mathrm{Hom}(m/m^{2}\otimes_{\mathbb{Q}_{p}}E,E)$ over $E$,
which sends a point $(p^{r},(1+p)^{k-1}-1,a:b)\in\widetilde{U}_{r}(E)$ to the
class in $\mathbb{P}(\mathrm{Hom}(m/m^{2}\otimes_{\mathbb{Q}_{p}}E,E))$
represented by $v$, where

$v(\overline{f_{1}}\otimes 1)=a\text{ and }v(\overline{f_{2}}\otimes 1)=b,$

with bar denoting image modulo $m^{2}$.

###### Proof.

Let $P=(p^{r},(1+p)^{k-1}-1,a:b)$ be an $E$-valued point in the fiber above
the exceptional point. Assume $b\not=0$. Write
$P=(p^{r},(1+p)^{k-1}-1,a/b:1)$. From the construction of $\widetilde{U}_{r}$,
we see that $P\in\widetilde{V}_{2}(E)$. Therefore $P$ is an $E$-valued point
of $\text{Sp }B_{i}$, for some $i\geq 0$. Let $g_{P}$ be the homomorphism
$B_{i}\to E$ associated with $P$. The discussion at the end of Section 3.1
implies that $g_{P}$ is defined by $g_{P}(S_{1})=p^{r}$,
$g_{P}(S_{2})=(1+p)^{k-1}-1$ and $g_{P}(Q_{2})=a/b$. Let $m_{P}$ be the kernel
of $g_{P}$. The map $\mathcal{O}(U_{r})\to B_{i}$ induces an $E$-linear
surjection

$(m/m^{2})\otimes_{\mathbb{Q}_{p}}E\to
f_{2}B_{i}/f_{2}B_{i}m_{P}\otimes_{B_{i},g_{P}}E,$

given by

$\displaystyle\overline{f_{1}}\otimes 1$ $\displaystyle\mapsto$
$\displaystyle\overline{f_{2}}\otimes(a/b)$
$\displaystyle\overline{f_{2}}\otimes 1$ $\displaystyle\mapsto$
$\displaystyle\overline{f_{2}}\otimes 1,$

where the bars over the elements on the left denote their images modulo
$m^{2}$ and the bars over the elements on the right denote their images modulo
$f_{2}B_{i}m_{P}$, and where we have used $\overline{f_{1}}\otimes
1=\overline{f_{2}}\overline{Q_{2}}\otimes 1=\overline{f_{2}}\otimes(a/b)$ in
the codomain. The codomain of this map can be identified with $E$ up to
multiplication by a non-zero scalar. The class of the above map in
$\mathbb{P}(\mathrm{Hom}(m/m^{2}\otimes_{\mathbb{Q}_{p}}E,E))$ is the same as
that of the map $v$, where $v(\overline{f_{1}}\otimes 1)=a$ and
$v(\overline{f_{2}}\otimes 1)=b$. A similar construction works if $a\neq 0$.
One checks immediately that the resulting assignment $P\mapsto v$ is well-
defined and a bijection. ∎

##  4 Explicit bases of $H^{1}(\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi))$ and
$\mathcal{L}$-invariants

Let $\Gamma=\mathrm{Gal}(\mathbb{Q}_{p}(\mu_{p^{\infty}})/\mathbb{Q}_{p})$ and
think of the cyclotomic character $\chi$ as a character
$\chi:\Gamma\to\mathbb{Z}_{p}^{*}$. In this section, we study two explicit
bases of the first Fontaine-Herr cohomology group
$H^{1}(\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi))$ and the corresponding
formulas for the $\mathcal{L}$-invariant. Here
$\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi)$ is the Robba ring over
$\mathbb{Q}_{p}$ in the variable $T$, with the standard actions of $\varphi$
and $\Gamma$ twisted by $x^{r}\chi$.

Colmez has constructed two power series $G(|x|,r+1)$ and $G^{\prime}(|x|,r+1)$
in $\mathcal{R}_{\mathbb{Q}_{p}}$ which he uses to define a basis of
$H^{1}(\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi))$ [Col08, Proposition 2.19]. In
the first subsection, we explicitly compute $G(|x|,r+1)$. In the second
subsection, we give a partial description of $G^{\prime}(|x|,r+1)$. In the
third subsection, we state Colmez’s formula for the $\mathcal{L}$-invariant of
a non-zero element of $H^{1}(\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi))$
expressed as a linear combination in this basis. In the same subsection, we
describe another basis of $H^{1}(\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi))$
studied by Benois [Ben11, Proposition 1.5.4]. Finally, we find the change of
basis matrix between these two bases and restate the formula for the
$\mathcal{L}$-invariant in terms of Benois’s basis (Definition 4.6).

We recall some standard facts about Robba rings. For $M>0$, let
$\mathcal{E}_{E}^{]0,M]}$ be the set of bidirectional power series in the
variable $T$ with coefficients in $E$ that converge on the elements of
$\overline{\mathbb{Q}}_{p}$ with valuation belonging to $(0,M]$. More
precisely, it is the set of the series $\sum_{n=-\infty}^{\infty}a_{n}T^{n}$
satisfying

$\lim_{n\to\pm\infty}v_{p}(a_{n})+ns\to\infty,\text{ for all }0<s\leq M.$

For each $0<s\leq M$, there is a valuation $v_{p}(\>\>,s)$ on
$\mathcal{E}_{E}^{]0,M]}$ defined by
$v_{p}(\sum_{n=-\infty}^{\infty}a_{n}T^{n},s)=\inf\\{v_{p}(a_{n})+ns\text{
}|\text{ }n\in\mathbb{Z}\\}$. Any sequence in $\mathcal{E}_{E}^{]0,M]}$ that
is Cauchy with respect to $v_{p}(\>\>,s)$, for each $0<s\leq M$ is convergent
in $\mathcal{E}_{E}^{]0,M]}$ (see [Ked06, Definition 2.5.1]). Moreover,
choosing a sequence $0<r_{l}\leq M$ converging to $0$ with $r_{1}=M$, we get a
countable family of valuations $v_{p}(\>\>,r_{l})$ on
$\mathcal{E}_{E}^{]0,M]}$. Note that if $r_{l}\leq s\leq M$, for some $l\geq
1$, then $v_{p}(f(T),s)\geq\inf\\{v_{p}(f(T),r_{l}),v_{p}(f(T),M)\\}$.
Therefore to check if a sequence in $\mathcal{E}_{E}^{]0,M]}$ is convergent,
we only need to check that it converges to a common limit with respect to the
valuations $v_{p}(\>\>,r_{l})$, for $l\geq 1$. In particular,
$\mathcal{E}_{E}^{]0,M]}$ is a Fréchet space with respect to the valuations
$v_{p}(\>\>,r_{l})$. For the space $\mathcal{E}_{E}^{]0,\frac{1}{p-1}]}$, we
set $r_{l}=\dfrac{1}{\phi(p^{l})}$, where $\phi$ is Euler’s totient function.

The Robba ring over $E$ is defined to be
$\mathcal{R}_{E}=\cup_{M>0}\mathcal{E}_{E}^{]0,M]}$. It is endowed with
commuting actions of an operator $\varphi$ and the group $\Gamma$ via $\varphi
T=(1+T)^{p}-1$ and $\gamma T=(1+T)^{\chi(\gamma)}-1$, for $\gamma\in\Gamma$,
respectively.

We recall some facts from [Laz62]. Let $\phi_{n}(T)$ for $n\geq 1$ be the
$p^{n}$-th cyclotomic polynomial defined by

$\phi_{n}(T)=\frac{(1+T)^{p^{n}}-1}{(1+T)^{p^{n-1}}-1}=1+(1+T)^{p^{n-1}}+\cdots+(1+T)^{(p-1)p^{n-1}}.$
(3)

The polynomials $\phi_{n}(T)$ are $\frac{1}{\phi(p^{n})}$-extremal in the
sense of [Laz62, Definition 2.7]. This means that
$v_{p}(\phi_{n}(T),1/\phi(p^{n}))$ is attained at the constant and leading
terms:

$v_{p}(a_{0})=v_{p}\left(\phi_{n}(T),\frac{1}{\phi(p^{n})}\right)=v_{p}(a_{p^{n-1}(p-1)})+\frac{p^{n-1}(p-1)}{\phi(p^{n})}=1,$
(4)

where $\phi_{n}(T)=a_{0}+a_{1}T+\cdots+a_{p^{n-1}(p-1)}T^{p^{n-1}(p-1)}$, with
$a_{0}=p$ and $a_{p^{n-1}(p-1)}=1$. For $n\geq 1$, we also have

$\varphi(\phi_{n}(T))=\phi_{n+1}(T).$ (5)

Let $t=\log{(1+T)}\in\mathcal{E}^{]0,\frac{1}{p-1}]}$. The following formula
relates $t$ and the cyclotomic polynomials:

$t=T\prod_{n\geq 1}\frac{\phi_{n}(T)}{p}.$ (6)

We also have

$\varphi(t)=pt,$ (7)

and

$\gamma(t)=\chi(\gamma)t,$ (8)

for all $\gamma\in\Gamma$.

Let $r\geq 0$. By [Col08, Proposition 2.16 (i)], there is an isomorphism

$\mathcal{E}^{]0,\frac{1}{p-1}]}/t^{r+1}\to\prod_{n\geq
1}\mathcal{E}^{]0,\frac{1}{p-1}]}/\phi_{n}^{r+1}.$ (9)

By [Col08, Proposition 2.16 (ii)], for $n\geq 1$, there is a
$\Gamma$-equivariant isomorphism

$\iota_{n}:\mathcal{E}^{]0,\frac{1}{p-1}]}/\phi_{n}^{r+1}\to\mathbb{Q}_{p}(\zeta_{p^{n}})[t]/t^{r+1}$

obtained by sending $T$ to $\zeta_{p^{n}}e^{t/p^{n}}-1$. Thus the
$\Gamma$-equivariant homomorphism

$\mathcal{E}^{]0,\frac{1}{p-1}]}/t^{r+1}\to\prod_{n\geq
1}\mathbb{Q}_{p}(\zeta_{p^{n}})[t]/t^{r+1}$ (10)

induced by the maps $\iota_{n}$ for $n\geq 1$ is an isomorphism.

For the rest of the paper, we fix a topological generator $\gamma$ of $\Gamma$
such that $\chi(\gamma)=\zeta_{p-1}^{a}(1+p)$ for some fixed integer $a$.

###  4.1 An explicit description of $G(|x|,r+1)$

Let $r\geq 1$. The first basis vector of
$H^{1}(\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi))$ constructed by Colmez is
represented by the element

$c_{1}=(t^{-(r+1)}(p^{-1}\varphi-1)G(|x|,r+1),t^{-(r+1)}(\gamma-1)G(|x|,r+1))$

in
$\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi)\oplus\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi)$
(for the untwisted action of $\varphi$ and $\gamma$), see [Col08, Proposition
2.19]. Here $G(|x|,r+1)$ is any power series $f(T)$ in
$\mathcal{E}^{]0,\frac{1}{p-1}]}$ mapping to $\prod_{n\geq 1}\frac{1}{p^{n}}$
under the map (9) (cf. [Col08, Section 2.6]). In other words, $G(|x|,r+1)$
satisfies the system of congruences

$\begin{array}[]{rcl}f(T)&\equiv&\dfrac{1}{p}\text{
}\mod{\phi_{1}(T)^{r+1}}\\\\[8.0pt]
f(T)&\equiv&\dfrac{1}{p^{2}}\mod{\phi_{2}(T)^{r+1}}\\\ &\vdots&\\\
f(T)&\equiv&\dfrac{1}{p^{n}}\mod{\phi_{n}(T)^{r+1}}\\\ &\vdots&\\\
\end{array}$

The goal of this section is to write down $G(|x|,r+1)$ explicitly.

######  Definition 4.1.

For each $n\geq 1$ and $r\geq 1$, define

$G_{n,\,r+1}(T):=\left[1-\left(\frac{\phi_{n}(T)}{p}\right)^{r+1}\right]^{r+1}\frac{1}{p^{n}}\prod_{i>n}\left[1-\left(1-\frac{\phi_{i}(T)}{p}\right)^{r+1}\right]^{r+1}.$

To check $G_{n,\,r+1}(T)$ is well defined, we need to show that the infinite
product converges. We prove that for any $n,r\geq 1$, the product
$\prod_{i>n}(1-(1-(\phi_{i}(T)/p))^{r+1})$ converges.

######  Lemma 4.2.

For any $c=1,2,\cdots,p-1$ and any $j=1,2,\cdots,cp^{i-1}$, we have

$v_{p}\left({cp^{i-1}\choose j}\right)=i-1-v_{p}(j).$

###### Proof.

Exercise. ∎

######  Lemma 4.3.

For any $i\geq 1$, we have
$v_{p}\left(1-\phi_{i}(T)/p,\frac{1}{\phi(p^{i})}\right)=0$ and for $l\geq 1$,
we have $v_{p}\left(1-\phi_{i}(T)/p,\frac{1}{\phi(p^{l})}\right)>i-l-1$.

###### Proof.

Since $\phi_{i}(T)$ is $\frac{1}{\phi(p^{i})}$-extremal, we see that by (4),
$v_{p}\left(\phi_{i}(T),\frac{1}{\phi(p^{i})}\right)=1$. Since all the terms
except the constant terms of the polynomials $\phi_{i}(T)$ and $\phi_{i}(T)-p$
are the same, we get
$v_{p}\left(\phi_{i}(T)-p,\frac{1}{\phi(p^{i})}\right)=1$. Therefore
$v_{p}\left(1-\phi_{i}(T)/p,\frac{1}{\phi(p^{i})}\right)=0$.

Write $\phi_{i}(T)-p=a_{1}T+a_{2}T^{2}+\cdots+a_{\phi(p^{i})}T^{\phi(p^{i})}$.
By expanding the powers of $(1+T)$ in $\phi_{i}(T)$, we see that

$a_{j}={p^{i-1}\choose j}+{2p^{i-1}\choose j}+\cdots+{(p-1)p^{i-1}\choose j}.$

By Lemma 4.2,

$v_{p}(a_{j})+\frac{j}{\phi(p^{l})}\geq
i-1-v_{p}(j)+\frac{p^{v_{p}(j)}}{\phi(p^{l})}=i-1-v_{p}(j)+\frac{p^{v_{p}(j)-(l-1)}}{p-1}>i-1-v_{p}(j)+v_{p}(j)-(l-1)=i-l.$

Therefore
$v_{p}\left(1-\phi_{i}(T)/p,\frac{1}{\phi(p^{l})}\right)=v_{p}\left(\phi_{i}(T)-p,\frac{1}{\phi(p^{l})}\right)-1>i-l-1$.
∎

We now prove that the product $\prod_{i>n}(1-(1-(\phi_{i}(T)/p))^{r+1})$
converges in $\mathcal{E}^{]0,\frac{1}{p-1}]}$ for $n,r\geq 1$. Consider the
polynomials $1-(1-(\phi_{i}(T)/p))^{r+1}$ as elements of
$\mathcal{E}^{]0,\infty]}$. By [Laz62, Proposition 4.11], the product
$\prod_{i>n}(1-(1-\phi_{i}(T)/p)^{r+1})$ converges in
$\mathcal{E}^{]0,\infty]}$ if and only if

$v_{p}((1-\phi_{i}(T)/p)^{r+1},s)\to\infty\>\text{ as }\>i\to\infty,$ (11)

for each $s>0$. Fix $s>0$ and pick $l_{s}\geq 1$ such that
$\frac{1}{\phi(p^{l_{s}})}<s$. As $(1-\phi_{i}(T)/p)^{r+1}$ contains no
negative power of $T$, we have

$v_{p}((1-\phi_{i}(T)/p)^{r+1},s)\geq(r+1)v_{p}\left(1-\phi_{i}(T)/p,\frac{1}{\phi(p^{l_{s}})}\right)>(r+1)(i-l_{s}-1),$

by Lemma 4.3, whence (11) holds. Therefore the product
$\prod_{i>n}(1-(1-(\phi_{i}(T)/p))^{r+1})$ converges in
$\mathcal{E}^{]0,\infty]}\subseteq\mathcal{E}^{]0,\frac{1}{p-1}]}$.

Our candidate for $G(|x|,r+1)$ is $\sum_{n=1}^{\infty}G_{n,\,r+1}(T)$. To see
that the infinite sum is well defined, we prove the following theorem.

######  Theorem 4.4.

For any positive integer $l$, the sequence $G_{n,\,r+1}(T)$ converges to $0$
with respect to the valuation $v_{p}\left(\quad,\frac{1}{\phi(p^{l})}\right)$.

###### Proof.

Fix $l\geq 1$. We need to show
$v_{p}\left(G_{n,\,r+1}(T),\frac{1}{\phi(p^{l})}\right)\to\infty$ as
$n\to\infty$. Consider
$v_{p}\left(G_{n,\,r+1}(T),\frac{1}{\phi(p^{l})}\right)=$

$\begin{array}[]{rcl}(r+1)v_{p}\left(1-\left(\frac{\phi_{n}(T)}{p}\right)^{r+1},\frac{1}{\phi(p^{l})}\right)-n+(r+1)\sum_{i>n}v_{p}\left(1-\left(1-\frac{\phi_{i}(T)}{p}\right)^{r+1},\frac{1}{\phi(p^{l})}\right).\end{array}$

Note that
$1-\left(\frac{\phi_{n}(T)}{p}\right)^{r+1}=\left(1-\frac{\phi_{n}(T)}{p}\right)\left(1+\frac{\phi_{n}(T)}{p}+\cdots+\left(\frac{\phi_{n}(T)}{p}\right)^{r}\right)$.
Now if $n\geq l$, then
$v_{p}\left(\phi_{n}(T),\frac{1}{\phi(p^{l})}\right)=1$. Indeed, since
$\phi_{n}(T)=a_{0}+a_{1}T+\cdots+a_{\phi(p^{n})}T^{\phi(p^{n})}$ is
$\frac{1}{\phi(p^{n})}$-extremal, we have
$v_{p}(a_{i})+\frac{i}{\phi(p^{l})}\geq v_{p}(a_{i})+\frac{i}{\phi(p^{n})}\geq
v_{p}(a_{0})=1$, for any $i$. Hence
$v_{p}\left(\phi_{n}(T),\frac{1}{\phi(p^{l})}\right)=v_{p}(a_{0})=1$. Thus
$v_{p}\left((\phi_{n}(T)/p)^{j},\frac{1}{\phi(p^{l})}\right)=0$, for any
$j\geq 0$. Therefore

$v_{p}\left(1-\left(\frac{\phi_{n}(T)}{p}\right)^{r+1},\frac{1}{\phi(p^{l})}\right)\geq
v_{p}\left(1-\frac{\phi_{n}(T)}{p},\frac{1}{\phi(p^{l})}\right)>n-l-1,$

where the last inequality is a consequence of Lemma 4.3. On the other hand,
writing
$1-(1-(\phi_{i}(T)/p))^{r+1}=a^{\prime}_{0}+a^{\prime}_{1}T+\cdots+a^{\prime}_{(r+1)\phi(p^{i})}T^{(r+1)\phi(p^{i})}$,
we observe that $a^{\prime}_{0}=1$ and
$a^{\prime}_{1}T+\cdots+a^{\prime}_{(r+1)\phi(p^{i})}T^{(r+1)\phi(p^{i})}=-(1-(\phi_{i}(T)/p))^{r+1}$.
By Lemma 4.3,
$v_{p}\left(-(1-\phi_{i}(T)/p)^{r+1},\frac{1}{\phi(p^{l})}\right)>0$ if $i>l$.
We also know that $v_{p}(a^{\prime}_{0})=0$. Therefore
$v_{p}\left(1-(1-\phi_{i}(T)/p)^{r+1},\frac{1}{\phi(p^{l})}\right)=0$.

Using these estimates together, we see that for $n\geq l$,

$v_{p}\left(G_{n,\,r+1}(T),\frac{1}{\phi(p^{l})}\right)>(r+1)(n-l-1)-n=nr-(r+1)l-(r+1).$

Letting $n\to\infty$, we see that
$v_{p}\left(G_{n,\,r+1}(T),\frac{1}{\phi(p^{l})}\right)\to\infty$, as desired.
Therefore the sequence $G_{n,\,r+1}(T)$ converges to $0$ with respect to the
valuation $v_{p}\left(\quad,\frac{1}{\phi(p^{l})}\right)$ for each $l>0$. ∎

Using the theorem above, we see that the series
$G(|x|,r+1)=\sum_{n=1}^{\infty}G_{n,\,r+1}(T)$ converges. It is easily
checked, using $\phi_{i}(T)\equiv p\mod\phi_{n}(T)$ for $i>n$, that it is a
solution to the system of congruences that we started with:

$G(|x|,r+1)\equiv\frac{1}{p^{n}}\mod{\phi_{n}(T)^{r+1}},\mathrm{\text{ for all
}}n\in\mathbb{N}.$

Therefore, we have written down $G(|x|,r+1)$ explicitly.

###  4.2 A partial description of $G^{\prime}(|x|,r+1)$

Let $r\geq 1$. The second basis vector of
$H^{1}(\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi))$ constructed by Colmez is
represented by the element

$c_{2}=(t^{-(r+1)}(p^{-1}\varphi-1)(\log
T-G^{\prime}(|x|,r+1)),t^{-(r+1)}(\gamma-1)(\log T-G^{\prime}(|x|,r+1)))$

in
$\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi)\oplus\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi)$
(for the untwisted action of $\varphi$ and $\gamma$), see [Col08, Proposition
2.19]. Here $G^{\prime}(|x|,r+1)$ is an element of
$\mathcal{E}^{]0,\frac{1}{p-1}]}$ mapping to $\prod_{n\geq
1}\log(\zeta_{p^{n}}e^{t/p^{n}}-1)$ under the map (10) (cf. [Col08, Section
2.7]).

We wish to compute $G^{\prime}(|x|,r+1)$ explicitly, just as we did for
$G(|x|,r+1)$. We are only able to determine $G^{\prime}(|x|,r+1)$ modulo $t$
but this is sufficient for our purposes.

Consider the following element

$g^{\prime}(T)=\frac{\log\gamma}{\gamma-1}\log T-\log
T-\frac{\log\chi(\gamma)t}{\chi(\gamma)-1}\in\mathcal{E}^{]0,\frac{1}{p-1}]}.$
(12)

In the proof of [Ben11, Theorem 1.5.7], Benois relates $g^{\prime}(T)$ (which
he calls $y$) to the element
$d_{1}=-t^{-1}\nabla_{0}(\log{T})+(\chi(\gamma)-1)^{-1}$ of
$\mathcal{R}_{\mathbb{Q}_{p}}[\log T,1/t]$ using the following equation

$d_{1}=-\log(\chi(\gamma))^{-1}t^{-1}(\log T+g^{\prime}(T)),$ (13)

where $\nabla_{0}=\frac{1}{\log{\chi(\gamma)}}\frac{\log{\gamma}}{\gamma-1}$.

We claim that
$\iota_{n}(-g^{\prime}(T))\equiv\log(\zeta_{p^{n}}e^{t/p^{n}}-1)\mod{t}$.
Indeed,

$\begin{array}[]{ll}\iota_{n}(g^{\prime}(T))&=\iota_{n}\left(\frac{\log\gamma}{\gamma-1}\log
T-\log T-\frac{\log\chi(\gamma)t}{\chi(\gamma)-1}\right)\\\
&\equiv\frac{\log\gamma}{\gamma-1}\log{(\zeta_{p^{n}}-1)}-\log(\zeta_{p^{n}}-1)\mod{t}\\\
&\equiv\frac{1+\gamma+\cdots+\gamma^{p^{n-1}(p-1)-1}}{p^{n-1}(p-1)}\frac{\log\gamma^{p^{n-1}(p-1)}}{\gamma^{p^{n-1}(p-1)}-1}\log(\zeta_{p^{n}}-1)-\log(\zeta_{p^{n}}-1)\mod{t}.\end{array}$

Since $\gamma^{p^{n-1}(p-1)}\log(\zeta_{p^{n}}-1)=\log(\zeta_{p^{n}}-1)$, we
get

$\begin{array}[]{ll}\iota_{n}(g^{\prime}(T))&\equiv\frac{1+\gamma+\cdots+\gamma^{p^{n-1}(p-1)-1}}{p^{n-1}(p-1)}\log(\zeta_{p^{n}}-1)-\log(\zeta_{p^{n}}-1)\mod{t}\\\
&\equiv\frac{1}{p^{n-1}(p-1)}\log(N_{\mathbb{Q}_{p}(\zeta_{p^{n}})/\mathbb{Q}_{p}}(\zeta_{p^{n}}-1))-\log(\zeta_{p^{n}}-1)\mod{t}\\\
&\equiv-\log(\zeta_{p^{n}}-1)\mod{t}\\\
&\equiv-\log(\zeta_{p^{n}}e^{t/p^{n}}-1)\mod{t}.\end{array}$

Since the images of $-g^{\prime}(T)$ and $G^{\prime}(|x|,r+1)$ under the
isomorphism (10) are equal (with $r$ there equal to zero!), there exists
$g^{\prime\prime}(T)\in\mathcal{E}^{]0,\frac{1}{p-1}]}$ such that

$G^{\prime}(|x|,r+1)=-g^{\prime}(T)+tg^{\prime\prime}(T).$ (14)

This determines $G^{\prime}(|x|,r+1)$ explicitly modulo $t$.

###  4.3 Another basis of $H^{1}(\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi))$

Let $r\geq 1$. In this section, we consider a different basis
$\\{\overline{\alpha}_{r+1},\overline{\beta}_{r+1}\\}$ of
$H^{1}(\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi))$ described by Benois in
[Ben11, Proposition 1.5.4] which is more suitable for computations. Here bar
denotes class in $H^{1}(\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi))$. By [Ben11,
Corollary 1.5.5], for any class in
$H^{1}(\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi))$ represented by
$(a,b)\in\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi)\oplus\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi)$,
we have

$\overline{(a,b)}=\lambda^{\prime}\cdot\overline{\alpha}_{r+1}+\mu^{\prime}\cdot\overline{\beta}_{r+1},\text{
where }\lambda^{\prime}=\mathrm{res}(at^{r}dt)\text{ and
}\mu^{\prime}=\mathrm{res}(bt^{r}dt).$ (15)

We want to compute the change of basis relation between Colmez’s basis and
Benois’s basis. This allows us to state the formula for the
$\mathcal{L}$-invariant of a non-zero cohomology class in terms of certain
residues.

######  Proposition 4.5.

Let

$\begin{array}[]{rcl}c_{1}&=&(t^{-(r+1)}(p^{-1}\varphi-1)G(|x|,r+1),\>t^{-(r+1)}(\gamma-1)G(|x|,r+1)),\\\\[3.0pt]
c_{2}&=&(t^{-(r+1)}(p^{-1}\varphi-1)(\log
T-G^{\prime}(|x|,r+1)),\>t^{-(r+1)}(\gamma-1)(\log
T-G^{\prime}(|x|,r+1)))\end{array}$

be elements of
$\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi)\oplus\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi)$
representing Colmez’s basis of
$H^{1}(\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi))$. Then

$\overline{c}_{1}=-\frac{p-1}{p}\cdot\overline{\alpha}_{r+1},\qquad\overline{c}_{2}=\log(\chi(\gamma))\cdot\overline{\beta}_{r+1}.$

###### Proof.

We prove the proposition using $(\ref{eq:10})$. We therefore need to compute
the four residues

$\begin{array}[]{lll}\mathrm{res}(t^{-1}(p^{-1}\varphi-1)G(|x|,r+1)dt),&&\mathrm{res}(t^{-1}(\gamma-1)G(|x|,r+1)dt),\\\\[3.0pt]
\mathrm{res}(t^{-1}(p^{-1}\varphi-1)(\log
T-G^{\prime}(|x|,r+1))dt),&&\mathrm{res}(t^{-1}(\gamma-1)(\log
T-G^{\prime}(|x|,r+1))dt).\end{array}$

We shall use the following formulas: for any
$f(T)\in\mathcal{R}_{\mathbb{Q}_{p}}$, we have

$\displaystyle\ \mathrm{res}(\varphi(f(T))dt)$ $\displaystyle=$
$\displaystyle\mathrm{res}(f(T)dt),$ (16)
$\displaystyle\mathrm{res}(\gamma(f(T))dt)$ $\displaystyle=$
$\displaystyle\chi(\gamma)^{-1}\mathrm{res}(f(T)dt).$ (17)

  * •

Recall that $G(|x|,r+1)=\sum_{n=1}^{\infty}G_{n,\,r+1}(T)$, where

$\begin{array}[]{ll}&G_{n,\,r+1}(T)=\Big{[}1-\Big{(}\frac{\phi_{n}(T)}{p}\Big{)}^{r+1}\Big{]}^{r+1}\frac{1}{p^{n}}\prod_{i>n}\Big{[}1-\Big{(}1-\frac{\phi_{i}(T)}{p}\Big{)}^{r+1}\Big{]}^{r+1}.\end{array}$

Using equation $(\ref{eq:3})$, we see that
$(p^{-1}\varphi)G_{n,\,r+1}(T)=G_{n+1,\,r+1}(T)$. Therefore

$(p^{-1}\varphi-1)G(|x|,r+1)=-G_{1,\,r+1}(T).$

Hence
$\mathrm{res}\left(t^{-1}\left(p^{-1}\varphi-1\right)G(|x|,r+1)dt\right)=-\mathrm{res}(t^{-1}G_{1,\,r+1}(T)dt)$.
Now

$\begin{array}[]{ll}\mathrm{res}(t^{-1}G_{1,\,r+1}(T)dt)&=\mathrm{res}\Big{(}t^{-1}\Big{[}1-\Big{(}\frac{\phi_{1}(T)}{p}\Big{)}^{r+1}\Big{]}^{r+1}\frac{1}{p}{\prod_{i>1}}\Big{[}1-\Big{(}1-\frac{\phi_{i}(T)}{p}\Big{)}^{r+1}\Big{]}^{r+1}dt\Big{)}\\\
&\stackrel{{\scriptstyle(\ref{eq:4})}}{{=}}\mathrm{res}\Big{(}T^{-1}\Big{(}\frac{\phi_{1}(T)}{p}\Big{)}^{-1}\Big{[}1-\Big{(}\frac{\phi_{1}(T)}{p}\Big{)}^{r+1}\Big{]}^{r+1}\frac{1}{p}f(T)dt\Big{)},\end{array}$

where
$f(T)=\prod_{i>1}\Big{(}\frac{\phi_{i}(T)}{p}\Big{)}^{-1}\Big{[}1-\Big{(}1-\frac{\phi_{i}(T)}{p}\Big{)}^{r+1}\Big{]}^{r+1}$.
An argument similar to those above shows that
$f(T)\in\mathcal{E}^{]0,\frac{1}{p-1}]}$. Moreover, since the value of
$\phi(T)/p$ at $T=0$ is $1$, we see that $f(T)\in 1+T\mathbb{Q}_{p}[[T]]$.
Using equation (5), we see that there exists $g(T)\in
1+T\mathbb{Q}_{p}[[T]]\cap\mathcal{E}^{]0,\frac{1}{p-1}]}$ such that
$f(T)=\varphi(g(T))$. We therefore have

$\mathrm{res}(t^{-1}G_{1,\,r+1}(T)dt)=\mathrm{res}\Big{(}T^{-1}\Big{(}\frac{\phi_{1}(T)}{p}\Big{)}^{-1}\Big{[}1-\Big{(}\frac{\phi_{1}(T)}{p}\Big{)}^{r+1}\Big{]}^{r+1}\frac{1}{p}\varphi(g(T))dt\Big{)}.$

Adding and subtracting $1$ from the term
$[1-(\frac{\phi_{1}(T)}{p})^{r+1}]^{r+1}$, we get

$\begin{array}[]{ll}&\mathrm{res}(t^{-1}G_{1,\,r+1}(T)dt)\\\
&\stackrel{{\scriptstyle(\ref{eq:2})}}{{=}}\mathrm{res}\Big{(}\frac{1}{(1+T)^{p}-1}\varphi(g(T))dt\Big{)}+\mathrm{res}\Big{(}T^{-1}\Big{(}\frac{\phi_{1}(T)}{p}\Big{)}^{-1}\Big{(}\Big{[}1-\Big{(}\frac{\phi_{1}(T)}{p}\Big{)}^{r+1}\Big{]}^{r+1}-1\Big{)}\frac{1}{p}\varphi(g(T))dt\Big{)}.\end{array}$

The first term is equal to
$\mathrm{res}\Big{(}\varphi\Big{(}\frac{g(T)}{T}\Big{)}dt\Big{)}$, which by
(16), equals $\mathrm{res}\Big{(}\frac{g(T)}{T}dt\Big{)}=1$, since the
constant term of $g(T)$ is $1$. For the second term, we see that
$\frac{\phi_{1}(T)}{p}$ divides
$\Big{[}1-\Big{(}\frac{\phi_{1}(T)}{p}\Big{)}^{r+1}\Big{]}^{r+1}-1$ as
polynomials. Also, the constant term of
$\Big{(}\frac{\phi_{1}(T)}{p}\Big{)}^{-1}\Big{(}\Big{[}1-\Big{(}\frac{\phi_{1}(T)}{p}\Big{)}^{r+1}\Big{]}^{r+1}-1\Big{)}\varphi(g(T))$
is $-1$. Hence, the second residue is $-1/p$. Therefore,

$\mathrm{res}\left(t^{-1}\left(p^{-1}\varphi-1\right)G(|x|,r+1)dt\right)=\frac{1}{p}-1.$

  * •

We prove $\mathrm{res}\left(t^{-1}(\gamma-1)G(|x|,r+1)dt\right)=0$, by showing
that for all $n\geq 1$,

$\mathrm{res}\left(t^{-1}(\gamma-1)G_{n,\,r+1}(T)dt\right)=0.$

Again by (5), we see that $(p^{-1}\varphi)G_{n,\,r+1}(T)=G_{n+1,\,r+1}(T)$.
Therefore

$\begin{array}[]{rcl}\mathrm{res}\left(t^{-1}(\gamma-1)G_{n,\,r+1}(T)dt\right)&=&\mathrm{res}\left(t^{-1}(\gamma-1)(p^{-1}\varphi)^{n-1}G_{1,\,r+1}(T)dt\right)\\\
&\stackrel{{\scriptstyle(\ref{eq:5})}}{{=}}&\mathrm{res}\left(\varphi^{n-1}t^{-1}(\gamma-1)G_{1,\,r+1}(T)dt\right)\\\
&\stackrel{{\scriptstyle(\ref{eq:6})}}{{=}}&\mathrm{res}\left(\varphi^{n-1}(\chi(\gamma)\gamma-1)(t^{-1}G_{1,\,r+1}(T))dt\right),\end{array}$

which vanishes by (16) and (17).

  * •

Using (14), we get
$\log{T}-G^{\prime}(|x|,r+1)=\log{T}+g^{\prime}(T)-tg^{\prime\prime}(T)$,
whereas rearranging the terms in equation (13), we get $\log
T+g^{\prime}(T)=-\log(\chi(\gamma))td_{1}$. These two equations imply

$\log{T}-G^{\prime}(|x|,r+1)=-\log{(\chi(\gamma))}td_{1}-tg^{\prime\prime}(T).$

Therefore we have

$t^{-1}(p^{-1}\varphi-1)(\log{T}-G^{\prime}(|x|,r+1))\stackrel{{\scriptstyle(\ref{eq:5})}}{{=}}-\log{(\chi(\gamma))}(\varphi-1)d_{1}-(\varphi-1)g^{\prime\prime}(T).$

By the discussion on [Ben11, p. 1604, p. 1601],
$\mathrm{res}(((\varphi-1)d_{1})dt)=0$. By (16), we get

$\mathrm{res}(t^{-1}(p^{-1}\varphi-1)(\log T-G^{\prime}(|x|,r+1))dt)=0.$

  * •

Similarly, we have

$t^{-1}(\gamma-1)(\log{T}-G^{\prime}(|x|,r+1))\stackrel{{\scriptstyle(\ref{eq:6})}}{{=}}-\log{(\chi(\gamma))}(\chi(\gamma)\gamma-1)d_{1}-(\chi(\gamma)\gamma-1)g^{\prime\prime}(T).$

Using $(\chi(\gamma)\gamma-1)d_{1}=-\frac{1}{T}$ (see the proof of [Ben11,
Theorem 1.5.7 iiib)] and using equation (17), we get

$\mathrm{res}(t^{-1}(\gamma-1)(\log
T-G^{\prime}(|x|,r+1))dt)=\log{\chi(\gamma)}.$

Using equation (15), we see that $\overline{c}_{1}$ is
$-\frac{p-1}{p}\cdot\overline{\alpha}_{r+1}$ and $\overline{c}_{2}$ is
$\log{(\chi(\gamma))}\cdot\overline{\beta}_{r+1}$. ∎

Let us state the formula for the $\mathcal{L}$-invariant using [Col08,
Definition $2.20$]. Let $(a,b)$ represent a non-zero element of
$H^{1}(\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi))$. Writing
$\overline{(a,b)}=\lambda\cdot\overline{c}_{1}+\mu\cdot\overline{c}_{2}$,
where the bar over an element denotes its class in
$H^{1}(\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi))$, Colmez defines the
$\mathcal{L}$-invariant associated to the class of $(a,b)$ to be
$\frac{p-1}{p}\frac{\lambda}{\mu}\in\mathbb{P}^{1}(\mathbb{Q}_{p})$.333 The
sign is the opposite of the one in [Col08, Definition 2.20]. We believe that
this choice of sign is correct. Indeed, consider the image of $p(1+p)$ under
the Kummer map $\kappa:\mathbb{Q}_{p}^{*}\to
H^{1}(\mathbb{Q}_{p},\mathbb{Q}_{p}(1))$. By the discussion in [Ben11, Section
1.5.6], we see that with the original choice of sign, the
$\mathcal{L}$-invariant associated to the image of $\kappa(p(1+p))$ under the
canonical isomorphism $H^{1}(\mathbb{Q}_{p},\mathbb{Q}_{p}(1))\simeq
H^{1}(\mathcal{R}_{\mathbb{Q}_{p}}(x|x|))$ is $-\log(1+p)$. However, using
Tate’s formula, the $\mathcal{L}$-invariant associated to $\kappa(p(1+p))$ is
equal to $\frac{\log(p(1+p))}{v_{p}(p(1+p))}=\log(1+p)$. The original sign is
also incompatible with our results in the Introduction. Using Proposition 4.5,
we can restate the formula for the $\mathcal{L}$-invariant in terms of
Benois’s basis.

######  Definition 4.6.

If
$\overline{(a,b)}=\lambda^{\prime}\cdot\overline{\alpha}_{r+1}+\mu^{\prime}\cdot\overline{\beta}_{r+1}\neq
0$, where $\overline{(a,b)}$ is the class of $(a,b)$ in
$H^{1}(\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi))$, then the
$\mathcal{L}$-invariant associated to this class is

$\mathcal{L}=-\mathrm{log(\chi(\gamma))}\cdot\frac{\lambda^{\prime}}{\mu^{\prime}}.$

##  5 Cohomology of $m\mathcal{R}_{\mathcal{O}(U_{r})}(\delta_{U_{r}})$

The aim of this section is to find a generator of
$H^{1}(m\mathcal{R}_{\mathcal{O}(U_{r})}(\delta_{U_{r}}))$ for the
tautological character
$\delta_{U_{r}}:\mathbb{Q}_{p}^{*}\to\mathcal{O}(U_{r})^{*}$, which will be
defined in the first subsection. This generator is an essential ingredient in
the proof of Theorem 6.1.

###  5.1 The tautological character
$\delta_{U_{r}}:\mathbb{Q}_{p}^{*}\to\mathcal{O}(U_{r})^{*}$

Define $\delta_{U_{r}}$ by the following equations

$\begin{array}[]{rcl}\delta_{U_{r}}(p)&=&S_{1},\\\
\delta_{U_{r}}(\zeta_{p-1})&=&\zeta_{p-1}^{r+1},\\\
\delta_{U_{r}}(1+p)&=&1+S_{2}.\end{array}$ (18)

This character is tautological in the following sense. If $\delta$ is an
$E$-valued character of $\mathbb{Q}_{p}^{*}$ and if
$(\delta(p),\delta(\zeta_{p-1}),\delta(1+p)-1)\in U_{r}(E)$, then we have the
following commutative diagram

${\mathcal{O}(U_{r})^{*}}$${\mathbb{Q}_{p}^{*}}$${E^{*},}$$\scriptstyle{\delta_{U_{r}}}$$\scriptstyle{\delta}$

where the vertical map is induced by the affinoid algebra map
$\mathcal{O}(U_{r})\to E$ associated to the $E$-valued point
$(\delta(p),\delta(\zeta_{p-1}),\delta(1+p)-1)$ of $U_{r}$.

We prove that $\delta_{U_{r}}$ is bien plaçé in the sense of [Che13,
Definition $2.31$].

######  Lemma 5.1.

$\delta_{U_{r}}$ is bien plaçé.

###### Proof.

  * •

Given any $i\in\mathbb{Z}$, we have to check that
$1-\delta_{U_{r}}(p)p^{i}=1-S_{1}p^{i}$ is not a zero divisor in
$\mathcal{O}(U_{r})$. But this follows immediately from the fact that
$\mathbb{Q}_{p}\langle S_{1},S_{2}\rangle$ is a domain and
$\mathbb{Q}_{p}\langle S_{1},S_{2}\rangle\to\mathcal{O}(U_{r})$ is flat
([BGR84, Corollary 7.3.2/6]).

  * •

Given any $i\geq 0$, we have to check that the image of
$1-\delta_{U_{r}}(\chi(\gamma))\chi(\gamma)^{1-i}=1-\zeta_{p-1}^{a(r+2-i)}(1+S_{2})(1+p)^{1-i}$
in the quotient $\mathcal{O}(U_{r})/(1-\delta_{U_{r}}(p)p^{-i})$ is not a zero
divisor. This again follows from the facts that $\mathbb{Q}_{p}\langle
S_{1},S_{2}\rangle/(1-\delta_{U_{r}}(p)p^{-i})\to\mathcal{O}(U_{r})/(1-\delta_{U_{r}}(p)p^{-i})$
is flat ([BGR84, Corollary 7.3.2/6]), that the above element is non-zero on
the left (easy to check) and that the ring on the left is a domain (indeed, if
$f(S_{1},S_{2})g(S_{1},S_{2})=(1-S_{1}p^{-i})h(S_{1},S_{2})$, then applying
the Weierstrass division theorem ([BGR84, Corollary 5.2.1/2]), we have
$f=q(1-S_{1}p^{-i})+r$, $g=q^{\prime}(1-S_{1}p^{-i})+r^{\prime}$, for some
$q,q^{\prime}\in\mathbb{Q}_{p}\langle S_{1},S_{2}\rangle$ and
$r,r^{\prime}\in\mathbb{Q}_{p}\langle S_{2}\rangle$, so that
$(1-S_{1}p^{-i})|rr^{\prime}$ and hence one of $r$ or $r^{\prime}$ must be
zero). ∎

###  5.2 Computation of
$H^{1}(m\mathcal{R}_{\mathcal{O}(U_{r})}(\delta_{U_{r}}))$

In this section, we make explicit some facts in [Che13] about the first
Fontaine-Herr cohomology groups of certain ‘big’ $(\varphi,\Gamma)$-modules.

Let $\mathcal{R}_{\mathcal{O}(U_{r})}(\delta_{U_{r}})$ be the Robba ring over
$\mathcal{O}(U_{r})$ in the variable $T$, with the standard actions of
$\varphi$ and $\Gamma$ twisted by $\delta_{U_{r}}$: $\varphi
T=\delta_{U_{r}}(p)((1+T)^{p}-1)$ and $\gamma
T=\delta_{U_{r}}(\chi(\gamma))((1+T)^{\chi(\gamma)}-1)$. Let $\psi$ be the
usual left inverse of $\varphi$, twisted by $\delta_{U_{r}}(p^{-1})$. Let
$\mathcal{R}_{\mathcal{O}(U_{r})}^{+}(\delta_{U_{r}})$ be the power series in
$\mathcal{R}_{\mathcal{O}(U_{r})}(\delta_{U_{r}})$ consisting of non-negative
powers of $T$.

First, we compute an $\mathcal{O}(U_{r})$-generator of
$H^{1}(\mathcal{R}_{\mathcal{O}(U_{r})}(\delta_{U_{r}}))$. By [Che13, Theorem
$2.33$], the inclusion
$m\mathcal{R}_{\mathcal{O}(U_{r})}(\delta_{U_{r}})\to\mathcal{R}_{\mathcal{O}(U_{r})}(\delta_{U_{r}})$
induces an isomorphism
$H^{1}(m\mathcal{R}_{\mathcal{O}(U_{r})}(\delta_{U_{r}}))\to
H^{1}(\mathcal{R}_{\mathcal{O}(U_{r})}(\delta_{U_{r}}))$. Second, we use this
isomorphism to lift the $\mathcal{O}(U_{r})$-generator of
$H^{1}(\mathcal{R}_{\mathcal{O}(U_{r})}(\delta_{U_{r}}))$ to an
$\mathcal{O}(U_{r})$-generator of
$H^{1}(m\mathcal{R}_{\mathcal{O}(U_{r})}(\delta_{U_{r}}))$ (see Definition
5.5).

In Section 5.1, we showed that $\delta_{U_{r}}$ is bien plaçé. By [Che13,
Proposition 2.32] we have the following isomorphisms

${C(D^{+})/(\gamma-1)}$${C(D)/(\gamma-1)}$${D^{\psi=1}/(\gamma-1)}$${H^{1}(D),}$

where
$D=\mathcal{R}_{\mathcal{O}(U_{r})}(\delta_{U_{r}}),C(D)=(1-\varphi)D^{\psi=1},D^{+}=\mathcal{R}_{\mathcal{O}(U_{r})}^{+}(\delta_{U_{r}})$
and $C(D^{+})=(1-\varphi)(D^{+})^{\psi=1}$, and where we recall that the
actions of $\varphi$ and $\psi$ are the usual ones twisted by
$\delta_{U_{r}}$. The first map is induced by the canonical injection of
$D^{+}$ into $D$. The second map is induced by $f(T)\mapsto(1-\varphi)f(T)$.
The third map is induced by $f(T)\mapsto(0,f(T))$. We clarify that the
cohomology group appearing in the display above is in the sense of
$(\psi,\Gamma)$-modules.

We wish to find an $\mathcal{O}(U_{r})$-generator of $H^{1}(D)$. We do so by
finding an $\mathcal{O}(U_{r})$-generator of $C(D^{+})/(\gamma-1)$ and then
using the isomorphisms above to get an $\mathcal{O}(U_{r})$-generator of
$H^{1}(D)$. To find an $\mathcal{O}(U_{r})$-generator of
$C(D^{+})/(\gamma-1)$, we prove a couple of lemmas.

######  Lemma 5.2.

$C(D^{+})=(D^{+})^{\psi=0}$.

###### Proof.

By definition, $C(D^{+})$ is the image of the map
$1-\varphi:(D^{+})^{\psi=1}\to(D^{+})^{\psi=0}$. Applying [Che13, Lemma
2.9(vi)] with $\lambda=S_{1}$ and $N=0$ ($r\geq 1\Rightarrow|S_{1}|<1$), we
see that this map is a bijection. The lemma follows. ∎

######  Lemma 5.3.

$S_{1}(1+T)$ generates $(D^{+})^{\psi=0}/(\gamma-1)$ as a free
$\mathcal{O}(U_{r})$-module of rank $1$.

###### Proof.

In order to prove this, we apply [Che13, Proposition $2.14$]. We know that
$D^{+}$ is a free $\mathcal{R}_{\mathcal{O}(U_{r})}^{+}$-module of rank one,
say with basis $w$. To check that $D^{+}$ is $\Gamma$-bounded in the sense of
[Che13, Section 1.8], we choose the following model of $\mathcal{O}(U_{r})$

$\mathcal{A}=\mathbb{Z}_{p}\langle
S_{1},S_{2},T_{1},T_{2},T_{3}\rangle/(p^{r}T_{1}-S_{1},1-T_{1}T_{2},pT_{3}-S_{2}).$

Then for $\gamma^{\prime}\in\Gamma$ we have
$\mathrm{Mat}(\gamma^{\prime})=[\delta_{U_{r}}(\chi(\gamma^{\prime}))]\in
M_{1}(\mathcal{A}[[T]])$, so $D^{+}$ is $\Gamma$-bounded. Using [Che13,
Proposition $2.14$], we see that $\\{S_{1}(1+T)w\\}$ is a basis of
$(D^{+})^{\psi=0}$ over $\mathcal{R}_{\mathcal{O}(U_{r})}^{+}(\Gamma)$ (see
[Che13, Section 2.12] for the definition of the last ring). Now
$\mathcal{R}_{\mathcal{O}(U_{r})}^{+}(\Gamma)/(\gamma-1)\cong\mathcal{O}(U_{r})$.
Therefore $(D^{+})^{\psi=0}/(\gamma-1)$ is a free $\mathcal{O}(U_{r})$-module
of rank one generated by $S_{1}(1+T)w$. ∎

For the remainder of section, we drop $w$ when we talk about the
$\mathcal{O}(U_{r})$-generator $S_{1}(1+T)w$ of $(D^{+})^{\psi=0}/(\gamma-1)$.
Using these lemmas, we see that the quotient $C(D^{+})/(\gamma-1)$ is
generated by $S_{1}(1+T)$ as a free $\mathcal{O}(U_{r})$-module. Pushing this
generator to $C(D)/(\gamma-1)$ under the first isomorphism
$C(D^{+})/(\gamma-1)\to C(D)/(\gamma-1)$, we see that $S_{1}(1+T)$ is an
$\mathcal{O}(U_{r})$-generator of $C(D)/(\gamma-1)$. Now we want a generator
of $D^{\psi=1}/(\gamma-1)$.

######  Lemma 5.4.

The element $y=S_{1}\sum_{n=0}^{\infty}S_{1}^{n}(1+T)^{p^{n}}$ is the preimage
of $S_{1}(1+T)$ under the isomorphism $1-\varphi:D^{\psi=1}/(\gamma-1)\to
C(D)/(\gamma-1)$.

###### Proof.

Note $y\in D^{\psi=1}$ (for the twisted $\psi$). To see this, write (for the
twisted $\varphi$)

$\begin{array}[]{rcl}S_{1}\sum_{n=0}^{\infty}S_{1}^{n}(1+T)^{p^{n}}&=&\varphi(S_{1}\sum_{n=0}^{\infty}S_{1}^{n}(1+T)^{p^{n}})+(1+T)\varphi(1).\end{array}$

From the definition of $\psi$ (cf. [Che13, Section 2.1]), we see that $y\in
D^{\psi=1}$. Clearly $(1-\varphi)y=S_{1}(1+T)$. ∎

Finally, consider the third isomorphism $D^{\psi=1}/(\gamma-1)\to H^{1}(D)$
induced by sending $f(T)$ to $(0,f(T))$. Using this isomorphism, we see that
the class of $(0,y)$ is a generator of $H^{1}(D)$ as a free
$\mathcal{O}(U_{r})$-module. We repeat that this cohomology group is as a
$(\psi,\Gamma)$-modules. In order to get a generator of the
$(\varphi,\Gamma)$-cohomology group, we note that $(\gamma-1)$ is bijective on
$D^{\psi=0}$ by [Che13, Corollary 2.5], so

${H^{1}_{(\varphi,\Gamma)}(D)}$${H^{1}_{(\psi,\Gamma)}(D)}$($\scriptstyle-\psi$,
id)

is an isomorphism of cohomology groups (cf. [Che13, Section 2.1]). Since $y\in
D^{\psi=1}$, we have $(\varphi-1)y\in D^{\psi=0}$. Since $(\gamma-1)$ is
bijective on $D^{\psi=0}$, there exists a unique $x\in D^{\psi=0}$ such that
$(\gamma-1)x=(\varphi-1)y$. Therefore $(x,y)$ represents a class in
$H^{1}_{(\varphi,\Gamma)}(D)$. Moreover, the class of $(x,y)$ maps to the
class of $(0,y)$ under the isomorphism above. Therefore, the class of $(x,y)$
generates $H^{1}_{(\varphi,\Gamma)}(D)$ as a free $\mathcal{O}(U_{r})$-module.
From now on, every cohomology group that we write will be as
$(\varphi,\Gamma)$-modules.

We conclude this section by describing an $\mathcal{O}(U_{r})$-generator of
$H^{1}(m\mathcal{R}_{\mathcal{O}(U_{r})}(\delta_{U_{r}}))$.

Using the isomorphism $H^{1}(mD)\to H^{1}(D)$, we see that the class of
$(x,y)$ is represented by elements of $mD$: there exists a $d\in D$ such that
$x-(\varphi-1)d,\,y-(\gamma-1)d\in mD$.

######  Definition 5.5.

We denote by $e$ the class of $(x-(\varphi-1)d,\,y-(\gamma-1)d)$ in
$H^{1}(mD)$.

Thus $e$ is an explicit generator of the free $\mathcal{O}(U_{r})$-module
$H^{1}(m\mathcal{R}_{\mathcal{O}(U_{r})}(\delta_{U_{r}}))$ of rank $1$.

##  6 The $\mathcal{L}$-invariant of the limit point

Recall that the fiber in the blow-up $\widetilde{U}_{r}$ of $U_{r}$ over the
exceptional point $(p^{r},(1+p)^{r+1}-1)$ corresponding to the maximal ideal
$m=(f_{1},f_{2})$ is parametrized by
$\mathcal{L}\in\mathbb{P}^{1}(\overline{\mathbb{Q}}_{p})$. The following
theorem gives us a formula for the $\mathcal{L}$-invariant associated to an
$E$-valued point $(p^{r},(1+p)^{r+1}-1,a:b)$ in the exceptional fiber in terms
of $a$ and $b$.

######  Theorem 6.1.

The $\mathcal{L}$-invariant of the $(\varphi,\Gamma)$-module associated to the
$E$-valued point in the exceptional fiber with coordinates
$(p^{r},(1+p)^{r+1}-1,\>a:b)$ is

$\mathcal{L}=-\frac{(1+p)^{r+1}\log(1+p)}{p^{r}}\cdot\frac{a}{b}\in\mathbb{P}^{1}(E).$

###### Proof.

Let $v\in\mathrm{Hom}(m/m^{2}\otimes_{\mathbb{Q}_{p}}E,E)$ be a tangent vector
representing the tangent direction associated with the point
$(p^{r},(1+p)^{r+1}-1,\>a:b)$ by Proposition 3.5. Recall that
$D=\mathcal{R}_{\mathcal{O}(U_{r})}(\delta_{U_{r}})$. Consider the
specialization map
$v^{*}:E\otimes_{\mathbb{Q}_{p}}mD\to\mathcal{R}_{E}(x^{r}\chi)$ induced by
the following composition of maps

$\qquad mD\to
m/m^{2}\otimes_{\mathbb{Q}_{p}}D/mD\xrightarrow{v}E\otimes_{\mathbb{Q}_{p}}\mathcal{R}_{\mathbb{Q}_{p}}(x^{r}\chi)=\mathcal{R}_{E}(x^{r}\chi)$

given by

$(f_{1}g_{1}+f_{2}g_{2})\mapsto\overline{f_{1}}\otimes\overline{g_{1}}+\overline{f_{2}}\otimes\overline{g_{2}}\mapsto
v(\overline{f_{1}})\overline{g_{1}}+v(\overline{f_{2}})\overline{g_{2}},\qquad$

where $g_{1},g_{2}\in D$ are arbitrary, $\overline{g_{1}},\overline{g_{2}}$
are their images in $D/mD$, and $\overline{f_{1}},\overline{f_{2}}$ are the
images of $f_{1}=S_{1}-p^{r}$ and $f_{2}=1+S_{2}-(1+p)^{r+1}$ in $m/m^{2}$,
respectively.

The map $v^{*}$ yields a specialization map
$H^{1}(v^{*}):E\otimes_{\mathbb{Q}_{p}}H^{1}(mD)\to
H^{1}(\mathcal{R}_{E}(x^{r}\chi))$ induced by

$H^{1}(v^{*})(\eta_{1},\eta_{2})=(v^{*}(\eta_{1}),v^{*}(\eta_{2})),$

for $(\eta_{1},\eta_{2})\in mD\oplus mD$ representing a cohomology class in
$H^{1}(mD)$. Recall that $e$ is represented by
$(x-(\delta_{U_{r}}(p)\varphi-1)d,\,y-(\delta_{U_{r}}(\chi(\gamma))\gamma-1)d)\in
mD\oplus mD$ and is an $\mathcal{O}(U_{r})$-generator of $H^{1}(mD)$, where
now $\varphi$ and $\gamma$ are the usual untwisted operators (see Definition
5.5). The image of $e$ under $H^{1}(v^{*})$ is the $(\varphi,\Gamma)$-module
associated to the point $(p^{r},(1+p)^{r+1}-1,\>a:b)$ (cf. the proof of
[Che13, Proposition 3.9] and also [Che13, Theorem 2.33]). Note that the
generator $e$ and the tangent vector $v$ are only well defined up to scalars,
but the isomorphism class of the corresponding $(\varphi,\Gamma)$-module is
unchanged under multiplication by scalars in
$H^{1}(\mathcal{R}_{E}(x^{r}\chi))$. In any case, the corresponding
$\mathcal{L}$-invariant only depends on the projective image of the cohomology
class $H^{1}(v^{*})(e)$.

Let us compute the $\mathcal{L}$-invariant of the $(\varphi,\Gamma)$-module
given by $H^{1}(v^{*})(e)$ using Definition 4.6. For convenience of notation,
let $\mathrm{Res}(f(T))=\mathrm{res}(f(T)dt)$ for
$f(T)\in\mathcal{R}_{E},\,\mathcal{R}_{\mathcal{O}(U_{r})}$. Write

$H^{1}(v^{*})(e)=\lambda_{v}\cdot\overline{\alpha}_{r+1}+\mu_{v}\cdot\overline{\beta}_{r+1},$

where, by (15),

$\begin{array}[]{r}\lambda_{v}=\mathrm{Res}(t^{r}v^{*}(x-(\delta_{U_{r}}(p)\varphi-1)d)),\quad\mu_{v}=\mathrm{Res}(t^{r}v^{*}(y-(\delta_{U_{r}}(\chi(\gamma))\gamma-1)d)).\end{array}$

To compute these residues, we first note that the following square commutes

${mD\otimes_{\mathbb{Q}_{p}}E}$${\mathcal{R}_{E}(x^{r}\chi)}$${m\otimes_{\mathbb{Q}_{p}}E}$${E,}$$\scriptstyle{v^{*}}$$\scriptstyle{\mathrm{Res}(t^{r}\\_)}$$\scriptstyle{\mathrm{Res}({t^{r}}\\_)}$$\scriptstyle{v}$

where we write $v$ again for the map $m\otimes
E\xrightarrow{\overline{\>\>}}m/m^{2}\otimes E\xrightarrow{v}E$. Indeed, given
any element $f_{1}g_{1}+f_{2}g_{2}\in mD$, we have

$\begin{array}[]{rcl}\mathrm{Res}(t^{r}v^{*}(f_{1}g_{1}+f_{2}g_{2}))&=&\mathrm{Res}(t^{r}(v(\overline{f_{1}})\overline{g_{1}}+v(\overline{f_{2}})\overline{g_{2}}))\\\
&=&v(\overline{f_{1}})\mathrm{Res}(t^{r}\overline{g_{1}})+v(\overline{f_{2}})\mathrm{Res}(t^{r}\overline{g_{2}})\\\
&=&v(\mathrm{Res}(t^{r}g_{1})f_{1}+\mathrm{Res}(t^{r}g_{2})f_{2})\\\
&=&v(\mathrm{Res}(t^{r}(f_{1}g_{1}+f_{2}g_{2}))).\end{array}$

Now we prove that $\mathrm{Res}(t^{r}x)=0$ and $\mathrm{Res}(t^{r}y)=0$. We
use the following formulas, which are similar to (16) and (17). For any
$f(T)\in\mathcal{R}_{\mathcal{O}(U_{r})}$, we have

$\displaystyle\mathrm{Res}(\varphi(f(T)))$ $\displaystyle=$
$\displaystyle\mathrm{Res}(f(T)),$ (19)
$\displaystyle\mathrm{Res}(\gamma(f(T)))$ $\displaystyle=$
$\displaystyle\chi(\gamma)^{-1}\mathrm{Res}(f(T)).$ (20)

Using the definition of $x$ and $y$ (see the discussion before Definition
5.5), we get

$(\delta_{U_{r}}(\chi(\gamma))\gamma-1)x=(\delta_{U_{r}}(p)\varphi-1)y=-S_{1}(1+T),$

by Lemma 5.4. Multiplying by $t^{r}$ and taking residues, by (8) and (7), we
get

$\begin{array}[]{lcl}\mathrm{Res}((\delta_{U_{r}}(\chi(\gamma))\chi(\gamma)^{-r}\gamma-1)t^{r}x)&=&\mathrm{Res}(-t^{r}S_{1}(1+T))=0,\\\
\mathrm{Res}((\delta_{U_{r}}(p)p^{-r}\varphi-1)t^{r}y)&=&\mathrm{Res}(-t^{r}S_{1}(1+T))=0.\end{array}$

Applying (20) and (19) to the terms on the left, we get
$\mathrm{Res}(t^{r}x)=0=\mathrm{Res}(t^{r}y)$.

Thus, using the commutativity of the diagram above, we get

$\begin{array}[]{lcl}\lambda_{v}&=&v(\mathrm{Res}(t^{r}(x-(\delta_{U_{r}}(p)\varphi-1)d)))\\\
&=&-v(\mathrm{Res}(t^{r}(\delta_{U_{r}}(p)\varphi-1)d))\\\
&\stackrel{{\scriptstyle(\ref{eq:5})}}{{=}}&-v(\mathrm{Res}((\delta_{U_{r}}(p)p^{-r}\varphi-1)t^{r}d)).\end{array}$

Using equation (19), we get

$\begin{array}[]{lcl}\lambda_{v}&=&-v((\delta_{U_{r}}(p)p^{-r}-1)\mathrm{Res}(t^{r}d))\\\
&\stackrel{{\scriptstyle(\ref{def of taut
char})}}{{=}}&-v((S_{1}-p^{r})p^{-r}\mathrm{Res}(t^{r}d))\\\
&=&-v(f_{1}p^{-r}\mathrm{Res}(t^{r}d))\\\
&=&-p^{-r}v(f_{1}\mathrm{Res}(t^{r}d)).\end{array}$

The expression in the brackets belongs to $m$. Recall that $v:m\otimes E\to E$
is the composition of the maps ${\>}^{-}:m\to m/m^{2}$ and $v:m/m^{2}\otimes
E\to E$. We therefore get

$\lambda_{v}=-p^{-r}\overline{\mathrm{Res}(t^{r}d)}\cdot
v(\overline{f_{1}})=-p^{-r}\overline{\mathrm{Res}(t^{r}d)}\cdot a.$

Similarly,

$\begin{array}[]{rcl}\mu_{v}&=&v(\mathrm{Res}(t^{r}(y-(\delta_{U_{r}}(\chi(\gamma))\gamma-1)d)))\\\
&=&-v(\mathrm{Res}(t^{r}(\delta_{U_{r}}(\chi(\gamma))\gamma-1)d))\\\
&\stackrel{{\scriptstyle(\ref{eq:6})}}{{=}}&-v(\mathrm{Res}((\delta_{U_{r}}(\chi(\gamma))\chi(\gamma)^{-r}\gamma-1)t^{r}d)).\end{array}$

Using equation (20), we get

$\mu_{v}=-v((\delta_{U_{r}}(\chi(\gamma))\chi(\gamma)^{-(r+1)}-1)\mathrm{Res}(t^{r}d)).$

Since $\chi(\gamma)=\zeta_{p-1}^{a}(1+p)$, for some $a$, we get

$\begin{array}[]{rcl}\mu_{v}&\stackrel{{\scriptstyle(\ref{def of taut
char})}}{{=}}&-v((\zeta_{p-1}^{a(r+1)}(1+S_{2})\zeta_{p-1}^{-a(r+1)}(1+p)^{-(r+1)}-1)\mathrm{Res}(t^{r}d))\\\
&=&-v(f_{2}(1+p)^{-(r+1)}\mathrm{Res}(t^{r}d))\\\
&=&-(1+p)^{-(r+1)}v(f_{2}\mathrm{Res}(t^{r}d)).\end{array}$

The expression inside the brackets is an element of $m$. As above, we get

$\mu_{v}=-(1+p)^{-(r+1)}\overline{\mathrm{Res}(t^{r}d)}\cdot
v(\overline{f_{2}})=-(1+p)^{-(r+1)}\overline{\mathrm{Res}(t^{r}d)}\cdot b.$

Both $\lambda_{v}$ and $\mu_{v}$ cannot be equal to $0$ simultaneously because
$H^{1}(v^{*})(e)\neq 0$. So $\overline{\mathrm{Res}(t^{r}d)}\not=0$. By
Definition 4.6, the $\mathcal{L}$-invariant of $H^{1}(v^{*})(e)$ is

$-\log(\chi(\gamma))\cdot\frac{\lambda_{v}}{\mu_{v}}=-\frac{(1+p)^{r+1}\log(1+p)}{p^{r}}\cdot\frac{a}{b}.\qed$

##  7 Proof of Theorem 1.1

Let $k\geq 3$ and let $r=k-2$. Let $(k_{n},a_{n})$ for $n\geq 1$ be as in (1).
These quantities depend on
$\mathcal{L}\in{\mathbb{P}}^{1}(\overline{\mathbb{Q}}_{p})$. We prove that the
sequence of crystalline representations $V^{*}_{k_{n},a_{n}}$ converges to the
semi-stable representation $V^{*}_{k,\mathcal{L}}$.

We recall some facts from Section 2. Recall that the ordered pair of
characters $(\delta_{1,n},\delta_{2,n})$ is associated to the representation
$V^{*}_{k_{n},a_{n}}$ and the sequence of characters
$\delta_{1,n}\delta_{2,n}^{-1}=\mu_{y_{n}^{2}}\chi^{k_{n}-1}$, where $y_{n}$
is as in (2), converges to the exceptional character $x^{r}\chi$. Moreover,
the character $\mu_{y_{n}^{2}}\chi^{k_{n}-1}$ corresponds to the following
point of $\widetilde{U}_{r}$

$(y_{n}^{2},(1+p)^{k_{n}-1}-1,\>y_{n}^{2}-p^{r}:(1+p)^{k_{n}-1}-(1+p)^{k-1}).$
(21)

We compute the limit of the above points in $\widetilde{U}_{r}$ as
$n\to\infty$ using the following lemmas.

######  Lemma 7.1.

We have

$\lim_{n\to\infty}\frac{y_{n}^{2}-p^{r}}{p^{n}}=\begin{cases}\mathcal{L}p^{r}(p-1),&\text{if
$\mathcal{L}\not=\infty$}\\\ 2p^{r},&\text{if
$\mathcal{L}=\infty$}.\end{cases}$

###### Proof.

Assume $\mathcal{L}\neq\infty$. We have

$\begin{array}[]{rcl}y_{n}^{2}-p^{r}&=&\left(\dfrac{a_{n}+\sqrt{a_{n}^{2}-4p^{k_{n}-1}}}{2}\right)^{2}-p^{r}=a_{n}^{2}\left(\dfrac{1+\sqrt{1-4p^{k_{n}-1}a_{n}^{-2}}}{2}\right)^{2}-p^{r}\end{array}$

For large $n$, the valuation of $a_{n}=p^{r/2}+\mathcal{L}p^{n+r/2}(p-1)/2$ is
equal to $r/2$. Therefore for large $n$, the expression inside the second
radical sign is an element of $1+p^{1+p^{n}(p-1)}\mathcal{O}_{E}$, where
$\mathcal{O}_{E}$ is the ring of integers of $E$. Since taking square roots is
an automorphism of this group, we see that $\sqrt{1-4p^{k_{n}-1}a_{n}^{-2}}\in
1+p^{1+p^{n}(p-1)}\mathcal{O}_{E}$. Therefore we can write

$\frac{1+\sqrt{1-4p^{k_{n}-1}a_{n}^{-2}}}{2}=1+up^{1+p^{n}(p-1)},$

for some $u\in\mathcal{O}_{E}$. Substituting this in the previous equation, we
get

$\begin{array}[]{rcl}y_{n}^{2}-p^{r}&=&(p^{r}+\mathcal{L}p^{n+r}(p-1)+\mathcal{L}^{2}p^{2n+r}(p-1)^{2}/4)(1+2up^{1+p^{n}(p-1)}+u^{2}p^{2+2p^{n}(p-1)})-p^{r}\\\
&=&p^{r}+\mathcal{L}p^{n+r}(p-1)+p^{2n}(u^{\prime})-p^{r},\end{array}$

for some $u^{\prime}$ whose valuation is bounded below. So

$\lim_{n\to\infty}\frac{y_{n}^{2}-p^{r}}{p^{n}}=\mathcal{L}p^{r}(p-1).$

The limit in the case $\mathcal{L}=\infty$ is computed similarly. ∎

######  Lemma 7.2.

We have

$\lim_{n\to\infty}\frac{(1+p)^{k_{n}-1}-(1+p)^{k-1}}{p^{n}}=\begin{cases}(1+p)^{k-1}(p-1)\log{(1+p)},&\text{if
$\mathcal{L}\not=\infty$}\\\ 0,&\text{if $\mathcal{L}=\infty$}.\end{cases}$

###### Proof.

Assume $\mathcal{L}\neq\infty$. Since

$(1+p)^{k_{n}-1}-(1+p)^{k-1}=(1+p)^{k-1}[(1+p)^{p^{n}(p-1)}-1],$

we see that

$\displaystyle\lim_{n\to\infty}\frac{(1+p)^{k_{n}-1}-(1+p)^{k-1}}{p^{n}}$
$\displaystyle=$
$\displaystyle(1+p)^{k-1}\lim_{n\to\infty}\frac{(1+p)^{p^{n}(p-1)}-1}{p^{n}}$
$\displaystyle=$
$\displaystyle(1+p)^{k-1}\lim_{n\to\infty}\frac{\\{1+[(1+p)^{(p-1)}-1]\\}^{p^{n}}-1}{p^{n}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{eq:4})}}{{=}}$
$\displaystyle(1+p)^{k-1}\log{(1+p)^{p-1}}.\qquad\qquad\qquad\qquad\qquad\qquad\quad$

The limit in the case $\mathcal{L}=\infty$ is proved similarly. ∎

Write the sequence (21) as follows

$\left(y_{n}^{2},(1+p)^{k_{n}-1}-1,\>\frac{y_{n}^{2}-p^{r}}{p^{n}}:\frac{(1+p)^{k_{n}-1}-(1+p)^{k-1}}{p^{n}}\right).$

Using Lemma 7.1 and Lemma 7.2, we see that the sequence above converges to the
point

$\begin{cases}(p^{r},(1+p)^{k-1}-1,\>\mathcal{L}p^{r}(p-1):(1+p)^{k-1}(p-1)\log{(1+p)}),&\text{if
$\mathcal{L}\not=\infty$}\\\ (p^{r},(1+p)^{k-1}-1,\>1:0),&\text{if
$\mathcal{L}=\infty$}.\end{cases}$

Assume $\mathcal{L}\neq\infty$. By Theorem 6.1, the $\mathcal{L}$-invariant of
the $(\varphi,\Gamma)$-module associated to this limit point is
$-\mathcal{L}$. The corresponding Galois representation is
$V(\mu_{p^{r/2}},\mu_{1/p^{r/2}}\chi^{1-k},-\mathcal{L})$ in the notation of
[Col08]. Comparing the filtered $(\varphi,N)$-module associated to
$V_{k,\mathcal{L}}^{*}$ given in the Introduction with the one associated to
$V(\mu_{p^{r/2}},\mu_{1/p^{r/2}}\chi^{1-k},-\mathcal{L})$ using [Col08,
Section 4.6] (more specifically, [Col08, Proposition 4.18] with $a=1-k$,
$b=0$, $\alpha=\mu_{p^{k/2}}$, and $\mathcal{L}$ replaced by $-\mathcal{L}$),
we see that $V(\mu_{p^{r/2}},\mu_{1/p^{r/2}}\chi^{1-k},-\mathcal{L})\simeq
V_{k,\mathcal{L}}^{*}$. Thus the sequence of crystalline representations
$V^{*}_{k_{n},a_{n}}$ converges to the semi-stable representation
$V^{*}_{k,\mathcal{L}}$ for $\mathcal{L}\in E$.

Now assume we are in the $\mathcal{L}=\infty$ case. By Theorem 6.1, the
$\mathcal{L}$-invariant associated to the above limit point is $\infty$. The
corresponding Galois representation is
$V(\mu_{p^{r/2}},\mu_{1/p^{r/2}}\chi^{1-k},\infty)$ in the notation of
[Col08]. Comparing the filtered $\varphi$-module associated to
$V_{k,\infty}^{*}$ in the Introduction with the one associated to
$V(\mu_{p^{r/2}},\mu_{1/p^{r/2}}\chi^{1-k},\infty)$ in [Col08, Section 4.5]
(more precisely, take $a=1-k$, $b=0$, $\beta=\mu_{p^{r/2}}$ and
$\alpha=\mu_{p^{k/2}}$ in the second isomorphism in [Col08, Proposition
4.13]), we see that $V(\mu_{p^{r/2}},\mu_{1/p^{r/2}}\chi^{1-k},\infty)\simeq
V_{k,\infty}^{*}$. (Alternatively, the corresponding Galois representation is
$V(\mu_{p^{r/2}},\mu_{1/p^{r/2}}\chi^{1-k})$ in the notation of [Ber12] and,
by [Ber12, Proposition 3.1], this last representation is isomorphic to the
crystalline representation $V_{k,a_{p}}^{*}$ with $a_{p}=p^{k/2}+p^{k/2-1}$,
which as mentioned in the Introduction, is isomorphic to $V^{*}_{k,\infty}$.)
Thus, the sequence of crystalline representations $V^{*}_{k_{n},a_{n}}$ again
converges to the (crystalline) representation $V^{*}_{k,\infty}$.

##  8 Proof of Theorem 1.2

In this section, we use Theorem 1.1 along with Chenevier’s local constancy
result and the zig-zag conjecture to compute the reductions of semi-stable
representations $V_{k,\mathcal{L}}$ with weights $k$ in the range $[3,p+1]$
for $p$ odd.

Consider the crystalline representation $V_{k_{n},a_{n}}$, with
$(k_{n},a_{n})$ as in (1). Since $v_{p}(a_{n})=\frac{r}{2}$ for $r=k-2$, for
$n$ sufficiently large, the weight $k_{n}$ is an exceptional weight with
respect to the half-integral slope $0<v=\frac{r}{2}\leq\frac{p-1}{2}$ in the
sense of [Gha21, Section 1], namely $k_{n}\equiv 2v+2\mod(p-1)$, for $n$
large. Therefore, we can apply444The conjecture excludes some weights that are
$p$-adically close to some small weights, but these small weights are always
strictly larger than $p+1$, so do not interfere with the argument here. the
zig-zag conjecture [Gha21, Conjecture 1.1] to compute the reduction of
$V_{k_{n},a_{n}}$ for large $n$. The zig-zag conjecture specifies the
reduction in terms of an alternating sequence of irreducible and reducible mod
$p$ representations depending on the relative size of a rational parameter
$\tau$ with respect to (certain integer shifts of) another integer parameter
$t$.

For $V_{k_{n},a_{n}}$, the formula for $\tau$ (cf. [Gha21, (1.1)]), which we
call $\tau_{n}$, is

$\tau_{n}=v_{p}\left(\dfrac{a_{n}^{2}-{k_{n}-2-v_{-}\choose
v_{+}}{k_{n}-2-v_{+}\choose v_{-}}p^{r}}{pa_{n}}\right),$

where $v_{-}$ and $v_{+}$ are the largest and the smallest integers,
respectively, such that $v_{-}<v<v_{+}$. Let us simplify this. We first treat
the case $\mathcal{L}\neq\infty$. We have

$\tau_{n}=v_{p}\left(\dfrac{p^{r}(1+p^{n}(p-1)\mathcal{L}/2)^{2}-{k+p^{n}(p-1)-2-v_{-}\choose
v_{+}}{k+p^{n}(p-1)-2-v_{+}\choose
v_{-}}p^{r}}{p^{1+r/2}(1+p^{n}(p-1)\mathcal{L}/2)}\right).$

For large $n$, we have $v_{-}+v_{+}=k-2$, so

$\displaystyle{k+p^{n}(p-1)-2-v_{-}\choose v_{+}}$ $\displaystyle=$
$\displaystyle\frac{(1+p^{n}(p-1))(2+p^{n}(p-1))\cdots(v_{+}+p^{n}(p-1))}{v_{+}!}$
$\displaystyle=$ $\displaystyle 1+p^{n}(p-1)H_{+}+\text{ terms involving
}p^{2n},$

for the harmonic sum555Thus, the provenance of the harmonic sums that are
prevalent in all the computation of the reductions of semi-stable
representations in the literature can now be traced back to the $p$-adic
expansions of the binomial coefficients appearing in the zig-zag conjecture.
$H_{+}=\sum_{i=1}^{v_{+}}\frac{1}{i}$. Similarly,

${k+p^{n}(p-1)-2-v_{+}\choose v_{-}}=1+p^{n}(p-1)H_{-}+\text{ terms involving
}p^{2n},$

for $H_{-}=\sum_{i=1}^{v_{-}}\frac{1}{i}$ (if $v_{-}\geq 1$; $H_{-}=0$ if
$v_{-}=0$). Substituting, we get

$\displaystyle\tau_{n}$ $\displaystyle=$ $\displaystyle
r+v_{p}((1+\mathcal{L}p^{n}(p-1)/2)^{2}-[1+p^{n}(p-1)(H_{-}+H_{+})+\text{
terms involving }p^{2n}])$ $\displaystyle\qquad-
v_{p}(p^{1+r/2}(1+\mathcal{L}p^{n}(p-1)/2))$ $\displaystyle=$ $\displaystyle
r+v_{p}(\mathcal{L}p^{n}(p-1)-(H_{-}+H_{+})p^{n}(p-1)+\text{terms involving
}p^{2n})$ $\displaystyle\qquad-(1+r/2)-v_{p}(1+\mathcal{L}p^{n}(p-1)/2).$

Thus, for large $n$,

$\tau_{n}=r/2-1+n+v_{p}(\mathcal{L}-H_{-}-H_{+}),$

if $\mathcal{L}\neq H_{-}+H_{+}$ (and $\tau_{n}\geq r/2-1+2n+c$ for some
rational number $c$ independent of $n$ if $\mathcal{L}=H_{-}+H_{+}$).

As mentioned above, the zig-zag conjecture involves another parameter $t$
which for $V_{k_{n},a_{n}}$ we call $t_{n}$. We have

$t_{n}=v_{p}(k_{n}-2-r)=n.$

These formulas for $\tau_{n}$ and $t_{n}$ show that for large $n$ the
parameter $\tau_{n}-t_{n}$ lies, independently of $n$, in one of the intervals
(which may possibly be a point) that appear in the statement of [Gha21,
Conjecture 1.1]. This interval is determined by the size of
$\nu=v_{p}(\mathcal{L}-H_{{-}}-H_{{+}})$ (and is the rightmost one when
$\nu=\infty$). The conjecture accordingly specifies the exact shape of the
reductions of the crystalline representations $V_{k_{n},a_{n}}$ for large $n$
in terms of the size of $\nu$. Since taking duals commutes with taking
reduction, we obtain the reductions of the $V^{*}_{k_{n},a_{n}}$ for large $n$
in terms of the size of $\nu$. Using Theorem 1.1 and Chenevier’s local
constancy result, we get the reduction of the limiting semi-stable
representation $V^{*}_{k,\mathcal{L}}$ in terms of the size of $\nu$. Finally,
taking duals again, we obtain Theorem 1.2 for $\mathcal{L}$ finite.

Now assume we are in the case $\mathcal{L}=\infty$. Then a computation similar
to the one above shows that $\tau_{n}=r/2-1+n$ and $t_{n}=n^{2}$, so that for
large $n$, we have $\tau_{n}<t_{n}$ (and so $\tau_{n}-t_{n}$ is in the
leftmost interval appearing in the conjecture). Using [Gha21, Conjecture 1.1]
and Theorem 1.1 again, we see that
$\overline{V}_{k,\infty}\sim\mathrm{ind}(\omega_{2}^{k-1})$, at least on the
inertia subgroup $I_{\mathbb{Q}_{p}}$. Thus Theorem 1.2 also holds for
$\mathcal{L}=\infty$ as $\nu=-\infty$ (but as remarked after the theorem, the
result in this case is classical by [Edi92] and even holds without assuming
zig-zag!).

##  9 Bounded Hodge-Tate weights

In this section, we use the techniques of this paper to give a proof of the
fact that the limit of a sequence of two-dimensional (irreducible) crystalline
representations $V_{n}$ for $n\geq 1$ with Hodge-Tate weights in an interval
$[a,b]$ such that the difference of the Hodge-Tate weights is at least two
infinitely often is also (irreducible) crystalline with Hodge-Tate weights in
the interval $[a,b]$. However, note that Berger [Ber04, Théorème 1] has proved
more generally that the limit of subquotients of crystalline representations
with bounded Hodge-Tate weights belonging to $[a,b]$ is also crystalline with
Hodge-Tate weights in $[a,b]$.

Twisting by a fixed power of the cyclotomic character, we may assume that
infinitely often the Hodge-Tate weights of $V_{n}$ are $(0,k_{n}-1)$ with
$k_{n}\geq 3$. Then there exists a sequence of unramified characters $\mu_{n}$
for $n\geq 1$ such that $V_{n}\simeq V_{k_{n},a_{n}}\otimes\mu_{n}$, for some
$a_{n}$ with $v_{p}(a_{n})>0$. For each $n\geq 1$, consider the ordered pair
of characters $(\delta_{1,n},\delta_{2,n})$ associated with $V_{n}^{*}$ under
triangulation. We have $\delta_{1,n}=\mu_{y_{n}}\cdot\mu_{n}^{-1}$ and
$\delta_{2,n}=\mu_{1/y_{n}}\chi^{1-k_{n}}\cdot\mu_{n}^{-1}$, where $y_{n}$ is
as in (2). The assumption that the sequence $V_{n}$ converges means that the
associated sequence of points in $\widetilde{\mathcal{T}}_{2}$ converge to a
point in $\widetilde{\mathcal{T}}_{2}$. Say that the sequence
$(\delta_{1,n},\delta_{2,n})$ converges to $(\delta_{1},\delta_{2})$ in
$(\mathcal{T}\times\mathcal{T})\setminus F^{\prime}$.

First assume that $\delta_{1}\delta_{2}^{-1}$ is not of the form $x^{r}\chi$
for $r\geq 0$. Then, by convention, the $\mathcal{L}$-invariant associated to
the limit point is $\infty$. Using the convergence of (the unramified parts
of) $\delta_{1,n}\delta_{2,n}$ and $\delta_{1,n}\delta_{2,n}^{-1}$, we see
that $\mu_{n}^{-2}$ converges to $\mu_{\alpha^{2}}$ for some
$\alpha\in\overline{\mathbb{Q}}_{p}^{*}$ and $\mu_{y_{n}^{2}}$ converges to
$\mu_{y^{2}}$ for some $y\in\overline{\mathbb{Q}}_{p}^{*}$. Therefore
$\delta_{1,n}=\mu_{y_{n}}\cdot\mu_{n}^{-1}$ converges to
$\delta_{1}=\mu_{\pm\alpha y}$. Similarly, $\delta_{2,n}$ converges to
$\delta_{2}=\mu_{\pm\alpha/y}\chi^{1-k}$ for some $k$. So the sequence
$V_{n}^{*}$ converges to $V(\mu_{\pm\alpha
y},\mu_{\pm\alpha/y}\chi^{1-k},\infty)$, for some $k$, in the notation of
[Col08]. Now $k_{n}$ is a bounded sequence of integers converging to $k$.
Therefore $k_{n}=k$ for large $n$. In particular, $k\geq 3$. But
$V(\mu_{\pm\alpha y},\mu_{\pm\alpha/y}\chi^{1-k},\infty)$ is equal to the
crystalline representation $V_{k,y+p^{k-1}/y}^{*}\otimes\mu_{\pm\alpha}$.
Taking duals, we see that the sequence $V_{n}$ converges to the crystalline
representation $V_{k,y+p^{k-1}/y}\otimes\mu_{\pm\alpha}^{-1}$. Untwisting by
the fixed power of the cyclotomic character, the original sequence $V_{n}$
also converges to a crystalline representation with Hodge-Tate weights in
$[a,b]$.

Now assume that $\delta_{1}\delta_{2}^{-1}=x^{r}\chi$, for some $r\geq 0$.
This means that $\delta_{1,n}\delta_{2,n}^{-1}=\mu_{y_{n}^{2}}\chi^{k_{n}-1}$
converges to $x^{r}\chi$ in the neighborhood $U_{r}$ of $x^{r}\chi$ in
$\mathcal{T}$. Let $k=r+2$. Since $k_{n}$ is a bounded sequence of integers,
the convergence implies that $k_{n}=k$ for all but finitely many $n$. In
particular, we have $k\geq 3$. The point corresponding to the character
$\mu_{y_{n}^{2}}\chi^{k-1}$ in the blow-up $\widetilde{U}_{r}$ of $U_{r}$ is

$(y_{n}^{2},\>(1+p)^{k-1}-1,\>y_{n}^{2}-p^{r}:0)=(y_{n}^{2},\>(1+p)^{k-1}-1,\>1:0),$

at least if $y_{n}^{2}\neq p^{r}$ (if $y_{n}^{2}=p^{r}$, then it is also the
last point since this is the only crystalline point in the fiber above
$x^{r}\chi$). Since $y_{n}^{2}$ converges to $p^{r}$, this sequence converges
to the point $(p^{r},\>(1+p)^{k-1}-1,1:0)$ in $\widetilde{U}_{r}$. Applying
Theorem 6.1, we see that the $\mathcal{L}$-invariant associated with this
limit point is $\infty$. Therefore the limit of the sequence $V_{n}^{*}$ is
$V(\delta_{1},\delta_{2},\infty)$ in the notation of [Col08].

Now
$\delta_{1}\delta_{2}^{-1}=x^{r}\chi=\mu_{p^{r/2}}\cdot(\mu_{1/p^{r/2}}\chi^{1-k})^{-1}$
implies that there is a character $\delta$ such that
$\delta_{1}=\delta\cdot\mu_{p^{r/2}}$ and
$\delta_{2}=\delta\cdot\mu_{1/p^{r/2}}\chi^{1-k}$. Using [Col08, Proposition
4.4 (i)], we see that $(\delta_{1}\delta_{2})(p)$ is a unit. Hence $\delta(p)$
is a unit. Consequently, $V(\delta_{1},\delta_{2},\infty)\simeq
V(\mu_{p^{r/2}},\mu_{1/p^{r/2}}\chi^{1-k},\infty)\otimes\delta$. Therefore the
limit of the crystalline representations $V^{*}_{n}$ is the crystalline
representation $V_{k,\infty}^{*}\otimes\delta$. Taking duals, we see that the
limit of the $V_{n}$ is $V_{k,\infty}\otimes\delta^{-1}$. Note that the
$\delta_{1,n}=\mu_{y_{n}}\cdot\mu_{n}^{-1}$ are unramified characters
converging to $\delta_{1}=\delta\cdot\mu_{p^{r/2}}$. This forces $\delta$ to
be unramified and hence crystalline. Therefore
$V_{k,\infty}\otimes\delta^{-1}$ is crystalline. The last representation has
Hodge-Tate weights $(0,k-1)$. Untwisting by the fixed power of the cyclotomic
character, we see that the Hodge-Tate weights of the limit representation of
the original sequence again lie in $[a,b]$.

Acknowledgements: The first and second authors acknowledge support of the
Department of Atomic Energy under project number 12-R&D-TFR-5.01-0500. During