\input{preamble.tex}
\unittitle{Quasiunipotent differential modules}

In this unit, we construct a class of examples of differential modules
on open annuli which are solvable at a boundary. In the process, we illustrate
a numerical relationship between wild ramification in positive characteristic
and convergence of solutions of $p$-adic differential equations. We also
state the $p$-adic local monodromy theorem, for differential modules 
with Frobenius structure on an annulus, and prove the rank $1$ case.

Throughout this unit, we assume that our complete nonarchimedean field carries
a discrete valuation (e.g., finite extensions of $\QQ_p$ are okay but not
$\CC_p$). Getting rid of this assumption throws in a number of subtleties which
we will not address here.

Notation: for $E/F$ a Galois extension of fields, write
$G_{E/F}$ for $\Gal(E/F)$. If $E = F^{\sep}$, write $G_F$ instead, to mean
the absolute Galois group.

Also, note that I haven't added all references and details; I plan to put a
bit more when I fold this into the compiled notes. (That will be true for
the remainder of the course.)

\section{Some key rings}

Recall that we defined the ring $\calE$ as
the completion of $\gotho_K ((t)) \otimes_{\gotho_K} K$
for the 1-Gauss norm
\[
\left| \sum_{i \in \ZZ} c_i t^i \right|_1 = \sup_i \{|c_i|\}.
\]
Besides the $p$-adic topology, it is natural to consider also the
\emph{weak topology} on $\calE$, in which a sequence converges to 0
if it does so in the $t$-adic topology on $\calE / \gothm_K^m \gotho_{\calE}$
for each $m \in \ZZ$. Note that $\calE$ is complete for both topologies.

Because $K$ carries a discrete valuation, the supremum defining the Gauss
norm of a nonzero element $x = \sum x_i t^i \in \calE$ is achieved by some $i$. 
If $j$ 
is the least such index, then the sum
\[
x_j^{-1} t^{-j} \sum_{l=0}^\infty (1-x_j^{-1} t^{-j} x)^l
\]
converges in the weak topology (but not in the $p$-adic topology!)
to an inverse of $x$. That is,
$\calE$ is a discrete complete nonarchimedean field with residue field $\kappa_K((t))$.

Put 
\[
\calE^\dagger = \bigcup_{\alpha \in (0,1)}
K \langle \alpha/t, t \rrbracket_0;
\]
that is, $\calE^\dagger$ consists of formal sums $\sum c_i t^i$ which
have bounded coefficients and converge in some range $\alpha \leq |t| < 1$.

\begin{lemma}
\begin{enumerate}
\item[(a)]
The ring $\calE^\dagger$ is a field.
\item[(b)]
Under the norm $|\cdot|_1$, the valuation ring
$\gotho_{\calE^\dagger}$ is a local ring with maximal ideal
$\gothm_K \gotho_{\calE^\dagger}$.
\item[(c)]
The pair $(\gotho_{\calE^\dagger}, \gothm_K \gotho_{\calE^\dagger})$
is henselian.
\end{enumerate}
\end{lemma}
This last property implies that 
finite separable extensions of $\kappa_{\calE^\dagger} = \kappa_K((t))$
lift functorially to finite \'etale extensions of $\gotho_{\calE^\dagger}$
(and to unramified extensions of $\calE^\dagger$).
In particular, the maximal unramified extension
$\calE^{\dagger,\unr}$ carries an action of
$G_{\kappa_K((t))}$.
\begin{proof}
The proof of (a) uses the same construction as for $\calE$, except that 
the series converges under $|\cdot|_\alpha$ for some $\alpha < 1$. From this,
(b) is straightforward. The proof of (c) is to reduce to working in
some $K \langle \alpha/t, t \rrbracket_0$ and use the fact that the latter
ring is complete for the Fr\'echet topology generated by $|\cdot|_\alpha$
and $|\cdot|_1$.
\end{proof}

\section{Finite representations and differential modules}

Let $V$ be a finite dimensional vector space over $K$,
and let $\tau: G_{\kappa_K((t))} \to \GL(V)$ be a continuous homomorphism for the
\emph{discrete} topology on $\GL(V)$. That is, $\tau$ factors through
$G_{L/\kappa_K((t)))}$ for some finite separable extension $L$
of $\kappa_K((t))$.

Let $\calE_L^{\dagger}$ be the finite unramified extension of $\calE^\dagger$
corresponding to $L$; then $G_{\kappa_K((t))}$ 
acts on $\calE_L^{\dagger}$ with fixed field
$\calE^\dagger$. 
(Minor weirdness: by the Cohen structure theorem, $L$ can always be written
as a power series field $\lambda((u))$, and similarly for $\calE^\dagger_L$.
But if $L$ induces an inseparable residue field extension, then you can't
ensure that $\kappa_K$ can be contained in $\lambda$. I recommend
not worrying about this unless you really have to.)

Let us view $V \otimes_K \calE_L^{\dagger}$
as a $G_{\kappa_K((t))}$-module with the action on the first factor coming from $\tau$
and the action on the second factor as above. Put
\[
D^\dagger(V) = (V \otimes_K \calE_L^{\dagger})^{G_{\kappa_K((t))}}.
\]

\begin{lemma}
The space $D^\dagger(V)$ is an $\calE^\dagger$-vector space of dimension
$\dim_K(V)$.
\end{lemma}
\begin{proof}
This is a consequence of the nonabelian version of Hilbert's Theorem 90:
for any finite Galois extension $E/F$ of fields, the nonabelian cohomology set
$H^1(G_{E/F}, \GL_n(E))$ is trivial.
\end{proof}

Note that $\frac{d}{dt}$ extends uniquely to $\calE_L^\dagger$, and hence to
$D^\dagger(V)$ by taking the action on $V$ to be trivial. Since the action
of $\frac{d}{dt}$ commutes with the Galois action, we also obtain an action on
$D^\dagger(V)$. That is, $D^\dagger(V)$ is a differential module over 
$\calE^\dagger$.

Note that there is a sense in which 
it makes sense to compute the subsidiary radii of $D^\dagger(V) \otimes F_\rho$
for $\rho \in (0,1)$ sufficiently close to 1. Namely, realize $D^\dagger(V)$
as a differential module over $K \langle \alpha/t, t \rrbracket_0$ 
for some $\alpha$ and
compute there. Beware that any two such realizations for a given
$\alpha$ need only become
isomorphic over $K \langle \beta/t, t \rrbracket_0$ for some
$\beta \in [\alpha,1)$. However, the following statement is unambiguous.
\begin{prop} \label{P:solvable}
The generic radius of convergence of $D^\dagger(V) \otimes \calE$ is equal to $1$.
Consequently (by continuity of generic radius of convergence),
$D^\dagger(V)$ is solvable at $1$.
\end{prop}
\begin{proof}
This can be shown directly, but it also follows from the existence of
a Frobenius structure on $D^\dagger(V)$. Namely, fix any Frobenius lift
$\phi$ on $\calE^\dagger$; then $\phi$ extends uniquely to
$\calE^\dagger_L$. Let $\phi$ act on 
$V \otimes_K \calE_L^{\dagger}$ using the trivial action on the first factor;
this action commutes with the  Galois action, so we get a $\phi$-action on
$D^\dagger(V)$ compatible with the derivation.
\end{proof}
Note that the Frobenius structure constructed in the previous proof is
pure of slope 1 (i.e., is \emph{unit-root}), because one can pick a 
Galois-stable lattice in $V$ and do everything integrally. This will allow
us to form a converse assertion; see below.

\section{Ramification and differential slopes}

There is a close relationship between $R(D^\dagger(V) \otimes F_\rho)$
and wild ramification of the representation $V$. To explain this,
I need to recall a bit of classical ramification theory for local fields
(as in Serre's \emph{Local Fields}, Chapter~IV).

Let $F$ be a complete discrete nonarchimedean field whose residue
field $\kappa_F$ is \emph{perfect} (this hypothesis is crucial!).
Let $E$ be a finite Galois extension of $F$. The \emph{lower numbering filtration}
of $G_{E/F}$ is defined as follows: for $i \geq -1$ an integer.
\[
G_{E/F,i} = \ker(G_{E/F} \to \Aut(\gotho_E/\gothm_E^{i+1})).
\]
For $i \geq -1$ real, we define $G_{E/F,i} = G_{E/F,\lceil i \rceil}$.
The lower numbering filtration behaves nicely with respect to subgroups of
$G_{E/F}$ but not quotients; it thus cannot be defined on the absolute Galois
group $G_F$.

The \emph{upper numbering filtration} of $G_{E/F}$ is defined by the relation
$G_{E/F}^{\phi_{E/F}(i)} = G_{E/F,i}$, where
\[
\phi_{E/F}(i) = \int_0^i [G_{E/F,0}:G_{E/F,t}]^{-1}\,dt.
\]
Note that the indices where the filtration jumps are now rational numbers,
but not necessarily integers. In any case, one has the following.
\begin{prop}[Herbrand]
Let $E'$ be a Galois subextension of $E/F$, and put $H = \Gal(E/E')$,
so that $H$ is normal in $G_{E/F}$
and $G_{E/F}/H = G_{E'/F}$. Then $G_{E'/F}^i = (G_{E/F}^i H)/H$;
that is, the upper numbering filtration is compatible with forming
quotients of $G_{E/F}$.
\end{prop}
Consequently, we obtain a filtration $G_F^i$ on $G_F$ which induces
the upper numbering filtration on each $G_{E/F}^i$.

If we take $F = \kappa_K((t))$, we then obtain the following.
(The attribution is somewhat complicated, involving Crew, Matsuda, Tsuzuki,
Christol-Mebkhout, Andr\'e, etc.; see the compiled notes.)
\begin{theorem} \label{T:compare filt}
Assume that $\kappa_K$ is perfect.
Let $V$ be a finite dimensional vector space over $K$,
and let $\tau: G_{\kappa_K((t))} \to \GL(V)$ be a continuous homomorphism 
for the
\emph{discrete} topology on $\GL(V)$.
Then for $\rho \in (0,1)$ sufficiently close to $1$,
\[
R(D^\dagger(V) \otimes F_\rho) = \rho^b, \qquad
b = \max\{i \geq 1: G_{\kappa_K((t))}^i \not\subseteq \ker(\tau)\}.
\]
\end{theorem}
\begin{cor}
Let $V_1,\dots,V_m$ be the constituents of $V$, and let $\tau_j:
G_{\kappa_K((t))} \to \GL(V_j)$ be the corresponding homomorphisms.
For $\rho \in (0,1)$ sufficiently close to $1$, the 
multiset of subsidiary radii
of $D^\dagger(V) \otimes F_\rho$ consists of $\max\{i \geq 1: 
G_{\kappa_K((t)),i} \not\subseteq \ker(\tau_j)\}$
with multiplicity $\dim(V_j)$, for $j=1,\dots,m$.
\end{cor}
Using the integrality properties of subsidiary radii, we may deduce that
for $\rho \in (0,1)$ sufficiently close to 1,
the the product of the subsidiary radii is an integral power of $\rho$;
this amounts to verifying the \emph{Hasse-Arf theorem} for $V$
(integrality of the Artin conductor).

One might reasonably wonder whether there is a good analogue of
Theorem~\ref{T:compare filt} in case the residue field of $K$ is not perfect.
There are several difficulties, one of which is to decide upon a good analogue
of the upper numbering filtration. Such an analogue has been constructed by
Abbes and Saito; the resulting analogue of Theorem~\ref{T:compare filt}
was proved recently by Chiarellotto and Pulita for $\dim(V) = 1$,
and more recently, by Liang Xiao in general.

\section{Representations with finite image of inertia}

Let $\tau: G_{\kappa_K((t))}
 \to \GL(V)$ be a homomorphism which is now continuous for the
$p$-adic topology on $V$, rather than the discrete topology. One can form
a differential module over $\calE$ by taking
\[
D(V) = (V \otimes_K \widehat{\calE^{\unr}})^{G_{\kappa_K((t))}}
\]
but this in general does not descend to $\calE^\dagger$.

Suppose, however, that the image of $G_{\kappa_K((t)),1}
\cong G_{\kappa_K^{\sep}((t))}$ (the inertia subgroup) is finite; 
that is, $\tau$ has \emph{finite
local monodromy}. 
Let $\calE^\dagger_{\kappa_K^{\sep}((t))}$ 
be the ring defined in the same fashion as $\calE^\dagger$
but using $\widehat{K^{\unr}}$ on the coefficients; let
$G_{\kappa_K((t))}$ act on this ring via its unramified quotient.
We can then define
\[
D^\dagger(V) = (V \otimes_K (\calE^\dagger_{\kappa_K^{\sep}((t))})^{\unr})^{G_{\kappa_K((t))}}
\]
and this will be a differential module over $\calE^\dagger$
of the right dimension, again carrying a unit-root Frobenius structure.

\section{Unit-root Frobenius structures}

If $M$ is a differential module over $\calE^\dagger$, we
say $M$ is \emph{quasiconstant} if $M \otimes \calE^\dagger_L$ admits
a basis of horizontal sections for some $L$.

\begin{theorem}[Tsuzuki] \label{T:unit-root plmt}
Let $M$ be a finite differential module over $\calE^\dagger$ admitting 
a unit-root Frobenius structure for some Frobenius lift. 
Then $M$ is quasiconstant.
\end{theorem}
It is important to note that the existence of a unit-root Frobenius
structure for one Frobenius lift implies the same for any other Frobenius lift.
In fact, for the proof we need to make a more precise observation,
which is easy to check from the change of Frobenius construction.
Let us say that for $c \in [0,1)$, a basis $e_1,\dots,e_n$ of $M$
is \emph{$c$-constant} if $\Phi$ acts on this basis via a matrix
$A = \sum_i A_i t^i$ satisfying $|A|_1 = |A^{-1}| = 1$ and $|A-A_0|_1 \leq c$.

\begin{lemma}
Let $M$ be a finite differential module over $\calE^\dagger$ admitting 
a unit-root Frobenius structure for some Frobenius lift $\phi_1$.
Suppose that $e_1,\dots,e_n$ is a $c$-constant basis. Then
$e_1,\dots, e_n$ is also $c$-constant for the Frobenius structure corresponding
to any other Frobenius lift $\phi_2$.
\end{lemma}

Given this lemma, the proof can be broken into three steps; we refer to
Tsuzuki's original paper (\textit{Amer. J. Math.} 1998) for the proofs.
\begin{lemma}
Let $M$ be a finite differential module over $\calE^\dagger$ admitting 
a unit-root Frobenius structure. Then there exists a positive integer $m$
coprime to $p$ such that $M \otimes \calE^\dagger[t^{1/m}]$ admits
a $c$-constant basis for some $c \in (0,1)$.
\end{lemma}
\begin{lemma}
Let $M$ be a finite differential module over $\calE^\dagger$ admitting 
a unit-root Frobenius structure and a $c$-constant basis for some
$c \in (0,1)$. Then for some finite extension $L$ of $\kappa_K((t))$,
$M \otimes \calE^\dagger_L$ admits a $c'$-constant basis for some
$c' \in (0,c)$.
\end{lemma}

These first two lemmas are easy for an absolute Frobenius lift, because
you can actually choose the basis to be fixed modulo some power of $\gothm_K$.
This is the only case Tsuzuki originally addressed; however, in the compiled
notes, I will explain how to modify the argument slightly to work for
a general Frobenius lift.

\begin{lemma}
Let $M$ be a finite differential module over $\calE^\dagger$ admitting 
a unit-root Frobenius structure. 
Suppose that $M$ admits a $c$-constant basis for some $c < p^{-1/(p-1)}$.
Then $M$ is constant.
\end{lemma}
As noted by Christol, this last lemma  can be proved elegantly  using Frobenius 
antecedents.

\section{Quasiunipotent differential modules}

Define the \emph{Robba ring} to be
\[
\calR = \cup_{\alpha \in (0,1)} K \langle \alpha/t, t \}\};
\]
that is, $\calR$ consists of formal sums $\sum c_i t^i$ which
converge in some range $\alpha \leq |t| < 1$, but need not have bounded
coefficients. 
Unlike its subring $\calE^\dagger$, $\calR$ is not a field; for instance,
the element
\[
\log(1+t) = \sum_{i=1}^{\infty} \frac{(-1)^{i-1}}{i} t^i
\]
is not invertible (because its Newton polygon has infinitely many slopes).
More generally, we have the following easy fact.
\begin{lemma}
We have $\calR^\times = (\calE^\dagger)^\times$.
\end{lemma}
In particular, $\calR$ does not have a natural $p$-adic topology. The most useful
topology on $\calR$ is the \emph{LF topology}, which is the direct limit of the
Fr\'echet topology on each $K \langle \alpha/t, t\}\}$ defined by
the $|\cdot|_\rho$ for $\rho \in [\alpha,1)$.

In fact, the ring $\calR$ is not even noetherian (this is related to an earlier
exercise), but the following useful facts are true, essentially by work of
Lazard.
(These depend on $K$ being spherically
complete, which follows from our hypothesis that $K$ is in fact discretely valued.)
\begin{prop}
For an ideal $I$ of $\calR$, the following are equivalent.
\begin{enumerate}
\item[(a)] The ideal $I$ is closed in the LF topology.
\item[(b)] The ideal $I$ is finitely generated.
\item[(c)] The ideal $I$ is principal.
\end{enumerate}
\end{prop}
\begin{prop}
Any finite free module on the half-open annulus with closed inner radius
$\alpha$ and open outer radius $1$ is represented by a finite free module
over $K \langle \alpha/t, t \}\}$, and so corresponds to a finite free module
over $\calR$. (The first part generalizes to half-open and open annuli
with arbitrary boundary radii.)
\end{prop}

For $L$ a finite separable extension of $\kappa_K((t))$, put
\[
\calR_L = \calR \otimes_{\calE^\dagger} \calE^\dagger_L.
\]
We say a finite differential module $M$ over $\calR$ is \emph{quasiconstant} if 
there exists $L$ such that $M \otimes \calR_L$ is trivial. We say $M$
is \emph{quasiunipotent} if it is a successive extension of quasiconstant
modules; it is equivalent to ask that $M \otimes \calE^\dagger_L$
be unipotent (i.e., an extension of trivial differential modules)
for some $L$ (exercise).

Quasiunipotent differential modules have many useful properties.
For instance, by Proposition~\ref{P:solvable}, they are all solvable at $1$.
Another important property is the following.
\begin{prop} \label{P:perfect pairing}
Let $M$ be a quasiunipotent differential module over $\calR$. Then the spaces
$H^0(M), H^1(M)$ are finite dimensional, and there is a
perfect pairing
\[
H^0(M) \times H^1(M^\dual) \to H^1(M \otimes M^\dual) \to H^1(\calR) 
\cong K \frac{dt}{t}.
\]
\end{prop}
\begin{proof}
This can be reduced to the unipotent case, for which it is an exercise.
\end{proof}

The following important theorem asserts that many naturally occurring
differential modules, including Picard-Fuchs modules, are quasiunipotent.
See the notes for further discussion.
\begin{theorem}[$p$-adic local monodromy theorem] \label{T:plmt}
Let $M$ be a finite differential module over $\calR$ admitting a Frobenius
structure for some Frobenius lift. Then $M$ is quasiunipotent.
\end{theorem}

We will have more to say about this theorem later. 

\notes

The weak topology on $\calE$ is called the \emph{levelwise topology} in
[Ked04].

The statement of the $p$-adic local monodromy theorem (Theorem~\ref{T:plmt})
was originally known under the name
\emph{Crew's conjecture}, because it emerged from 
the work of Crew on finite dimensionality of rigid cohomology with
coefficients in an overconvergent $F$-isocrystal. 
The original conjecture only concerned modules such that the differential
and Frobenius structures were both defined over $\calE^\dagger$;
this form was restated in a more geometric form by de Jong.

The restricted case of Crew's conjecture just described is the one
that appears in applications to $p$-adic cohomology. However, the
general form is in many ways more natural; this was illustrated by 
the work of Tsuzuki, who explained how 
for an absolute Frobenius lift, Theorem~\ref{T:plmt} 
would follow from a slope filtration theorem.
Moreover, the proof by Berger that Crew's conjecture implies Fontaine's
conjecture $C_{pst}$ (that de Rham representations are potentially
semistable) requires the unrestricted form of Crew's conjecture.

There are essentially two methods for proving Theorem~\ref{T:plmt},
each with its own merits. One method is to follow Tsuzuki's suggestion
to construct slope filtrations for 
difference modules; this was carried out by Kedlaya.
(Beware that Tsuzuki's original reduction argument only applies in the case of 
an absolute Frobenius lift; for the general case, you have to modify it
as sketched in these notes.)

The second method is to use various results of Christol-Mebkhout
to analyze differential modules which are solvable at 1.
This method was carried out by Andr\'e 
and Mebkhout (independently of each other and of Kedlaya).

\exercises

\begin{enumerate}
\item
Let $M$ be a differential module over $\calR$ such that for some
finite separable extension of $L$, $M \otimes \calE^\dagger_L$
is unipotent. Prove that $M$ is quasiunipotent.
\item
Prove Proposition~\ref{P:perfect pairing} in the case where $M$ is unipotent.
\end{enumerate}

\end{document}

In the meantime, let
us verify it in the case of rank $1$, assuming that $\kappa_K$ is perfect
and $K$ contains a $p$-th root of unity $\zeta_p$. (The last two assumptions are easy
to get rid of.)
\begin{proof}[Proof of Theorem~\ref{T:plmt} for $\rank(M) = 1$]
Since it does not matter which Frobenius lift we use, we take a standard Frobenius
$\phi(t) = t^q$ for some power $q$ of $p$ (and any isometry $\phi_K$).
Also, we will write everything in terms of the derivation $t \frac{d}{dt}$
rather than $\frac{d}{dt}$.

Choose a generator $v$ of $M$. We then have $\Phi(v) = av$ for some
$a \in \calR$ which must be a unit, forcing $a \in \calE^{\dagger}$;
there is no harm in normalizing so that $|a|_1 = 1$.
We also have $D(v) = nv$ for some $n \in \calR$ satisfying
\[
na + a' = q a\phi(n),
\]
or $n - q \phi(n) = a'/a$. This forces $n \in \calE^\dagger$:
if $n = \sum n_i t^i$, we deduce
$|n_i| \leq |a'/a|_1$ first for $i$ not divisible by $q$, then for $i$
not divisible by $q^2$, and so on.

We also claim that $n_0 \in \ZZ_p \cap \QQ$. Namely, write $a = \sum a_i t^i$ and
choose the least index $j$ for which $|a_j| = |a|_1$. By the master
factorization theorem, we can factor $a_j^{-1} t^{-j} a = a_+ a_-$ where
$a_+ \in \gotho_K \llbracket t \rrbracket^\times$
and $a_- \in (\gotho_K \llbracket t^{-1} \rrbracket \cap K \langle \alpha/t
\rangle)^\times$ for some $\alpha \in (0,1)$. We then have
\[
a'/a = j + a_+'/a_+ + a_-'/a_-
\]
and the terms $a_+'/a_+$ and $a_-'/a_-$ contribute zero to the constant term.
So the constant term $m$ of $a'/a$ is an integer, and $n_0$ satisfies
$n_0 - q \phi_K(n_0) = m$. Since $\phi_K$ must fix $\QQ_p$, we deduce
$n_0 = m + qm + q^2m + \cdots \in \ZZ_p \cap \QQ$.

It thus suffices to consider the case $n_0 = 0$ hereafter,
as we can reduce to this case by 
adjoining a suitable prime-to-$p$ root of $t$, relabeling that root as 
$t$, and adjusting $a$ by a power of said new $t$.
In this case, we observe that $M^{\otimes p^h}$ is constant for some $h$;
namely, we have
$D(v^{\otimes p^h}) = p^h n v^{\otimes p^h}$, and for $h$ sufficiently large,
the series $u = \exp(-\int (n/t)\,dt)$ converges to give
the horizontal section $u v^{\otimes p^h}$ of $M^{\otimes p^h}$.

It thus suffices to check that if $M^{\otimes p}$ is constant, then
$M$ is quasi-constant. That is, we must show that for some $u \in \calE^\dagger$
with $pn = u'/u$, we can find a $p$-th root of $u$ in some
$\calE^\dagger_L$.

Choose any $u = \sum u_i t^i \in \calE^\dagger$ with $pn = u'/u$, normalized so that
$|u|_1 = 1$. Since $u' \equiv 0 \pmod{p}$, 
we know that $|u_i| \leq p^{-1}$ if $i \notequiv 0 \pmod{p}$.
Now put $w = \sum_{i \in \ZZ} w_i t^i$ where
$w_i$ lifts the $p$-th root of the image of $u_{pi}$ in $\kappa_K$.
Then 
\[
u \equiv w^p \pmod{p},
\]

$w_i^p \equiv u_{pi} \pmod{p}$.
we can write $u$ as the sum of something of norm at most $p^{-1}$ plus
something involving only $p$-th powers. The latter is cong
Note that $n \equiv a'/a \pmod{p}$, so $u'/u \equiv pa'/a \pmod{p^2}$.

we must show that $u$ has a $p$-th root in $\calE^\dagger_L$ for some $L$.

\end{proof}