$\displaystyle\tfrac{1}{2}\big{(}\deg(\underline{\zeta})-\deg(\underline{\xi})\big{)}+\Delta_{k_{0},n-\frac{1}{2}d_{k_{0}}^{\mathrm{Iw}}}+\tfrac{k_{0}-2}{2}(n-\tfrac{1}{2}d_{k_{0}}^{\mathrm{Iw}})-v_{p}\big{(}g_{n,\hat{k},\hat{k}_{0}}(w_{k_{0}})\big{)}-m_{n}(k)\cdot
v_{p}(w_{k_{0}}-w_{k})$ $\displaystyle=\ $
$\displaystyle\tfrac{1}{2}\big{(}\deg(\underline{\zeta})-\deg(\underline{\xi})\big{)}+\Delta_{k_{0},\frac{1}{2}d_{k_{0}}^{\mathrm{Iw}}-n}-\Delta^{\prime}_{k_{0},\frac{1}{2}d_{k_{0}}^{\mathrm{Iw}}-n}.$

This is the same as (5.5.2) (when $i=m_{n}(k_{0})$).

The case when $i>m_{n}(k_{0})$ is easier (compared to the proof of Proposition
4.7). In this case, we will prove an inequality stronger than (5.5.2) without
the $\Delta-\Delta^{\prime}$ at the end. The estimate (5.2.1) we assumed
implies that, for $j=0,\dots,m_{n}(k)-1$,

$\displaystyle
v_{p}\big{(}A_{k,j}^{(\underline{\zeta}\times\underline{\xi})}\big{)}$
$\displaystyle\geq$
$\displaystyle\tfrac{1}{2}\big{(}\deg(\underline{\zeta})-\deg(\underline{\xi})\big{)}+\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-j}-\Delta^{\prime}_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-m_{n}(k)}$
$\displaystyle\geq$
$\displaystyle\tfrac{1}{2}\big{(}\deg(\underline{\zeta})-\deg(\underline{\xi})\big{)}+1+\tfrac{1}{2}(\tfrac{1}{2}d_{k}^{\mathrm{new}}-j)^{2}-\tfrac{1}{2}(\tfrac{1}{2}d_{k}^{\mathrm{new}}-m_{n}(k))^{2},$

where the second inequality follows from Proposition 2.19. So we need to prove
the following inequality:

$\displaystyle\tfrac{1}{2}(\tfrac{1}{2}d_{k}^{\mathrm{new}}-m_{n}(k_{0}))^{2}-\tfrac{1}{2}(\tfrac{1}{2}d_{k}^{\mathrm{new}}-i)^{2}+1+\tfrac{1}{2}(\tfrac{1}{2}d_{k}^{\mathrm{new}}-j)^{2}-\tfrac{1}{2}(\tfrac{1}{2}d_{k}^{\mathrm{new}}-m_{n}(k))^{2}$
$\displaystyle\geq(1+v_{p}(k_{\bullet}-k_{0\bullet}))\cdot\big{(}(i-m_{n}(k_{0}))+(m_{n}(k)-j)\big{)}.$

To simplify notations, we set $\gamma:=v_{p}(k_{\bullet}-k_{0\bullet})$,
$x=\frac{1}{2}d_{k_{0}}^{\mathrm{new}}-m_{n}(k_{0})>y=\frac{1}{2}d_{k_{0}}^{\mathrm{new}}-i\geq
0$, and
$z=\frac{1}{2}d_{k}^{\mathrm{new}}-j>w=\frac{1}{2}d_{k}^{\mathrm{new}}-m_{n}(k)\geq
0$. We need to prove that

(5.5.3) $x^{2}-y^{2}+2+z^{2}-w^{2}\geq 2(1+\gamma)(x-y+z-w).$

When $\gamma=0$, the inequality (5.5.3) is straightforward. So we assume
$\gamma\geq 1$ below. Note that $|k_{\bullet}-k_{0\bullet}|\geq p^{\gamma}$
and so
$|\frac{1}{2}d_{k}^{\mathrm{Iw}}-\frac{1}{2}d_{k_{0}}^{\mathrm{Iw}}|\geq
p^{\gamma}$. So we have $x+w\geq p^{\gamma}$. We separate several cases:

  * •

If $x+y\geq 2\gamma+2$ and $z+w\geq 2\gamma+2$, the inequality (5.5.3) is
clear.

  * •

If $z+w\leq 2\gamma+1$, then $w\leq\gamma$. The condition $x+w\geq p^{\gamma}$
implies that $x\geq p^{\gamma}-w$. So

$\displaystyle(z-w)(2+2\gamma-z-w)$ $\displaystyle\leq$
$\displaystyle\Big{(}\frac{(z-w)+2+2\gamma-
z-w}{2}\Big{)}^{2}=(\gamma+1-w)^{2},$ $\displaystyle(x-y)(x+y-(2+2\gamma))$
$\displaystyle\geq$ $\displaystyle p^{\gamma}-w-2-2\gamma.$

The difference of the term (plus $2$) is

$p^{\gamma}-w-2\gamma-(\gamma+1-w)^{2}\geq
p^{\gamma}-3\gamma-(\gamma+1)^{2}\geq 0\textrm{ if }\gamma\geq 2.$

When $\gamma=1$ and $w=1$, the left hand side is also larger than $0$. When
$\gamma=1$ and $w=0$, we have $x\geq p$, and so

$(x-y)(x+y-4)\geq 2x-5\geq 5\quad\textrm{and}\quad(z-w)(4-z-w)\leq 4.$

(5.5.3) still holds.

  * •

If $x+y\leq 2\gamma+1$, a similar argument proves (5.5.3); we leave this as an
exercise for interested readers.

This completes the proof of (2) assuming (1).

We now turn to prove (1) of Proposition 5.4. The proof resembles the proof of
Claim 1 in Proposition 4.4. By (5.2.1) and Proposition 2.19,

$\displaystyle v_{p}(A_{k,i}^{(\underline{\zeta}\times\underline{\xi})})\geq\
$
$\displaystyle\tfrac{1}{2}\big{(}\deg(\underline{\zeta})-\deg(\underline{\xi})\big{)}+\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i}-\Delta^{\prime}_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-m_{n}(k)}$
$\displaystyle\geq\ $
$\displaystyle\tfrac{1}{2}\big{(}\deg(\underline{\zeta})-\deg(\underline{\xi})\big{)}+m_{n}(k)-i$
$\displaystyle A_{k}^{(\underline{\zeta}\times\underline{\xi})}(w)$
$\displaystyle\,\cdot g_{n,\hat{k}}(w)\in
p^{\frac{1}{2}(\deg(\zeta)-\deg(\xi))+\deg g_{n}(w)}\cdot\mathcal{O}[w/p].$

So if we can prove that

(5.5.4)
$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}\in
p^{\frac{1}{2}(\deg(\underline{\zeta})-\deg(\underline{\xi}))+\deg
g_{n}(w)}\cdot\mathcal{O}\langle w/p\rangle,$

then we would deduce that $h_{\underline{\zeta}\times\underline{\xi}}(w)\cdot
g_{n}(w)\in p^{\frac{1}{2}(\deg(\underline{\zeta})-\deg(\underline{\xi}))+\deg
g_{n}(w)}\cdot\mathcal{O}\langle w/p\rangle$, from this it would follow that
$h_{\underline{\zeta}\times\underline{\xi}}(w)\in
p^{\frac{1}{2}(\deg(\underline{\zeta})-\deg(\underline{\xi}))}\cdot\mathcal{O}\langle
w/p\rangle$. So we now focus on proving (5.5.4).

We go back to the discussion of halo estimate near the end of Section 3.
Recall the matrix $\mathrm{U}_{\mathbf{C}}$ and $\mathrm{Y}$ from Notation
3.17. For each ordered tuple
$\underline{\eta}=(\eta_{1},\dots,\eta_{n})\in\mathbb{N}^{n}$, write
$\mathrm{U}_{\mathbf{C}}(\underline{\zeta}\times\underline{\eta})$ for the
submatrices with rows in $\underline{\zeta}$ and columns in
$\underline{\eta}$. Then the equality
$\mathrm{U}^{\dagger}=\mathrm{Y}\mathrm{U}_{\mathbf{C}}\mathrm{Y}^{-1}$ of
(3.17.1) implies that

(5.5.5)
$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}=\sum_{\underline{\lambda},\underline{\eta}}\Big{(}\mathrm{Y}_{\mathbf{e}_{\zeta_{1}},\mathbf{e}_{\lambda_{1}}}\cdots\mathrm{Y}_{\mathbf{e}_{\zeta_{n}},\mathbf{e}_{\lambda_{n}}}\cdot\mathrm{det}\big{(}\mathrm{U}_{\mathbf{C}}(\underline{\lambda}\times\underline{\eta})\big{)}\cdot(\mathrm{Y}^{-1})_{\mathbf{e}_{\eta_{1}},\mathbf{e}_{\xi_{1}}}\cdots(\mathrm{Y}^{-1})_{\mathbf{e}_{\eta_{n}},\mathbf{e}_{\xi_{n}}}\Big{)},$

where the sum runs over all ordered tuples
$\underline{\lambda}=(\lambda_{1},\dots,\lambda_{n}),\underline{\eta}=(\eta_{1},\dots,\eta_{n})\in\mathbb{N}^{n}$.
It is enough to check (5.5.4) for each term above when
$\lambda_{i}\geq\zeta_{i}$ and $\eta_{i}\leq\xi_{i}$ for every $i$ (as
$\mathrm{Y}$ and $\mathrm{Y}^{-1}$ are upper triangular by Lemma 3.16).

Note that the condition $\lambda_{i}\geq\zeta_{i}$ and Lemma 3.16 imply that

$\displaystyle
v_{p}(\mathrm{Y}_{\mathbf{e}_{\zeta_{i}},\mathbf{e}_{\lambda_{i}}})+\frac{1}{2}\big{(}\deg\mathbf{e}_{\lambda_{i}}-\deg\mathbf{e}_{\zeta_{i}}\big{)}+v_{p}(\deg\mathbf{e}_{\lambda_{i}}!)$
$\displaystyle\geq\ \ $
$\displaystyle\frac{\deg\mathbf{e}_{\lambda_{i}}-\deg\mathbf{e}_{\zeta_{i}}}{2}+v_{p}\Big{(}\frac{\deg\mathbf{e}_{\lambda_{i}}!}{\deg\mathbf{e}_{\zeta_{i}}!}\Big{)}+\Big{\lfloor}\frac{\deg\mathbf{e}_{\zeta_{i}}}{p}\Big{\rfloor}-\Big{\lfloor}\frac{\deg\mathbf{e}_{\lambda_{i}}}{p}\Big{\rfloor}-\Big{\lfloor}\frac{\deg\mathbf{e}_{\lambda_{i}}-\deg\mathbf{e}_{\zeta_{i}}}{p^{2}-p}\Big{\rfloor}$
$\displaystyle\geq\ \ $
$\displaystyle\frac{\deg\mathbf{e}_{\lambda_{i}}-\deg\mathbf{e}_{\zeta_{i}}}{2}+v_{p}\Big{(}\Big{\lfloor}\frac{\deg\mathbf{e}_{\lambda_{i}}}{p}\Big{\rfloor}!\Big{)}-v_{p}\Big{(}\Big{\lfloor}\frac{\deg\mathbf{e}_{\zeta_{i}}}{p}\Big{\rfloor}!\Big{)}-\Big{\lfloor}\frac{\deg\mathbf{e}_{\lambda_{i}}-\deg\mathbf{e}_{\zeta_{i}}}{p^{2}-p}\Big{\rfloor}\geq
0$

By a similar argument, the condition $\xi_{i}\geq\eta_{i}$ and Lemma 3.16
imply that

$v_{p}\big{(}(\mathrm{Y}^{-1})_{\mathbf{e}_{\eta_{i}},\mathbf{e}_{\xi_{i}}}\big{)}\geq\frac{1}{2}\big{(}\deg\mathbf{e}_{\eta_{i}}-\deg\mathbf{e}_{\xi_{i}}\big{)}+v_{p}(\deg\mathbf{e}_{\eta_{i}}!).$

So to prove that each term of the right hand side of (5.5.5) belongs to
$p^{\frac{1}{2}(\deg(\underline{\zeta})-\deg(\underline{\xi}))+\deg
g_{n}(w)}\cdot\mathcal{O}\langle w/p\rangle$, it suffices to show the
following

Claim: Assume that $\lambda_{1}<\cdots<\lambda_{n}$ and
$\eta_{1}<\cdots<\eta_{n}$. We have
$v_{p}\big{(}\mathrm{det}\big{(}\mathrm{U}_{\mathbf{C}}(\underline{\lambda}\times\underline{\eta})\big{)}\big{)}$
(meaning the $p$-adic valuation in the ring $\mathcal{O}\langle w/p\rangle$)
is greater than or equal to

$\displaystyle\tfrac{\deg(\underline{\zeta})-\deg(\underline{\xi})}{2}+\deg
g_{n}(w)+\sum_{i=1}^{n}\Big{(}\tfrac{\deg\mathbf{e}_{\lambda_{i}}-\deg\mathbf{e}_{\zeta_{i}}}{2}+v_{p}(\deg\mathbf{e}_{\lambda_{i}}!)+\tfrac{\deg\mathbf{e}_{\xi_{i}}-\deg\mathbf{e}_{\eta_{i}}}{2}-v_{p}(\deg\mathbf{e}_{\eta_{i}}!)\Big{)}$
$\displaystyle=\ $ $\displaystyle\deg
g_{n}(w)+\frac{\deg(\underline{\lambda})-\deg(\underline{\eta})}{2}+\sum_{i=1}^{n}v_{p}\Big{(}\frac{\deg\mathbf{e}_{\lambda_{i}}!}{\deg\mathbf{e}_{\eta_{i}}!}\Big{)}.$

To be extremely careful about the boundary case, we set

(5.5.6) $\boldsymbol{\delta}:=\deg
g_{n}(w)-\sum_{i=1}^{n}\Big{(}\deg\mathbf{e}_{i}-\Big{\lfloor}\frac{\deg\mathbf{e}_{i}}{p}\Big{\rfloor}\Big{)}\stackrel{{\scriptstyle\eqref{E:degree
approx halo bound}}}{{\in}}\\{0,1\\}.$

Moreover, we point out that $\boldsymbol{\delta}=1$ can only possibly happen
when $\deg\mathbf{e}_{n+1}-\deg\mathbf{e}_{n}=p-1-a$. We first treat two
special cases of the Claim, representing different strategies, which are
introduced to treat with the subtlety that $\boldsymbol{\delta}$ might be $1$.

  * (i)

when $\underline{\lambda}=\underline{n}$ and
$\underline{\eta}\neq\underline{n}$, Proposition 3.18 implies that

$v_{p}\big{(}\mathrm{det}\mathrm{U}_{\mathbf{C}}(\underline{n}\times\underline{\eta})\big{)}\geq\sum_{i=1}^{n}\Big{(}\deg\mathbf{e}_{i}-\Big{\lfloor}\frac{\deg\mathbf{e}_{\eta_{i}}}{p}\Big{\rfloor}\Big{)}\stackrel{{\scriptstyle\eqref{E:bold
delta}}}{{=}}\deg
g_{n}(w)-\boldsymbol{\delta}-\sum_{i=1}^{n}\Big{(}\Big{\lfloor}\frac{\deg\mathbf{e}_{\eta_{i}}}{p}\Big{\rfloor}-\Big{\lfloor}\frac{\deg\mathbf{e}_{i}}{p}\Big{\rfloor}\Big{)}.$

Comparing this with the Claim, it suffices to prove

$\sum_{i=1}^{n}\bigg{(}\frac{\deg\mathbf{e}_{\eta_{i}}-\deg\mathbf{e}_{i}}{2}+v_{p}\Big{(}\frac{\deg\mathbf{e}_{\eta_{i}}!}{\deg\mathbf{e}_{i}!}\Big{)}-\Big{\lfloor}\frac{\deg\mathbf{e}_{\eta_{i}}}{p}\Big{\rfloor}+\Big{\lfloor}\frac{\deg\mathbf{e}_{i}}{p}\Big{\rfloor}\bigg{)}\geq\boldsymbol{\delta}.$

We may assume that $\eta_{i}\geq i$ for each $i$ and $\eta_{n}\geq n+1$. Then
the needed inequality follows from combining the following two inequalities:

$\frac{\deg\mathbf{e}_{n+1}-\mathbf{e}_{n}}{2}\geq\boldsymbol{\delta}\quad\textrm{and}\quad
v_{p}\Big{(}\frac{\deg\mathbf{e}_{\eta_{i}}!}{\deg\mathbf{e}_{i}!}\Big{)}-\Big{\lfloor}\frac{\deg\mathbf{e}_{\eta_{i}}}{p}\Big{\rfloor}+\Big{\lfloor}\frac{\deg\mathbf{e}_{i}}{p}\Big{\rfloor}=v_{p}\Big{(}\frac{\lfloor\deg\mathbf{e}_{\eta_{i}}/p\rfloor!}{\lfloor\deg\mathbf{e}_{i}/p\rfloor!}\Big{)}\geq
0,$

where the first inequality uses the precise condition when
$\boldsymbol{\delta}$ could be equal to $1$ as discussed above.

  * (ii)

When $\underline{\lambda}=\\{1,\dots,n-1,n+1\\}$ and
$\underline{\eta}=\underline{n}$. Let $\gamma$ be the largest nonnegative
integer such that $p^{\gamma}$ divides a number in
$\\{\deg\mathbf{e}_{n}+1,\dots,\deg\mathbf{e}_{n+1}\\}$. Then
$v_{p}(\deg\mathbf{e}_{n+1}!)-v_{p}(\deg\mathbf{e}_{n}!)=\gamma$. We need to
prove

(5.5.7)
$v_{p}\big{(}\mathrm{det}\big{(}\mathrm{U}_{\mathbf{C}}(\underline{\lambda},\underline{n})\big{)}\big{)}\geq\deg
g_{n}(w)+\frac{\deg\mathbf{e}_{n+1}-\deg\mathbf{e}_{n}}{2}+\gamma.$

By Corollary 3.27, we have

$v_{p}\big{(}\mathrm{det}\big{(}\mathrm{U}_{\mathbf{C}}(\underline{\lambda},\underline{n})\big{)}\big{)}\geq
D(\underline{\lambda},\underline{n})+\sum_{i=1}^{n}\Big{(}\deg\mathbf{e}_{i}-\Big{\lfloor}\frac{\deg\mathbf{e}_{i}}{p}\Big{\rfloor}\Big{)}+\big{(}\deg\mathbf{e}_{n+1}-\deg\mathbf{e}_{n}\big{)}.$

It suffices to prove that

$D(\underline{\lambda},\underline{n})+\frac{\deg\mathbf{e}_{n+1}-\deg\mathbf{e}_{n}}{2}\geq\boldsymbol{\delta}+\gamma.$

As $\boldsymbol{\delta}=1$ only happens when
$\deg\mathbf{e}_{n+1}-\mathbf{e}_{n}=p-1-a$, the condition $2\leq a\leq p-5$
is enough to imply that
$\frac{\deg\mathbf{e}_{n+1}-\deg\mathbf{e}_{n}}{2}\geq\boldsymbol{\delta}+1$.
On the other hand, we use Lemma 3.29 to note that
$D_{=0}(\underline{n},0)=\cdots=D_{=0}(\underline{n},\gamma-1)$, so for every
$j=0,\dots,\gamma-2$

$D_{=0}(\underline{\lambda},j)=D_{=0}(\underline{n},j+1)+1.$

It follows that $D(\underline{\lambda},\underline{n})\geq\gamma-1$. This Claim
in this case is proved.

We remark that, the proof of (i) follows from standard halo estimate
Proposition 3.18. On the other hand, as shown by the proof of (ii), the usual
halo bound Proposition 3.18 cannot be used to control the $\gamma$ on the
right hand side of (5.5.7). The subtle improvement of halo estimate in
Corollary 3.27 is essential for this proof.

We now prove Claim under the assumption that
$\underline{\lambda}\neq\underline{n}$, which share certain similarity with
the proof of (ii). By Corollary 3.27, it suffices to show that

$D(\underline{\lambda},\underline{\eta})+\sum_{i=1}^{n}\Big{(}\deg\mathbf{e}_{\lambda_{i}}-\Big{\lfloor}\frac{\deg\mathbf{e}_{\eta_{i}}}{p}\Big{\rfloor}\Big{)}\geq\deg
g_{n}(w)+\sum_{i=1}^{n}\bigg{(}\frac{\deg\mathbf{e}_{\lambda_{i}}-\deg\mathbf{e}_{\eta_{i}}}{2}+v_{p}\Big{(}\frac{\deg\mathbf{e}_{\lambda_{i}}!}{\deg\mathbf{e}_{\eta_{i}}!}\Big{)}\bigg{)},$

or equivalently, to show that

(5.5.8)
$D(\underline{\lambda},\underline{\eta})+\sum_{i=1}^{n}\bigg{(}\frac{\deg\mathbf{e}_{\lambda_{i}}+\deg\mathbf{e}_{\eta_{i}}}{2}+v_{p}\Big{(}\Big{\lfloor}\frac{\deg\mathbf{e}_{\eta_{i}}}{p}\Big{\rfloor}!\Big{)}\bigg{)}\geq\deg
g_{n}(w)+\sum_{i=1}^{n}v_{p}(\deg\mathbf{e}_{\lambda_{i}}!).$

We first reduce to the case when $\underline{\eta}=\underline{n}$. For this,
it suffices to show that for a subset
$\underline{\eta}^{\prime}\subset\mathbb{N}$ of size $n$ with
$\eta^{\prime}_{i}=\eta_{i}$ for all $i$ except some $i=i_{0}$ when
$\eta^{\prime}_{i_{0}}-\eta_{i_{0}}=1$, we have

(5.5.9)
$D(\underline{\lambda},\underline{\eta}^{\prime})+\frac{\deg\mathbf{e}_{\eta^{\prime}_{i_{0}}}-\deg\mathbf{e}_{\eta_{i_{0}}}}{2}+v_{p}\bigg{(}\frac{\lfloor\deg\mathbf{e}_{\eta^{\prime}_{i_{0}}}/p\rfloor!}{\lfloor\deg\mathbf{e}_{\eta_{i_{0}}}/p\rfloor!}\bigg{)}\geq
D(\underline{\lambda},\underline{\eta}).$

The condition $2\leq a\leq p-5$ implies that
$\deg\mathbf{e}_{\eta_{i_{0}}}-\deg\mathbf{e}_{\eta^{\prime}_{i_{0}}}\geq 2$;
so (5.5.9) follows from (3.26.1).

Now, we assume $\underline{\eta}=\underline{n}$. It remains to show for any
subset $\underline{\lambda}\subseteq\mathbb{N}$ (but
$\underline{\lambda}\neq\underline{n}$),

(5.5.10)
$D(\underline{\lambda},\underline{n})+\sum_{i=1}^{n}\frac{\deg\mathbf{e}_{\lambda_{i}}-\deg\mathbf{e}_{i}}{2}\geq\boldsymbol{\delta}+\sum_{i=1}^{n}v_{p}\Big{(}\frac{\deg\mathbf{e}_{\lambda_{i}}!}{\deg\mathbf{e}_{i}!}\Big{)}.$

Moreover, we may assume that $\underline{\lambda}\neq\\{1,\dots,n-1,n+1\\}$ as
it has been treated in (ii).

For this, we make an induction on $\underline{\lambda}$; each time, we replace
the largest element in $\underline{\lambda}$, say $\lambda_{n}$, by the
smallest element in $\underline{n}$ but not in $\underline{\lambda}$, say
$n_{-}$. Since we have ruled out two special cases of $\underline{\lambda}$,
we must have $\lambda_{n}-n_{-}\geq 2$.

Write
$\underline{\lambda}^{\prime}=\underline{\lambda}\cup\\{n_{-}\\}\backslash\\{\lambda_{n}\\}$.
We need to prove

$D(\underline{\lambda},\underline{n})+\frac{\deg\mathbf{e}_{\lambda_{n}}-\deg\mathbf{e}_{n_{-}}}{2}\geq\boldsymbol{\delta}+D(\underline{\lambda}^{\prime},\underline{n})+v_{p}\Big{(}\frac{\deg\mathbf{e}_{\lambda_{n}}!}{\deg\mathbf{e}_{n_{-}}!}\Big{)}.$

Indeed, if $\underline{\lambda}^{\prime}\neq\underline{n}$, we did not even
need the $\boldsymbol{\delta}$ on the right hand side to complete the
induction. If $\underline{\lambda}^{\prime}=\underline{n}$, the above
inequality is equivalent to (5.5.10) by Lemma 3.29.

Write $\gamma$ for the maximal $p$-adic valuation for integers between
$\deg\mathbf{e}_{n_{-}}+1$ and $\deg\mathbf{e}_{\lambda_{n}}$; so that we must
have
$v_{p}\big{(}\frac{\deg\mathbf{e}_{\lambda_{n}}!}{\deg\mathbf{e}_{n_{-}}!}\big{)}\leq\gamma+\big{\lfloor}\frac{\deg\mathbf{e}_{\lambda_{n}}-\deg\mathbf{e}_{n_{-}}-2}{p-1}\big{\rfloor}$.
We need to prove

(5.5.11)
$D(\underline{\lambda},\underline{n})+\frac{\deg\mathbf{e}_{\lambda_{n}}-\deg\mathbf{e}_{n_{-}}}{2}-\Big{\lfloor}\frac{\deg\mathbf{e}_{\lambda_{n}}-\deg\mathbf{e}_{n_{-}}-2}{p-1}\Big{\rfloor}\geq\boldsymbol{\delta}+D(\underline{\lambda}^{\prime},\underline{n})+\gamma.$

Put $\delta$ to be unique integer such that
$\deg\mathbf{e}_{\lambda_{n}}-\deg\mathbf{e}_{n_{-}}\in\big{(}(p-1)p^{\delta-1},(p-1)p^{\delta}\big{]}$;
it is clear that $\gamma\geq\delta-1$. The case when $\gamma\leq\delta$ is
easier, which we discuss first. In this case, when changing
$\deg\mathbf{e}_{\lambda_{n}}$ to $\deg\mathbf{e}_{n_{-}}$, only the last
$\gamma$ digits in the $p$-adic expansion may change from some nonzero number
to $0$. So $D(\underline{\lambda},\underline{n})\geq
D(\underline{\lambda}^{\prime},\underline{n})-\gamma$. We need to prove that

(5.5.12)
$\frac{\deg\mathbf{e}_{\lambda_{n}}-\deg\mathbf{e}_{n_{-}}}{2}-\Big{\lfloor}\frac{\deg\mathbf{e}_{\lambda_{n}}-\deg\mathbf{e}_{n_{-}}-2}{p-1}\Big{\rfloor}\geq
2\gamma+\boldsymbol{\delta}.$

If $\gamma=0$, then $\deg\mathbf{e}_{\lambda_{n}}-\deg\mathbf{e}_{n_{-}}=p-1$;
so (5.5.12) says $\frac{p-1}{2}\geq\boldsymbol{\delta}$, which is obvious. If
$\delta\geq\gamma=1$, the condition $2\leq a\leq p-5$ implies that the left
hand side $\geq\frac{p-1}{2}\geq 2+\boldsymbol{\delta}$. If
$\delta\geq\gamma\geq 2$, it is clear that the left hand side
$\geq\frac{1}{4}(p-1)p^{\gamma-1}>2\gamma+\boldsymbol{\delta}$. This completes
the proof of (5.5.12) when $\gamma\leq\delta$.

From now on, we assume that $\gamma>\delta$; in this case, the $p$-adic
expansions of $\deg\mathbf{e}_{\lambda_{n}}$ and $\deg\mathbf{e}_{n_{-}}$ look
like

(5.5.13)
$\displaystyle\deg\mathbf{e}_{\lambda_{n}}=\overline{\cdots\cdots\,\alpha_{\gamma+2}\,\alpha_{\gamma+1}\alpha_{\gamma}\,0\,0\,\cdots\cdots\cdots\,0\,0\,\alpha_{\delta}\,\cdots\alpha_{0}},$
$\displaystyle\deg\mathbf{e}_{n_{-}}=\overline{\cdots\cdots\,\alpha_{\gamma+2}\,\alpha_{\gamma+1}(\alpha_{\gamma}-1)\,(p-1)\,\cdots\,(p-1)\,\alpha^{\prime}_{\delta}\,\cdots\alpha^{\prime}_{0}}.$

Here each $\alpha_{j}$ and $\alpha^{\prime}_{j}$ belong to $\\{0,\dots,p-1\\}$
(and $\alpha_{\gamma}\geq 1$), and the two numbers
$\deg\mathbf{e}_{\lambda_{n}}$ and $\deg\mathbf{e}_{n_{-}}$ agree beyond the
$(\gamma+1)$th digits. The condition
$\deg\mathbf{e}_{\lambda_{n}}-\deg\mathbf{e}_{n_{-}}\leq(p-1)p^{\delta}$
implies that $\alpha^{\prime}_{\delta}\neq 0$.

We need to trace back to the definition of
$D(\underline{\lambda},\underline{n})$ in Notation 3.26 to compute

(5.5.14) $\displaystyle
D(\underline{\lambda},\underline{n})-D(\underline{\lambda}^{\prime},\underline{n})$
$\displaystyle=\sum_{j\geq 0}\;$
$\displaystyle\Big{(}\max\big{\\{}D_{=0}(\underline{\lambda},j)-D_{=0}(\underline{n},j+1),\,0\big{\\}}-\max\big{\\{}D_{=0}(\underline{\lambda}^{\prime},j)-D_{=0}(\underline{n},j+1),\,0\big{\\}}\Big{)}.$

By Lemma 3.29 and the fact $n_{-}\leq n\leq\lambda_{n}$,

$D_{=0}(\underline{n},\delta+1)=\dots=D_{=0}(\underline{n},\gamma-1).$

It is not hard to see that for every $j\in\\{\delta+1,\dots,\gamma-2\\}$

$D_{=0}(\underline{\lambda}^{\prime},j)\geq
D_{=0}(\underline{n},j)=D_{=0}(\underline{n},j+1).$

Yet (5.5.13) implies that for each such $j$,
$D_{=0}(\underline{\lambda},j)=D_{=0}(\underline{\lambda}^{\prime},j)+1.$ The
contribution of $j$th term to (5.5.14) is $1$. For $j=\gamma$, Lemma 3.29
implies that

$D_{=0}(\underline{\lambda}^{\prime},\gamma)\leq
D_{=0}(\underline{n},\gamma)\leq D_{=0}(\underline{n},\gamma+1).$

So the $(j=\gamma)$th term in (5.5.14) is zero.

Summarizing the above discussion, we have

(5.5.15) $D(\underline{\lambda},\underline{n})\geq
D(\underline{\lambda}^{\prime},\underline{n})+\max\\{\gamma-\delta-2,\,0\\}-\delta,$

where the term $\gamma-\delta-2$ comes from $j$th term with
$j\in\\{\delta+1,\dots,\gamma-2\\}$ and the term $-\delta$ corresponds to
$j\in\\{0,\dots,\delta-1\\}$. The terms with $j=\delta$ or $\gamma-1$ has
nonnegative contribution to (5.5.14). To prove (5.5.11), it suffices to show
that

(5.5.16)
$\frac{\deg\mathbf{e}_{\lambda_{n}}-\deg\mathbf{e}_{n_{-}}}{2}-\Big{\lfloor}\frac{\deg\mathbf{e}_{\lambda_{n}}-\deg\mathbf{e}_{n_{-}}-2}{p-1}\Big{\rfloor}\geq
2\delta+2+\boldsymbol{\delta}.$

If $\delta=0$, then $\deg\mathbf{e}_{\lambda_{n}}-\deg\mathbf{e}_{n_{-}}=p-1$;
so (5.5.15) says $\frac{p-1}{2}\geq 2+\boldsymbol{\delta}$, which is obvious.
If $\delta\geq 2$, it is clear that the left hand side
$\geq\frac{1}{4}(p-1)p^{\delta-1}>2\delta+2+\boldsymbol{\delta}$. If
$\delta=1$ and $\lambda_{n}-n_{-}\geq 4$, the left hand side $\geq p-2\geq
4+\boldsymbol{\delta}$.

For the remaining case $\lambda_{n}-n_{-}=3$, we have $\delta=1$. Now (5.5.16)
becomes

$\frac{\deg\mathbf{e}_{\lambda_{n}}-\deg\mathbf{e}_{n_{-}}}{2}-\Big{\lfloor}\frac{\deg\mathbf{e}_{\lambda_{n}}-\deg\mathbf{e}_{n_{-}}-2}{p-1}\Big{\rfloor}\geq
4+\boldsymbol{\delta}.$

The left hand side is $\geq\frac{p-1}{2}\geq 4+\boldsymbol{\delta}$ as we have
assumed $p\geq 11$. This finishes the proof of Claim under the assumption that
$\underline{\lambda}\neq\underline{n}$.

Finally we prove Claim for
$\underline{\lambda}=\underline{\xi}=\underline{n}$. By (5.5.5) and the fact
that $Y^{-1}$ is uppertriangular, we have

$\displaystyle\mathrm{det}(\mathrm{U}^{\dagger}(\underline{n}))$
$\displaystyle=\sum_{\underline{\lambda},\underline{\eta}\in\mathbb{N}^{n}}\mathrm{det}(\mathrm{U}_{\mathbf{C}}(\underline{\lambda}\times\underline{\eta}))\prod_{i=1}^{n}Y_{\mathbf{e}_{i},\mathbf{e}_{\lambda_{i}}}Y^{-1}_{\mathbf{e}_{\eta_{i}},\mathbf{e}_{i}}$
$\displaystyle=\sum_{\underline{\lambda}\in\mathbb{N}^{n}}\mathrm{det}(\mathrm{U}_{\mathbf{C}}(\underline{\lambda}\times\underline{n}))\prod_{i=1}^{n}Y_{\mathbf{e}_{i},\mathbf{e}_{\lambda_{i}}}Y^{-1}_{\mathbf{e}_{i},\mathbf{e}_{i}}$

As Claim has been proved for all
$\mathrm{U}_{\mathbf{C}}(\underline{\lambda}\times\underline{n})$’s with
$\underline{\lambda}\neq\underline{n}$, if we write
$f(w)=\mathrm{det}(\mathrm{U}^{\dagger}(\underline{n}))-\mathrm{det}(\mathrm{U}_{\mathbf{C}}(\underline{n}))$,
we have $f(w)\in p^{\deg g_{n}(w)}\mathcal{O}\langle\frac{w}{p}\rangle$. By
Corollary 3.10, there exists $h(w)=\sum\limits_{n\geq
0}h_{n}\cdot(\frac{w}{p})^{n}\in\mathcal{O}\langle\frac{w}{p}\rangle$,
$h_{n}\in\mathcal{O}$ for all $n$, such that
$\mathrm{det}(\mathrm{U}^{\dagger}(\underline{n}))=p^{-\deg
g_{n}(w)}g_{n}(w)h(w)$. For simplicity, we set $d=\deg g_{n}(w)$ and
$g_{n}(w)=\sum_{i=0}^{d}p^{i}c_{i}w^{d-i}$ with $c_{0}=1$ and
$c_{i}\in\mathbb{Z}_{p}$, $i=1,\dots,d$. If there exists an integer $M$
satisfying $v_{p}(h_{M})<d$, let $m$ be the largest integer with this property
(it exists as $h(w)\in\mathcal{O}\langle\frac{w}{p}\rangle$). The
$w^{d+m}$-coefficient of
$\mathrm{det}(\mathrm{U}^{\dagger}(\underline{n}))=p^{-\deg
g_{n}(w)}g_{n}(w)h(w)$ is
$p^{-d}\sum_{i=0}^{d}p^{i}c_{i}\frac{h_{m+i}}{p^{m+i}}$, which has $p$-adic
valuation $-d+v_{p}(\frac{h_{m}}{p^{m}})<-m$. On the other hand, it follows
from Lemma 3.14 that
$\mathrm{det}(\mathrm{U}_{\mathbf{C}}(\underline{n}))\in\mathcal{O}\llbracket
w\rrbracket$, and we see from the equality
$\mathrm{det}(\mathrm{U}^{\dagger}(\underline{n}))=\mathrm{det}(\mathrm{U}_{\mathbf{C}}(\underline{n}))+f(w)$
that the $p$-adic valuation of the $w^{d+m}$-coefficient of
$\mathrm{det}(\mathrm{U}_{\mathbf{C}}(\underline{n}))$ is greater or equal to
$-m$, which is a contradiction. Hence $v_{p}(h_{m})\geq d$ for all $m$ and
$\mathrm{det}(\mathrm{U}^{\dagger}(\underline{n}))\in
g_{n}(w)\mathcal{O}\langle\frac{w}{p}\rangle\subset p^{\deg
g_{n}(w)}\mathcal{O}\langle\frac{w}{p}\rangle$. So we also have
$\mathrm{det}(\mathrm{U}_{\mathbf{C}}(\underline{n}))\in p^{\deg
g_{n}(w)}\mathcal{O}\langle\frac{w}{p}\rangle$. This concludes the proof of
Claim as well as the proof of Proposition 5.4. ∎

######  Remark 5.6.

We point out that the proof of this proposition is where the condition
$a\notin\\{1,p-4\\}$ and $p\geq 11$ are used. The problem is rooted in the
number $\boldsymbol{\delta}=\deg
g_{n}(w)-\sum_{i=1}^{n}\deg\mathbf{e}_{i}-\big{\lfloor}\frac{\deg\mathbf{e}_{i}}{p}\big{\rfloor}\in\\{0,1\\}$
measuring the error from halo estimate in Corollary 3.27.

##  6\. Proof of local ghost conjecture III: cofactor expansions

Now, we come to explain Step II as outlined at the beginning of Section 4,
which aims to reduce Theorem 5.2 to the estimate we have proved in Proposition
5.4 for subminors. We conclude the proof of Theorem 5.2 and hence Theorem 2.7
at the end of this section.

Keep the notations from the previous section, and recall that a relevant
character $\varepsilon$ is fixed throughout yet suppressed from the notation.

######  Notation 6.1.

We fix $n\in\mathbb{N}$ and a weight $k=k_{\varepsilon}+(p-1)k_{\bullet}$ such
that $m_{n}(k)\neq 0$.

Similar to Proposition 3.6(2), let
$\mathrm{L}_{k}\in\mathrm{M}_{\infty}(\mathcal{O})$ denote the following
infinite matrix:

  * •

The upper-left $(d_{k}^{\mathrm{Iw}}\times d_{k}^{\mathrm{Iw}})$-block of
$\mathrm{L}_{k}$ is the Atkin–Lehner operator
$\mathrm{AL}_{(k,\tilde{\varepsilon}_{1})}$ acting on the power basis
$\mathbf{B}_{k}$; it is an antidiagonal matrix whose
$(i,d_{k}^{\mathrm{Iw}}+1-i)$-entry is $p^{\deg\mathbf{e}_{i}}$.

  * •

Entries of $\mathrm{L}_{k}$ other than the upper-left
$d_{k}^{\mathrm{Iw}}\times d_{k}^{\mathrm{Iw}}$ are the same as the
corresponding entries of $\mathrm{U}^{\dagger}$ evaluated at $w=w_{k}$.

This matrix $\mathrm{L}_{k}$ is block upper triangular by (2.11.2) of
Proposition 2.11(1). Set

$\mathrm{T}_{k}:=\mathrm{U}^{\dagger}-\mathrm{L}_{k}\;\in\;\mathrm{M}_{\infty}(\mathcal{O}\langle
w/p\rangle).$

For two subsets of integers $\underline{\zeta}=(\zeta_{1}<\dots<\zeta_{n})$
and $\underline{\xi}=(\xi_{1}<\dots<\xi_{n})$ of cardinality $n$, we let
$\mathrm{L}_{k}(\underline{\zeta}\times\underline{\xi})$ (resp.
$\mathrm{T}_{k}(\underline{\zeta}\times\underline{\xi})$) denote the
submatrices of $\mathrm{L}_{k}$ (resp. $\mathrm{T}_{k}$) with rows in
$\underline{\zeta}$ and columns in $\underline{\xi}$. Then Definition-
Proposition 3.21 says that the corank of
$\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})|_{w=w_{k}}$ is
at least
$n-d_{k}^{\mathrm{ur}}-r_{\underline{\zeta}\times\underline{\xi}}(k)-s_{\underline{\xi}}(k)$.
In the following discussion, we will use the sets $\underline{\zeta}$ and
$\underline{\xi}$ as the natural row and column index sets of the matrices
$\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})$,
$\mathrm{L}_{k}(\underline{\zeta}\times\underline{\xi})$ and
$\mathrm{T}_{k}(\underline{\zeta}\times\underline{\xi})$.

We also need a sign convention: when computing the determinant of a matrix
like $\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})$, we write
the row and columns in increasing order of the numbers in $\underline{\zeta}$
and $\underline{\xi}$. For a subset $I\subseteq\underline{\zeta}$, we write
$\mathrm{sgn}(I,\underline{\zeta})$ to mean the sign of permutation that sends
$\underline{\zeta}$ to the _ordered_ disjoint union of
$I\sqcup(\underline{\zeta}-I)$, where elements in each of $I$ and
$\underline{\zeta}-I$ are in increasing order.

The following key linear algebra result roughly states that, modulo an
appropriate power of $w-w_{k}$, we may express the determinant of
$\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})$ as the linear
combination of determinants of minors of smaller sizes.

######  Lemma 6.2.

Let $k$, $\mathrm{U}^{\dagger}$, $\mathrm{T}_{k}$, $\mathrm{L}_{k}$,
$\underline{\zeta}$, and $\underline{\xi}$ be as above. Fix
$J_{0}\subseteq\underline{\xi}$ a subset of cardinality $j_{0}$. We write
$\mathrm{T}_{k}(\underline{\zeta}\times\underline{\xi};J_{0})$ for the
$\underline{\zeta}\times\underline{\xi}$-matrix whose
$(\underline{\xi}-J_{0})$-columns are given by that of $\mathrm{U}^{\dagger}$
and whose $J_{0}$-columns are given by that of $\mathrm{T}_{k}$. Then

(6.2.1)
$\displaystyle\mathrm{det}\big{(}\mathrm{T}_{k}(\underline{\zeta}\times\underline{\xi};J_{0})\big{)}=\sum_{J\subseteq
J_{0}}$ $\displaystyle\sum_{\begin{subarray}{c}I\subseteq\underline{\zeta}\\\
\\#I=\\#J\end{subarray}}(-1)^{\\#J}\mathrm{sgn}(I,\underline{\zeta})\mathrm{sgn}(J,\underline{\xi})$
$\displaystyle\cdot\mathrm{det}\big{(}\mathrm{L}_{k}(I\times
J)\big{)}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}((\underline{\zeta}-I)\times(\underline{\xi}-J))\big{)}.$

In particular, as power series in $E\llbracket w-w_{k}\rrbracket$, we have the
following congruence

(6.2.2)
$\displaystyle\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times$
$\displaystyle\underline{\xi})\big{)}\equiv\sum_{\begin{subarray}{c}J\subseteq
J_{0}\\\
J\neq\emptyset\end{subarray}}\sum_{\begin{subarray}{c}I\subseteq\underline{\zeta}\\\
\\#I=\\#J\end{subarray}}(-1)^{\\#J-1}\mathrm{sgn}(I,\underline{\zeta})\mathrm{sgn}(J,\underline{\xi})\cdot\mathrm{det}\big{(}\mathrm{L}_{k}(I\times
J)\big{)}$
$\displaystyle\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}((\underline{\zeta}-I)\times(\underline{\xi}-J))\big{)}\qquad\bmod(w-w_{k})^{\operatorname{corank}\mathrm{T}_{k}(\underline{\zeta}\times\underline{\xi};J_{0})|_{w=w_{k}}}.$

###### Proof.

The equality (6.2.1) is a purely formal linear algebra equality and it does
not need the special properties of the matrices $\mathrm{U}^{\dagger}$,
$\mathrm{T}_{k}$, and $\mathrm{L}_{k}$ beyond the equality
$\mathrm{L}_{k}+\mathrm{T}_{k}=\mathrm{U}^{\dagger}$. Indeed, we may write
$\mathrm{T}_{k}(\underline{\zeta}\times\underline{\xi};J_{0})=\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})+(-\mathrm{L}_{k}(\underline{\zeta}\times
J_{0}))$, where we view $\mathrm{L}_{k}(\underline{\zeta}\times J_{0})$ as a
$\underline{\zeta}\times\underline{\xi}$-matrix whose
$(\underline{\xi}-J_{0})$-columns are zero. Since taking determinant is
(multi)-linear with respect to the columns, taking the cofactor expansion with
respect to the expression above exactly gives (6.2.1). For example, if
$\mathrm{L}_{k}(\underline{\zeta}\times\underline{\xi})$ has only four nonzero
entries, at the (upper left)
$\\{\zeta_{1},\zeta_{2}\\}\times\\{\xi_{1},\xi_{2}\\}$-minor, and
$J_{0}=\\{\xi_{1},\xi_{2}\\}$, then
$\mathrm{T}_{k}(\underline{\zeta}\times\underline{\xi};J_{0})=\mathrm{T}_{k}(\underline{\zeta}\times\underline{\xi})$
and the formula (6.2.1) reads

$\displaystyle\mathrm{det}\big{(}\mathrm{T}_{k}(\underline{\zeta}\times\underline{\xi};J_{0})\big{)}=\
$
$\displaystyle\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}-\sum_{i,j=1}^{2}(-1)^{i-j}L_{\zeta_{i},\xi_{j}}\mathrm{det}\big{(}\mathrm{U}^{\dagger}((\underline{\zeta}-\zeta_{i})\times(\underline{\xi}-\xi_{j})\big{)}$
$\displaystyle+\mathrm{det}{\begin{pmatrix}L_{\zeta_{1},\xi_{1}}&L_{\zeta_{1},\xi_{2}}\\\
L_{\zeta_{2},\xi_{1}}&L_{\zeta_{2},\xi_{2}}\end{pmatrix}}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}((\underline{\zeta}-\\{\zeta_{1},\zeta_{2}\\})\times(\underline{\xi}-\\{\xi_{1},\xi_{2}\\})\big{)}.$

Here $L_{\zeta_{i},\xi_{j}}$ is the $(\zeta_{i},\xi_{j})$-entry of $L_{k}$.

The congruence relation (6.2.2) follows immediately from (6.2.1) and the
observation that
$\mathrm{T}_{k}(\underline{\zeta}\times\underline{\xi};J_{0})$ is divisible by
$(w-w_{k})^{\operatorname{corank}\mathrm{T}_{k}(\underline{\zeta}\times\underline{\xi};J_{0})|_{w=w_{k}}}$
in $E\llbracket w-w_{k}\rrbracket$. ∎

######  Notation 6.3.

Now, we apply Lemma 6.2 to the situation of Theorem 5.2 with the fixed integer
$n\geq 2$, a ghost zero $w_{k}$ of $g_{n}(w)$, and subsets $\underline{\zeta}$
and $\underline{\xi}$ of cardinality $n$. Then we have
$\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})$,
$\mathrm{T}_{k}(\underline{\zeta}\times\underline{\xi})$,
$\mathrm{L}_{k}(\underline{\zeta}\times\underline{\xi})$,
$r_{\underline{\zeta}\times\underline{\xi}}(k)$, and $s_{\underline{\xi}}(k)$
as defined above. Let $J_{\underline{\zeta}\times\underline{\xi}}$ denote the
set consisting of _all_ $\xi_{j}\in\underline{\xi}$ such that either
$\xi_{j}>d_{k}^{\mathrm{Iw}}$ or
$d_{k}^{\mathrm{Iw}}+1-\xi_{j}\in\underline{\zeta}$. Then
$\\#J_{\underline{\zeta}\times\underline{\xi}}=r_{\underline{\zeta}\times\underline{\xi}}(k)+s_{\underline{\xi}}(k)$.

We introduce the following notation to reorganize the congruence relation from
Lemma 6.2. For every $j\leq
r_{\underline{\zeta}\times\underline{\xi}}(k)+s_{\underline{\xi}}(k)$, we
denote

(6.3.1)
$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{j}:=\sum_{\begin{subarray}{c}I\subseteq\underline{\zeta}\\\
\\#I=j\end{subarray}}\sum_{\begin{subarray}{c}J\subseteq
J_{\underline{\zeta}\times\underline{\xi}}\\\
\\#J=j\end{subarray}}\mathrm{sgn}(I,\underline{\zeta})\mathrm{sgn}(J,\underline{\xi})\cdot\mathrm{det}\big{(}\mathrm{L}_{k}(I\times
J)\big{)}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}((\underline{\zeta}-I)\times(\underline{\xi}-J))\big{)}.$

This is a signed sum of the products of the determinants of some minors of
$\mathrm{U}^{\dagger}$ of size $n-j$, with the determinants of the complement
minors in $\mathrm{L}_{k}$. In particular,
$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{0}=\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}$,
and the Lemma 6.2 above (applied to the case
$J_{0}=J_{\underline{\zeta}\times\underline{\xi}}$) in particular implies that

(6.3.2)
$\displaystyle\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}$
$\displaystyle\times\underline{\xi})\big{)}\equiv\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{1}-\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{2}+\cdots$
$\displaystyle+(-1)^{r_{\underline{\zeta}\times\underline{\xi}}(k)+s_{\underline{\xi}}(k)-1}\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{r_{\underline{\zeta}\times\underline{\xi}}(k)+s_{\underline{\xi}}(k)}\qquad\bmod(w-w_{k})^{n-d_{k}^{\mathrm{ur}}}.$

(Note that by Definition-Proposition 3.21 implies that
$\mathrm{T}_{k}(\underline{\zeta}\times\underline{\xi};J_{\underline{\zeta}\times\underline{\xi}})\big{|}_{w=w_{k}}$
has corank at least $n-d_{k}^{\mathrm{ur}}$.)

Our argument needs an elaborated version of (6.3.2), with one goal: we try to
write
$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}$
as a linear combination of minors of smallest possible size (after modulo an
appropriate power of $w-w_{k}$). This is the following.

######  Lemma 6.4.

Keep the notation as above. For a fixed nonnegative integer $j_{0}\leq
r_{\underline{\zeta}\times\underline{\xi}}(k)+s_{\underline{\xi}}(k)-1$, we
have the following congruence of power series in $E\llbracket
w-w_{k}\rrbracket$:

(6.4.1)
$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}\equiv\sum_{j>j_{0}}(-1)^{j-j_{0}-1}\binom{j-1}{j_{0}}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{j}\quad\bmod(w-w_{k})^{\max\\{0,n-d_{k}^{\mathrm{ur}}-j_{0}\\}}.$

More generally, for every nonnegative integers $\ell\leq
j_{0}<r_{\underline{\zeta}\times\underline{\xi}}(k)+s_{\underline{\xi}}(k)$,
we have the following congruence of power series in $E\llbracket
w-w_{k}\rrbracket$:

(6.4.2)
$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{\ell}\equiv\sum_{j>j_{0}}(-1)^{j-j_{0}-1}\binom{j-\ell-1}{j_{0}-\ell}\binom{j}{\ell}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{j}\quad\bmod(w-w_{k})^{\max\\{0,n-d_{k}^{\mathrm{ur}}-j_{0}\\}}.$

###### Proof.

We first prove (6.4.2) in the special case when $\ell=j_{0}$. When
$\ell=j_{0}=0$, this is exactly (6.3.2). To treat the general case with
$\ell=j_{0}$, we apply Lemma 6.2 (especially (6.2.2)) to each factor
$\mathrm{det}\big{(}\mathrm{U}^{\dagger}((\underline{\zeta}-I)\times(\underline{\xi}-J))\big{)}$
appearing in (6.3.1), to deduce the following:

$\displaystyle\mathrm{det}$
$\displaystyle\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{j_{0}}:=\sum_{\begin{subarray}{c}I\subseteq\underline{\zeta}\\\
\\#I=j_{0}\end{subarray}}\sum_{\begin{subarray}{c}J\subseteq
J_{\underline{\zeta}\times\underline{\xi}}\\\
\\#J=j_{0}\end{subarray}}\mathrm{sgn}(I,\underline{\zeta})\mathrm{sgn}(J,\underline{\xi})\cdot\mathrm{det}\big{(}\mathrm{L}_{k}(I\times
J)\big{)}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}((\underline{\zeta}-I)\times(\underline{\xi}-J))\big{)}$
$\displaystyle\equiv\ $
$\displaystyle\sum_{\begin{subarray}{c}I\subseteq\underline{\zeta}\\\
\\#I=j_{0}\end{subarray}}\sum_{\begin{subarray}{c}J\subseteq
J_{\underline{\zeta}\times\underline{\xi}}\\\
\\#J=j_{0}\end{subarray}}\mathrm{sgn}(I,\underline{\zeta})\mathrm{sgn}(J,\underline{\xi})\cdot\mathrm{det}\big{(}\mathrm{L}_{k}(I\times
J)\big{)}\cdot\sum_{\begin{subarray}{c}J^{\prime}\subseteq
J_{\underline{\zeta}\times\underline{\xi}}-J\\\
J^{\prime}\neq\emptyset\end{subarray}}\sum_{\begin{subarray}{c}I^{\prime}\subseteq\underline{\zeta}-I\\\
\\#I^{\prime}=\\#J^{\prime}\end{subarray}}(-1)^{\\#J^{\prime}-1}$
$\displaystyle\
\mathrm{sgn}(I^{\prime},\underline{\zeta}-I)\mathrm{sgn}(J^{\prime},\underline{\xi}-J)\cdot\mathrm{det}\big{(}\mathrm{L}_{k}(I^{\prime}\times
J^{\prime})\big{)}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}((\underline{\zeta}-I-I^{\prime})\times(\underline{\xi}-J-J^{\prime}))\big{)}$

modulo $(w-w_{k})^{\max\\{0,n-d_{k}^{\mathrm{ur}}-j_{0}\\}}$. Here we make use
of Definition-Proposition 3.21 to deduce that
$\operatorname{rank}\,\mathrm{T}_{k}\big{(}(\underline{\zeta}-I)\times(\underline{\xi}-J);J_{\underline{\zeta}\times\underline{\xi}}-J\big{)}|_{w=w_{k}}$
is at most $d_{k}^{\mathrm{ur}}$ and hence its corank is at least
$n-j_{0}-d_{k}^{\mathrm{ur}}$.

We set $I^{\prime\prime}=I\sqcup I^{\prime}$ and $J^{\prime\prime}=J\sqcup
J^{\prime}$ and set $j:=\\#I^{\prime\prime}=\\#J^{\prime\prime}>j_{0}$. Then
the above long expression is equal to

$\displaystyle\sum_{j>j_{0}}$
$\displaystyle(-1)^{j-j_{0}-1}\sum_{\begin{subarray}{c}I^{\prime\prime}\subseteq\underline{\zeta}\\\
\\#I^{\prime\prime}=j\end{subarray}}\sum_{\begin{subarray}{c}J^{\prime\prime}\subseteq
J_{\underline{\zeta}\times\underline{\xi}}\\\
\\#J^{\prime\prime}=j\end{subarray}}\sum_{\begin{subarray}{c}I\subseteq
I^{\prime\prime}\\\
\\#I=j_{0}\end{subarray}}\sum_{\begin{subarray}{c}J\subseteq
J^{\prime\prime}\\\
\\#J=j_{0}\end{subarray}}\mathrm{sgn}(I,\underline{\zeta})\mathrm{sgn}(J,\underline{\xi})\mathrm{sgn}(I^{\prime\prime}-I,\underline{\zeta}-I)\mathrm{sgn}(J^{\prime\prime}-J,\underline{\xi}-J)$
$\displaystyle\cdot\mathrm{det}\big{(}\mathrm{L}_{k}(I\times
J)\big{)}\cdot\mathrm{det}\big{(}\mathrm{L}_{k}((I^{\prime\prime}-I)\times(J^{\prime\prime}-J))\big{)}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}((\underline{\zeta}-I^{\prime\prime})\times(\underline{\xi}-J^{\prime\prime}))\big{)}.$

Using the sign equality

$\mathrm{sgn}(I,\underline{\zeta})\mathrm{sgn}(I^{\prime\prime}-I,\underline{\zeta}-I)=\mathrm{sgn}(I^{\prime\prime},\underline{\zeta})\mathrm{sgn}(I,I^{\prime\prime})$

and the similar sign equality for $\xi$, $J$, $J^{\prime}$, and
$J^{\prime\prime}$, we may rewrite the above sum as

$\displaystyle\sum_{j>j_{0}}(-1)^{j-j_{0}-1}\sum_{\begin{subarray}{c}I^{\prime\prime}\subseteq\underline{\zeta}\\\
\\#I^{\prime\prime}=j\end{subarray}}\sum_{\begin{subarray}{c}J^{\prime\prime}\subseteq
J_{\underline{\zeta}\times\underline{\xi}}\\\
\\#J^{\prime\prime}=j\end{subarray}}\mathrm{sgn}(I^{\prime\prime},\underline{\zeta})\mathrm{sgn}(J^{\prime\prime},\underline{\xi})\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}((\underline{\zeta}-I^{\prime\prime})\times(\underline{\xi}-J^{\prime\prime}))\big{)}$
$\displaystyle\cdot\sum_{\begin{subarray}{c}I\subseteq I^{\prime\prime}\\\
\\#I=j_{0}\end{subarray}}\sum_{\begin{subarray}{c}J\subseteq
J^{\prime\prime}\\\
\\#J=j_{0}\end{subarray}}\mathrm{sgn}(I,I^{\prime\prime})\mathrm{sgn}(J,J^{\prime\prime})\cdot\mathrm{det}\big{(}\mathrm{L}_{k}(I\times
J)\big{)}\cdot\mathrm{det}\big{(}\mathrm{L}_{k}((I^{\prime\prime}-I)\times(J^{\prime\prime}-J))\big{)}.$

But the sum in the second line is simply $\binom{j}{j_{0}}$ times
$\mathrm{det}\big{(}\mathrm{L}_{k}(I^{\prime\prime}\times
J^{\prime\prime})\big{)}$ by Laplace expansion theorem on determinants, where
the factor $\binom{j}{j_{0}}$ corresponds to the number of choices of the
subset $I\subseteq I^{\prime\prime}$. From this, we deduce that

$\displaystyle\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{j_{0}}\equiv\
$
$\displaystyle\sum_{j>j_{0}}(-1)^{j-j_{0}-1}\sum_{\begin{subarray}{c}I^{\prime\prime}\subseteq\underline{\zeta}\\\
\\#I^{\prime\prime}=j\end{subarray}}\sum_{\begin{subarray}{c}J^{\prime\prime}\subseteq
J_{\underline{\zeta}\times\underline{\xi}}\\\
\\#J^{\prime\prime}=j\end{subarray}}\mathrm{sgn}(I^{\prime\prime},\underline{\xi})\mathrm{sgn}(J^{\prime\prime},\underline{\xi})$
$\displaystyle\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}((\underline{\xi}-I^{\prime\prime})\times(\underline{\xi}-J^{\prime\prime}))\big{)}\cdot\binom{j}{j_{0}}\cdot\mathrm{det}\big{(}\mathrm{L}_{k}(I^{\prime\prime}\times
J^{\prime\prime})\big{)}$

$\mod(w-w_{k})^{\max\\{0,n-d_{k}^{\mathrm{ur}}-j_{0}\\}}$. This is exactly
(6.4.2) when $\ell=j_{0}$.

We now prove the general case by induction on the difference $j_{0}-\ell$. The
base case when $\ell=j_{0}$ is just treated. Assume that we have proved
(6.4.2) with smaller $j_{0}-\ell$. Then we have the following congruences
(corresponding to the cases of $(\ell,j_{0}-1)$ and $(j_{0},j_{0})$).

$\displaystyle\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{\ell}$
$\displaystyle\,\equiv\sum_{j>j_{0}-1}(-1)^{j-j_{0}}\binom{j-\ell-1}{j_{0}-\ell-1}\binom{j}{\ell}\cdot\mathrm{det}\big{(}\mathrm{U}^{\prime\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{j}\
\bmod(w-w_{k})^{\max\\{0,n-d_{k}^{\mathrm{ur}}-j_{0}+1\\}},$
$\displaystyle\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{j_{0}}$
$\displaystyle\,\equiv\sum_{j>j_{0}}(-1)^{j-j_{0}-1}\binom{j}{j_{0}}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{j}\quad\bmod(w-w_{k})^{\max\\{0,n-d_{k}^{\mathrm{ur}}-j_{0}\\}}.$

Plugging the second congruence into the first one (and modulo the smaller
power $(w-w_{k})^{\max\\{0,n-d_{k}^{\mathrm{ur}}-j_{0}\\}}$), we immediate
deduce (6.4.2) by noting

$\binom{j_{0}}{\ell}\binom{j}{j_{0}}-\binom{j-\ell-1}{j_{0}-\ell-1}\binom{j}{\ell}=\binom{j-\ell-1}{j_{0}-\ell}\binom{j}{\ell}.$

∎

######  Remark 6.5.

We point out a variant of the above lemma that we will use later. Fix any
power series $\eta(w)\in 1+(w-w_{k})E\llbracket w-w_{k}\rrbracket$. For
$J_{0}\subseteq J_{\underline{\zeta}\times\underline{\xi}}$, write

$\widetilde{\mathrm{T}}_{k}(\underline{\zeta}\times\underline{\xi};J_{0}):=\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})-\eta(w)^{-1}\cdot\mathrm{L}_{k}(\underline{\zeta}\times
J_{0})\in\mathrm{M}_{\infty}(E\llbracket w-w_{k}\rrbracket);$

then we obtain a formula of
$\mathrm{det}\big{(}\widetilde{\mathrm{T}}_{k}(\underline{\zeta}\times\underline{\xi};J_{0})\big{)}$
analogous to (6.2.1), with additional factor $\eta(w)^{-\\#J}$ on the right
hand side. Yet
$\widetilde{\mathrm{T}}_{k}(\underline{\zeta}\times\underline{\xi};J_{0})|_{w=w_{k}}=\mathrm{T}_{k}(\underline{\zeta}\times\underline{\xi};J_{0})|_{w=w_{k}}$
have the same corank. So if we define the analogue of (6.3.1) to be

(6.5.1)
$\displaystyle\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}^{\sim}_{j}:=\eta(w)^{-j}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{j}$
$\displaystyle=\ $
$\displaystyle\sum_{\begin{subarray}{c}I\subseteq\underline{\zeta}\\\
\\#I=j\end{subarray}}\sum_{\begin{subarray}{c}J\subseteq
J_{\underline{\zeta}\times\underline{\xi}}\\\
\\#J=j\end{subarray}}\mathrm{sgn}(I,\underline{\zeta})\mathrm{sgn}(J,\underline{\xi})\cdot\eta(w)^{-j}\cdot\mathrm{det}\big{(}\mathrm{L}_{k}(I\times
J)\big{)}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}((\underline{\zeta}-I)\times(\underline{\xi}-J))\big{)},$

exactly the same argument in Lemmas 6.2 and 6.4 shows that, for every
nonnegative integers $\ell\leq
j_{0}<r_{\underline{\zeta}\times\underline{\xi}}(k)+s_{\underline{\xi}}(k)$,
we have the following congruence of power series in $E\llbracket
w-w_{k}\rrbracket$:

(6.5.2)
$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{\ell}^{\sim}\equiv\sum_{j>j_{0}}(-1)^{j-j_{0}-1}\binom{j-\ell-1}{j_{0}-\ell}\binom{j}{\ell}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{j}^{\sim}\
\bmod(w-w_{k})^{\max\\{0,n-d_{k}^{\mathrm{ur}}-j_{0}\\}}.$

######  Notation 6.6.

To further simplify notations later, we normalize

(6.6.1)
$B_{k,i}^{(\underline{\zeta}\times\underline{\xi})}:=p^{\frac{1}{2}(\deg(\underline{\xi})-\deg(\underline{\zeta}))}\cdot
A_{k,i}^{(\underline{\zeta}\times\underline{\xi})}\cdot g_{n,\hat{k}}(w_{k}).$

So condition (5.2.1) is equivalent to, for $i=0,1,\dots,m_{n}(k)-1$,

(6.6.2)
$v_{p}\big{(}B_{k,i}^{(\underline{\zeta}\times\underline{\xi})}\big{)}\geq\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i}-\tfrac{k-2}{2}(\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n).$

Further, we normalize the minors appearing in the formula (6.4.2) as follows
and consider their expansions as power series in $E\llbracket
w-w_{k}\rrbracket$:

(6.6.3)
$p^{\frac{1}{2}(\deg(\underline{\xi})-\deg(\underline{\zeta}))}\cdot\frac{\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{\ell}}{g_{n-\ell,\hat{k}}(w)/g_{n-\ell,\hat{k}}(w_{k})}=\sum_{i\geq
0}B_{k,i}^{(\underline{\zeta}\times\underline{\xi},\ell)}(w-w_{k})^{i}.$

This normalization has in mind that the natural way to understand each sum of
minor determinants appearing in
$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{\ell}$
is through its Lagrange interpolation along $g_{n-\ell}(w)$. In particular,
when $\ell=0$, $B_{k,i}^{(\underline{\zeta}\times\underline{\xi},0)}$ is equal
to $B_{k,i}^{(\underline{\zeta}\times\underline{\xi})}$ in (6.6.1) for
$i=0,\dots,m_{n}(k)-1$.

As a convention, if $i<0$, we set
$B_{k,i}^{(\underline{\zeta}\times\underline{\xi},\ell)}=0$.

The following estimate on
$B_{k,i}^{(\underline{\zeta}\times\underline{\xi},\ell)}$ can be harvested
from the inductive hypothesis and Proposition 5.4.

######  Lemma 6.7.

Assume that $p\geq 11$ and $2\leq a\leq p-5$. Keep the notation as above and
assume that Theorem 5.2 holds for minors of size strictly smaller than $n$.
Assume that
$\ell\in\big{\\{}1,2,\dots,\min\\{n-d_{k}^{\mathrm{ur}},\,r_{\underline{\zeta}\times\underline{\xi}}(k)+s_{\underline{\xi}}(k)\\}\big{\\}}$
is taken so that $m_{n-\ell}(k)\leq m_{n}(k)-1$ (the latter condition is
equivalent to requiring $\ell\geq 2n-d_{k}^{\mathrm{Iw}}+1$ when
$n\geq\frac{1}{2}d_{k}^{\mathrm{Iw}}$). We have, for every
$i\in\\{m_{n-\ell}(k),\dots,m_{n}(k)-1\\}$,

(6.7.1) $\displaystyle
v_{p}\big{(}B_{k,i}^{(\underline{\zeta}\times\underline{\xi},\ell)}\big{)}\geq\
$
$\displaystyle\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-m_{n-\ell}(k)}-\tfrac{k-2}{2}\big{(}\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n\big{)}-\tfrac{1}{2}\big{(}(\tfrac{1}{2}d_{k}^{\mathrm{new}}-m_{n-\ell}(k))^{2}-(\tfrac{1}{2}d_{k}^{\mathrm{new}}-i)^{2}\big{)}$
(6.7.2) $\displaystyle\geq\ $
$\displaystyle\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i}-\tfrac{k-2}{2}\big{(}\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n\big{)}.$

Later, we will refer (6.7.1) as the strong estimate and (6.7.2) the weak
estimate.

###### Proof.

Here the second inequality follows from Proposition 2.19. We now prove the
first one. Recall that
$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{\ell}$
is a $\mathbb{Z}$-linear combination of

$\mathrm{det}\big{(}\mathrm{L}_{k}(I\times
J)\big{)}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}((\underline{\zeta}-I)\times(\underline{\xi}-J))$

over subsets $I\subseteq\underline{\zeta}$ and $J\subseteq
J_{\underline{\zeta}\times\underline{\xi}}$ of cardinality $\ell$. Consider
the following Taylor expansion in $E\llbracket w-w_{k}\rrbracket$:

(6.7.3) $\frac{\mathrm{det}\big{(}\mathrm{L}_{k}(I\times
J)\big{)}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}((\underline{\zeta}-I)\times(\underline{\xi}-J))}{g_{n-\ell,\hat{k}}(w)}=\sum_{i\geq
0}A^{(\underline{\zeta}\times\underline{\xi},I,J)}_{k,i}(w-w_{k})^{i};$

comparing to (6.6.3), we did not multiply the left hand side with
$p^{\frac{1}{2}(\deg(\underline{\xi})-\deg(\underline{\zeta}))}\cdot
g_{n-\ell,\hat{k}}(w_{k})$. As
$v_{p}(g_{n-\ell,\hat{k}}(w_{k}))=\Delta^{\prime}_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-m_{n-\ell}(k)}-\frac{k-2}{2}\big{(}\frac{1}{2}d_{k}^{\mathrm{new}}-m_{n-\ell}(k)\big{)}$,
to prove condition (6.7.1), it is enough to show that

(6.7.4) $\displaystyle
v_{p}\big{(}A_{k,i}^{(\underline{\zeta}\times\underline{\xi},I,J)}\big{)}\geq\
$
$\displaystyle\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-m_{n-\ell}(k)}-\Delta^{\prime}_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-m_{n-\ell}(k)}+\tfrac{k-2}{2}\cdot\ell$
$\displaystyle-\tfrac{1}{2}\big{(}(\tfrac{1}{2}d_{k}^{\mathrm{new}}-m_{n-\ell}(k))^{2}-(\tfrac{1}{2}d_{k}^{\mathrm{new}}-i)^{2}\big{)}+\tfrac{1}{2}(\deg(\underline{\zeta})-\deg(\underline{\xi})).$

(Here we secretly used the condition that $m_{n-\ell}(k)<m_{n}(k)-1$.)

Using the notation from Proposition 5.4(2) to write

$\mathrm{det}\big{(}\mathrm{U}^{\dagger}((\underline{\zeta}-I)\times(\underline{\xi}-J))\big{)}\big{/}g_{n,\hat{k}}(w)=\sum_{i\geq
0}a_{k,i}^{((\underline{\zeta}-I)\times(\underline{\xi}-J))}(w-w_{k})^{i},$

then Proposition 5.4(2) and the inductive assumption of the lemma shows that

$\displaystyle
v_{p}\big{(}a_{k,i}^{((\underline{\zeta}-I)\times(\underline{\xi}-J))}\big{)}\geq\
$
$\displaystyle\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-m_{n-\ell}(k)}-\Delta^{\prime}_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-m_{n-\ell}(k)}$
$\displaystyle-\tfrac{1}{2}\big{(}-\deg(\underline{\zeta}-I)+\deg(\underline{\xi}-J)+(\tfrac{1}{2}d_{k}^{\mathrm{new}}-m_{n-\ell}(k))^{2}-(\tfrac{1}{2}d_{k}^{\mathrm{new}}-i)^{2}\big{)}.$

Therefore, to prove (6.7.4) and hence the lemma, it is enough to show that

(6.7.5) $\displaystyle v_{p}\big{(}\mathrm{det}(\mathrm{L}_{k}(I\times
J))\big{)}\geq\ $
$\displaystyle\tfrac{k-2}{2}\cdot\ell+\tfrac{1}{2}\big{(}\deg(\underline{\zeta})-\deg(\underline{\xi})\big{)}-\tfrac{1}{2}\big{(}\deg(\underline{\zeta}-I)-\deg(\underline{\xi}-J)\big{)}$
$\displaystyle=\ $
$\displaystyle\tfrac{k-2}{2}\cdot\ell+\tfrac{1}{2}(\deg(I)-\deg(J)).$

Write $J=J^{\prime}\sqcup J^{\prime\prime}$ with
$J^{\prime}=J\cap\underline{d_{k}^{\mathrm{Iw}}}$. For each $\xi\in
J^{\prime}$, write
$\xi^{\mathrm{op}}:=d_{k}^{\mathrm{Iw}}+1-\xi\in\underline{\zeta}$ (since
$\xi\in J_{\underline{\zeta},\underline{\xi}}$). Define
$I^{\prime}:=\\{\xi^{\mathrm{op}}\,|\,\xi\in J^{\prime}\\}$ and
$I^{\prime\prime}=I\backslash I^{\prime}$. Then the $\xi$th column of
$\mathrm{L}_{k}(I\times J)$ has only one nonzero entry at
$(\xi^{\mathrm{op}},\xi)$, which is $p^{\deg\mathbf{e}_{\xi^{\mathrm{op}}}}$
as introduced in Notation 6.1. So

$\mathrm{det}(\mathrm{L}_{k}(I\times J))=p^{\sum_{\xi\in
J^{\prime}}\deg\mathbf{e}_{\xi^{\mathrm{op}}}}\cdot\mathrm{det}(\mathrm{L}_{k}(I^{\prime\prime}\times
J^{\prime\prime})).$

Note that for each $\xi\in J^{\prime}$, there is a tautological equality
$\deg\mathbf{e}_{\xi^{\mathrm{op}}}=\tfrac{k-2}{2}+\tfrac{1}{2}\big{(}\deg\mathbf{e}_{\xi^{\mathrm{op}}}-\deg\mathbf{e}_{\xi}\big{)}$.
So (6.7.5) is equivalent to the following inequality

(6.7.6) $v_{p}\big{(}\mathrm{det}(\mathrm{L}_{k}(I^{\prime\prime}\times
J^{\prime\prime}))\big{)}\geq\tfrac{k-2}{2}\cdot\\#J^{\prime\prime}+\tfrac{1}{2}(\deg(I^{\prime\prime})-\deg(J^{\prime\prime})).$

But this is clear as every element $\xi\in J^{\prime\prime}$ satisfies
$\deg\mathbf{e}_{\xi}>k-2$; so
$\frac{k-2}{2}\\#J^{\prime\prime}\leq\tfrac{1}{2}\deg(J^{\prime\prime})$, and
every entry of $\mathrm{U}^{\dagger}(I^{\prime\prime}\times J^{\prime\prime})$
in the $\zeta$’s row belongs to
$p^{\frac{1}{2}\deg(\mathbf{e}_{\zeta})}\mathcal{O}\langle w/p\rangle$ by
Proposition 3.2; so its evaluation at $w=w_{k}$ belongs to
$p^{\frac{1}{2}\deg(\mathbf{e}_{\zeta})}\mathcal{O}$. (6.7.6) clearly follows
from this weak Hodge bound estimate. The lemma is proved. ∎

###  6.8. Proof of Theorem 5.2

We are now ready to start the proof of Theorem 5.2, by induction on $n$, that
is, we assume that Theorem 5.2 has been proved for all minors of size strictly
smaller than $n$, and we hope to prove Theorem 5.2 for all $n\times n$ minors.
The case of $n=1$ has been handled in § 5.3.

We quickly recall the setup: we have fixed a relevant character $\varepsilon$
(and suppressed it from all notations), an integer $n\geq 2$, two finite
subsets $\underline{\zeta}$ and $\underline{\xi}$ of cardinality $n$, and an
integer $k=k_{\varepsilon}+(p-1)k_{\bullet}$ such that $m_{n}(k)\neq 0$. The
elements $B^{(\underline{\zeta}\times\underline{\xi})}_{k,i}$ for
$i=1,\dots,m_{n}(k)-1$ are defined in Notation 6.6 by the Lagrange
interpolation of
$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}$
along $g_{n}(w)$ (after an appropriate normalization), or equivalently
determined by the Taylor expansion of
$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}$
as a power series in $E\llbracket w-w_{k}\rrbracket$. We will prove
inductively the following.

######  Statement 6.9.

Keep the notation as above. For every $i\leq m_{n}(k)-1$ and every
$\ell\in\big{\\{}0,1,\dots,\min\\{n-d_{k}^{\mathrm{ur}},\,r_{\underline{\zeta}\times\underline{\xi}}(k)+s_{\underline{\xi}}(k)\\}\big{\\}}$,
such that $m_{n-\ell}(k)\leq m_{n}(k)$ (which, when
$n\geq\frac{1}{2}d_{k}^{\mathrm{Iw}}$, is equivalent to requiring that
$\ell\geq 2n-d_{k}^{\mathrm{Iw}}$ or $\ell=0$), we have

(6.9.1)
$v_{p}\big{(}B_{k,i}^{(\underline{\zeta}\times\underline{\xi},\ell)}\big{)}\geq\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i}-\tfrac{k-2}{2}\big{(}\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n\big{)}.$

Then condition (6.6.2) or equivalently Theorem 5.2 is the special case of
Statement 6.9 when $\ell=0$.

_For the rest of this section, as $k$ is already fixed, we will write
$r_{\underline{\zeta},\underline{\xi}}$, $s_{\underline{\xi}}$, and
$m_{\underline{\zeta},\underline{\xi}}$ for
$r_{\underline{\zeta},\underline{\xi}}(k)$, $s_{\underline{\xi}}(k)$, and
$m_{\underline{\zeta},\underline{\xi}}(k)$, respectively._

###  6.10. First stab at Statement 6.9

Definition-Proposition 3.21 says that
$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}$
and more generally every
$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{\ell}$
is divisible by
$(w-w_{k})^{\max\\{0,n-d_{k}^{\mathrm{ur}}-r_{\underline{\zeta}\times\underline{\xi}}-s_{\underline{\xi}}\\}}$
in $E\llbracket w-w_{k}\rrbracket$. So if
$i<n-d_{k}^{\mathrm{ur}}-r_{\underline{\zeta}\times\underline{\xi}}-s_{\underline{\xi}}$,
$B_{k,i}^{(\underline{\zeta}\times\underline{\xi},\ell)}=0$ and the
corresponding condition (6.9.1) automatically holds.

Now consider the next easiest case when
$i=n-d_{k}^{\mathrm{ur}}-r_{\underline{\zeta}\times\underline{\xi}}-s_{\underline{\xi}}=m_{\underline{\zeta}\times\underline{\xi}}$.
We may assume that $i\geq 0$, otherwise there is nothing to prove. In this
case,
$m_{n-r_{\underline{\zeta}\times\underline{\xi}}-s_{\underline{\xi}}}(k)=m_{\underline{\zeta}\times\underline{\xi}}(k)=i$.
So in the particular case when
$\ell=r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}}$, the
weak estimate (6.7.2) exactly gives (6.9.1). Now we assume that
$\ell\in\\{0,\dots,r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}}-1\\}$.
Applying Lemma 6.4 to the case when
$j_{0}=r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}}-1$, we
deduce that

$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{\ell}\equiv\binom{r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}}}{\ell}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}}}\quad\bmod(w-w_{k})^{i+1}.$

Comparing the coefficients of $(w-w_{k})^{i}$, we immediately deduce that

$v_{p}\big{(}B_{k,i}^{(\underline{\zeta}\times\underline{\xi},\ell)}\big{)}=v_{p}\Big{(}\binom{r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}}}{\ell}B_{k,i}^{(\underline{\zeta}\times\underline{\xi},r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}})}\Big{)}\stackrel{{\scriptstyle\eqref{E:estimate
of smaller minors
weak}}}{{\geq}}\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i}-\tfrac{k-2}{2}\big{(}\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n\big{)}.$

This proves Statement 6.9 in the corresponding situation.

Since the situation in general is more complicated, we consider another case
when
$i=m_{\underline{\zeta}\times\underline{\xi}}+1=n-d_{k}^{\mathrm{ur}}-r_{\underline{\zeta}\times\underline{\xi}}-s_{\underline{\xi}}+1$,
to illustrate the new phenomenon by spelling out all the terms involved. First
of all, in the special cases
$\ell=r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}}$ and
$\ell=r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}}-1$,
Statement 6.9 just restates the weak estimate (6.7.2). So we assume below that
$\ell\in\\{0,\dots,r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}}-2\\}$.
We apply Lemma 6.4 to the case when
$j_{0}=r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}}-2$ to
deduce that, modulo $(w-w_{k})^{i+1}$

$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{\ell}\equiv\binom{j_{0}+1}{\ell}\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{j_{0}+1}-(j_{0}-\ell+1)\binom{j_{0}+2}{\ell}\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{j_{0}+2}.$

Dividing both sides by
$p^{\frac{1}{2}(\deg(\underline{\xi})-\deg(\underline{\zeta}))}\cdot
g_{n-\ell,\hat{k}}(w)/g_{n-\ell,\hat{k}}(w_{k})$ and further by
$(w-w_{k})^{i}$ (to kill the auxiliary powers), we arrive at, modulo
$(w-w_{k})^{2}$,

(6.10.1) $\displaystyle
B^{(\underline{\zeta}\times\underline{\xi},\ell)}_{k,i-1}+B^{(\underline{\zeta}\times\underline{\xi},\ell)}_{k,i}(w-w_{k})\hskip
120.0pt$ $\displaystyle\equiv\
\binom{j_{0}+1}{\ell}\frac{g_{n-j_{0}-1,\hat{k}}(w)/g_{n-j_{0}-1,\hat{k}}(w_{k})}{g_{n-\ell,\hat{k}}(w)/g_{n-\ell,\hat{k}}(w_{k})}\Big{(}B^{(\underline{\zeta}\times\underline{\xi},j_{0}+1)}_{k,i-1}+B^{(\underline{\zeta}\times\underline{\xi},j_{0}+1)}_{k,i}(w-w_{k})\Big{)}\qquad$
$\displaystyle-(j_{0}-\ell+1)\binom{j_{0}+2}{\ell}\frac{g_{n-j_{0}-2,\hat{k}}(w)/g_{n-j_{0}-2,\hat{k}}(w_{k})}{g_{n-\ell,\hat{k}}(w)/g_{n-\ell,\hat{k}}(w_{k})}\Big{(}B^{(\underline{\zeta}\times\underline{\xi},j_{0}+2)}_{k,i-1}+B^{(\underline{\zeta}\times\underline{\xi},j_{0}+2)}_{k,i}(w-w_{k})\Big{)}.$

Suggested by this, we consider the following.

######  Notation 6.11.

For every $j\geq 0$, we write the following power series expansion:

(6.11.1)
$\eta_{j}(w):=\frac{g_{n-j,\hat{k}}(w)/g_{n-j,\hat{k}}(w_{k})}{g_{n,\hat{k}}(w)/g_{n,\hat{k}}(w_{k})}=1+\eta_{j,1}(w-w_{k})+\eta_{j,2}(w-w_{k})^{2}+\cdots\in
E\llbracket w-w_{k}\rrbracket.$

Comparing the $(w-w_{k})$-coefficients in (6.10.1), we deduce

$\displaystyle B_{k,i}^{(\underline{\zeta}\times\underline{\xi},\ell)}\ $
$\displaystyle=\binom{j_{0}+1}{\ell}B_{k,i}^{(\underline{\zeta}\times\underline{\xi},j_{0}+1)}-(j_{0}-\ell+1)\binom{j_{0}+2}{\ell}B_{k,i}^{(\underline{\zeta}\times\underline{\xi},j_{0}+2)}$
$\displaystyle+\;\binom{j_{0}+1}{\ell}(\eta_{j_{0}+1,1}-\eta_{\ell,1})B_{k,i-1}^{(\underline{\zeta}\times\underline{\xi},j_{0}+1)}-(j_{0}-\ell+1)\binom{j_{0}+2}{\ell}(\eta_{j_{0}+2,1}-\eta_{\ell,1})B_{k,i-1}^{(\underline{\zeta}\times\underline{\xi},j_{0}+2)}.$

By the weak estimate (6.7.2), the first two terms have $p$-adic valuation
greater than or equal to
$\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i}-\frac{1}{2}(d_{k}^{\mathrm{Iw}}-n)$.
But we need to show the sum of the latter two terms does not interfere here.
Our strategy is to show that _the power series $\eta_{j}(w)$ is
“approximately” the same as $\eta_{1}(w)^{j}$_, and thus each $\eta_{j,1}$ is
“approximately” equal to $j\cdot\eta_{1,1}$, and thus we are reduced to prove

(6.11.2) $\binom{j_{0}+1}{\ell}\cdot(j_{0}-\ell+1)\cdot
B_{k,i-1}^{(\underline{\zeta}\times\underline{\xi},j_{0}+1)}=(j_{0}-\ell+2)(j_{0}-\ell+1)\binom{j_{0}+2}{\ell}\cdot
B_{k,i-1}^{(\underline{\zeta}\times\underline{\xi},j_{0}+2)},$

which follows from what we just proved in the case of
$i=m_{\underline{\zeta}\times\underline{\xi}}(k)$, namely

$B_{k,m_{\underline{\zeta}\times\underline{\xi}}(k)}^{(\underline{\xi},j_{0}+1)}=(j_{0}+2)\cdot
B_{k,m_{\underline{\zeta}\times\underline{\xi}}(k)}^{(\underline{\xi},j_{0}+2)}.$

######  Remark 6.12.

Note that it is important to cancel major terms in different $\eta$-functions,
especially when $i$ is almost as large as $\frac{1}{2}d_{k}^{\mathrm{new}}$;
in this case, the difference
$\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-(i-1)}-\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i}\approx\frac{p-1}{2}(\frac{1}{2}d_{k}^{\mathrm{new}}-i)$,
yet the term $\eta_{\ell,1}$ roughly has $p$-adic valuation equal to the
maximal $v_{p}(w_{k^{\prime}}-w_{k})$, for all $k^{\prime}$ running over the
zeros of $g_{n}(w)$, which is about $\ln k/\ln p$. As we will show below that
the terms that do not get canceled through (6.11.2) have relatively large
$p$-adic valuation, controlled by the difference
$\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-(i-1)}-\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i}$.

To implement this strategy in the special case is not particularly easier than
the general case. So we now proceed directly to prove Statement 6.9 (in the
general case).

###  6.13. Proof of Statement 6.9

The proof is by induction on $i$, starting with the smallest case
$i=m_{\underline{\zeta}\times\underline{\xi}}=n-d_{k}^{\mathrm{ur}}-r_{\underline{\zeta}\times\underline{\xi}}-s_{\underline{\xi}}$
already treated in § 6.10 (and when
$i<m_{\underline{\zeta}\times\underline{\xi}}$, Statement 6.9 also holds
automatically.) Now, let
$i_{0}\in\\{m_{\underline{\zeta}\times\underline{\xi}}+1,\dots,m_{n}(k)-1\\}$,
and suppose that Statement 6.9 has been proved for all nonnegative integers
$i<i_{0}$. We may clearly assume that $i_{0}\geq 0$, as otherwise there is
nothing to prove. We set

$j_{0}:=r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}}-(i_{0}-m_{\underline{\zeta}\times\underline{\xi}}+1)=n-d_{k}^{\mathrm{ur}}-i_{0}-1.$

Then when $\ell>j_{0}$, one can check that $i_{0}\geq m_{n-\ell}(k)$ and thus
Statement 6.9 just repeats (6.7.1). We henceforth assume
$\ell\in\\{0,\dots,j_{0}\\}$. First, we apply Lemma 6.4 to deduce that

(6.13.1)
$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\xi})\big{)}_{\ell}\equiv\sum_{j>j_{0}}(-1)^{j-j_{0}-1}\binom{j-\ell-1}{j_{0}-\ell}\binom{j}{\ell}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\xi})\big{)}_{j}\quad\bmod(w-w_{k})^{i_{0}+1}.$

We point out that, the condition $j>j_{0}=n-d_{k}^{\mathrm{ur}}-i_{0}-1$
implies that

(6.13.2) $m_{n-j}(k)=n-j-d_{k}^{\mathrm{ur}}=i_{0}+1-(j-j_{0})<m_{n}(k).$

Instead of using the the numbers
$B_{k,i}^{(\underline{\zeta}\times\underline{\xi},j)}$’s to express the Taylor
expansion of above in $E\llbracket w-w_{k}\rrbracket$, we define the
following:

(6.13.3) $\Big{(}\sum_{i\geq
0}B_{k,i}^{(\underline{\zeta}\times\underline{\xi},j)}(w-w_{k})^{i}\Big{)}\cdot\frac{\eta_{j}(w)}{\eta_{1}(w)^{j}}=\sum_{i\geq
0}C_{k,i}^{(\underline{\zeta}\times\underline{\xi},j)}(w-w_{k})^{i}\quad\in\
E\llbracket w-w_{k}\rrbracket.$

Or equivalently by (6.6.3), in $E\llbracket w-w_{k}\rrbracket$, we have an
equality

(6.13.4)
$p^{\frac{1}{2}(\deg(\underline{\xi})-\deg(\underline{\zeta}))}\cdot\frac{\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{j}}{g_{n,\hat{k}}(w)/g_{n,\hat{k}}(w_{k})}\cdot\eta_{1}(w)^{-j}=\sum_{i\geq
0}C_{k,i}^{(\underline{\zeta}\times\underline{\xi},j)}(w-w_{k})^{i}.$

In fact, changing to $B_{k,i}^{(\underline{\zeta}\times\underline{\xi},j)}$ to
$C_{k,i}^{(\underline{\zeta}\times\underline{\xi},j)}$ is “harmless” for the
purpose of our proof.

######  Proposition 6.14.

Fix
$j\in\\{0,\dots,\min\\{n-d_{k}^{\mathrm{ur}},r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}}\\}\\}$
such that $m_{n-j}(k)\leq m_{n}(k)$ (which, when
$n\geq\frac{1}{2}d_{k}^{\mathrm{Iw}}$, is equivalent to requiring that $j\geq
2n-d_{k}^{\mathrm{Iw}}$ or $j=0$), and assume that Statement 6.9 holds true
for all nonnegative integers $i<i_{0}$, then

$v_{p}\big{(}B_{k,i_{0}}^{(\underline{\zeta}\times\underline{\xi},j)}\big{)}\geq\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i_{0}}-\tfrac{k-2}{2}\big{(}\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n\big{)}$
$\Longleftrightarrow\quad
v_{p}\big{(}C_{k,i_{0}}^{(\underline{\zeta}\times\underline{\xi},j)}\big{)}\geq\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i_{0}}-\tfrac{k-2}{2}\big{(}\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n\big{)}.$

We temporarily assume this technical result, whose proof will be given later
in § 6.18.

######  Remark 6.15.

For the rest of the inductive proof of Statement 6.9, we will only need the
analogue of the seemingly weaker version of Lemma 6.7, namely
$v_{p}\big{(}C_{k,i}^{(\underline{\zeta}\times\underline{\xi},\ell)}\big{)}\geq\Delta^{\prime}_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i}-\tfrac{k-2}{2}\big{(}\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n\big{)}$
when $i\geq m_{n-j}(k)$. The stronger inequality in Lemma 6.7 is only used to
enable transferring estimates between
$B_{k,i}^{(\underline{\zeta}\times\underline{\xi},\ell)}$’s and
$C_{k,i}^{(\underline{\zeta}\times\underline{\xi},\ell)}$’s.

######  Lemma 6.16.

For every nonnegative integer $\ell^{\prime}\leq
j^{\prime}_{0}<r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}}$,
we have

(6.16.1)
$C_{k,n-d_{k}^{\mathrm{ur}}-j^{\prime}_{0}-1}^{(\underline{\zeta}\times\underline{\xi},\ell^{\prime})}=\sum_{j^{\prime}=j^{\prime}_{0}+1}^{r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}}}(-1)^{j^{\prime}-j^{\prime}_{0}-1}\binom{j^{\prime}-\ell^{\prime}-1}{j^{\prime}_{0}-\ell^{\prime}}\binom{j^{\prime}}{\ell^{\prime}}C_{k,n-d_{k}^{\mathrm{ur}}-j^{\prime}_{0}-1}^{(\underline{\zeta}\times\underline{\xi},j^{\prime})}$

###### Proof.

Applying Remark 6.5 to the case $\eta(w):=\eta_{1}(w)$, then (6.5.2) implies
that for every nonnegative integer $\ell^{\prime}\leq
j^{\prime}_{0}<r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}}$,
modulo $(w-w_{k})^{\max\\{0,n-d_{k}^{\mathrm{ur}}-j_{0}\\}}$ in $E\llbracket
w-w_{k}\rrbracket$,

$\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{\ell^{\prime}}\cdot\eta_{1}(w)^{-\ell^{\prime}}\equiv\sum_{j^{\prime}>j^{\prime}_{0}}(-1)^{j^{\prime}-j^{\prime}_{0}-1}\binom{j^{\prime}-\ell^{\prime}-1}{j^{\prime}_{0}-\ell^{\prime}}\binom{j^{\prime}}{\ell^{\prime}}\cdot\mathrm{det}\big{(}\mathrm{U}^{\dagger}(\underline{\zeta}\times\underline{\xi})\big{)}_{j^{\prime}}\cdot\eta_{1}(w)^{-j^{\prime}}.$

(6.16.1) follows from dividing the above congruence by
$p^{\tfrac{1}{2}(\deg(\underline{\zeta})-\deg(\underline{\xi}))}\cdot
g_{n,\hat{k}}(w)/g_{n,\hat{k}}(w_{k})$ and then taking the coefficient of
$(w-w_{k})^{n-d_{k}^{\mathrm{ur}}-j^{\prime}_{0}-1}$. ∎

###  6.17. Proof of Statement 6.9 assuming Proposition 6.14

Continuing with the inductive proof of Statement 6.9 initiated in § 6.13.
Recall that the inductive hypothesis says that Statement 6.9 holds for all
$i<i_{0}$ for some
$i_{0}\in\\{m_{\underline{\zeta}\times\underline{\xi}}+1,\dots,m_{n}(k)-1\\}$
By the inductive hypothesis, the assumption for Proposition 6.14 holds, and
thus
$v_{p}\big{(}C_{k,i_{0}}^{(\underline{\zeta}\times\underline{\xi},j)}\big{)}\geq\Delta^{\prime}_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i_{0}}-\tfrac{k-2}{2}\big{(}\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n\big{)}$
for every $j>j_{0}$ by Lemma 6.7 and Proposition 6.14, where
$j_{0}=r_{\underline{\zeta}\times\underline{\xi}}+s_{\underline{\xi}}-(i_{0}-m_{\underline{\zeta}\times\underline{\xi}}+1)$.
(In particular, if $n\geq\frac{1}{2}d_{k}^{\mathrm{Iw}}$, $j_{0}\geq
2n-d_{k}^{\mathrm{Iw}}$.)

Then using the formula (6.16.1) in the case when $j^{\prime}_{0}=j_{0}$,
$\ell^{\prime}=\ell$ (and thus
$n-d_{k}^{\mathrm{ur}}-j^{\prime}_{0}-1=i_{0}$), we deduce that
$C_{k,i_{0}}^{(\underline{\zeta}\times\underline{\xi},\ell)}$ is a
$\mathbb{Z}$-linear combination of
$C_{k,i_{0}}^{(\underline{\zeta}\times\underline{\xi},j)}$’s with $j>j_{0}$.
Thus,
$v_{p}\big{(}C_{k,i_{0}}^{(\underline{\zeta}\times\underline{\xi},\ell)}\big{)}\geq\Delta^{\prime}_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i_{0}}-\tfrac{k-2}{2}\big{(}\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n\big{)}$.
By Proposition 6.14 again, we deduce that
$B_{k,i_{0}}^{(\underline{\zeta}\times\underline{\xi},\ell)}$ has the same
estimate; this then completes the inductive proof of Statement 6.9, and hence
conclude the proof of the local ghost Theorem 2.7 (assuming Proposition 6.14).

###  6.18. Proof of Proposition 6.14

We now come back to prove this last missing piece for the proof of Statement
6.9 and the local ghost Theorem 2.7. We claim that if we expand

$\frac{\eta_{j}(w)}{\eta_{1}(w)^{j}}=1+\eta_{(j),1}(w-w_{k})+\eta_{(j),2}(w-w_{k})^{2}+\cdots\in
E\llbracket w-w_{k}\rrbracket,$

then for every $t\in\\{1,\dots,m_{n}(k)-1\\}$, setting
$q_{t}:=\min\\{m_{n}(k)-t,m_{n-j}(k)\\}$, we have

(6.18.1)
$v_{p}(\eta_{(j),t})>\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t)}-\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t}}+\tfrac{1}{2}\big{(}(\tfrac{1}{2}d_{k}^{\mathrm{new}}-q_{t})^{2}-(\tfrac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t))^{2}\big{)}.$

We first prove the statement of this lemma assuming this claim: from the
definition of $C_{k,i}^{(\underline{\zeta}\times\underline{\xi},j)}$ in
(6.13.3), we see that,

$C_{k,i_{0}}^{(\underline{\zeta}\times\underline{\xi},j)}=B_{k,i_{0}}^{(\underline{\zeta}\times\underline{\xi},j)}+\sum_{i=0}^{i_{0}-1}B_{k,i}^{(\underline{\zeta}\times\underline{\xi},j)}\cdot\eta_{(j),i_{0}-i}.$

When $i<m_{n-j}(k)$ (and $i<i_{0}$), set $t:=i_{0}-i$ so that
$q_{t}=\min\\{m_{n}(k)+i-i_{0},m_{n-j}(k)\\}$. In particular, $q_{t}+t>i_{0}$
as $i<m_{n-j}(k)$ and $i_{0}<m_{n}(k)$. In either case, we have

$\displaystyle
v_{p}\big{(}B_{k,i}^{(\underline{\zeta}\times\underline{\xi},j)}\eta_{(j),i_{0}-i}\big{)}\geq\
$
$\displaystyle\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i}-\tfrac{k-2}{2}\big{(}\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n\big{)}+\big{(}\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t)}-\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t}}\big{)}$
$\displaystyle\geq\ $
$\displaystyle\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i}-\tfrac{k-2}{2}\big{(}\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n\big{)}+\big{(}\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i_{0}}-\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i}\big{)}$
$\displaystyle=\ $
$\displaystyle\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i_{0}}-\tfrac{k-2}{2}\big{(}\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n\big{)}.$

Here, for first inequality, we used Statement 6.9 to estimate
$v_{p}\big{(}B_{k,i}^{(\underline{\zeta}\times\underline{\xi},j)}\big{)}$ and
used (6.18.1) (but forgetting the term
$\tfrac{1}{2}\big{(}(\tfrac{1}{2}d_{k}^{\mathrm{new}}-q_{t})^{2}-(\tfrac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t))^{2}\big{)}$)
to estimate $\eta_{(j),i_{0}-i}$. The second inequality follows from the
convexity of $\underline{\Delta}_{k}$ and that $q_{t}+t>i_{0}$.

When $i\geq m_{n-j}(k)$ (and $i<i_{0}<m_{n}(k)$), we need the strong estimate
in (6.7.1). Set $t=i_{0}-i$ and in this case,

$q_{t}+t=\min\\{m_{n}(k),m_{n-j}(k)+i_{0}-i\\}=m_{n-j}(k)+i_{0}-i\leq i_{0}.$

In this case, we have $m_{n-j}(k)=q_{t}\leq q_{t}+t\leq i_{0}<m_{n}(k)$.

By the stronger estimate (6.7.1) and (6.18.1), we have

$\displaystyle
v_{p}\big{(}B_{k,i}^{(\underline{\zeta}\times\underline{\xi},j)}\eta_{(j),i_{0}-i}\big{)}$
$\displaystyle\geq\ $
$\displaystyle\Big{(}\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-m_{n-j}(k)}-\tfrac{1}{2}\big{(}(\tfrac{1}{2}d_{k}^{\mathrm{new}}-m_{n-j}(k))^{2}-(\tfrac{1}{2}d_{k}^{\mathrm{new}}-i)^{2}\big{)}-\tfrac{k-2}{2}\big{(}\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n\big{)}\Big{)}$
$\displaystyle+\Big{(}\Delta_{k,\tfrac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t)}-\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-m_{n-j}(k)}+\tfrac{1}{2}\big{(}(\tfrac{1}{2}d_{k}^{\mathrm{new}}-m_{n-j}(k))^{2}-(\tfrac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t))^{2}\big{)}\Big{)}$
$\displaystyle=\ $
$\displaystyle\Delta_{k,\tfrac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t)}-\tfrac{1}{2}\big{(}(\tfrac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t))^{2}-(\tfrac{1}{2}d_{k}^{\mathrm{new}}-i)^{2}\big{)}-\tfrac{k-2}{2}\big{(}\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n\big{)}$
$\displaystyle\geq\ $
$\displaystyle\Delta_{k,\tfrac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t)}-\tfrac{1}{2}\big{(}(\tfrac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t))^{2}-(\tfrac{1}{2}d_{k}^{\mathrm{new}}-i_{0})^{2}\big{)}-\tfrac{k-2}{2}\big{(}\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n\big{)}$
$\displaystyle\geq\ $
$\displaystyle\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-i_{0}}-\tfrac{k-2}{2}\big{(}\tfrac{1}{2}d_{k}^{\mathrm{Iw}}-n\big{)},$

where the last inequality follows from Proposition 2.19. The lemma then
follows from the two estimates above.

It remains to prove the claim, namely the inequality (6.18.1). By the
definition of $\eta_{j}$ in Notation 6.11, we may rewrite

$\eta_{j}(w)=\prod_{\begin{subarray}{c}k^{\prime}\equiv
k_{\varepsilon}\bmod(p-1)\\\ k^{\prime}\neq
k\end{subarray}}\Big{(}1+\frac{w-w_{k}}{w_{k}-w_{k^{\prime}}}\Big{)}^{m_{n-j}(k^{\prime})-m_{n}(k^{\prime})},$
(6.18.2) $\displaystyle\frac{\eta_{j}(w)}{\eta_{1}(w)^{j}}\ $
$\displaystyle=\prod_{\begin{subarray}{c}k^{\prime}\equiv
k_{\varepsilon}\bmod(p-1)\\\ k^{\prime}\neq
k\end{subarray}}\Big{(}1+\frac{w-w_{k}}{w_{k}-w_{k^{\prime}}}\Big{)}^{m_{n-j}(k^{\prime})-m_{n}(k^{\prime})-j(m_{n-1}(k^{\prime})-m_{n}(k^{\prime}))}$
$\displaystyle=1+\eta_{(j),1}(w-w_{k})+\eta_{(j),2}(w-w_{k})^{2}+\cdots.$

Set
$m_{n,j}(k^{\prime}):=m_{n-j}(k^{\prime})-m_{n}(k^{\prime})-j(m_{n-1}(k^{\prime})-m_{n}(k^{\prime}))$.
The weight $k^{\prime}$ term in the product expression of
$\frac{\eta_{j}(w)}{\eta_{1}(w)^{j}}$ is not $1$ (or equivalently
$m_{n,j}(k^{\prime})\neq 0$) only when the function $n^{\prime}\mapsto
m_{n^{\prime}}(k^{\prime})$ for $n^{\prime}\in[n-j,n]$ fails to be linear, or
equivalently, at least one of $d_{k^{\prime}}^{\mathrm{ur}}$,
$d_{k^{\prime}}^{\mathrm{Iw}}-d_{k^{\prime}}^{\mathrm{ur}}$, or
$\frac{1}{2}d_{k^{\prime}}^{\mathrm{Iw}}$ belongs to $(n-j,n)$. We call those
weights $k^{\prime}$ _bad weights_.

The upshot of the proof is the following: (6.18.2) implies that for
$t\in\\{1,\dots,m_{n}(k)-1\\}$, $\eta_{(j),t}$ is the sum of terms of the form

(6.18.3) $\prod_{\alpha=1}^{t}\frac{1}{w_{k}-w_{k^{\prime}_{\alpha}}},$

where each $k^{\prime}_{\alpha}$ is a bad weights, satisfying certain
constraints: if $m_{n,j}(k^{\prime}_{\alpha})>0$, the multiplicity of
$k^{\prime}_{\alpha}$ appearing in (6.18.3) is less than or equal to
$m_{n,j}(k^{\prime}_{\alpha})$; if $m_{n,j}(k^{\prime}_{\alpha})<0$, the
expansion (6.18.2) is considered as a Taylor expansion; so there is no
constraint in the multiplicity of $k^{\prime}_{\alpha}$ appearing in (6.18.3).
Roughly speaking, we will prove that, among all these bad weights, there is at
most one $k^{\prime}_{\alpha}$ such that
$v_{p}(w_{k}-w_{k^{\prime}_{\alpha}})$ is extraordinarily large. When we cite
Proposition 2.19 later, most of the $v_{p}(w_{k}-w_{k^{\prime}_{\alpha}})$
will be controlled by the term
$(\ell^{\prime}-\ell)\cdot\big{\lfloor}\frac{\ln((p+1)\ell)}{\ln
p}+1\big{\rfloor}$ and the exceptional $w_{k^{\prime}_{\alpha}}$ corresponds
to the distinguished weight $k^{\prime}_{\alpha}$ there in.

We now make this proof more precise. Write $n^{*}:=n$ if
$n\leq\frac{1}{2}d_{k}^{\mathrm{Iw}}$ and $n^{*}=d_{k}^{\mathrm{Iw}}-n$ if
$n\geq\frac{1}{2}d_{k}^{\mathrm{Iw}}$. In particular, $m_{n^{*}}(k)=m_{n}(k)$
and $n^{*}\leq n$. From the above discussion, we reduce the proof to the
following:

let $\mathcal{S}=\\{k_{\alpha}^{\prime}|\alpha=1,\dots,t\\}$ be a set of bad
weights (not necessarily distinct) satisfying that if
$m_{n,j}(k_{\alpha}^{\prime})>0$ for some $\alpha\in[1,t]$, the multiplicity
of $k_{\alpha}^{\prime}$ in $\mathcal{S}$ is less or equal to
$m_{n,j}(k_{\alpha}^{\prime})$. Then we have

(6.18.4)
$\sum_{\alpha=1}^{t}v_{p}(w_{k}-w_{k_{\alpha}^{\prime}})\leq\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t}}-\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t)}-\frac{1}{2}\big{(}(\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t})^{2}-(\frac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t))^{2}\big{)}$

Suppose that there exists some $\alpha\in[1,t]$ such that either
$d_{k_{\alpha}^{\prime}}^{\mathrm{ur}}$ or
$d_{k_{\alpha}^{\prime}}^{\mathrm{Iw}}-d_{k_{\alpha}^{\prime}}^{\mathrm{ur}}$
belongs to $[n^{\ast},d_{k}^{\mathrm{Iw}}-n^{\ast})$. Without loss of
generality, we can assume $\alpha=t$.

  1. (1)

When $t\leq m_{n}(k)-m_{n-j}(k)$, we have $q_{t}=q_{t-1}=m_{n-j}(k)$. The
number $s\coloneqq\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t}-t+1$ satisfies
$\frac{1}{2}d_{k}^{\mathrm{Iw}}-n^{\ast}\leq s-1$, and hence
$[n^{\ast},d_{k}^{\mathrm{Iw}}-n^{\ast})\subset[\frac{1}{2}d_{k}^{\mathrm{Iw}}-(s-1),\frac{1}{2}d_{k}^{\mathrm{Iw}}+(s-1)]$.
Proposition 2.19 (with $\ell=\ell^{\prime}=s-1<\ell^{\prime\prime}=s$) implies
$v_{p}(w_{k}-w_{k_{t}^{\prime}})\leq\Delta_{k,s}-\Delta_{k,s-1}-\frac{1}{2}(s^{2}-(s-1)^{2})$.
To prove (6.18.4), it suffices to prove

(6.18.5)
$\sum_{\alpha=1}^{t-1}v_{p}(w_{k}-w_{k_{\alpha}^{\prime}})\leq\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t-1}}-\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-(q_{t-1}+t-1)}-\frac{1}{2}\big{(}(\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t-1})^{2}-(\frac{1}{2}d_{k}^{\mathrm{new}}-(q_{t-1}+t-1))^{2}\big{)}.$

  2. (2)

When $t>m_{n}(k)-m_{n-j}(k)$, we have $q_{t}=m_{n}(k)-t$ and
$q_{t-1}=q_{t}+1$. The number
$s\coloneqq\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t}$ satisfies
$\frac{1}{2}d_{k}^{\mathrm{Iw}}-n^{\ast}\leq s-1$. A similar argument as in
$(1)$ implies that we can reduce to prove (6.18.5).

Repeating the above argument, we can assume that none of the bad weights
$k_{\alpha}^{\prime}$ in (6.18.4) satisfies that either
$d_{k_{\alpha}^{\prime}}^{\mathrm{ur}}$ or
$d_{k_{\alpha}^{\prime}}^{\mathrm{Iw}}-d_{k_{\alpha}^{\prime}}^{\mathrm{ur}}$
belongs to $[n^{\ast},d_{k}^{\mathrm{Iw}}-n^{\ast})$.

Set
$\gamma\coloneqq\lfloor\frac{\ln((p+1)(\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t}))}{\ln
p}+1\rfloor$. We first remark that if some $k_{\alpha}^{\prime}$ satisfies
$\frac{1}{2}d_{k_{\alpha}^{\prime}}^{\mathrm{Iw}}\in(n-j,n)$, then
$|k^{\prime}_{\alpha\bullet}-k_{\bullet}|<j$. By our assumption
$m_{n-j}(k)\leq m_{n}(k)$ we always have
$\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t}\geq\frac{j}{2}$ and hence
$v_{p}(w_{k}-w_{k_{\alpha}^{\prime}})\leq 1+\lfloor\frac{\ln j}{\ln
p}\rfloor\leq\gamma$.

  1. (1)

If $v_{p}(w_{k}-w_{k_{\alpha}^{\prime}})\leq\gamma$ for all $\alpha\in[1,t]$,
since
$(n-j,n)\subset[\frac{1}{2}d_{k}^{\mathrm{Iw}}-(\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t}),\frac{1}{2}d_{k}^{\mathrm{Iw}}+(\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t})]$,
we can apply Proposition 2.19 to
$l=\frac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t)<l^{\prime}=l^{\prime\prime}=\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t}$
and $k^{\prime}=k_{\alpha}^{\prime}$ for all $\alpha\in[1,t]$, and have

$\sum_{\alpha=1}^{t}v_{p}(w_{k}-w_{k_{\alpha}^{\prime}})\leq
t\cdot\gamma\leq\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t}}-\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t)}-\frac{1}{2}\big{(}(\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t})^{2}-(\frac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t))^{2}\big{)}.$

  2. (2)

If $v_{p}(w_{k}-w_{k^{\prime}})\geq\gamma+1$ for some
$k^{\prime}\in\mathcal{S}$, we assume that the multiplicity of $k^{\prime}$ in
$\mathcal{S}$ is $M>0$ and $k^{\prime}=k_{\alpha}^{\prime}$ for
$\alpha\in[t-M+1,t]$. It follows from Remark 2.20 and our assumption on
$\mathcal{S}$ that $k^{\prime}$ is the unique element in $\mathcal{S}$ with
the property $v_{p}(w_{k}-w_{k^{\prime}})\geq\gamma+1$. Moreover we have that
either $d_{k^{\prime}}^{\mathrm{ur}}$ or
$d_{k^{\prime}}^{\mathrm{Iw}}-d_{k^{\prime}}^{\mathrm{ur}}$ belongs to
$(n-j,n^{\ast})$ while $\frac{1}{2}d_{k^{\prime}}^{\mathrm{Iw}}\notin(n-j,n)$.

When $d_{k^{\prime}}^{\mathrm{ur}}\in(n-j,n^{\ast})$, we have
$\frac{1}{2}d_{k^{\prime}}^{\mathrm{Iw}}\geq n$ and hence
$m_{n-j}(k^{\prime})=0$,$m_{n}(k^{\prime})=n-d_{k^{\prime}}^{\mathrm{ur}}$ and
$m_{n-1}(k^{\prime})=m_{n}(k^{\prime})-1$. It follows that
$m_{n,j}(k^{\prime})=d_{k^{\prime}}^{\mathrm{ur}}-n+j>0$ and

(6.18.6)
$\frac{1}{2}d_{k}^{\mathrm{Iw}}-d_{k^{\prime}}^{\mathrm{ur}}=\frac{1}{2}d_{k}^{\mathrm{Iw}}-n+j-m_{n,j}(k)\leq\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t}-m_{n,j}(k^{\prime}).$

When
$d_{k^{\prime}}^{\mathrm{Iw}}-d_{k^{\prime}}^{\mathrm{ur}}\in(n-j,n^{\ast})$,
we have $\frac{1}{2}d_{k^{\prime}}^{\mathrm{Iw}}\leq n-j$ and hence
$m_{n-j}(k^{\prime})=d_{k^{\prime}}^{\mathrm{Iw}}-d_{k^{\prime}}^{\mathrm{ur}}-(n-j)>0$
and $m_{n-1}(k^{\prime})=m_{n}(k^{\prime})=0$. It follows that
$m_{n,j}(k^{\prime})=d_{k^{\prime}}^{\mathrm{Iw}}-d_{k^{\prime}}^{\mathrm{ur}}-(n-j)$
and

(6.18.7)
$\frac{1}{2}d_{k}^{\mathrm{Iw}}-(d_{k^{\prime}}^{\mathrm{Iw}}-d_{k^{\prime}}^{\mathrm{ur}})=\frac{1}{2}d_{k}^{\mathrm{Iw}}-n+j-m_{n,j}(k^{\prime})\leq\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t}-m_{n,j}(k^{\prime}).$

Let $\ell=\frac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t)$,
$\ell^{\prime\prime}=\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t}$ and
$\ell^{\prime}=\ell^{\prime\prime}-M\in[\ell,\ell^{\prime\prime}]$. It follows
from (6.18.6) and (6.18.7) that either $d_{k^{\prime}}^{\mathrm{ur}}$ or
$d_{k^{\prime}}^{\mathrm{Iw}}-d_{k^{\prime}}^{\mathrm{ur}}$ belongs to
$[\frac{1}{2}d_{k}^{\mathrm{Iw}}-\ell^{\prime},\frac{1}{2}d_{k}^{\mathrm{Iw}}+\ell^{\prime}]$.
We apply Proposition 2.19 to $\ell$, $\ell^{\prime}$, $\ell^{\prime\prime}$
and $k^{\prime}$, and have

$\displaystyle\sum_{\alpha=1}^{t}v_{p}(w_{k}-w_{k^{\prime}_{\alpha}})$
$\displaystyle\leq(t-M)\cdot\gamma+M\cdot
v_{p}(w_{k}-w_{k^{\prime}})\leq\Delta_{k,\ell^{\prime\prime}}-\Delta_{k,\ell}-\frac{1}{2}(\ell^{\prime\prime
2}-\ell^{2})$
$\displaystyle=\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t}}-\Delta_{k,\frac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t)}-\frac{1}{2}\big{(}(\frac{1}{2}d_{k}^{\mathrm{new}}-q_{t})^{2}-(\frac{1}{2}d_{k}^{\mathrm{new}}-(q_{t}+t))^{2}\big{)}.$

This completes the proof of the claim and thus Proposition 6.14 and Theorem
2.7. $\Box$

##  7\. Triangulline deformation space and crystalline slopes

In this section, we recall the triangulline deformation space defined by
Breuil–Hellman–Schraen [BHS17] and then compare it with the eigenvariety
attached to Paškūnas’ universal deformation of representations of
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$ [Pa13]. This together with the known
$p$-adic local Langlands correspondence for
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$ allows us to transport the local ghost
theorem to results regarding slopes on triangulline deformation spaces.

The argument in this section is relatively well known to experts, but some of
the awkward arguments are inserted to treat central characters for
completeness.

######  Notation 7.1.

As in previous sections, let $p$ be an odd prime, and let
$E,\mathcal{O},\mathbb{F}$ be coefficient rings as in § 1.26. For a formal
$\mathcal{O}$-scheme $\operatorname{Spf}(R)$, let
$\operatorname{Spf}(R)^{\mathrm{rig}}$ denote the associated rigid analytic
space over $E$. We will later frequently write $E^{\prime}$ to mean a finite
extension of $E$, typically in the situation of referring to a point of
$\operatorname{Spf}(R)^{\mathrm{rig}}$ over $E^{\prime}$; we will freely do so
without defining $E^{\prime}$, and in such case, we use
$\mathcal{O}^{\prime}$, $\varpi^{\prime}$, and $\mathbb{F}^{\prime}$ denote
the corresponding ring of integers, the uniformizer, and the residue field,
respectively.

For a crystabelian representation $V$ of $\operatorname{Gal}_{\mathbb{Q}_{p}}$
(with coefficients in $E^{\prime}$), write $\mathbb{D}_{\mathrm{pcrys}}(V)$
for the limit of the crystalline functor for $\mathbb{Q}_{p}(\mu_{p^{n}})$
with $n$ sufficiently large.

We normalize the local class field theory so that the Artin map
$\mathbb{Q}_{p}^{\times}\to\operatorname{Gal}_{\mathbb{Q}_{p}}^{\mathrm{ab}}$
sends $p$ to the _geometric_ Frobenius. In what follows, we will practically
identify characters of $\mathbb{Q}_{p}^{\times}$ (with values in
$\mathcal{O}^{\times}$ or $\mathbb{F}^{\times}$) and characters of
$\operatorname{Gal}_{\mathbb{Q}_{p}}$.

We use the following notations for local Galois representations:

  * •

For $\alpha\in\mathbb{F}^{\times}$ or $\mathcal{O}^{\times}$, write
$\mathrm{ur}(\alpha)$ for the one-dimensional unramified representation of
$\operatorname{Gal}_{\mathbb{Q}_{p}}$ sending the geometric Frobenius element
to $\alpha$.

  * •

Let
$\omega_{1}:\operatorname{Gal}_{\mathbb{Q}_{p}}\to\operatorname{Gal}(\mathbb{Q}_{p}(\mu_{p})/\mathbb{Q}_{p})\cong\mathbb{F}_{p}^{\times}$
denote the _first fundamental character_ of
$\operatorname{Gal}_{\mathbb{Q}_{p}}$.

  * •

Let
$\chi_{\mathrm{cycl}}:\mathbb{Q}_{p}^{\times}\subset\operatorname{Gal}_{\mathbb{Q}_{p}}^{\mathrm{ab}}\to\operatorname{Gal}(\mathbb{Q}_{p}(\mu_{p^{\infty}})/\mathbb{Q}_{p})\cong\mathbb{Z}_{p}^{\times}$
denote the cyclotomic character; its reduction modulo $p$ is precisely
$\omega_{1}$.

Recall $\Delta:=\mathbb{F}_{p}^{\times}$, the isomorphism
$\mathcal{O}\llbracket(1+p\mathbb{Z}_{p})^{\times}\rrbracket\cong\mathcal{O}\llbracket
w\rrbracket$, and the universal character
$\chi_{\mathrm{univ}}^{(\varepsilon)}:\Delta\times\mathbb{Z}_{p}^{\times}\to\mathcal{O}\llbracket
w\rrbracket^{(\varepsilon),\times}$ associated to a character $\varepsilon$ of
$\Delta^{2}$ from § 2.4(1). For each $\varepsilon$, call
$\mathcal{W}^{(\varepsilon)}:=(\operatorname{Spf}\mathcal{O}\llbracket
w\rrbracket^{(\varepsilon)})^{\mathrm{rig}}$ the _weight space labeled by
$\varepsilon$_. Put
$\mathcal{W}:=\bigcup_{\varepsilon}\mathcal{W}^{(\varepsilon)}$; it
parametrizes continuous characters of $\Delta\times\mathbb{Z}_{p}^{\times}$.
Write
$\chi_{\mathrm{univ}}:\Delta\times\mathbb{Z}_{p}^{\times}\to\mathcal{O}_{\mathcal{W}}^{\times}$
for the universal character. Put
$\mathcal{W}_{0}:=(\operatorname{Spf}\mathcal{O}\llbracket
w\rrbracket)^{\mathrm{rig}}$, parametrizing continuous characters of
$(1+p\mathbb{Z}_{p})^{\times}$.

Let
$\widetilde{\mathcal{W}}:=(\operatorname{Spf}\mathcal{O}\llbracket(\mathbb{Z}_{p}^{\times})^{2}\rrbracket)^{\mathrm{rig}}$
be the rigid analytic space parametrizing continuous characters of
$(\mathbb{Z}_{p}^{\times})^{2}$. There is a natural isomorphism

(7.1.1)
${\mathcal{W}\times\mathcal{W}_{0}}$${\widetilde{\mathcal{W}}}$${(\chi,\eta)}$${\big{(}\
(\alpha,\delta)\mapsto\alpha\cdot\chi(\bar{\delta},\alpha)\cdot\eta(\alpha\delta\omega(\bar{\alpha}\bar{\delta})^{-1})\
\textrm{ for
}\alpha,\delta\in\mathbb{Z}_{p}^{\times}\big{)}.}$$\scriptstyle{\cong}$

_Here, we used $\chi(\bar{\delta},\alpha)$ as opposed to
$\chi(\bar{\alpha},\delta)$ because our later convention uses the lower
triangular matrix local analytic Jacquet functor. The additional factor
$\alpha$ at the beginning indicates a twist by cyclotomic character in our
convention._ Under this isomorphism, we may view $\mathcal{W}$ as a subspace
of $\widetilde{\mathcal{W}}$ where the universal character is trivial on
$\\{1\\}\times(1+p\mathbb{Z}_{p})^{\times}$; and at the same time, we have a
projection map $\mathrm{pr}_{W}:\widetilde{\mathcal{W}}\to\mathcal{W}$, along
$\mathcal{W}_{0}$.

Later, we often consider a rigid analytic space $\mathcal{X}$ and the morphism
$\mathrm{id}_{\mathcal{X}}\times\mathrm{pr}_{W}:\mathcal{X}\times\widetilde{\mathcal{W}}\to\mathcal{X}\times\mathcal{W}$;
we write $\mathrm{pr}_{W}$ for it when no confusion arises.

######  Notation 7.2.

For the rest of this paper, we use
$\bar{r}_{p}:\operatorname{Gal}_{\mathbb{Q}_{p}}\to\operatorname{GL}_{2}(\mathbb{F})$
to denote a reducible and generic residual local Galois representation

$\bar{r}_{p}=\begin{pmatrix}\mathrm{ur}(\bar{\alpha}_{1})\omega_{1}^{a+b+1}&*\\\
0&\mathrm{ur}(\bar{\alpha}_{2})\omega_{1}^{b}\end{pmatrix}:\operatorname{Gal}_{\mathbb{Q}_{p}}\to\operatorname{GL}_{2}(\mathbb{F})$

with $a\in\\{1,\dots,p-4\\}$, $b\in\\{0,\dots,p-2\\}$, and
$\bar{\alpha}_{1},\bar{\alpha}_{2}\in\mathbb{F}^{\times}$. We say
$\bar{r}_{p}$ is _split_ if $*=0$ and _nonsplit_ if $*\neq 0$. The condition
on $a$ ensures that there is a unique such nontrivial extension when
$\bar{r}_{p}$ is nonsplit, because
$\mathrm{H}^{1}(\operatorname{Gal}_{\mathbb{Q}_{p}},\mathrm{ur}(\bar{\alpha}_{2}^{-1}\bar{\alpha}_{1})\omega^{a+1})$
is one-dimensional.

We often write
$\bar{\rho}:\mathrm{I}_{\mathbb{Q}_{p}}\to\operatorname{GL}_{2}(\mathbb{F})$
for the corresponding residual inertia representation:

  * •

(nonsplit case) $\bar{\rho}={\begin{pmatrix}\omega_{1}^{a+b+1}&*\neq 0\\\
0&\omega_{1}^{b}\end{pmatrix}}$, where $*$ is the _unique_ nontrivial
extension (up to isomorphism) in the class
$\mathrm{H}^{1}(\mathrm{I}_{\mathbb{Q}_{p}},\omega_{1}^{a+1})^{\operatorname{Gal}_{\mathbb{F}_{p}}}=\mathrm{H}^{1}(\operatorname{Gal}_{\mathbb{Q}_{p}},\omega_{1}^{a+1})$;
and

  * •

(split case)
$\bar{\rho}^{\mathrm{ss}}=\omega_{1}^{a+b+1}\oplus\omega_{1}^{b}$.

We occasionally use a companion representation $\bar{\rho}^{\prime}$ for the
same construction with parameters $(a,b)$ changed to
$(a^{\prime},b^{\prime})=(p-3-a,a+b+1)$, or equivalently,
$\bar{\rho}^{\prime}={\begin{pmatrix}\omega_{1}^{a+b+1}&0\\\ {*}\neq
0&\omega_{1}^{b}\end{pmatrix}}$.

_These notations $\bar{\rho}$, $\bar{\rho}^{\prime}$ and
$\bar{\rho}^{\mathrm{ss}}$ are fixed throughout the rest of this paper._

###  7.3. Triangulline deformation spaces

Let $\mathcal{T}$ denote the rigid analytic space parametrizing continuous
characters of $(\mathbb{Q}_{p}^{\times})^{2}$, or more precisely,

(7.3.1)
$\mathcal{T}=\big{(}\mathbb{G}_{m}^{\mathrm{rig}}\times(\operatorname{Spf}\mathbb{Z}_{p}\llbracket\mathbb{Z}_{p}^{\times}\rrbracket)^{\mathrm{rig}}\big{)}^{2}\cong(\mathbb{G}_{m}^{\mathrm{rig}})^{2}\times\widetilde{\mathcal{W}},$

where
$\mathbb{G}_{m}^{\mathrm{rig}}=\cup_{n\in\mathbb{N}}\operatorname{Spm}\big{(}\mathbb{Q}_{p}\langle\frac{u}{p^{n}},\frac{p^{n}}{u}\rangle\big{)}$
is the rigid analytic $\mathbb{G}_{m}$. The point on $\mathcal{T}$ associated
to a character
$(\delta_{1},\delta_{2}):(\mathbb{Q}_{p}^{\times})^{2}\to\mathbb{C}_{p}^{\times}$
is
$(\delta_{1}(p),\delta_{2}(p),\delta_{1}|_{\mathbb{Z}_{p}^{\times}},\delta_{2}|_{\mathbb{Z}_{p}^{\times}})$.
There is a natural _weight map_
$\mathrm{wt}:\mathcal{T}\to\widetilde{\mathcal{W}}$. Define
$\mathcal{T}_{\mathrm{reg}}$ to be the Zariski open subspace of $\mathcal{T}$,
where neither $\delta_{1}/\delta_{2}$ nor $\delta_{2}/\delta_{1}$ is a
character of $\mathbb{Q}_{p}^{\times}$ in the following list:

$x\mapsto x^{n}\textrm{ and }x\mapsto x^{n}\chi_{\mathrm{cycl}}(x)\textrm{
with }n\in\mathbb{Z}_{\geq 0}.$

Let
$\bar{r}_{p}:\operatorname{Gal}_{\mathbb{Q}_{p}}\to\operatorname{GL}_{2}(\mathbb{F})$
be a residual Galois representation. Let $R_{\bar{r}_{p}}^{\square}$ denote
the framed deformation ring of $\bar{r}_{p}$ parametrizing deformations of
$\bar{r}_{p}$ into matrix representations of
$\operatorname{Gal}_{\mathbb{Q}_{p}}$ with coefficients in noetherian complete
local $\mathcal{O}$-algebras. Then the Krull dimension of
$R_{\bar{r}_{p}}^{\square}$ is $9$. Let $V_{\mathrm{univ}}^{\square}$ denote
the universal (matrix) representation over $R_{\bar{r}_{p}}^{\square}$.

Let $\mathcal{X}_{\bar{r}_{p}}^{\square}$ denote the rigid analytic space over
$E$ associated to the formal scheme
$\operatorname{Spf}R_{\bar{r}_{p}}^{\square}$; it has dimension $8$. Write
$\mathcal{V}_{\mathrm{univ}}^{\square}$ for the associated universal
representation over $\mathcal{X}_{\bar{r}_{p}}^{\square}$. For a point
$x\in\mathcal{X}_{\bar{r}_{p}}^{\square}$ over $E^{\prime}$, write
$\mathcal{V}_{x}$ for universal Galois representation of
$\operatorname{Gal}_{\mathbb{Q}_{p}}$ over $E^{\prime}$ at $x$.

Following [BHS17, Definition 2.4], we define the triangulline deformation
space as follows.

######  Definition 7.4.

Let $U_{\bar{r}_{p},\mathrm{reg}}^{\square,\mathrm{tri}}$ denote the set of
closed points
$(x,\delta_{1},\delta_{2})\in\mathcal{X}_{\bar{r}_{p}}^{\square}\times\mathcal{T}_{\mathrm{reg}}$
(with some residue field $E^{\prime}$) such that the associated
$(\varphi,\Gamma)$-module
$\mathbb{D}_{\mathrm{rig}}^{\dagger}(\mathcal{V}_{x})$ sits in an exact
sequence

(7.4.1)
$0\to\mathcal{R}_{E^{\prime}}(\delta_{1})\to\mathbb{D}_{\mathrm{rig}}^{\dagger}(\mathcal{V}_{x})\to\mathcal{R}_{E^{\prime}}(\delta_{2})\to
0,$

where $\mathcal{R}_{E^{\prime}}$ is the Robba ring for $\mathbb{Q}_{p}$ with
coefficients in $E^{\prime}$; see [KPX14, § 6] and [Li15] for the notation
$\mathcal{R}_{E^{\prime}}(-)$ and related discussions on triangulations of
$(\varphi,\Gamma)$-modules.

The _triangulline deformation space of $\bar{r}_{p}$_, denoted by
$\mathcal{X}_{\bar{r}_{p}}^{\square,\mathrm{tri}}$, is the Zariski closure of
$U_{\bar{r}_{p},\mathrm{reg}}^{\square,\mathrm{tri}}$ inside the product
$\mathcal{X}_{\bar{r}_{p}}^{\square}\times\mathcal{T}$.

######  Proposition 7.5.

  1. (1)

The space $\mathcal{X}_{\bar{r}_{p}}^{\square,\mathrm{tri}}$ is a subspace of
$\mathcal{X}_{\bar{r}_{p}}^{\square}\times\mathcal{T}$ consisting of points
$(x,\delta_{1},\delta_{2})$ for which $\mathrm{det}(\mathcal{V}_{x})$
corresponds to $\delta_{1}\delta_{2}$ under local class field theory.
Moreover, set
$\mathcal{X}_{\bar{r}_{p}}^{\square,\mathrm{tri},\circ}:=\mathcal{X}_{\bar{r}_{p}}^{\square,\mathrm{tri}}\cap\big{(}\mathcal{X}_{\bar{r}_{p}}\times(\mathbb{G}_{m}^{\mathrm{rig}})^{2}\times\mathcal{W}\big{)}$,
then (7.1.1) induces an isomorphism

${\mathcal{X}_{\bar{r}_{p}}^{\square,\mathrm{tri},\circ}\times\mathcal{W}_{0}}$${\mathcal{X}_{\bar{r}_{p}}^{\square,\mathrm{tri}}}$${\big{(}(\mathcal{V}_{x},\delta_{1},\delta_{2}),\eta\big{)}}$${(\mathcal{V}_{x}\otimes\eta,\delta_{1}\otimes\eta,\delta_{2}\otimes\eta),}$

which is compatible with projections to the factor
$(\mathbb{G}_{m}^{\mathrm{rig}})^{2}$.

  2. (2)

The set $U_{\bar{r}_{p},\mathrm{reg}}^{\square,\mathrm{tri}}$ is the set of
closed points of a Zariski open and dense subspace
$\mathcal{U}_{\bar{r}_{p},\mathrm{reg}}^{\square,\mathrm{tri}}$ of
$\mathcal{X}_{\bar{r}_{p}}^{\square,\mathrm{tri}}$. The space
$\mathcal{X}_{\bar{r}_{p}}^{\square,\mathrm{tri}}$ is equidimensional of
dimension $7$.

###### Proof.

(1) obviously holds for points in
$U_{\bar{r}_{p},\mathrm{reg}}^{\square,\mathrm{tri}}$ and hence for
$\mathcal{X}_{\bar{r}_{p}}^{\square,\mathrm{tri}}$. (2) is proved in [BHS17,
Théorèm 2.6]. ∎

The main theorem of this section is the following.

######  Theorem 7.6.

Assume that $p\geq 11$. Let
$\bar{r}_{p}:\operatorname{Gal}_{\mathbb{Q}_{p}}\to\operatorname{GL}_{2}(\mathbb{F})$
be a residual local Galois representation as in Notation 7.2 with $2\leq a\leq
p-5$, and let $\bar{\rho}$ be as defined therein. Let
$\mathcal{X}_{\bar{r}_{p}}^{\square,\mathrm{tri}}$ be the triangulline
deformation space defined above. Let $\underline{x}=(x,\delta_{1},\delta_{2})$
be an $E^{\prime}$-point of
$\mathcal{X}_{\bar{r}_{p}}^{\square,\mathrm{tri}}$. Then the character
$\varepsilon=\delta_{2}|_{\Delta}\times\delta_{1}|_{\Delta}\cdot\omega^{-1}$
is relevant to $\bar{r}_{p}|_{\mathrm{I}_{\mathbb{Q}_{p}}}$. Put
$w_{\star}:=(\delta_{1}\delta_{2}^{-1}\chi_{\mathrm{cycl}}^{-1})(\exp(p))-1$
(for the image of $\underline{x}$ in $\mathcal{W}$ under $\mathrm{pr}_{W}$).

  1. (1)

If $v_{p}(\delta_{1}(p))=-v_{p}(\delta_{2}(p))>0$, then $v_{p}(\delta_{1}(p))$
is equal to a slope appearing in
$\operatorname{NP}\big{(}G_{\bar{\rho}}^{(\varepsilon)}(w_{\star},-)\big{)}$.

  2. (2)

If $v_{p}(\delta_{1}(p))=0$, then either
$\varepsilon=\omega^{b}\times\omega^{a+b}$, or
$\varepsilon=\omega^{a+b+1}\times\omega^{b-1}$ and
$\bar{r}_{p}|_{\mathrm{I}_{\mathbb{Q}_{p}}}\simeq\bar{\rho}^{\mathrm{ss}}$.

  3. (3)

If $v_{p}(\delta_{1}(p))=\frac{k}{2}-1$ and
$\delta_{1}|_{\mathbb{Z}_{p}^{\times}}=\chi_{\mathrm{cycl}}^{k-1}\delta_{2}|_{\mathbb{Z}_{p}^{\times}}$
for some integer $k\in\mathbb{Z}_{\geq 2}$, then
$\delta_{1}(p)=p^{k-2}\delta_{2}(p)$.

Conversely, fix characters $\delta_{1}|_{\mathbb{Z}_{p}^{\times}}$ and
$\delta_{2}|_{\mathbb{Z}_{p}^{\times}}$ such that $\varepsilon$ defined above
is relevant to $\bar{r}_{p}|_{\mathrm{I}_{\mathbb{Q}_{p}}}$. Then every
nonzero slope of
$\operatorname{NP}\big{(}G_{\bar{\rho}}^{(\varepsilon)}(w_{\star},-)\big{)}$
for
$w_{\star}:=(\delta_{1}\delta_{2}^{-1}\chi_{\mathrm{cycl}}^{-1})(\exp(p))-1$,
appears as $v_{p}(\delta_{1}(p))$ at some closed point
$\underline{x}=(x,\delta_{1},\delta_{2})\in\mathcal{X}_{\bar{r}_{p}}^{\square,\mathrm{tri}}$
(for some continuous characters $\delta_{1},\delta_{2}$ of
$\mathbb{Q}_{p}^{\times}$ extending the given
$\delta_{1}|_{\mathbb{Z}_{p}^{\times}}$ and
$\delta_{2}|_{\mathbb{Z}_{p}^{\times}}$).

The proof of this theorem will occupy the rest of this section, and is
concluded in § 7.22. We quickly remark that case (1) corresponds to the case
when $\mathcal{V}_{x}$ is reducible, and case (3) mostly concerns the case
when $\mathcal{V}_{x}$ is semistable and noncrystalline (after a twist).

Temporarily admitting this theorem, we first deduce a couple of corollaries
that partially answer a conjecture of Breuil–Buzzard–Emerton on crystalline
slopes of Kisin’s crystabelian deformation spaces and a conjecture of Gouvêa
on slopes of crystalline deformation spaces.

###  7.7. Kisin’s crystabelian deformation space

Let $\bar{r}_{p}$, $R_{\bar{r}_{p}}^{\square}$, and
$V_{\mathrm{univ}}^{\square}$ be as above. Let
$\underline{\psi}=\psi_{1}\times\psi_{2}:(\mathbb{Z}_{p}^{\times})^{2}\to
E^{\times}$ be a finite character, and let
$\underline{k}=(k_{1},k_{2})\in\mathbb{Z}^{2}$ with $k_{1}<k_{2}$ be a pair of
Hodge–Tate weights. (In our convention, $\chi_{\mathrm{cycl}}$ has Hodge–Tate
weight $-1$.) In [Kis08], Kisin proved that there is a unique
$\mathcal{O}$-flat quotient
$R_{\bar{r}_{p}}^{\square,\underline{k},\underline{\psi}}$ of
$R^{\square}_{\bar{r}_{p}}$, called _the Kisin’s crystabelian deformation
ring_ , such that every homomorphism $x^{*}:R_{\bar{r}_{p}}^{\square}\to
E^{\prime}$ factors through
$R_{\bar{r}_{p}}^{\square,\underline{k},\underline{\psi}}$ if and only if
$\mathcal{V}_{x}$ is potentially crystalline with Hodge–Tate weights
$(k_{1},k_{2})$ and the action of $\mathrm{I}_{\mathbb{Q}_{p}}$ on
$\mathbb{D}_{\mathrm{pcrys}}(\mathcal{V}_{x})$ is isomorphic to
$\psi_{1}\oplus\psi_{2}$. (Here $\mathbb{D}_{\mathrm{pcrys}}(-)$ is defined in
Notation 7.1.) When $R_{\bar{r}_{p}}^{\square,\underline{k},\underline{\psi}}$
is nonempty, each of its irreducible component has Krull dimension is $6$.
Moreover, the associated rigid analytic space
$\mathcal{X}_{\bar{r}_{p}}^{\square,\underline{k},\underline{\psi}}:=\big{(}\operatorname{Spf}R_{\bar{r}_{p}}^{\square,\underline{k},\underline{\psi}}\big{)}^{\mathrm{rig}}$
is smooth of dimension $5$ over $E$.

######  Corollary 7.8.

Assume that $p\geq 11$. Let
$\bar{r}_{p}:\operatorname{Gal}_{\mathbb{Q}_{p}}\to\operatorname{GL}_{2}(\mathbb{F})$
be a residual local Galois representation as in Notation 7.2 with $2\leq a\leq
p-5$. Let $\underline{\psi}$ and $\underline{k}$ be as above, and let $x$ be
an $E^{\prime}$-point of
$\mathcal{X}_{\bar{r}_{p}}^{\square,\underline{k},\underline{\psi}}$. Let
$\alpha_{x}$ be an eigenvalue of the $\phi$-action on the subspace of
$\mathbb{D}_{\mathrm{pcrys}}(\mathcal{V}_{x})$ where
$\operatorname{Gal}(\mathbb{Q}_{p}(\mu_{p^{\infty}})/\mathbb{Q}_{p})$ acts
through $\psi_{1}$. Write
$w_{\star}:=(\psi_{1}\psi_{2}^{-1})(\exp(p))\exp(p(k_{2}-k_{1}-1))-1$ (for the
image of $x$ in $\mathcal{W}$ under $\mathrm{pr}_{W}$). Then the character
$\varepsilon=\psi_{2}|_{\Delta}\cdot\omega^{-k_{2}}\times\psi_{1}|_{\Delta}\cdot\omega^{-k_{1}-1}$
is relevant to $\bar{r}_{p}|_{\mathrm{I}_{\mathbb{Q}_{p}}}$, and

  1. (1)

if $v_{p}(\alpha_{x})-k_{1}\notin\\{0,k_{2}-k_{1}\\}$, then it is equal to a
slope appearing in
$\operatorname{NP}\big{(}G_{\bar{\rho}}^{(\varepsilon)}(w_{\star},-)\big{)}$;

  2. (2)

if $v_{p}(\alpha_{x})\in\\{k_{1},k_{2}\\}$, then $\mathcal{V}_{x}$ is
reducible; and

  3. (3)

in the special case $\psi_{1}=\psi_{2}$, we have
$v_{p}(\alpha_{x})\neq\frac{k_{2}-k_{1}}{2}-1$.

Conversely, every slope of
$\operatorname{NP}\big{(}G_{\bar{\rho}}^{(\varepsilon)}(w_{\star},-)\big{)}$
belonging to $(0,k_{2}-k_{1})$ (but not equal to $\frac{k_{2}-k_{1}}{2}-1$
when $\psi_{1}=\psi_{2}$) appears as the $v_{p}(\alpha_{x})-k_{1}$ at some
point
$x\in\mathcal{X}_{\bar{r}_{p}}^{\square,\underline{k},\underline{\psi}}$.

###### Proof.

If $v_{p}(\alpha_{x})\in\\{k_{1},k_{2}\\}$, the standard $p$-adic Hodge theory
implies that $\mathcal{V}_{x}$ is reducible.

We henceforth assume that we are in situation (1), i.e.
$v_{p}(\alpha_{x})\notin\\{k_{1},k_{2}\\}$. This essentially follows from
Theorem 1.5 because all crystabelian representations are triangulline. More
precisely, let
$x\in\mathcal{X}_{\bar{r}_{p}}^{\square,\underline{k},\underline{\psi}}(E^{\prime})$
be a closed point. By possibly replacing $E^{\prime}$ by a quadratic
extension, the action of crystalline Frobenius $\phi$ and
$\operatorname{Gal}(\mathbb{Q}_{p}(\mu_{p^{\infty}})/\mathbb{Q}_{p})$ on
$\mathbb{D}_{\mathrm{pcrys}}(\mathcal{V}_{x})$ have two (generalized)
eigencharacters: $(\alpha_{1},\psi_{1})$ and $(\alpha_{2},\psi_{2})$, with
$\psi_{1},\psi_{2}$ in the data defining the deformation space and
$\alpha_{1},\alpha_{2}\in E^{\prime\times}$. We can also always assume that
$(\alpha_{1},\psi_{1})$ is a genuine eigencharacter. Define characters
$\delta_{i}:\mathbb{Q}_{p}^{\times}\to E^{\prime\times}$ with $i=1,2$ by

$\delta_{i}(p)=p^{-k_{i}}\alpha_{i},\quad\delta_{i}|_{\mathbb{Z}_{p}^{\times}}=x^{-k_{i}}\psi_{i}.$

Standard facts of Berger’s functor provides a triangulation

(7.8.1)
$0\to\mathcal{R}_{E^{\prime}}(\delta_{1})\to\mathbb{D}_{\mathrm{rig}}(\mathcal{V}_{x})\to\mathcal{R}_{E^{\prime}}(\delta_{2})\to
0.$

Indeed, if this fails, it must be that the eigenspace for
$(\alpha_{1},\psi_{1})$ agrees with
$\mathrm{Fil}^{k_{2}}\mathbb{D}_{\mathrm{pcrys}}(\mathcal{V}_{x})$; then the
admissibility condition for $\mathbb{D}_{\mathrm{pcrys}}(\mathcal{V}_{x})$
forces $v_{p}(\alpha_{1})=k_{2}$, contradicting our assumption.

Now, (7.8.1) upgrades $x$ to a point $(x,\delta_{1},\delta_{2})$ of
$\mathcal{X}_{\bar{r}_{p}}^{\square,\mathrm{tri}}$, for which
$v_{p}(\delta_{1}(p))=v_{p}(\alpha_{1})-k_{1}$. (1) now follows from Theorem
7.6, with

(7.8.2)
$w_{\star}:=(\delta_{1}\delta_{2}^{-1}\chi_{\mathrm{cycl}}^{-1})(\exp(p))-1=(\psi_{1}\psi_{2}^{-1})(\exp((p))\exp(p(k_{2}-k_{1}-1))-1.$

It remains to prove (3). Assume that $\psi_{1}=\psi_{2}$. Suppose that the
subspace $\mathcal{Y}$ of
$\mathcal{X}_{\bar{r}_{p}}^{\square,\underline{k},\underline{\psi}}$ where
$v_{p}(\alpha_{x})=\frac{k_{2}-k_{1}}{2}-1$ is nonempty. Then this is a rigid
analytic subspace, so in particular, $\dim\mathcal{Y}=5$. For each
$x\in\mathcal{Y}$,
$\delta_{1}|_{\mathbb{Z}_{p}^{\times}}=\chi_{\mathrm{cycl}}^{k_{2}-k_{1}}\delta_{2}|_{\mathbb{Z}_{p}^{\times}}$.
Theorem 7.6(3) implies that $\delta_{1}(p)=p^{k_{2}-k_{1}-2}\delta_{2}(p)$.
This means that $\mathcal{Y}$ is confined in the subspace where the ratio of
two Frobenius eigenvalues on $\mathbb{D}_{\mathrm{pcrys}}(\mathcal{V}_{x})$ is
precisely $p$. Let $x$ be a point of $\mathcal{Y}$. The dimension of the
tangent space of
$\mathcal{X}_{\bar{r}_{p}}^{\square,\underline{k},\underline{\psi}}$ at $x$ is
equal to
$1+3+\dim\mathrm{H}^{1}_{f}(\operatorname{Gal}_{\mathbb{Q}_{p}},\mathrm{Ad}(\mathcal{V}_{x}))$,
where $1$ comes from infinitesimal central twist of $\mathcal{V}_{x}$ by an
unramified representation, $3$ comes from the framing variables, and the one-
dimensional
$\mathrm{H}^{1}_{f}(\operatorname{Gal}_{\mathbb{Q}_{p}},\mathrm{Ad}(\mathcal{V}_{x}))$
corresponds to varying the ratio of two Frobenius eigenvalues. But our earlier
discussion shows that the ratio of two Frobenius eigenvalues on
$\mathbb{D}_{\mathrm{pcrys}}(\mathcal{V}_{x})$ is fixed to be $p$. (3) is
proved.

Conversely, given a slope of
$\operatorname{NP}\big{(}G_{\bar{\rho}}^{(\varepsilon)}(w_{\star},-)\big{)}$
belonging to $(0,k_{2}-k_{1})$ (and not being equal to
$\frac{k_{2}-k_{1}}{2}-1$ when $\psi_{1}=\psi_{2}$), Theorem 7.6 defines a
triangulation (7.8.1) with $\mathcal{V}_{x}$ having the reduction
$\bar{r}_{p}$. The slope condition implies that (7.8.1) belongs to the type
$\mathscr{S}^{\mathrm{cris}}_{+}$ in [Colm08]. So $\mathcal{V}_{x}$ is
crystabelian. ∎

######  Remark 7.9.

  1. (1)

We omitted a full discussion when $\alpha_{x}\in\\{k_{1},k_{2}\\}$, which is a
standard exercise in $p$-adic Hodge theory.

  2. (2)

(Possibly up to replacing $E$ by a degree $2$ extension when
$\psi_{1}=\psi_{2}$), it is possible embed
$\mathcal{X}_{\bar{r}_{p}}^{\square,\underline{k},\underline{\psi}}$ into
$\mathcal{X}_{\bar{r}_{p}}^{\square,\mathrm{tri}}$ as a rigid analytic
subspace, but this construction is a little messy to present, in the ordinary,
critical, or Frobenius non-semisimple cases. We content ourselves with a
pointwise description and leave the “global” argument to interested readers.

The following answers positively a conjecture by Breuil–Buzzard–Emerton, and a
conjecture of Gouvêa, when the residual Galois representation is reducible and
generic. We refer to § 1.8 and § 1.11 for the discussion on their history, and
Remarks 1.10 and 1.13 for comments on prior related works.

######  Corollary 7.10.

Assume that $p\geq 11$. Let
$\bar{r}_{p}:\operatorname{Gal}_{\mathbb{Q}_{p}}\to\operatorname{GL}_{2}(\mathbb{F})$
be a residual local Galois representation as in Notation 7.2 with $2\leq a\leq
p-5$. Let $\underline{\psi}$, $\underline{k}$, $x$, $\alpha_{x}$ be as in
Corollary 7.8.

  1. (1)

If $m$ denotes the minimal _positive_ integer such that
$\psi_{1}\psi_{2}^{-1}$ is trivial on $(1+p^{m}\mathbb{Z}_{p})^{\times}$, then

$v_{p}(\alpha_{x})\in\begin{cases}\big{(}\frac{a}{2}+\mathbb{Z}\big{)}\cup\mathbb{Z}&\textrm{
when }m=1,\\\ \frac{1}{(p-1)p^{m-1}}\mathbb{Z}&\textrm{ when }m\geq
2.\end{cases}$

  2. (2)

If $\psi_{1}=\psi_{2}$, then

$v_{p}(\alpha_{x})-k_{1}\textrm{ or }k_{2}-1-v_{p}(\alpha_{x})\textrm{ belongs
to }\Big{[}0,\,\frac{k_{2}-k_{1}-1-\min\\{a+1,p-2-a\\}}{p+1}\Big{]}.$

###### Proof.

(1) When $m=1$, this follows from Corollary 7.8 and Proposition 2.18(6). When
$m\geq 2$, $v_{p}(w_{\star})=\frac{1}{(p-1)p^{m-1}}$, and the slopes of
$\operatorname{NP}\big{(}G^{(\varepsilon)}_{\bar{\rho}}(w_{\star},-)\big{)}$
are precisely $v_{p}(w_{\star})\cdot\big{(}\deg g_{n}^{(\varepsilon)}(w)-\deg
g_{n-1}^{(\varepsilon)}(w)\big{)}$ for some $n\in\mathbb{N}$ with multiplicity
one, by the last line of Definition-Proposition 2.12(4). In this case, (1)
follows from this and Corollary 7.8.

(2) If $\psi_{1}=\psi_{2}$, then $v_{p}(\alpha_{1})-k_{1}$ is a slope of
$\operatorname{NP}\big{(}G_{\bar{\rho}}^{(\varepsilon)}(w_{k_{2}-k_{1}},-)\big{)}$
which is not $\frac{k_{2}-k_{1}}{2}-1$. By Proposition 2.16(3)(4), either
$v_{p}(\alpha_{x})-k_{1}$ belongs to
$\big{[}0,\,\frac{k_{2}-k_{1}-1-\min\\{a+1,p-2-a\\}}{p+1}\big{]}$, or
$(k_{2}-k_{1}-1)-(v_{p}(\alpha_{x})-k_{1})=k_{2}-1-v_{p}(\alpha_{x})$ belongs
to this set. ∎

The rest of this section is devoted to proving Theorem 7.6, which is completed
in § 7.22.

###  7.11. Reducing Theorem 7.6 to the nonsplit case

We first show that Theorem 7.6 for $\bar{r}_{p}$ nonsplit implies the theorem
for $\bar{r}_{p}$ split. This is essentially because, at least pointwise for
an irreducible triangulline representation, there are lattices with different
reductions.

To make this precise, we first note that the character
$\varepsilon=\delta_{2}|_{\Delta}\times\delta_{1}|_{\Delta}\cdot\omega^{-1}$
is always relevant to $\bar{r}_{p}|_{\mathrm{I}_{\mathbb{Q}_{p}}}$ by
considering the $\mathrm{det}\mathcal{V}_{x}$. Next, by twisting all
representations by
$\omega\circ\omega_{1}^{-b}:\operatorname{Gal}_{\mathbb{Q}_{p}}\to\mathbb{F}_{p}^{\times}\to\mathcal{O}^{\times}$,
we may reduce to the case when $b=0$.

Now suppose that Theorem 7.6 holds for nonsplit residual local Galois
representations. Let $\bar{r}_{p}$ be a split residual local Galois
representation as in Notation 7.2 with $*=0$ and $b=0$. Then there is a unique
nonsplit residual local Galois representation $\bar{r}^{\prime}_{p}$ which is
an extension of $\mathrm{ur}(\bar{\alpha}_{2})$ by
$\mathrm{ur}(\bar{\alpha}_{1})\omega^{a+1}$. In particular,
$\bar{r}^{\prime}_{p}|_{\mathrm{I}_{\mathbb{Q}_{p}}}\simeq\bar{\rho}={\big{(}\begin{smallmatrix}\omega_{1}^{a+1}&*\neq
0\\\ 0&1\end{smallmatrix}\big{)}}$ as in Notation 7.2.

Let $\underline{x}=(x,\delta_{1},\delta_{2})$ be an $E^{\prime}$-point of
$\mathcal{U}_{\bar{r}_{p},\mathrm{reg}}^{\square,\mathrm{tri}}$ . (By Zariski
density, it is enough to consider points in the regular locus.) We separate
two cases.

(1) If $\mathcal{V}_{x}$ is irreducible, then it is well known that, after
possibly enlarging $E^{\prime}$, $\mathcal{V}_{x}$ admits an
$\mathcal{O}^{\prime}$-lattice $\mathcal{V}^{\circ}_{x}$ such that
$\mathcal{V}_{x}^{\circ}/\varpi^{\prime}\mathcal{V}_{x}^{\circ}\simeq\bar{r}^{\prime}_{p}$.
It follows that
$\underline{x}^{\prime}:=(\mathcal{V}^{\circ}_{x},\delta_{1},\delta_{2})$ also
defines a point on
$\mathcal{U}_{\bar{r}^{\prime}_{p},\mathrm{reg}}^{\square,\mathrm{tri}}$.
Theorem 7.6 for $\underline{x}^{\prime}$ implies that for Theorem 7.6 for
$\underline{x}$.

(2) If $\mathcal{V}_{x}$ is reducible, i.e. there exists an exact sequence
$0\to\mathcal{V}_{x}^{+}\to\mathcal{V}_{x}\to\mathcal{V}_{x}^{-}\to 0$ of
representations of $\operatorname{Gal}_{\mathbb{Q}_{p}}$. There are two
possibilities:

  1. (2a)

If $\delta_{1}(p)\in\mathcal{O}^{\prime\times}$, then (7.4.1) produces an
exact sequence of Galois representations. In particular,
$\mathcal{R}_{E^{\prime}}(\delta_{1})$ is isomorphic to either
$\mathbb{D}_{\mathrm{rig}}(\mathcal{V}_{x}^{+})$ or
$\mathbb{D}_{\mathrm{rig}}(\mathcal{V}_{x}^{-})$. This will imply that
$\delta_{2}|_{\Delta}\times\delta_{1}|_{\Delta}\cdot\omega^{-1}=1\times\omega^{a}$
or $\omega^{a+1}\times\omega^{-1}$, proving (2a).

  2. (2b)

If $v_{p}(\delta_{1}(p))>0$, this falls in the case of
$\mathscr{S}_{+}^{\mathrm{ord}}$ per classification of triangulline
representations in [Colm08, § 1.2]. In particular,
$v_{p}(\delta_{1}(p))=w(\delta_{1}\delta_{2}^{-1})\in\mathbb{N}$, where

$w(\delta_{1}\delta_{2}^{-1}):=\lim_{\begin{subarray}{c}\gamma\in\mathbb{Z}_{p}^{\times}\\\
\gamma\to
1\end{subarray}}\frac{\log(\delta_{1}\delta_{2}^{-1})}{\log(\chi_{\mathrm{cycl}}(\gamma))}$

is the (negative of) generalized Hodge–Tate weight. (In [Colm08], Colmez calls
$w(\delta_{1}\delta_{2}^{-1})$ the Hodge–Tate weight because in his convention
the cyclotomic character has Hodge–Tate weight $1$.) Put
$k:=w(\delta_{1}\delta_{2}^{-1})+1$. In this case, there is another
triangulation

$0\to
t^{k-1}\mathcal{R}_{E^{\prime}}(\delta_{2})\to\mathbb{D}_{\mathrm{rig}}(\mathcal{V}_{x})\to
t^{1-k}\mathcal{R}_{E^{\prime}}(\delta_{1})\to 0,$

which produces precisely the exact sequence
$0\to\mathcal{V}_{x}^{+}\to\mathcal{V}_{x}\to\mathcal{V}_{x}^{-}\to 0$. This
in particularly shows that

$\varepsilon=\delta_{2}|_{\Delta}\times\delta_{1}|_{\Delta}\cdot\omega^{-1}=\omega^{a-k+2}\times\omega^{k-2}.$

We need to show that, $k-1$ is a slope in
$\operatorname{NP}\big{(}G_{\bar{\rho}}^{(\varepsilon)}(w_{\star},-)\big{)}$,
(by directly exhibiting such a slope). There are two subcases we need to
consider.

  3. (2bi)

If
$\delta_{1}|_{(1+p\mathbb{Z}_{p})^{\times}}=\delta_{2}|_{(1+p\mathbb{Z}_{p})^{\times}}$,
then
$w_{\star}=(\delta_{1}\delta_{2}^{-1}\chi_{\mathrm{cycl}}^{-1})(\exp(p))=w_{k}$.
We invoke the compatibility of Atkin–Lehner involution and $p$-stabilization
with ghost series in Proposition 2.16(2)(3): the
$d_{k}^{\mathrm{Iw}}(\omega^{a-k+2}\times 1)$th slope of
$\operatorname{NP}\big{(}G_{\bar{\rho}}^{(\varepsilon)}(w_{k},-)\big{)}$ is
precisely $k-1$ minus the first slope of
$\operatorname{NP}\big{(}G_{\bar{\rho}}^{(\varepsilon^{\prime\prime})}(w_{k},-)\big{)}$
with $s_{\varepsilon^{\prime\prime}}=k-2-a-(k-2-a)=0$. So the latter ghost
slope is $0$, and thus the former ghost slope is $k-1$, i.e.
$v_{p}(\delta_{1}(p))$ is a slope of
$\operatorname{NP}\big{(}G_{\bar{\rho}}^{(\varepsilon)}(w_{k},-)\big{)}$.

  4. (2bii)

If the minimal positive integer $m$ such that
$\delta_{1}|_{(1+p^{m}\mathbb{Z}_{p})^{\times}}=\delta_{2}|_{(1+p^{m}\mathbb{Z}_{p})^{\times}}$
satisfies $m\geq 2$, then we are in the “halo region”; in particular,
$v_{p}(w_{\star})=\frac{1}{p^{m-2}(p-1)}$. In this case, Definition-
Proposition 2.12(4) implies that the $n$th slope of
$\operatorname{NP}\big{(}G_{\bar{\rho}}^{(\varepsilon)}(w_{\star},-)\big{)}$
is just $\frac{1}{p^{m-2}(p-1)}\big{(}\deg g_{n}^{(\varepsilon)}(w)-\deg
g_{n-1}^{(\varepsilon)}(w)\big{)}$. We compute this explicitly using the
formulas in Definition-Proposition 2.12(4) with $s_{\varepsilon}=\\{k-a-2\\}$,

     * •

If $a+s_{\varepsilon}<p-1$, note that $p^{m-1}(k-1)-1\equiv k-2\equiv
a+s_{\varepsilon}\bmod(p-1)$. So for
$N=\frac{p^{m-1}(k-1)-1-\\{k-2\\}}{p-1}+1$, we have
$\mathbf{e}_{2N}^{(\varepsilon)}=e_{2}^{*}z^{p^{m-1}(k-1)-1}$. Moreover, we
have

$\tfrac{p^{m-1}(k-1)-1-\\{k-2\\}}{p-1}+1-\\{k-2-a\\}-(a+2)\equiv\\{k-2\\}-\\{k-2-a\\}-a=0\pmod{p}.$

This implies by (2.12.1) with $n=2N-1$ and the “otherwise case” (as
$2N-2s_{\varepsilon}\equiv 2a+4\pmod{p}$),

$\deg g_{2N}^{(\varepsilon)}(w)-\deg
g_{2N-1}^{(\varepsilon)}(w)=\deg\mathbf{e}_{2N}^{(\varepsilon)}-\Big{\lfloor}\frac{\deg\mathbf{e}_{2N}^{(\varepsilon)}}{p}\Big{\rfloor}=p^{m-2}(p-1)(k-1).$

So the $2N$th slope of
$\operatorname{NP}\big{(}G_{\bar{\rho}}^{(\varepsilon)}(w_{\star},-)\big{)}$
is $k-1$.

     * •

If $a+s_{\varepsilon}\geq p-1$, the argument is similar. Still, we put
$N=\frac{p^{m-1}(k-1)-1-\\{k-2\\}}{p-1}+1$, but
$\mathbf{e}_{2N-1}^{(\varepsilon)}=e_{2}^{*}z^{p^{m-1}(k-1)-1}$. We have a
similar congruence

$N+1-\\{k-2-a\\}-(a+3)\equiv\\{k-2\\}-\\{k-2-a\\}-a+3=p\equiv 0\pmod{p}.$

This implies by (2.12.2) with $n=2N-1$ and the “otherwise case” (as
$2N-1-2s_{\varepsilon}\equiv 2a+5\pmod{p}$) that

$\deg g_{2N-1}^{(\varepsilon)}(w)-\deg
g_{2N-2}^{(\varepsilon)}(w)=\deg\mathbf{e}_{2N-1}^{(\varepsilon)}-\Big{\lfloor}\frac{\deg\mathbf{e}_{2N-1}^{(\varepsilon)}}{p}\Big{\rfloor}=p^{m-2}(p-1)(k-1).$

This means that the $(2N-1)$th slope of
$\operatorname{NP}\big{(}G_{\bar{\rho}}^{(\varepsilon)}(w_{\star},-)\big{)}$
is $k-1$.

Up to now, we have checked (1)–(3) of Theorem 7.6. Conversely, if
$\delta_{1}|_{\mathbb{Z}_{p}^{\times}}$ and
$\delta_{2}|_{\mathbb{Z}_{p}^{\times}}$ are given as in Theorem 7.6. Put
$w_{\star}:=(\delta_{1}\delta_{2}^{-1}\chi_{\mathrm{cycl}}^{-1})(\exp(p))-1$.
Let $\lambda$ be a slope of
$\operatorname{NP}\big{(}G_{\bar{\rho}}^{(\varepsilon)}(w_{\star},-)\big{)}$.

  * (1)’

If $\lambda>0$, Theorem 7.6 for the nonsplit representation
$\bar{r}^{\prime}_{p}$ produces an $E^{\prime}$-point
$\underline{x}^{\prime}=(x^{\prime},\delta_{1},\delta_{2})\in\mathcal{X}_{\bar{r}^{\prime}_{p}}^{\square,\mathrm{tri}}$
with $v_{p}(\delta_{1}(p))=\lambda$. Reversing the argument in (1) gives the
needed point of $\mathcal{X}_{\bar{r}^{\prime}_{p}}^{\square,\mathrm{tri}}$.

  * (2)’

If $\lambda=0$, we must have $\varepsilon=1\times\omega^{a}$. We construct a
point on $\mathcal{X}_{\bar{r}_{p}}^{\square,\mathrm{tri}}$ directly. Lift
$\bar{\alpha}_{i}\in\mathbb{F}^{\times}$ for each $i=1,2$ to
$\delta_{i}(p)\in\mathcal{O}^{\times}$. Then
$\mathcal{R}_{E^{\prime}}(\delta_{1})\oplus\mathcal{R}_{E^{\prime}}(\delta_{2})$
is the $(\varphi,\Gamma)$-module of $\delta_{1}\oplus\delta_{2}$, which
reduces to $\bar{r}_{p}$ automatically, with the correct slope and characters.

This completes the reduction of Theorem 7.6 to the reducible, nonsplit, and
generic case.

######  Remark 7.12.

  1. (1)

Supposedly, the proof of (2bii) should also follow from an analogous
compatibility of Atkin–Lehner involution for ghost series with wild
characters. We leave that for interested readers.

  2. (2)

It is a very interesting question to ask whether the above correspondence of
points between $\mathcal{U}_{\bar{r}_{p},\mathrm{reg}}^{\square,\mathrm{tri}}$
and $\mathcal{U}_{\bar{r}^{\prime}_{p},\mathrm{reg}}^{\square,\mathrm{tri}}$
can be made “globally” at the level of rigid analytic spaces or even at the
level of formal schemes. This seems to be a rather subtle yet very interesting
question.

######  Assumption 7.13.

In view of § 7.11, we assume that $\bar{r}_{p}$ is nonsplit for the rest of
this section. In particular,
$\bar{r}_{p}|_{\mathrm{I}_{\mathbb{Q}_{p}}}\simeq\bar{\rho}$. We write
$\bar{r}_{p}:\operatorname{Gal}_{\mathbb{Q}_{p}}\to\operatorname{GL}_{2}(\mathbb{F})$
as

(7.13.1) $\bar{r}_{p}={\begin{pmatrix}\bar{\chi}_{1}&*\neq 0\\\
0&\bar{\chi}_{2}\end{pmatrix}},\quad\textrm{with
}\bar{\chi}_{1}=\mathrm{ur}(\bar{\alpha}_{1})\cdot\omega_{1}^{a+b+1}\textrm{
and }\bar{\chi}_{2}=\mathrm{ur}(\bar{\alpha}_{2})\cdot\omega_{1}^{b}.$

###  7.14. Paškūnas modules

To relate the study of local ghost series with the triangulline deformation
space, we make use of the Paškūnas modules in [Pa13] for deformation of
$p$-adic representations of $\operatorname{GL}_{2}(\mathbb{Q}_{p})$. As [Pa13]
mainly considers the case with a fixed central character, some of our
constructions later may be slightly awkward. Similar arguments to remove
central character constraints can be found in [BD20, Appendix A] and
[CEGGPS18]. Let
$\zeta:\operatorname{Gal}_{\mathbb{Q}_{p}}\to\mathcal{O}^{\times}$ be a
character that induces a character of $\mathbb{Q}_{p}^{\times}$ by local class
field theory.

  * •

Let $\mathrm{Mod}_{\operatorname{Gal}_{\mathbb{Q}_{p}}}^{\mathrm{pro}}$ be the
category of profinite $\mathcal{O}$-modules $V$ with continuous
$\operatorname{Gal}_{\mathbb{Q}_{p}}$-actions.

  * •

Let $\mathfrak{C}$ be the category of profinite $\mathcal{O}$-modules $M$ with
continuous _right_ $\operatorname{GL}_{2}(\mathbb{Q}_{p})$-actions for which

    * –

the right $\operatorname{GL}_{2}(\mathbb{Z}_{p})$-action on $M$ extends to a
right
$\mathcal{O}\llbracket\operatorname{GL}_{2}(\mathbb{Z}_{p})\rrbracket$-module
structure on $M$, and

    * –

for every vector $v$ in the Pontryagin dual
$M^{\vee}:=\operatorname{Hom}_{\mathcal{O}}(M,E/\mathcal{O})$ equipped with
the induced left $\operatorname{GL}_{2}(\mathbb{Q}_{p})$-action, the left
$\mathcal{O}[\operatorname{GL}_{2}(\mathbb{Q}_{p})]$-submodule generated by
$v$ is of finite length.

  * •

Let $\mathfrak{C}_{\zeta}$ be the subcategory of $\mathfrak{C}$ consisting of
objects on which $\mathbb{Q}_{p}^{\times}$ acts by $\zeta$.

We chose to work with right
$\mathcal{O}\llbracket\operatorname{GL}_{2}(\mathbb{Q}_{p})\rrbracket$-actions
on objects of $\mathfrak{C}$ to match our definition of
$\mathcal{O}\llbracket\mathrm{K}_{p}\rrbracket$-projective augmented modules
in Definition 2.2. This can be easily translated from references [Pa13, Pa15,
HP19, BD20] by considering the inverse action.

There is a natural covariant _modified Colmez functor_

$\check{\mathbf{V}}_{\zeta}:\mathfrak{C}_{\zeta}\to\mathrm{Mod}_{\operatorname{Gal}_{\mathbb{Q}_{p}}}^{\mathrm{pro}},$

which is compatible with taking projective limits and whose evaluation on
finite length objects $M$ is given by
$\check{\mathbf{V}}_{\zeta}(M):=\mathbf{V}(M^{\vee})^{\vee}(\chi_{\mathrm{cycl}}\zeta)$,
where $(-)^{\vee}=\operatorname{Hom}_{\mathcal{O}}(-,E/\mathcal{O})$ is the
Pontryagin duality and $\mathbf{V}(-)$ is the functor defined in [Colm10b]. In
particular, for two characters
$\bar{\eta}_{1},\bar{\eta}_{2}:\mathbb{Q}_{p}^{\times}\to\mathbb{F}^{\times}$
such that
$\bar{\eta}_{1}\bar{\eta}_{2}\bar{\chi}_{\mathrm{cycl}}^{-1}=\zeta\bmod\varpi$,

$\check{\mathbf{V}}_{\zeta}\Big{(}\operatorname{Ind}_{B(\mathbb{Q}_{p})}^{\operatorname{GL}_{2}(\mathbb{Q}_{p})}\big{(}\bar{\eta}_{1}\otimes\bar{\eta}_{2}\bar{\chi}_{\mathrm{cycl}}^{-1}\big{)}^{\vee}\Big{)}\cong\bar{\eta}_{1}.$

We note that for a different character
$\zeta^{\prime}:\operatorname{Gal}_{\mathbb{Q}_{p}}\to\mathcal{O}^{\times}$,

(7.14.1)
$\check{\mathbf{V}}_{\zeta\zeta^{\prime}}(M\otimes\zeta^{\prime}\circ\mathrm{det})\cong\check{\mathbf{V}}_{\zeta}(M)\otimes\zeta^{\prime}.$

We focus on the case of Assumption 7.13. Take the earlier $\zeta$ to satisfy
$\zeta\equiv\omega^{a+2b}\bmod\varpi$.

Let $\pi(\bar{r}_{p})$ denote the smooth representation of
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$ over $\mathbb{F}$ associated to
$\bar{r}_{p}$ by the mod $p$ Langlands correspondence. Explicitly,
$\pi(\bar{r}_{p})$ is the nontrivial extension $\bar{\pi}_{1}-\bar{\pi}_{2}$
with

$\bar{\pi}_{1}=\operatorname{Ind}_{B(\mathbb{Q}_{p})}^{\operatorname{GL}_{2}(\mathbb{Q}_{p})}\big{(}\bar{\chi}_{2}\otimes\bar{\chi}_{1}\bar{\chi}_{\mathrm{cycl}}^{-1}\big{)}\textrm{\quad
and\quad}\bar{\pi}_{2}=\operatorname{Ind}_{B(\mathbb{Q}_{p})}^{\operatorname{GL}_{2}(\mathbb{Q}_{p})}\big{(}\bar{\chi}_{1}\otimes\bar{\chi}_{2}\bar{\chi}_{\mathrm{cycl}}^{-1}\big{)}.$

In particular, we have

$\check{\mathbf{V}}_{\zeta}(\pi(\bar{r}_{p})^{\vee})\cong\check{\mathbf{V}}_{\zeta}(\bar{\pi}_{2}^{\vee}-\bar{\pi}_{1}^{\vee})\cong(\bar{\chi}_{1}-\bar{\chi}_{2})\cong\bar{r}_{p}.$

This is independent of the choice of $\zeta$ and agrees with [Pa13, § 8]; yet
[Pa15, § 6.1] seems to have a minor error by swapping the $\bar{\pi}_{1}$ with
$\bar{\pi}_{2}$, which is later corrected in [HP19].

Let $\boldsymbol{1}_{\mathrm{tw}}$ denote $\mathcal{O}\llbracket
u,v\rrbracket$ equipped with a $\mathbb{Q}_{p}^{\times}$-action where $p$ acts
by multiplication by $1+u$ and $a\in\mathbb{Z}_{p}^{\times}$ acts by
multiplication by $(1+v)^{\log(a/\omega(\bar{a}))/p}$; such action extends to
an action of $\operatorname{Gal}_{\mathbb{Q}_{p}}$ via local class field
theory.

As
$\operatorname{End}_{\operatorname{Gal}_{\mathbb{Q}_{p}}}(\bar{r}_{p})\cong\mathbb{F}$,
the deformation problem of $\bar{r}_{p}$ is representable by a noetherian
complete local $\mathcal{O}$-algebra $R_{\bar{r}_{p}}$. Let
$R_{\bar{r}_{p}}^{\zeta}$ denote the quotient parametrizing the deformations
of $\bar{r}_{p}$ with fixed determinant $\zeta$; let
$\mathfrak{m}_{R_{\bar{r}_{p}}^{\zeta}}$ denote its maximal ideal. Let
$V_{\mathrm{univ}}^{\zeta}$ denote the universal deformation of $\bar{r}_{p}$
over $R_{\bar{r}_{p}}^{\zeta}$. It is well known that there is a
(noncanonical) isomorphism

$R_{\bar{r}_{p}}^{\square}\cong
R_{\bar{r}_{p}}^{\zeta}\widehat{\otimes}_{\mathcal{O}}\mathcal{O}\llbracket
u,v,z_{1},z_{2},z_{3}\rrbracket,$

so that the framed and unframed universal deformations of $\bar{r}_{p}$
satisfy:

$V_{\mathrm{univ}}^{\zeta}\widehat{\boxtimes}_{\mathcal{O}}{\bf
1}_{\mathrm{tw}}\widehat{\otimes}_{\mathcal{O}}\mathcal{O}\llbracket
z_{1},z_{2},z_{3}\rrbracket\cong V_{\mathrm{univ}}^{\square}.$

Following [Pa13, § 8], we have the following.

######  Theorem 7.15.

Keep the notation as above. Let
$\widetilde{P}_{\zeta}\twoheadrightarrow\bar{\pi}_{1}^{\vee}$ be a projective
envelope of $\pi_{1}^{\vee}$ in $\mathfrak{C}_{\zeta}$ and put
$R_{\pi_{1},\zeta}:=\operatorname{End}_{\mathfrak{C}_{\zeta}}(\widetilde{P}_{\zeta})$.

  1. (1)

The $\check{\mathbf{V}}_{\zeta}(\widetilde{P}_{\zeta})$ can be viewed as a
$2$-dimensional representation of $\operatorname{Gal}_{\mathbb{Q}_{p}}$ over
$R_{\pi_{1},\zeta}$ lifting $\bar{r}_{p}$; this induces an isomorphism
$R_{\bar{r}_{p}}^{\zeta}\xrightarrow{\ \cong\ }R_{\pi_{1},\zeta}$, and
$\check{\mathbf{V}}_{\zeta}(\widetilde{P}_{\zeta})\cong
V_{\mathrm{univ}}^{\zeta}$.

  2. (2)

Define the following object in $\mathfrak{C}$:

(7.15.1)
$\widetilde{P}^{\square}:=\widetilde{P}_{\zeta}\widehat{\boxtimes}_{\mathcal{O}}{\bf
1}_{\mathrm{tw}}\widehat{\otimes}_{\mathcal{O}}\mathcal{O}\llbracket
z_{1},z_{2},z_{3}\rrbracket,$

equipped with the tensor product right
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$-action (which is
$\mathcal{O}\llbracket z_{1},z_{2},z_{3}\rrbracket$-linear). Then
$\widetilde{P}^{\square}$ carries a natural $R_{\bar{r}_{p}}^{\square}$-action
from the left that commutes with the right
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$-action. Moreover,
$\widetilde{P}^{\square}$ does not depend on the choice of $\zeta$.

  3. (3)

There exists
$x\in\mathfrak{m}_{R^{\zeta}_{\bar{r}_{p}}}\backslash\big{(}\mathfrak{m}^{2}_{R^{\zeta}_{\bar{r}_{p}}}+(\varpi)\big{)}$
such that $\widetilde{P}^{\square}$ is isomorphic to the projective envelope
of $\operatorname{Sym}^{a}\mathbb{F}^{\oplus 2}\otimes\mathrm{det}^{b}$ as a
right $\mathcal{O}\llbracket
u,x,z_{1},z_{2},z_{3}\rrbracket\llbracket\operatorname{GL}_{2}(\mathbb{Z}_{p})\rrbracket$-module.

###### Proof.

(1) is [Pa13, Corollary 8.7]. For (2), the left
$R_{\bar{r}_{p}}^{\square}$-action comes from the isomorphism
$R_{\bar{r}_{p}}^{\zeta}\cong R_{\pi_{,}\zeta}$ proved in (1). The uniqueness
follows from (7.14.1).

We now prove (3). For $A=\mathcal{O}$ or $\mathcal{O}\llbracket x\rrbracket$,
let
$\mathrm{Mod}_{A\llbracket\operatorname{GL}_{2}(\mathbb{Z}_{p})\rrbracket,\zeta}^{\mathrm{fg}}$
denote the category of finitely generated right
$A\llbracket\operatorname{GL}_{2}(\mathbb{Z}_{p})\rrbracket$-modules with the
scalar $\mathbb{Z}_{p}^{\times}$ acting by $\zeta$. By [Pa15, Theorem 5.2],
there exists $x\in\mathfrak{m}_{R^{\zeta}_{\bar{r}_{p}}}$ such that
$x:\widetilde{P}_{\zeta}\to\widetilde{P}_{\zeta}$ is injective and
$\widetilde{P}_{\zeta}/x\widetilde{P}_{\zeta}$ is the projective envelope of
$(\mathrm{soc}_{\operatorname{GL}_{2}(\mathbb{Z}_{p})}\bar{\pi}_{1})^{\vee}=\operatorname{Sym}^{a}\mathbb{F}^{\oplus
2}\otimes\mathrm{det}^{b}$ in
$\mathrm{Mod}_{\mathcal{O}\llbracket\operatorname{GL}_{2}(\mathbb{Z}_{p})\rrbracket,\zeta}^{\mathrm{fg}}$.
In addition, [HP19, Theorem 3.3(iii)] proves that
$x\notin\big{(}\mathfrak{m}^{2}_{R^{\zeta}_{\bar{r}_{p}}}+(\varpi)\big{)}$. It
then remains to show that $\widetilde{P}_{\zeta}$ is projective in the
$\mathrm{Mod}_{\mathcal{O}\llbracket
x\rrbracket\llbracket\operatorname{GL}_{2}(\mathbb{Z}_{p})\rrbracket,\zeta}^{\mathrm{fg}}$,
as the projectivity is preserved for tensor products of the form in (7.15.1).
(Note that the variable $v$ in $\widetilde{P}_{\zeta}$ measuring the central
twist of $(1+p\mathbb{Z}_{p})^{\times}$ is “absorbed” into the projective
envelope as an
$\mathcal{O}\llbracket\operatorname{GL}_{2}(\mathbb{Z}_{p})\rrbracket$-module.)
Choose a character $\eta$ of $(1+p\mathbb{Z}_{p})^{\times}$ such that
$\zeta|_{(1+p\mathbb{Z}_{p})^{\times}}=\eta^{2}$. Then it is enough to show
that $\widetilde{P}_{\zeta}\otimes\eta^{-1}\circ\mathrm{det}$ is a projective
right $\mathcal{O}\llbracket x\rrbracket\llbracket H\rrbracket$-module with
$H=\operatorname{GL}_{2}(\mathbb{Z}_{p})/(1+p\mathbb{Z}_{p})^{\times}$, or
equivalently,

$\operatorname{Tor}_{>0}^{\mathcal{O}\llbracket x\rrbracket\llbracket