2.1 Conventions

The following conventions will be used throughout the article. We let $\mathbb {F}_q$ be an extension of $\mathbb {F}_p$ with $a=\left [\mathbb {F}_q:\mathbb {F}_p\right ]$ . It is enough to prove Theorem 1.1 after replacing q with a larger power of p. In particular, we increase q throughout the article if it simplifies arguments. We will frequently have families of things indexed by $i=0,\dotsc , a-1$ (e.g., the p-adic digits $e_{Q,i}$ of $\epsilon _{Q}$ from Section 1.1). It will be convenient to have the indices ‘wrap around’ modulo a. That is, we take $e_{Q,a}$ to be $e_{Q,0}$ , $e_{Q,a+1}$ to be $e_{Q,1}$ and so forth.

Let $L_0$ be the unramified extension of $\mathbb {Q}_p$ whose residue field is $\mathbb {F}_q$ . Let E be a finite totally ramified extension of $\mathbb {Q}_p$ of degree e and set $L=E\otimes _{\mathbb {Q}_p} L_0$ . Define $\mathcal {O}_L$ (resp., $\mathcal {O}_E$ ) to be the ring of integers of L (resp., E) and let $\mathfrak m$ be the maximal ideal of $\mathcal {O}_L$ . We let $\pi _\circ $ be a uniformising element of E. Fix $\pi =(-p)^{\frac {1}{p-1}}$ , and for any positive rational number s we set $\pi _s=\pi ^{\frac {1}{s}}$ . We will assume that E is large enough to contain $\pi _{s_{\tau _i}}$ for each $i=1,\dotsc , \mathbf {m}$ . We also assume that E is large enough to contain the image of $\rho ^{wild}$ (i.e., E contains enough pth-power roots of unity). Define $\nu $ to be the endomorphism $\text {id}\otimes \text {Frob}$ of L, where $\text {Frob}$ is the p-Frobenius automorphism of $L_0$ . For any E-algebra R and $x \in R$ , we obtain an operator $R \to R$ sending $r \mapsto xr$ . By abuse of notation, we will refer to this operator as x. Finally, for any ring R with valuation $v:R \to \mathbb {R}$ and any $x \in R$ with $v(x)>0$ , we let $v_x(\cdot )$ denote the normalisation of v satisfying $v_x(x)=1$ .

2.2 Frobenius endomorphisms

Let $\overline {A}$ be an $\mathbb {F}_q$ -algebra, let A be an $\mathcal {O}_L$ -algebra with $A\otimes _{\mathcal {O}_L} \mathbb {F}_q = \overline {A}$ and let $\mathcal {A}=A \otimes _{\mathcal {O}_L} L$ . A p-Frobenius endomorphism (resp., q-Frobenius endomorphism) of A is a ring endomorphism $\nu :A \to A$ (resp., $\sigma : A \to A$ ) that extends the map $\nu $ (resp., $\nu ^a=id$ ) on $\mathcal {O}_L$ defined in Section 2.1 and reduces to the pth-power map (resp., qth-power map) of $\overline {A}$ . Note that $\nu $ (resp., $\sigma $ ) extends to a map $\nu : \mathcal {A} \to \mathcal {A}$ (resp., $\sigma : \mathcal {A} \to \mathcal {A}$ ), which we refer to as a p-Frobenius endomorphism (resp., q-Frobenius endomorphism) of $\mathcal {A}$ . For a square matrix $M=\left (m_{i,j}\right )$ with entries in $\mathcal {A}$ , we take $M^{\nu ^k}$ to mean the matrix $\left (m_{i,j}^{\nu ^k}\right )$ and we define $M^{\nu ^{a-1} + \dotsb + \nu + 1}$ by $M^{\nu ^{a-1}} \dotsm M^{\nu } M$ .

2.3 Definitions of local rings

We begin by defining some rings and modules which will be used throughout this article. Define the L-algebras

$$ \begin{align*} {{\mathcal{E}_t}} = \left\{ \sum_{-\infty}^\infty a_nt^n{\kern2pt} \middle | \begin{array} {l} \text{ We have } a_n\in L,\ \lim\limits_{n\to-\infty} v_p(a_n)=\infty, \\ \text{ and}\ v_p(a_n) \text{ is bounded below.} \end{array} \right \}, \\ {{\mathcal{E}_t^{\dagger}}} = \left\{ \sum_{-\infty}^\infty a_nt^n \in \mathcal{E}{\kern2pt} \middle | \begin{array} {l} \text{ There exists } m>0 \text{ such that} \\ v_p(a_n) \geq -mn \text{ for } n\ll 0. \end{array} \right \}. \end{align*} $$

We refer to $\mathcal {E}_t \ \left (\text {resp., } \mathcal {E}^{\dagger }_t\right )$ as the Amice ring (resp., the bounded Robba ring) over L with parameter t. We will often omit the t in the subscript if there is no ambiguity. Note that $\mathcal {E}^{\dagger }$ and $\mathcal {E}$ are local fields with residue field $\mathbb {F}_q((t))$ . The valuation $v_p$ on L extends to the Gauss valuation on each of these fields. We define $\mathcal {O}_{\mathcal {E}} \ \left (\text {resp., } \mathcal {O}_{\mathcal {E}^{\dagger }}\right )$ to be the subring of $\mathcal {E} \ \left (\text {resp., } \mathcal {E}^{\dagger }\right )$ consisting of formal Laurent series with coefficients in $\mathcal {O}_L$ . Let $u \in \mathcal {O}_{\mathcal {E}^{\dagger }}$ be such that the reduction of u in $\mathbb {F}_q((t))$ is a uniformising element. Then we have $\mathcal {E}_u = \mathcal {E} \ \left (\text {resp., } \mathcal {E}_u^{\dagger }=\mathcal {E}^{\dagger }\right )$ . In particular, we see that u is a different parameter of $\mathcal {E}$ . Note that if $\nu :\mathcal {E}\to \mathcal {E}$ is any p-Frobenius endomorphism, we have $\mathcal {E}^{\nu =1}=E$ . For $m \in \mathbb {Z}$ , we define the L-vector space of truncated Laurent series

$$ \begin{align*} \mathcal{E}^{\leq m} = \left\{ \sum_{-\infty}^\infty a_nt^n \in \mathcal{E}{\kern2pt} \middle |{\kern2pt} a_n=0\text{ for all }n>m \right \}. \end{align*} $$

The space $\mathcal {E}^{\leq 0}$ is a ring, and $\mathcal {E}^{\leq m}$ is an $\mathcal {E}^{\leq 0}$ -module. There is a natural projection $\mathcal {E} \to \mathcal {E}^{\leq m}$ given by truncating the Laurent series. Finally, we define the following $\mathcal {O}_L$ -algebra:

$$ \begin{align*} \mathcal{O}_{\mathcal{E}\left(0,r\right]} = \left\{ \sum_{-\infty}^\infty a_nt^n \in \mathcal{O}_{\mathcal{E}}{\kern2pt} \middle |{\kern2pt} \lim_{n\to-\infty} v_p(a_{n}) +rn =\infty \right \}. \end{align*} $$

Set ${ {\mathcal {E}(0,r]}}=\mathcal {O}_{\mathcal {E}\left (0,r\right ]}\otimes _{\mathcal {O}_L} L$ . Note that $\mathcal {E}(0,r]$ is the ring of bounded functions on the closed annulus $0<v_p(t)\leq r$ . In particular, we have $\mathcal {E}^{\dagger }= \bigcup \limits _{r>0} \mathcal {E}(0,r]$ .

2.4 Matrix notation

For any $c_0,\dotsc , c_{a-1} \in \mathcal {E}$ , we define the following $a\times a$ matrices:

$$ \begin{align*} \mathbf{diag}\left(c_0,\dotsc,c_{a-1}\right) &= \begin{pmatrix} c_{0} & & \\ & \ddots & \\ & & c_{a-1} \end{pmatrix}, \\ \mathbf{cyc}\left(c_0, \dotsc, c_{a-1}\right) & = \begin{pmatrix} & c_{0} & & \\ & & \ddots & \\ & & & c_{a-2} \\ c_{a-1} & && \end{pmatrix},\\ \mathbf{tcyc}\left(c_0, \dotsc, c_{a-1}\right) & = \mathbf{cyc}\left(c_0, \dotsc, c_{a-1}\right)^{\mathrm{T}}. \end{align*} $$

3.1 Mapping to $\mathbb {P}^1$

3.2 Basic setup

Write $\mathbb {P}^1_{\mathbb {F}_q}=\text {Proj}\left (\mathbb {F}_q[x_1,x_2]\right )$ and let $\overline {t}=\frac {x_1}{x_2}$ be a parameter at $0$ . Fix a morphism ${ {\eta }}$ as in Lemma 3.1. For $* \in \{0,1,\infty \}$ , we let $\left \{P_{*,1}, \dotsc , P_{*,r_*}\right \} = \eta ^{-1}(*)$ and set ${ {W}}= \eta ^{-1}(\{0,1,\infty \})$ . Again, we will increase q so that each $P_{*,i}$ is defined over $\mathbb {F}_q$ . Fix $Q=P_{*,i} \in W$ . We define ${ {e_Q}}$ to be the ramification index of Q over $*$ . From Lemma 3.1, if $*=1$ we have $e_{Q}=p-1$ for $1\leq i \leq r_1$ , so that $r_1(p-1)=\deg (\eta )$ . Also, by the Riemann–Hurwitz formula,

(4) $$ \begin{align} (g-1) + (r_0+r_1 + r_\infty) &= \deg(\eta)-g+1, \end{align} $$

where g denotes the genus of X. Let $U=\mathbb {P}^1_{\mathbb {F}_q}-\{0,1,\infty \}$ and ${ {V}}=X-W$ . Then $\eta : V \to U$ is a finite étale map of degree $\deg (\eta )$ . Let $\overline {B} \ \left (\text {resp., } \overline {A}\right )$ be the $\mathbb {F}_q$ -algebra such that $V=\text {Spec}\left (\overline {B}\right ) \ \left (\text {resp., } U=\text {Spec}\left (\overline {A}\right )\right )$ .

Let $\mathbb {P}^1_{\mathcal {O}_L}$ be the projective line over $\text {Spec}(\mathcal {O}_L)$ and let $\mathbf {P}^1_{\mathcal {O}_L}$ be the formal projective line over $\text {Spf}(\mathcal {O}_L)$ . Let t be a global parameter of $\mathbf {P}^1_{\mathcal {O}_L}$ lifting $\overline {t}$ . By the deformation theory of tame coverings [Reference Grothendieck and Murre11, Theorem 4.3.2], there exists a tame cover $\mathbf {X} \to \mathbf {P}^1_{\mathcal {O}_L}$ whose special fibre is $\eta $ , and by formal GAGA [26, Tag 09ZT] there exists a morphism of smooth curves $\mathbb {X} \to \mathbb {P}^1_{\mathcal {O}_L}$ whose formal completion is $\mathbf {X} \to \mathbf {P}^1_{\mathcal {O}_L}$ .

Define the functions $t_{0}=t$ , $t_\infty = \frac {1}{t}$ and $t_1=t-1$ . Let $[*]$ denote the $\mathcal {O}_L$ -point of $\mathbb {P}^1_{\mathcal {O}_L}$ given by $t_*=0$ . For $Q =P_{*,i}$ , let $[Q]$ be a point of $\eta ^{-1}([*])$ that reduces to Q in the special fibre. Note that such a point exists because $Q \in \eta ^{-1}(*)$ , but it is not necessarily unique. Let $\mathbb {U} = \mathbb {P}^1_{\mathcal {O}_L} - \{ [0], [1],[\infty ] \}$ and $\mathbb {V} = \mathbb {X}-\{[R]\}_{R\in W}$ . We define $\mathbf {U} = \mathbf {P}^1_{\mathcal {O}_L} - \{ 0,1,\infty \}$ and ${ {\mathbf {V}}} = \mathbf {X}-\{R\}_{R\in W}$ . Then $\mathbf {U}$ (resp., $\mathbf {V}$ ) is the formal completion of $\mathbb {U}$ (resp., $\mathbb {V}$ ). We let $\mathcal {U}^{rig} \ \left (\text {resp., } { {\mathcal {V}^{rig}}}\right )$ be the rigid analytic fibre of $\mathbf {U}$ (resp., $\mathbf {V}$ ). Let $\widehat {A} \ \left (\text {resp., } \widehat {\mathcal {A}}\right )$ be the ring of functions $\mathcal {O}_{\mathbf {U}}(\mathbf {U}) \ \left (\text {resp., } \mathcal {O}_{\mathcal {U}^{rig}}\left (\mathcal {U}^{rig}\right )\right )$ and let $\widehat {B}$ $\left (\text {resp., } \widehat {\mathcal {B}}\right )$ be the ring of functions $\mathcal {O}_{\mathbf {V}}(\mathbf {V}) \ \left (\text {resp., } \mathcal {O}_{\mathcal {V}^{rig}}\left (\mathcal {V}^{rig}\right )\right )$ .

3.3 Local parameters and overconvergent rings

For $Q=P_{*,i}$ , let $w_Q$ be a rational function on $\mathbb {X}$ that has a simple zero at Q. Let $\mathcal {E}_{*} \ \left (\text {resp., } \mathcal {E}_{Q}\right )$ be the Amice ring over L with parameter $t_{*} \ \left (\text {resp., } w_{Q}\right )$ . By expanding functions in terms of the $t_{*}$ and $w_{Q}$ , we obtain the following diagrams:

(5)

We let $A^{\dagger } \ \left (\text {resp., }{ {B^{\dagger }}}\right )$ be the subring of $\widehat {A} \ \left (\text {resp., } \widehat {B}\right )$ consisting of functions that are overconvergent in the tube $]*[$ for each $*\in \{0,1,\infty \}$ (resp., $]Q[$ for all $Q \in W$ ). In particular, $B^{\dagger }$ fits into the following Cartesian diagram:

(6)

Note that $A^{\dagger } \ \left (\text {resp., } B^{\dagger }\right )$ is the weak completion of A (resp., B) in the sense of [Reference Monsky and Washnitzer23, Section 2]. In particular, we have $A^{\dagger } =\mathcal {O}_L \Big < t,t^{-1}, \frac {1}{t-1} \Big>^{\dagger }$ and $B^{\dagger }$ is a finite étale $A^{\dagger }$ -algebra. Finally, we define $\mathcal {A}^{\dagger } \ \left (\text {resp., } \mathcal {B}^{\dagger }\right )$ to be $A^{\dagger } \otimes \mathbb {Q}_p \ \left (\text {resp., } B^{\dagger } \otimes \mathbb {Q}_p\right )$ . Then $\mathcal {A}^{\dagger } \ \left (\text {resp., } { {\mathcal {B}^{\dagger }}}\right )$ is equal to the functions in $\widehat {\mathcal {A}} \ \left (\text {resp., } \widehat {\mathcal {B}}\right )$ that are overconvergent in the tube $]*[$ for each $*\in \{0,1,\infty \}$ (resp., $]R[$ for all $R \in W$ ).

The extension $\mathcal {E}_Q^{\dagger }\Big /\mathcal {E}_*^{\dagger }$ is an unramified extension of local fields and thus completely determined by the residual extension. By our assumption on the tameness of $\eta $ , we know that this residual extension is tame and can be written as $\mathbb {F}_q\left (\left (t_*^{\frac {1}{e_Q}}\right )\right )\Bigg /\mathbb {F}_q((t_*))$ . Since $\mathcal {O}_{\mathcal {E}_Q^{\dagger }}$ is Henselian [Reference Matsuda19, Proposition 3.2], there exists a parameter $u_Q$ of $\mathcal {E}_Q^{\dagger }$ such that $u_Q^{e_Q}=t_*$ . We remark that $u_Q$ will be defined on an annulus inside the disc $]Q[$ , and in general it will not extend to a function on the whole disc.

We will need to consider functions in $\mathcal {B}^{\dagger }$ with a precise radius of overconvergence in terms of the parameters $u_Q$ . Let ${ {\mathbf {r}}}=\left (r_Q\right )_{Q \in W}$ be a tuple of positive rational numbers. We define ${ {\mathcal {B}(0,\mathbf {r}]}}$ to be the subring of functions in $\mathcal {B}^{\dagger }$ that overconverge in the annulus $0<v_p\left (u_Q\right )\leq r_Q$ .Footnote ¹ More precisely, $\mathcal {B}(0,\mathbf {r}]$ fits into the following Cartesian diagram:

(7)

Note that $\mathcal {B}^{\dagger }$ is the union over all $\mathcal {B}(0,\mathbf {r}]$ .

3.4 Global Frobenius and $U_p$ operators

Let $\nu :\mathcal {A}^{\dagger } \to \mathcal {A}^{\dagger }$ be the p-Frobenius endomorphism that restricts to $\nu $ on L and sends t to $t^p$ . Let $\sigma =\nu ^a$ . For $* \in \{0,1,\infty \}$ , we may extend $\nu $ to a p-Frobenius endomorphism of $\mathcal {E}_{*}^{\dagger }$ , which we refer to as $\nu _{*}$ . In terms of the parameters $t_*$ , these endomorphisms are given as follows:

$$ \begin{align*} t_0 \mapsto t_0^p, \quad t_\infty \mapsto t_\infty^p, \quad t_1\mapsto (t_1+1)^p-1. \end{align*} $$

Since the map $\widehat {A} \to \widehat {B}$ is étale and both rings are p-adically complete, we may extend both $\sigma $ and $\nu $ to maps $\sigma ,\nu :\widehat {B}\to \widehat {B}$ . This extends to a p-Frobenius endomorphism $\nu _{Q}$ of $\mathcal {E}_{Q}$ , which makes the diagrams (5) p-Frobenius equivariant. Furthermore, since $\nu _{Q}$ extends $\nu _{*}$ , we know that $\nu _{Q}$ induces a p-Frobenius endomorphism of $\mathcal {E}_{Q}^{\dagger }$ . It follows from diagram (6) that $\sigma $ and $\nu $ restrict to endomorphisms $\sigma , \nu : \mathcal {B}^{\dagger } \to \mathcal {B}^{\dagger }$ . The p-Frobenius endomorphisms $\nu _{Q}$ can be described as follows:

1. 1. When $*$ is $0$ or $\infty $ , have $u_{Q}^{\nu _Q}=u_{Q}^p$ , since $t_*^{\nu _*}=t_*^p$ and $u_{Q}^{e_Q}=t_*$ .

2. 2. When $*=1$ , we have $u_{Q}^{\nu _Q} = \sqrt [p-1]{\left (u_Q^{p-1}+1\right )^p-1}$ , since $t_1^{\nu _1}= (t_1+1)^p-1$ and $u_Q^{p-1}=t_1$ .

Following [Reference van der Put28, Section 3], there is a trace map $Tr_0: \mathcal {B}^{\dagger } \to \nu \left (\mathcal {B}^{\dagger }\right ) \ \left (\text {resp., } Tr: \mathcal {B}^{\dagger } \to \sigma \left (\mathcal {B}^{\dagger }\right )\right )$ . We may define the $U_p$ operator on $\mathcal {B}^{\dagger }$ :

$$ \begin{align*} U_p: \mathcal{B}^{\dagger} &\to \mathcal{B}^{\dagger} \\ x &\mapsto \frac{1}{p} \nu^{-1}(Tr_0(x)). \end{align*} $$

Similarly, we define $U_q=\frac {1}{q}\sigma ^{-1}\circ Tr$ , so that $U_p^a=U_q$ . Note that $U_p$ is E-linear and $U_q$ is L-linear. Both $U_p$ and $U_q$ extend to operators on $\mathcal {E}_Q^{\dagger }$ .

4.1 Type 1: $t \mapsto t^p$

First consider the p-Frobenius endomorphism $\nu :\mathcal {E}^{\dagger } \to \mathcal {E}^{\dagger }$ sending t to $t^p$ . We see that $U_p\left (t^i\right )=0$ if $p\nmid i$ and $U_p\left (t^i\right )=t^{\frac {i}{p}}$ if $p \mid i$ . Thus, for $s> 0$ we have

(8) $$ \begin{align} U_p\left(\mathcal{O}_{\mathcal{E}}^s\right) \subset \mathcal{O}_{\mathcal{E}}^{\frac{s}{p}} \quad \text{and} \quad U_p\left(\mathcal{O}_L\left[\!\left[ \pi_s t^{-1} \right]\!\right]\right)\subset \mathcal{O}_L\left[\!\left[ \pi_s^p t^{-1} \right]\!\right]. \end{align} $$

4.1.1 Local estimates

Let $R=(s,\mathbf {e},\epsilon ,\omega )$ be a ramification datum and let $e_0,\dotsc ,e_{a-1}$ be the p-adic digits of $\epsilon $ as in Section 1.1. For $j=0,\dotsc ,a-1$ , we define

$$ \begin{align*} q(\mathbf{e},j) &= - \sum_{i=0}^{a-1} (i+1) e_{i+j}. \end{align*} $$

Note that

(9) $$ \begin{align} q(\mathbf{e},j) - q(\mathbf{e},j+1) &= ae_j - \omega. \end{align} $$

Let $t_i^n \in \bigoplus \limits _{j=0}^{a-1} \mathcal {E}^{\dagger }$ denote the element that has $t^n$ in the ith coordinate and zero in the other coordinates. We then define the spaces

$$ \begin{align*} \mathcal{D}_{\mathbf{e},s}^{\left(j\right)}&= \pi_{as}^{q\left(\mathbf{e},j\right)} \pi_s^pt_j^{-1} \mathcal{O}_L\left[\!\left[ \pi_s^pt_j^{-1} \right]\!\right] \oplus \mathcal{O}_L \left[\!\left[ t_j \right]\!\right], \\ {{\mathcal{D}_{\mathbf{e},s}}} &= \bigoplus_{j=0}^{a-1} \mathcal{D}_{\mathbf{e},s}^{\left(j\right)} \subset \bigoplus_{j=0}^{a-1} \mathcal{E}^{\dagger}. \end{align*} $$

We know $-q(\mathbf {e},i) \leq a(p-1)$ , which implies $\pi _{as}^{q\left (\mathbf {e},j\right )} \pi _s^p \in \mathcal {O}_L$ . In particular,

$$ \begin{align*} \mathcal{D}_{\mathbf{e},s} &\subset \bigoplus_{j=0}^{a-1} \mathcal{O}_{\mathcal{E}^{\dagger}}. \end{align*} $$

Proposition 4.1. Let $\nu $ be the p-Frobenius endomorphism that sends $t \mapsto t^p$ . Set $\alpha \in \mathcal {O}_L\left[\!\left[ \pi _s t^{-1} \right]\!\right] $ and set $A=\mathbf {tcyc}\left (\alpha t^{-e_0}, \dotsc ,\alpha t^{-e_{a-1}}\right )$ . Then

(10) $$ \begin{align} U_p \circ A\left(\mathcal{D}_{\mathbf{e},s}\right) &\subset \mathcal{D}_{\mathbf{e},s}, \end{align} $$

(11) $$ \begin{align} U_p \circ A\left(\pi_{as}^{q\left(\mathbf{e},j\right)} \pi_s^{np} t_j^{-n}\right) &\subset \pi_s^{n\left(p-1\right)} \pi_{as}^{-\omega}\mathcal{D}_{\mathbf{e},s}, \end{align} $$

for $n\geq 1$ and $0\leq j \leq a-1$ .

Proof. For $n\geq 1$ we have $A\left (t_j^{-n}\right )=\alpha t_{j+1}^{-n-e_{j+1}}$ . Then from equation (9) we have

$$ \begin{align*} A\left(\pi_{as}^{q\left(\mathbf{e},j\right)}\pi_s^{pn} t_j^{-n}\right) &= \pi_{as}^{q\left(\mathbf{e},j\right)} \pi_s^{pn} t_{j+1}^{-n-e_{j+1}} \alpha \\ &= \pi_{as}^{q\left(\mathbf{e},j+1\right)} \pi_s^{n\left(p-1\right)}\pi_{as}^{-\omega} \cdot \left(\pi_s^{n + e_{j+1}}t_{j+1}^{-n-e_{j+1}}\alpha\right). \end{align*} $$

Note that $\pi _s^{n + e_{j+1}}t_{j+1}^{-n-e_{j+1}}\alpha \in \mathcal {O}_L\left[\!\left[ \pi _s t_{j+1}^{-1} \right]\!\right] $ . Then formula (11) follows from formula (8). To prove formula (10), we need to make sure $U_p \circ A\left (t_j^n\right ) \in \mathcal {D}_{\mathbf {e},s}$ for $n\geq 0$ , which can be done by a similar argument.

4.2 Type 2: $t \mapsto \sqrt [p-1]{\left (t^{p-1}+1\right )^p-1}$

Next, consider the p-Frobenius endomorphism $\nu :\mathcal {E}^{\dagger } \to \mathcal {E}^{\dagger }$ that sends t to $\sqrt [p-1]{\left (t^{p-1}+1\right )^p-1}$ . Define the following sequence of numbers:

$$ \begin{align*} b(n) & =\begin{cases} \left \lfloor \frac{-n-1}{p-1} \right \rfloor, & n\leq -1, \\ 0, & n \geq 0. \end{cases} \end{align*} $$

We then define the space

(12) $$ \begin{align} {{\mathcal{D}}} &= \prod_{n \in \mathbb{Z}} p^{b(n)}t^{n}\mathcal{O}_L, \end{align} $$

which we regard as a sub- $\mathcal {O}_L$ -module of $\mathcal {O}_{\mathcal {E}^{\dagger }}$ .

Proposition 4.2. Let $\nu $ be the p-Frobenius endomorphism of $\mathcal {E}^{\dagger }$ that sends t to $\sqrt [p-1]{\left (t^{p-1}+1\right )^p-1}$ . For all $n\in \mathbb {Z}_{\geq 0}$ and $0\leq k \leq p-1$ , we have

$$ \begin{align*} U_p\left(p^{b\left(-k-np\right)}t^{-k-np}\right) &\in p^n \mathcal{D}, \\ U_p(\mathcal{D}) &\subset \mathcal{D}. \end{align*} $$

5.1 F-crystals and p-adic representations

For this subsection, we let $\overline {S}$ be either $\text {Spec}\left (\mathbb {F}_q((t))\right )$ or a smooth, irreducible affine $\mathbb {F}_q$ -scheme $\text {Spec}\left (\overline {R}\right )$ . We let $S=\text {Spec}(R)$ be a flat $\mathcal {O}_L$ -scheme whose special fibre is $\overline {S}$ and assume that R is p-adically complete – for example, if $\overline {S}=\text {Spec}\left (\mathbb {F}_q((t))\right )$ , then we may take $R=\mathcal {O}_{\mathcal {E}}$ . Fix a p-Frobenius endomorphism $\nu $ on R (as in Section 2.2). Then $\sigma =\nu ^a$ is a q-Frobenius endomorphism.

Theorem 5.3 Katz [Reference Katz12, Section 4]

There is an equivalence of categories

$$ \begin{align*} \left\{\text{rank}\ d\ \text{unit-root }F\text{-crystals over } \overline{S}\right\} &\longleftrightarrow \left\{\text{continuous representations} \ \psi:\pi^{et}_1\left(\overline{S}\right) \to GL_d(\mathcal{O}_L)\right\}. \end{align*} $$

Let us describe a certain case of this correspondence. Let $\overline {S}_1 \to \overline {S}$ be a finite étale cover and assume that $\psi $ comes from a map $\psi _0:Gal\left (\overline {S}_1\Big /\overline {S}\right )\to GL_d(\mathcal {O}_E)$ . This cover deforms into a finite étale map of affine schemes $S_1=\text {Spec}(R_1) \to S$ . Both $\nu $ and $\sigma $ extend to $R_1$ and commute with the action of $Gal\left (\overline {S}_1\Big /\overline {S}\right )$ (see, e.g., [Reference Tsuzuki27, Section 2.6]). Let $V_0$ be a free $\mathcal {O}_E$ -module of rank d on which $Gal\left (\overline {S}_1\Big /\overline {S}\right )$ acts via $\psi _0$ and let $V=V_0 \otimes _{\mathcal {O}_E}\mathcal {O}_L$ . The unit-root F-crystal associated to $\psi $ is $M_\psi =\left (R_1 \otimes _{\mathcal {O}_L} V\right )^{Gal\left (\overline {S}_1\big /\overline {S}\right )}$ with $\varphi =\sigma \otimes _{\mathcal {O}_L} id$ . There is a map

$$ \begin{align*} \left(S_1 \otimes_{\mathcal{O}_{E}} V_0\right) \to \left(S_1 \otimes_{\mathcal{O}_L} V\right), \end{align*} $$

which is Galois equivariant. In particular, the map $\varphi $ has an ath root $\varphi _0=\nu \otimes _{\mathcal {O}_{E}} id$ .

Now make the additional assumption that $M_\psi $ is free as an R-module. Let $e_1,\dotsc , e_d$ be a basis of $M_\psi $ as an R-module and let $\mathbf {e}=[e_1,\dotsc ,e_d]$ . Then $\varphi (\mathbf {e})=\alpha \mathbf {e}$ (resp., $\varphi _0(\mathbf {e})=\alpha _0\mathbf {e}$ ), where $\alpha ,\alpha _0 \in GL_d(R)$ . We refer to the matrix $\alpha $ (resp., $\alpha _0$ ) as a Frobenius structure (resp., p-Frobenius structure) of M and to the matrix $\alpha ^{\mathrm {T}} \ \left (\text {resp., } \alpha _0^{\mathrm {T}}\right )$ as a dual Frobenius structure (resp., dual p-Frobenius structure) of M. We have the relation $\alpha ^{\mathrm {T}}=\left (\alpha _0^{\mathrm {T}}\right )^{\nu ^{a-1}+ \dotsb + \nu + 1}$ – recall from Section 2.2 that $\left (\alpha _0^{\mathrm {T}}\right )^{\nu ^{a-1}+ \dotsb + \nu + 1} = \left (\alpha _0^{\mathrm {T}}\right )^{\nu ^{a-1}} \dotsm \left (\alpha _0^{\mathrm {T}}\right )^{\nu } \alpha _0^{\mathrm {T}}$ . If $\mathbf {e}'=b\mathbf {e}$ and $\varphi (\mathbf {e}')=\alpha '\mathbf {e}'$ (resp., $\varphi (e_1)=\alpha _0'e_1$ ) with $\alpha ',\alpha _0',b \in GL_d(R)$ , then we have $(\alpha ')^{\mathrm {T}}=(b^\sigma )^{\mathrm {T}}\alpha ^{\mathrm {T}} \left (b^{-1}\right )^{\mathrm {T}} \ \left (\text {resp., } \left (\alpha _0'\right )^{\mathrm {T}}=(b^{\nu })^{\mathrm {T}}\alpha _0^{\mathrm {T}}\left (b^{-1}\right )^{\mathrm {T}}\right )$ . In particular, a dual Frobenius structure (resp., dual p-Frobenius structure) of M is unique up to $\sigma $ -skew conjugation (resp., $\nu $ -skew conjugation) by elements of $GL_d(R)$ . We remark that if $M_\psi $ has rank $1$ , then p-Frobenius structures (resp., Frobenius structures) are also dual p-Frobenius structures (resp., dual Frobenius structures).

5.2 Local Frobenius structures

We now restrict ourselves to the case when $\overline {S}=\text {Spec}\left (\mathbb {F}_q((t))\right )$ . In particular, unit-root F-crystals over $\overline {S}$ correspond to representations of $G_{\mathbb {F}_q((t))}$ , the absolute Galois group of $\mathbb {F}_q((t))$ . Note that since $\mathcal {O}_{\mathcal {E}}$ is a local ring, all locally free modules are free.

5.2.2 Wild Artin–Schreier–Witt characters

A global version over $\mathbb {G}_m$ of the following result is commonplace in the literature (see, e.g., [Reference Wan30, Section 4.1] for the exponential-sum situation or [Reference Liu and Wei18]). However, to the best of our knowledge, the local version presented here does not appear anywhere.

Proposition 5.5. Let $\nu $ be the p-Frobenius endomorphism of $\mathcal {O}_{\mathcal {E}}$ sending t to $t^p$ and let $\sigma =\nu ^{a}$ . Let $\psi :G_{\mathbb {F}_q((t))} \to \mathcal {O}_L^{\times }$ be a continuous character and let $M_\psi $ be the corresponding unit-root F-crystal. Assume that $Im(\psi )\cong \mathbb {Z}/p^n\mathbb {Z}$ . Let K be the fixed field of $\ker (\psi )$ and let s be the Swan conductor of $\psi $ . We assume that $\pi _s \in \mathcal {O}_E$ . Then there exists a p-Frobenius structure $E_r$ of $\psi $ such that $E_r \in \mathcal {O}_L\left[\!\left[ \pi _s t^{-1} \right]\!\right] $ and $E_r \equiv 1 \bmod \mathfrak m$ . Furthermore, if $c\in 1+\mathfrak m \mathcal {O}_{\mathcal {E}}$ is another p-Frobenius structure, there exists $b\in 1+\mathfrak m \mathcal {O}_{\mathcal {E}}$ with $E_r=\frac {b^{\nu }}{b}c$ .

Proof. The extension of $K/\mathbb {F}_q((t))$ corresponds to an equivalence class of $W_n\left (\mathbb {F}_q((t))\right )/(\textbf {Fr}-1)W_n\left (\mathbb {F}_q((t))\right )$ ; here $W_n\left (\mathbb {F}_q((t))\right )$ is the nth truncated Witt vectors and $\textbf {Fr}$ is the Frobenius map. Following [Reference Kosters and Wan14, Proposition 3.3], we may represent this equivalence class with

$$ \begin{align*} r(t) &= \sum_{i=0}^{n-1} \sum_{j=0}^{s_i} \left[r_{i,j}t^{-j}\right] p^i, \quad r_{i,s_i}\neq 0, \\ s&= \min_{i=0}^{n-1} \left\{p^{n-i}s_i\right\}, \end{align*} $$

where $r_{i,j} \in \mathbb {F}_q$ . Since $r(t) \in W_n\left (\mathbb {F}_q\left [t^{-1}\right ]\right )$ , the extension $K/\mathbb {F}_q((t))$ extends to finite étale $\mathbb {F}_q\left [t^{-1}\right ]$ -algebra B that fits into a commutative diagram

In particular, $\psi $ extends to a representation $\psi ^{ext}:Gal\left (B\big /\mathbb {F}_q\left [t^{-1}\right ]\right ) \to \mathcal {O}_L^{\times }$ . This extension is uniquely defined by the following property: For $k\geq 1$ and $x \in \mathbb {P}^1\left (\mathbb {F}_{q^k}\right )- \{0\}$ , we have

(13) $$ \begin{align} \psi^{ext}(Frob_x)&= \zeta_{p^n}^{Tr_{W_n\left(\mathbb{F}_{q^k}\right)/W_n\left(\mathbb{F}_p\right)}(r([x]))}, \end{align} $$

where $[x]$ denotes the Teichmüller lift of x in $W_n\left (\mathbb {F}_{q^k}\right )$ and $\zeta _{p^n}$ is a primitive $p^n$ th root of unity. Let $\mathcal {O}_L\left \langle t^{-1} \right \rangle \subset \mathcal {O}_{\mathcal {E}}$ be the Tate algebra in $t^{-1}$ with coefficients in $\mathcal {O}_L$ . Note that $\nu $ restricts to a p-Frobenius endomorphism of $\mathcal {O}_L\left \langle t^{-1} \right \rangle $ . All projective modules over $\mathcal {O}_L\left \langle t^{-1} \right \rangle $ are free, so that $M_{\psi ^{ext}}$ is isomorphic to $\mathcal {O}_L\left \langle t^{-1} \right \rangle $ as an $\mathcal {O}_L\left \langle t^{-1} \right \rangle $ -module. We see that $M_{\psi }=M_{\psi ^{ext}}\otimes _{\mathcal {O}_{L}\left \langle t^{-1} \right \rangle }\mathcal {O}_{\mathcal {E}}$ . In particular, any p-Frobenius structure of $M_{\psi ^{ext}}$ is a p-Frobenius structure of $M_{\psi }$ .

A series $\alpha _0 \in \mathcal {O}_L\left \langle t^{-1} \right \rangle $ is a p-Frobenius structure for $M_{\psi ^{ext}}$ if for every $x \in \mathbb {P}^1\left (\mathbb {F}_{q^k}\right )- \{0\}$ we have

$$ \begin{align*} \prod_{i=0}^{ak-1} \alpha_0([x])^{\nu^{i}} &= \psi^{ext}(Frob_x). \end{align*} $$

We let $E(x)$ denote the Artin–Hasse exponential and let $\gamma _i$ be an element of $\mathbb {Z}_p\left [\zeta _{p^n}\right ]$ with $E(\gamma _n)=\zeta _{p^n}^{p^{n-i}}$ . Note that $v_p(\gamma _i)=\frac {1}{p^{i-1}(p-1)}$ . Thus from equation (13) we see that

$$ \begin{align*} E_r &= \prod_{i=0}^{n-1} \prod_{j=0}^{s_i} E\left(\left[r_{i,j}\right] t^{-j} \gamma_{n-i}\right) \end{align*} $$

is a p-Frobenius structure of $M_{\psi ^{ext}}$ . Since $E(x) \in \mathbb {Z}_p\left[\!\left[ x\right]\!\right] $ , it is clear that $E_r \in \mathcal {O}_L\left[\!\left[ \pi _s t^{-1}\right]\!\right] $ .

5.3 The F-crystal associated to $\rho $

We now continue with $\rho $ from Section 1.1 and the setup from Section 3.

5.3.1 The Frobenius structure of $\rho ^{wild}$

Let $\mathcal {L}$ be a rank $1 \ \mathcal {O}_L$ -module on which $\pi _1^{et}(V)$ acts through $\rho ^{wild}$ . Let $f:C\to X$ be the $\mathbb {Z}/p^n\mathbb {Z}$ -cover that trivialises $\rho ^{wild}$ . Let $\overline {R}$ be the $\overline {B}$ -algebra with $C \times _X V=\text {Spec}\left (\overline {R}\right )$ . We may deform $\overline {B}\to \overline {R}$ to a finite étale map $\widehat {B} \to \widehat {R}$ . The F-crystal corresponding to $\rho $ is the $\widehat {B}$ -module $M=\left (\widehat {R} \otimes \mathcal {L}\right )^{Gal(C/X)}$ . For each $Q \in W$ and $P \in f^{-1}(Q)$ , we obtain a finite extension $\mathcal {E}_P^{\dagger }$ of $\mathcal {E}_Q^{\dagger }$ ; recall from Section 3.2 that $W=\eta ^{-1}(\{0,1,\infty \}$ ). As in Section 3.3, we may consider the ring of overconvergent functions $R^{\dagger }$ , which makes the following diagram Cartesian:

Since the action of $Gal(C/X)$ (resp., $\nu $ ) on $\bigoplus _{P \in f^{-1}(Q)} \mathcal {O}_{\mathcal {E}_P}$ preserves $\bigoplus _{P \in f^{-1}(Q)} \mathcal {O}_{\mathcal {E}_P^{\dagger }}$ , we see that $Gal(C/X)$ (resp., $\nu $ ) acts on $R^{\dagger }$ (see, e.g., [Reference Tsuzuki27, Section 2]). This gives the following proposition:

Proposition 5.7. Let $M^{\dagger } = (R^{\dagger } \otimes \mathcal {L})^{Gal(C/X)}$ . The map $M^{\dagger }\otimes _{B^{\dagger }} \widehat {B}\to M$ is a $\nu $ -equivariant isomorphism.

Lemma 5.8. The module $M^{\dagger }$ (resp., M) is a free $B^{\dagger }$ -module (resp., $\widehat {B}$ -module). Furthermore, M has a p-Frobenius structure $\alpha _0$ contained in $1 + \mathfrak m B^{\dagger }$ .

Proof. The proof of this is identical to [Reference Kramer-Miller15, Lemma 5.9].

5.3.2 The Frobenius structure of $ \bigoplus\limits_{j=0}^{a-1} \chi ^{\otimes p^{j}}$

By Kummer theory, there exists $\overline {f} \in \overline {B}^{\times }$ such that $\chi $ factors through the étale $\mathbb {Z}/(q-1)\mathbb {Z}$ -cover $\text {Spec}\left (\overline {B}\left [\overline {h}\right ]\right ) \to \text {Spec}\left (\overline {B}\right )$ , where $\overline {h}=\sqrt [q-1]{\overline {f}}$ . Let $f \in B^{\dagger }$ be a lift of $\overline {f}$ and set $h=\sqrt [q-1]{f}$ , so that $\text {Spec}\left (B^{\dagger }[h]\right ) \to \text {Spec}\left (B^{\dagger }\right )$ is an étale $\mathbb {Z}/(q-1)\mathbb {Z}$ -cover whose special fibre is $\text {Spec}\left (\overline {B}\left [\overline {h}\right ]\right ) \to \text {Spec}\left (\overline {B}\right )$ . There exists $0\leq \Gamma < q-1$ such that $\chi (g)=\frac {\left (h^\Gamma \right )^g}{h^\Gamma }$ for all $g \in \pi _1^{et}(V)$ . Write the p-adic expansion $\Gamma =\gamma _0 + \dotsb + \gamma _{a-1}p^{a-1}$ and define

$$ \begin{align*} \Gamma_j=\sum_{i=0}^{a-1} \gamma_{i+j} p^i. \\[-22pt]\end{align*} $$

Note that $\chi ^{\otimes p^{j}}(g)= \frac {\left (h^{\Gamma _j}\right )^g}{h^{\Gamma _j}}$ for each j. This gives the following proposition:

Proposition 5.9. The matrix $N=\mathbf {diag}\left (f^{-\Gamma _0}, \dotsc , f^{-\Gamma _{a-1}}\right )$ (resp., $N_0=\mathbf {tcyc}\left (f^{-\gamma _0}, \dotsc , f^{-\gamma _{a-1}}\right )$ ) is a dual Frobenius structure (resp., dual p-Frobenius structure) of $\bigoplus \limits _{j=0}^{a-1} \chi ^{\otimes p^{j}}$ and $N=N_0^{\nu ^{a-1} + \dotsb + 1}$ .

Set $Q\in W$ . Recall from Section 1.1 that we associate a tame ramification datum $T_Q=\left (\mathbf {e}_Q,\epsilon _Q,\omega _Q\right )$ to Q and write $\epsilon _Q=\sum e_{Q,i}p^i$ . The exponent of $\chi ^{\otimes p^j}$ at $Q \in W$ is

$$ \begin{align*} \frac{\epsilon_{Q,j}}{q-1} &\bmod \mathbb{Z} \text{, where } \\ \epsilon_{Q,j} &= \sum_{i=0}^{a-1} e_{Q,i+j} p^i.\\[-17pt] \end{align*} $$

By definition we have

$$ \begin{align*} -\text{Div}\left(\overline{f}^{\Gamma_j}\right) &= \sum_{Q \in W} \left(-\epsilon_{Q,j} + (q-1)n_{Q,j}\right)[Q], \\[-17pt]\end{align*} $$

with $n_{Q,j} \in \mathbb {Z}$ . Since $0\leq \epsilon _{Q,j}\leq q-2$ and $\sum _Q n_{Q,j} = \frac {\sum _Q\epsilon _{Q,j}}{q-1}$ , we know

(14) $$ \begin{align} \sum_{Q \in W} n_{Q,j} &\leq \mathbf{m} \leq r_0 + r_\infty, \\[-17pt]\nonumber\end{align} $$

where we recall that $\mathbf {m}$ is the number of points where $\rho $ is ramified. We also have

(15) $$ \begin{align} \begin{split} \sum_{j=0}^{a-1} \sum_{Q \in W} n_{Q,j} &= \frac{1}{q-1} \sum_{Q \in W} \sum_{j=0}^{a-1}\epsilon_{Q,j} \\ &= a\Omega_\rho, \end{split} \\[-17pt]\nonumber \end{align} $$

where $\Omega _\rho $ is the monodromy invariant introduced in Section 1.1.

5.3.3 Comparing local and global Frobenius structures

We fix ${ {\alpha _0}}$ as in Lemma 5.8 and set ${ {\alpha }}=\prod \limits _{i=0}^{a-1} \alpha _0^{\nu ^i}$ . We also let ${ {N}}$ and ${ {N_0}}$ be as in Proposition 5.9. In particular, $\alpha N$ (resp., $\alpha _0 N_0$ ) is a dual Frobenius structure (resp., dual p-Frobenius structure) of $\rho ^{wild}\otimes \bigoplus \limits _{j=0}^{a-1} \chi ^{\otimes p^{j}}$ . Set $Q \in W$ with $Q=P_{*,i}$ . There is a map $\overline {B} \to \mathbb {F}_q\left (\left (u_Q\right )\right )$ , where we expand each function on V in terms of the parameter $u_Q$ . This gives a point $\text {Spec}\left (\mathbb {F}_q\left (\left (u_{Q}\right )\right )\right ) \to V$ . By pulling back $\rho $ along this point we obtain a local representation $\rho _{Q}: G_{\mathbb {F}_q\left (\left (u_Q\right )\right )} \to \mathcal {O}_L^{\times }$ , where $G_{\mathbb {F}_q\left (\left (u_Q\right )\right )}$ is the absolute Galois group of $\mathbb {F}_q\left (\left (u_Q\right )\right )$ . We will compare $\alpha _0 N_0$ to the local dual p-Frobenius structures from Section 5.2.

There are three cases we need to consider. The first case is when $*=1$ . In this case $\rho _Q^{wild}$ and $\chi _Q$ are both unramified. This is because $\rho $ is only ramified at the points $\tau _1,\dotsc ,\tau _{\mathbf {m}}$ , and by Lemma 3.1 we have $\eta (\tau _i)\in \{0,\infty \}$ . The second case is when $*\in \{0,\infty \}$ and $\rho _Q^{wild}$ is unramified. The last case is when $* \in \{0,\infty \}$ and $\rho _Q^{wild}$ is ramified. In each case, we will describe a dual p-Frobenius structure ${ {C_Q}}$ of $\rho _Q^{wild} \otimes \bigoplus \limits _{j=0}^{a-1} \chi _Q^{\otimes p^{j}}$ , an element ${ {b_Q}} \in \mathcal {O}_{\mathcal {E}_Q}^{\dagger }$ and a diagonal matrix ${ {M_Q}} \in GL_{a}\left (\mathcal {O}_{\mathcal {E}^{\dagger }}\right )$ satisfying

(16) $$ \begin{align} \begin{split} \left(b_QM_Q\right)^{\nu}\alpha_0 N_0 \left(b_QM_Q\right)^{-1} &= C_Q, \\ \left(b_QM_Q\right)^{\sigma}\alpha N\left(b_QM_Q\right)^{-1}&=C_Q^{\nu^{a-1}+\nu^{a-2}+ \dotsb + 1}. \end{split} \end{align} $$

The dual p-Frobenius structure $C_Q$ will be closely related to the dual p-Frobenius structures studied in Section 5.2. It is helpful for us to introduce the following rings:

$$ \begin{align*} \mathcal{R}_Q &= \bigoplus_{j=0}^{a-1} \mathcal{E}_Q, & \mathcal{O}_{\mathcal{R}_Q} &= \bigoplus_{j=0}^{a-1} \mathcal{O}_{\mathcal{E}_Q}, \\ {{\mathcal{R}_Q^{\dagger}}} &= \bigoplus_{j=0}^{a-1} \mathcal{E}_Q^{\dagger}, & {{\mathcal{O}_{\mathcal{R}_Q^{\dagger}}}} &= \mathcal{R}_Q^{\dagger} \cap \mathcal{O}_{\mathcal{R}_Q}. \end{align*} $$

We define $u_{Q,j} \in \mathcal {R}_Q$ to have $u_Q$ in the jth coordinate and zero in the other coordinates. For each Q we will define a subspace ${ {\mathcal {O}_{\mathcal {R}_Q}^{con}}}\subset \mathcal {R}_Q^{\dagger }$ of elements satisfying some precise convergence conditions.

1. I. If $*=1$ , then $\nu _Q$ sends $u_Q \mapsto \sqrt [p-1]{\left (u_Q^{p-1}+1\right )^p-1}$ (see the end of Section 3.4).

   1. (Wild) As $\rho _Q$ is unramified, we know from Proposition 5.4 that there exists $b_Q \in 1 + \mathfrak m\mathcal {O}_{\mathcal {E}_Q}^{\dagger }$ such that the dual p-Frobenius structure $c_Q=\frac {b_Q^{\nu }}{b_Q}\alpha _0$ of $\rho _Q^{wild}$ lies in $1+\mathfrak m$ .

   2. (Tame) Since $\chi _Q$ is unramified, the exponent is zero. By Proposition 5.6, there exists $M_Q=\mathbf {diag}\left (m_{Q,0}, \dotsc , m_{Q,a-1}\right )$ with $v_{u_Q}\left (\overline {m_{Q,j}}\right )=n_{Q,j}$ such that $M_Q^{\sigma } N M_{Q}^{-1}=\mathbf {diag}\left (1, \dotsc , 1\right )$ and $M_Q^{\nu } N_0 M_{Q}^{-1}=\mathbf {tcyc}\left (1,\dotsc , 1\right )$ .

   3. (Both) We see that $C_Q=\mathbf {tcyc}\left (c_Q,\dotsc ,c_Q\right )$ is a dual p-Frobenius structure of $\rho _Q^{wild} \otimes \bigoplus \limits _{j=0}^{a-1} \chi _Q^{\otimes p^{j}}$ and that equation (16) holds. Define $\mathcal {O}_{\mathcal {R}_Q}^{con}$ to be $\bigoplus \limits _{j=0}^{a-1} \mathcal {D}$ viewed as a subspace of $\mathcal {O}_{\mathcal {R}_Q^{\dagger }}$ (see equation (12) for the definition of $\mathcal {D}$ ). From Proposition 4.2 we have

      (17) $$ \begin{align} \begin{split} U_p \circ C_Q \left(p^{b\left(k+pn\right)}u_{Q,j}^{-\left(k+pn\right)}\right) & \in p^{n}\mathcal{O}_{\mathcal{R}_Q}^{con}, \\ U_p \circ C_Q \left(\mathcal{O}_{\mathcal{R}_Q}^{con}\right) &\subset \mathcal{O}_{\mathcal{R}_Q}^{con}. \end{split} \end{align} $$

2. II. Next, consider the case where $*$ is $0$ or $\infty $ and $\rho ^{wild}_{Q}$ is unramified. Then $\nu _Q$ sends $u_Q \mapsto u_Q^p$ . We choose $\mathfrak {s}_Q \in \mathbb {Q}$ such that the following hold:

   (18) $$ \begin{align} \begin{split} \pi_{\mathfrak{s}_Q} &\in \mathcal{O}_E, \\ \frac{1}{\mathfrak{s}_Q} - \frac{\omega_Q}{a\mathfrak{s}_Q(p-1)}&\geq 1. \end{split} \end{align} $$

   1. (Wild) From Proposition 5.4 there exists $b_Q\in 1+\mathfrak m \mathcal {O}_{\mathcal {E}_Q}^{\dagger }$ such that $c_Q=\frac {b_Q^{\nu }}{b_Q} \alpha _0 \in 1+\mathfrak m$ is a dual p-Frobenius structure of $\rho _Q^{wild}$ .

   2. (Tame) By Proposition 5.6 there exists $M_Q=\mathbf {diag}\left (m_{Q,0}, \dotsc , m_{Q,a-1}\right )$ with $v_{u_Q}\left (\overline {m_{Q,j}}\right )=n_{Q,j}$ such that $M_Q^{\sigma } N M_{Q}^{-1}=\mathbf {diag}\left (u_Q^{-\epsilon _{Q,0}}, \dotsc , u_Q^{-\epsilon _{Q,a-1}}\right )$ and $M_Q^{\nu } N_0 M_{Q}^{-1}=\mathbf {tcyc}\left (u_Q^{-e_{Q,0}},\dotsc , u_Q^{-e_{Q,a-1}}\right )$ .

   3. (Both) We see that $C_Q=\mathbf {tcyc}\left (c_Qu_Q^{-e_{Q,0}},\dotsc ,c_Qu_Q^{-e_{Q,a-1}}\right )$ is a dual p-Frobenius structure of $\rho _Q^{wild}\otimes \bigoplus \limits _{j=0}^{a-1} \chi _Q^{\otimes p^{j}}$ and that equation (16) holds. Define $\mathcal {O}_{\mathcal {R}_Q}^{con}$ to be a copy of $\mathcal {D}_{\mathbf {e}_Q,\mathfrak {s}_Q}$ in $\mathcal {O}_{\mathcal {R}_Q^{\dagger }}$ (recall the definition of $\mathcal {D}_{\mathbf {e},s}$ from Section 4.1.1). From Proposition 4.1 we have

      (19) $$ \begin{align} \begin{split} U_p \circ C_Q \left(\pi_{a\mathfrak{s}_Q}^{q\left(\mathbf{e}_Q,j\right)} \pi_{\mathfrak{s}_Q}^{pn} u_{Q,j}^{-n}\right) &\in\pi_{\mathfrak{s}_Q}^{n\left(p-1\right)} \pi_{a\mathfrak{s}_Q}^{-\omega_Q} \mathcal{O}_{\mathcal{R}_Q}^{con}, \\ U_p \circ C_Q \left(\mathcal{O}_{\mathcal{R}_Q}^{con}\right) &\subset \mathcal{O}_{\mathcal{R}_Q}^{con}. \end{split} \end{align} $$

3. III. Finally, we consider the case when $*$ is $0$ or $\infty $ and $\rho ^{wild}_{Q}$ is ramified. Then $\nu _Q$ sends $u_Q \mapsto u_Q^p$ .

   1. (Wild) By Proposition 5.5 there is $b_Q\in 1+\mathfrak m \mathcal {O}_{\mathcal {E}_Q}^{\dagger }$ such that $c_Q=\frac {b_Q^{\nu }}{b_Q} \alpha _0 \in \mathcal {O}_L\left[\!\left[ \pi _{s_Q} u_Q^{-1}\right]\!\right] $ is a dual p-Frobenius structure of $\rho _Q^{wild}$ (recall that $s_Q$ is the Swan conductor of $\rho $ at Q). Note that $c_Q \equiv 1 \bmod \mathfrak m$ .

   2. (Tame) By Proposition 5.6 there exists $M_Q=\mathbf {diag}\left (m_{Q,0}, \dotsc , m_{Q,a-1}\right )$ with $v_{u_Q}\left (\overline {m_{Q,j}}\right )=n_{Q,j}$ such that $M_Q^{\sigma } N M_{Q}^{-1}=\mathbf {diag}\left (u_Q^{-\epsilon _{Q,0}}, \dotsc , u_Q^{-\epsilon _{Q,a-1}}\right )$ and $M_Q^{\nu } N_0 M_{Q}^{-1}=\mathbf {tcyc}\left (u_Q^{-e_{Q,0}},\dotsc , u_Q^{-e_{Q,a-1}}\right )$ .

   3. (Both) We see that $C_Q=\mathbf {tcyc}\left (c_Qu_Q^{-e_{Q,0}},\dotsc ,c_Qu_Q^{-e_{Q,a-1}}\right )$ is a dual p-Frobenius structure of $\rho _Q^{wild}\otimes \bigoplus \limits _{j=0}^{a-1} \chi _Q^{\otimes p^{j}}$ and that equation (16) holds. We define $\mathcal {O}_{\mathcal {R}_Q}^{con}$ to be a copy of $\mathcal {D}_{\mathbf {e}_Q,s_Q}$ in $\mathcal {O}_{\mathcal {R}_Q^{\dagger }}$ . From Proposition 4.1 we see that

      (20) $$ \begin{align} \begin{split} U_p \circ C_Q \left(\pi_{a s_Q}^{q\left(\mathbf{e}_Q,j\right)}\pi_{s_Q}^{pn} u_{Q,j}^{-n}\right) &\in \pi_{s_Q}^{n\left(p-1\right)}\pi_{as}^{-\omega_Q}\mathcal{O}_{\mathcal{R}_Q}^{con}, \\ U_p \circ C_Q \left(\mathcal{O}_{\mathcal{R}_Q}^{con}\right) &\subset \mathcal{O}_{\mathcal{R}_Q}^{con}. \end{split} \end{align} $$

5.3.4 Comparing global and semilocal Frobenius structures

We define the following spaces:

$$ \begin{align*} \begin{split} {{\mathcal{R}}} &= \bigoplus_{Q \in W} \mathcal{R}_Q, \qquad {{\mathcal{R}^{\dagger}}}=\bigoplus_{Q \in W} \mathcal{R}_Q^{\dagger},\\ {{\mathcal{R}^{trun}_Q}}&=\begin{cases} \bigoplus\limits_{j=0}^{a-1} \mathcal{E}_{Q}^{\leq -1}, & \eta(Q)=0,\infty, \\ \bigoplus\limits_{j=0}^{a-1} \mathcal{E}_{Q}^{\leq -p}, & \eta(Q)=1, \end{cases} \\ {{\mathcal{R}^{trun}}}&= \bigoplus_{Q \in W} \mathcal{R}^{trun}_Q, \qquad {{\mathcal{O}_{\mathcal{R}}^{con}}}= \bigoplus_{Q\in W} \mathcal{O}_{\mathcal{R}_Q}^{con} \subset \mathcal{R}^{\dagger}. \end{split} \end{align*} $$

Define $\mathcal {O}_{\mathcal {R}}$ to be $\bigoplus \limits _{Q \in W} \mathcal {O}_{\mathcal {R}_Q}$ and define $\mathcal {O}_{\mathcal {R}^{trun}}$ to be $\mathcal {R}^{trun} \cap \mathcal {O}_{\mathcal {R}}$ . Note that $\mathcal {O}_{\mathcal {R}}^{con}$ is contained in $\mathcal {O}_{\mathcal {R}}$ . There is a projection map $pr:\mathcal {R} \to \mathcal {R}^{trun}$ , which is the direct sum of the projection maps described in Section 2.3. By the definition of each summand of $\mathcal {O}_{\mathcal {R}}^{con}$ we see that

(21) $$ \begin{align} \ker(pr)\cap \mathcal{O}_{\mathcal{R}} &\subset \mathcal{O}_{\mathcal{R}}^{con}. \\[-17pt]\nonumber\end{align} $$

We may view $\bigoplus \limits _{j=0}^{a-1} \widehat {\mathcal {B}} \ \left (\text {resp., } \bigoplus \limits _{j=0}^{a-1} \mathcal {B}^{\dagger }\right )$ as a subspace of $\mathcal {R} \ \left (\text {resp., } \mathcal {R}^{\dagger }\right )$ using the maps (6). Let ${ {C}}$ (resp., ${ {M}}$ and ${ {b}}$ ) denote the endomorphism of $\mathcal {R}^{\dagger }$ that acts on the Q-coordinate by $C_Q \ \left (\text {resp., } M_Q \text { and } \mathbf {diag}\left (b_Q,\dotsc ,b_Q\right )\right )$ . This gives an operator $U_p \circ C: \mathcal {R}^{\dagger } \to \mathcal {R}^{\dagger }$ . From formulas (17), (19) and (20), we have

(22) $$ \begin{align} U_p \circ C \left(\mathcal{O}_{\mathcal{R}}^{con}\right) \subset \mathcal{O}_{\mathcal{R}}^{con}.\\[-17pt]\nonumber \end{align} $$

Also, by equation (16) know

(23) $$ \begin{align} \begin{split} (bM)^{\nu}\alpha_0 N_0(bM)^{-1}&= C, \\ (bM)^{\sigma}\alpha N(bM)^{-1}&=C^{\nu^{a-1}+\nu^{a-2}+ \dotsb + 1}. \end{split} \\[-17pt]\nonumber\end{align} $$

For each Q, we have

(24) $$ \begin{align} \begin{split} b_Q &\equiv 1 \bmod \mathfrak m, \\ M_Q &\equiv \mathbf{diag}\left(u_{Q,0}^{n_{Q,0}}g_0, \dotsc, u_{Q,a-1}^{n_{Q,a-1}} g_{a-1}\right) \bmod \mathfrak m, \end{split} \\[-17pt]\nonumber\end{align} $$

with $g_j \in \mathbb {F}_q\left[\!\left[ u_{Q,j}\right]\!\right] ^{\times }$ .

6.1 Normed vector spaces and Banach spaces

Let V be a vector space over L with a norm $\lvert \cdot \rvert $ compatible with the p-adic norm $\lvert \cdot \rvert _p$ on L. We will assume that for every $x \in V\backslash \{0\}$ , the norm $\lvert x\rvert $ lies in $\left \lvert L^{\times }\right \rvert _p$ , the norm group of $L^{\times }$ . We say V is a Banach space if it is also complete. Let $V_0\subset V$ denote the subset consisting of $x \in V$ satisfying $\lvert x\rvert \leq 1$ and let $\overline {V}=V_0\big /\mathfrak m V_0$ . If W is a subspace of V, we automatically give W the subspace norm unless otherwise specified.

6.1.1 Restriction of scalars to E

Let I be a set. Assume that $V \subset \mathbf {s}(I)$ has G as a formal basis. We may regard V as a vector space over E. Let $\zeta _1=1, \zeta _2,\dotsc ,\zeta _a \in \mathcal {O}_L$ be elements that reduce to a basis of $\mathbb {F}_q$ over $\mathbb {F}_p \bmod \pi _\circ $ and set $I_E=I \times \{1,\dotsc , a\}$ . We define

$$ \begin{align*} G_E &= \left\{ \zeta_j e_i\right\}_{\left(i,j\right)\in I_E}. \end{align*} $$

Note that $G_E$ is a formal basis of V over E.

6.2 Completely continuous operators and Fredholm determinants

6.2.2 Fredholm determinants

We continue with the notation from Section 6.2.1. We define the Fredholm determinant of u with respect to G to be the formal sum

(25) $$ \begin{align} \begin{split} \det(1-su\mid G) &= \sum_{n=0}^\infty c_ns^n, \\ c_n &= (-1)^n \sum_{\stackrel{S \subset I}{\lvert S\rvert=n}} \sum_{\sigma \in \text{Sym}(S)} \text{sgn}(\sigma) \prod_{i \in S} n_{i,\sigma(i)}. \end{split} \end{align} $$

We define the Fredholm determinant $\det \limits _E(1-sv\mid G_E)$ in an analogous manner using the matrix of v with respect to $G_E$ . Note that there is no reason a priori for the sums $c_n$ to converge. We will say that $\det (1-su\mid G)$ is well defined if each $c_n$ converges.

Lemma 6.7. Assume that V is a Banach space with orthonormal basis G and that u is completely continuous. Then $\det (1-su\mid G)$ is well defined and is an entire function in s. Furthermore, if $G'$ is another orthonormal basis of V, we have $\det (1-su\mid G)=\det (1-su\mid G')$ . The analogous result holds for v.

Proof. See [Reference Serre25, Proposition 7].

Definition 6.8. Continue with the notation from Lemma 6.7. We let $\det (1-su\mid V)$ denote the Fredholm determinant $\det (1-su\mid G)$ . By Lemma 6.7, this determinant does not depend on our choice of orthonormal basis. We define $\det \limits _E (1-sv\mid V)$ similarly.

6.2.3 Newton polygons of operators

Definition 6.9. Let $*$ be either p or q. Let $f(t)=\sum a_nt^n \in L\langle t\rangle ^{\times }$ be an entire function. We define the $*$ -adic Newton polygon $NP_*(f)$ to be the lower convex hull of the points $(n,v_*(a_n))$ . For $r>0$ , we let $NP_*(f)_{<r}$ denote the ‘subpolygon’ of $NP_*(f)$ consisting of all segments whose slope is less than r.

Definition 6.10. Adopt the notation from Section 6.2.2. Assume that ${\det (1-su\mid G)} \left (\text {resp., } \det \limits _E(1-sv\mid G_E)\right )$ is well defined and an entire function in s. Then we define $NP_*(u\mid G)$ (resp., $NP_*(v\mid G_E)$ ) to be $NP_*(\det (1-su\mid G)) \ \left (\text {resp., } NP_*\left (\det \limits _E(1-sv\mid G_E)\right )\right )$ . Further assume that V is a Banach space and u (resp., v) is completely continuous. Then by Lemma 6.7, the Fredholm determinant does not depend on the choice of orthonormal basis, so we define $NP_*(u\mid V)$ (resp., $NP_*(v\mid V)$ ) to be $NP_*(u\mid G)$ (resp., $NP_*(v\mid G_E)$ ).

Definition 6.11. Set $d \in \mathbb {Z}_{\geq 1} \cup \infty $ and let $A=\{c_n\}^{d}_{n\geq 1}$ be a nondecreasing sequence of real numbers. If $d=\infty $ we will make the assumption that $\lim \limits _{n \to \infty } c_n = \infty $ . Let $P_A$ be the ‘polygon’ of length d consisting of vertices $(0,0),(1,c_1),(2,c_1+c_2), \dotsc $ . We write

$$ \begin{align*} NP_*(f) &\succeq A\\[-17pt] \end{align*} $$

if the polygon $NP_*(f)$ lies above $P_A$ at every x-coordinate where both are defined.

The following lemma allows us to bound $NP_p(v\mid G_E)$ by estimating the columns of the matrix representing v.

Lemma 6.12. Assume that $\lim \limits _{i \in I} \mathbf {col}_{\left (i,1\right )}(v,G_E) = \infty $ . If v is $\nu ^{-1}$ -semilinear, then the Fredholm determinant $\det (1-sv\mid G_E)$ is well defined and we have

$$ \begin{align*} NP_p(v\mid G_E) &\succeq \left\{\mathbf{col}_{\left(i,1\right)}(v,G_E)\right\}^{\times a}_{i \in I},\\[-17pt] \end{align*} $$

where the superscript ‘ $\times a$ ’ means each slope is repeated a times.

Proof. Note that $v\left (\zeta _j e_i\right )=\zeta _j^{\nu ^{-1}}v(e_i)$ , which implies $\mathbf {col}_{\left (i,j\right )}(v,G_E)=\mathbf {col}_{\left (i,1\right )}(v,G_E)$ for each j. In particular, $\lim \limits _{\left (i,j\right ) \in I_E} \mathbf {col}_{\left (i,j\right )}(v,G_E) = \infty $ , so $\det \limits _E(1-sv\mid G_E)$ is well defined. By the definition of $c_n$ in equation (25), we see that

$$ \begin{align*} NP_p(v\mid G_E) &\succeq \left\{\mathbf{col}_{\left(i,j\right)}(v,G_E)\right\}_{\left(i,j\right) \in I_E} \\ &=\left\{\mathbf{col}_{\left(i,1\right)}(v,G_E)\right\}^{\times a}_{i \in I}. \\[-36pt] \end{align*} $$

7.1 The Monsky trace formula

Let us recall the Monsky trace formula in the case of curves. For a complete treatment, see [Reference Monsky22] or [Reference Wan29, Section 10]. Let $\Omega _{\mathcal {B}^{\dagger }}^i$ denote the space of i-forms of $\mathcal {B}^{\dagger }$ [Reference Monsky22, Section 4]. The map $\sigma $ induces a map $\sigma _i: \Omega ^i_{\mathcal {B}^{\dagger }} \to \Omega ^i_{\mathcal {B}^{\dagger }}$ sending $xdy$ to $x^\sigma d(y^\sigma )$ . As in [Reference van der Put28, Section 3], there exist trace maps $ \text {Tr}_i: \Omega _{\mathcal {B}^{\dagger }}^i \to \sigma \left (\Omega _{\mathcal {B}^{\dagger }}^i\right )$ . Let $\Theta _i$ denote the map $\sigma _i^{-1} \circ \text {Tr}_i$ . For $\omega \in \Omega ^1_{\mathcal {B}^{\dagger }}$ and $x \in \mathcal {B}^{\dagger }$ , we have

(26) $$ \begin{align} \Theta_1(x\omega^\sigma) &= \Theta_0(x)\omega.\\[-17pt]\nonumber \end{align} $$

Consider the L-function

(27) $$ \begin{align} L\left(\rho^{wild}\otimes \chi^{\otimes p^j},V,s\right)&= \prod_{x \in V} \frac{1}{1 - \rho^{wild}\otimes \chi^{\otimes p^j}(Frob_x) s^{\deg(x)}},\\[-17pt]\nonumber \end{align} $$

which is a slight modification of equation (1). Fix a tuple $\mathbf {r}=\left (r_Q\right )_{Q \in W}$ of positive rational numbers. Monsky shows that if the $r_Q$ are sufficiently small (so $\mathcal {B}(0,\mathbf {r}]$ consists of functions with sufficiently small radius of overconvergence), the operator $\Theta _i \circ \alpha f^{-\Gamma _j}$ is completely continuous on $\Omega _{\mathcal {B}\left (0,\mathbf {r}\right ]}^i$ . The Monsky trace formula states

(28) $$ \begin{align} L\left(\rho^{wild}\otimes \chi^{\otimes p^j},V,s\right) &= \frac{\det\left(1-s\Theta_1 \circ \alpha f^{-\Gamma_j}{\kern2pt}\middle |{\kern2pt} \Omega_{\mathcal{B}\left(0,\mathbf{r}\right]}^1\right)} {\det\left(1-s\Theta_0 \circ \alpha f^{-\Gamma_j}\middle| \mathcal{B}(0,\mathbf{r}]\right)},\\[-17pt]\nonumber \end{align} $$

where $\alpha $ , f, and $\Gamma _j$ are as in Sections 5.3.2 and 5.3.3. Thus, we may estimate $L\left (\rho ^{wild}\otimes \chi ^{\otimes p^j},V,s\right )$ by estimating operators on the space of $1$ -forms and $0$ -forms.

In our situation we may simplify equation (28). The map $\mathcal {A}^{\dagger } \to \mathcal {B}^{\dagger }$ is étale, which implies $\Omega _{\mathcal {B}^{\dagger }} = \pi ^* \Omega _{\mathcal {A}^{\dagger }}$ . Since $\Omega _{\mathcal {A}^{\dagger }}=\mathcal {A}^{\dagger } \frac {dt}{t}$ , we see that $\Omega _{\mathcal {B}^{\dagger }} = \mathcal {B}^{\dagger } \frac {dt}{t}$ . In particular, we have $\Omega _{\mathcal {B}\left (0,\mathbf {r}\right ]} = \mathcal {B}(0,\mathbf {r}] \frac {dt}{t}$ . Also, since $\frac {dt}{t}= \frac {1}{q}\left (\frac {dt}{t}\right )^\sigma $ , we know by equation (26) that $\Theta _1\left (x\frac {dt}{t}\right ) =\frac {1}{q} \Theta _0(x) \frac {dt}{t}$ . Thus, we have $\Theta _1=U_q$ and $\Theta _0=qU_q$ . Then equation (28) becomes

(29) $$ \begin{align} L\left(\rho^{wild}\otimes \chi^{\otimes p^j},V,s\right) &= \frac{\det\left(1-sU_q \circ \alpha f^{-\Gamma_j}\middle| \mathcal{B}(0,\mathbf{r}]\right)} {\det\left(1-sqU_q \circ \alpha f^{-\Gamma_j}\middle| \mathcal{B}(0,\mathbf{r}]\right)}.\\[-17pt]\nonumber \end{align} $$

As $\det \left (1-sU_q \circ \alpha f^{-\Gamma _j}{\kern2pt}\middle |{\kern2pt} \mathcal {B}(0,\mathbf {r}]\right ) \in 1+s\mathcal {O}_L\left[\!\left[ s\right]\!\right] $ , we know that $\frac {1}{\det \left (1-sqU_q \circ \alpha f^{-\Gamma _j}{\kern2pt}\middle |{\kern2pt} \mathcal {B}\left (0,\mathbf {r}\right ]\right )}$ lies in $ 1+qs\mathcal {O}_L\left[\!\left[ qs\right]\!\right] $ . This means each slope of $NP_q\left (\frac {1}{\det \left (1-sqU_q \circ \alpha f^{-\Gamma _j}{\kern2pt}\middle |{\kern2pt} \mathcal {B}\left (0,\mathbf {r}\right ]\right )}\right )$ is at least $1$ . In particular, we have

$$ \begin{align*} NP_q\left(L\left(\rho^{wild}\otimes \chi^{\otimes p^j},V,s\right)\right)_{<1} &= NP_q\left(U_q \circ \alpha f^{-\Gamma_j}\middle| \mathcal{B}(0,\mathbf{r}]\right)_{<1}.\\[-17pt] \end{align*} $$

Note that $\rho $ and $\rho ^{wild}\otimes \chi ^{\otimes p^j}$ are Galois conjugates. Thus, $L(\rho ,V,s)$ and $L\left (\rho ^{wild}\otimes \chi ^{\otimes p^j},V,s\right )$ are Galois conjugates. This gives

(30) $$ \begin{align} \begin{split} NP_q(L(\rho,V,s))_{<1} &= \frac{1}{a} NP_q\left(L\left(\rho^{wild}\otimes\bigoplus_{j=0}^{a-1} \chi^{\otimes p^j},V,s\right)\right)_{<1} \\ &= \frac{1}{a} NP_q\left(U_q \circ \alpha N{\kern2pt}\middle |{\kern2pt} \bigoplus_{j=0}^{a-1}\mathcal{B}(0,\mathbf{r}]\right)_{<1}, \end{split} \end{align} $$

where N is the dual Frobenius structure from Proposition 5.9.

7.2 Estimating $NP_q\left (U_q \circ \alpha N \middle | \bigoplus\limits_{j=0}^{a-1}\mathcal {B}(0,\mathbf {r}]\right )$

In this subsection we estimate the q-adic Newton polygon of $U_q\circ \alpha N $ acting on $ \bigoplus \limits _{j=0}^{a-1} \mathcal {B}(0,\mathbf {r}]$ .

Proposition 7.1. We have

$$ \begin{align*} \frac{1}{a}NP_q\left(U_q \circ \alpha N{\kern2pt}\middle |{\kern2pt} \bigoplus\limits_{j=0}^{a-1}\mathcal{B}(0,\mathbf{r}]\right)_{<1} &\succeq \{\underbrace{0,\dotsc,0}_{g-1+r_0+r_1+r_\infty-\Omega_\rho}\} \bigsqcup \left ( \bigsqcup_{i=1}^{\mathbf{m}} S_{\tau_i} \right ), \end{align*} $$

where $S_{\tau _i}$ is the slope set defined in Section 1.1 and $r_*$ is the cardinality of $\eta ^{-1}(*)$ defined in Section 3.2.

We break the proof up into several steps.

7.2.1 The twisted space and the ath root

We view $ \bigoplus \limits _{j=0}^{a-1} \widehat {\mathcal {B}}$ and $ \bigoplus \limits _{j=0}^{a-1} \mathcal {B}(0,\mathbf {r}]$ as subspaces of $\mathcal {R}$ , as described in Section 5.3.4. Unfortunately, the global Frobenius structure $\alpha N$ will not have nice growth properties like the local Frobenius structures studied in Section 5.1. Instead, we have to ‘twist’ this subspace using the matrices $bM$ defined in Section 5.3.4. Define the spaces

$$ \begin{align*} \widehat{V}&=bM\left( \bigoplus\limits_{j=0}^{a-1} \widehat{\mathcal{B}}\right),\\ V&=bM\left( \bigoplus\limits_{j=0}^{a-1} \mathcal{B}(0,\mathbf{r}]\right), \end{align*} $$

which we regard as subspaces of $\mathcal {R}$ . After decreasing $\mathbf {r}$ we may assume that

(31) $$ \begin{align} V &= \widehat{V} \bigcap \bigoplus_{Q \in W} \bigoplus_{j=0}^{a-1} \mathcal{E}_Q\left(0,r_Q\right]. \end{align} $$

In fact, equation (31) holds as long as $bM$ , viewed as a matrix with elements in $\bigoplus \limits _{Q \in W} \mathcal {E}_Q^{\dagger }$ , has entries contained in $\bigoplus \limits _{Q \in W} \mathcal {E}_Q\left (0,r_Q\right ]$ . From equation (23) we know that $U_q \circ C^{\nu ^{a-1}+ \dotsb + 1}$ and $U_p \circ C$ act on V. Since $U_q \circ C^{\nu ^{a-1}+ \dotsb + 1}=\left (U_p \circ C\right )^a$ , Lemma 6.13 tells us that

(32) $$ \begin{align} \begin{split} NP_q\left(U_q\circ \alpha N{\kern2pt}\middle |{\kern2pt}\bigoplus\limits_{j=0}^{a-1} \mathcal{B}(0,\mathbf{r}]\right)&=NP_q\left(U_q \circ C^{\nu^{a-1}+ \dotsb + 1}{\kern2pt}\middle |{\kern2pt}V\right) \\ &= \frac{1}{a} NP_p\left(U_p \circ C\mid V\right). \end{split} \end{align} $$

Proposition 7.2. The following hold:

1. 1. We have $pr\left (\widehat {V}_0\right )= \mathcal {O}_{\mathcal {R}^{trun}}$ , where $pr$ is the projection map defined in Section 5.3.4.

2. 2. Both $\ker \left (pr:V \to \mathcal {R}^{trun}\right )$ and $\ker \left (pr:\widehat {V} \to \mathcal {R}^{trun}\right )$ have dimension $a\left (g-1+r_0+r_1+r_\infty -\Omega _\rho \right )$ as vector spaces over L.

To prove Proposition 7.2 we need the following lemma:

Lemma 7.3. Let $f: R \to S$ be a continuous map of Banach spaces such that $f(R_0) \subset S_0$ . If $\overline {f}: \overline {R} \to \overline {S}$ is surjective, then f is surjective and $f(R_0)=S_0$ . Furthermore,

$$ \begin{align*} \overline{\ker(f)} &= \ker\left(\overline{f}\right). \end{align*} $$

Proof. This is proven by approximating the image and kernel of f. For more details see [Reference Kramer-Miller15, Lemma 7.3].

Proof of Proposition 7.2. Let us first consider $\widehat {V}$ . Define a function $\mu : W \to \mathbb {N}$ by

(33) $$ \begin{align} \mu(Q) &= \begin{cases} 1, & \eta(Q)\in \{0,\infty\}, \\ p, & \eta(Q) = 1. \end{cases} \end{align} $$

Let $\overline {M}$ be the reduction of $M \bmod \mathfrak m$ . By Lemma 7.3 and expression (24), we may prove the corresponding result for the map

(34) $$ \begin{align} \overline{pr}: \overline{M}\left(\bigoplus_{j=0}^{a-1} \overline{B}\right) \to \overline{\mathcal{R}^{trun}}=\bigoplus_{Q \in W} \bigoplus_{j=0}^{a-1} u_{Q,j}^{-\mu\left(Q\right)}\mathbb{F}_q\left[\!\left[ u_{Q,j}^{-1}\right]\!\right]. \end{align} $$

Define the divisor

$$ \begin{align*} D_j &= \sum_{i=1}^{r_1} (p-1)\left[P_{1,i}\right] - \sum_{Q \in W}n_{Q,j}[Q]. \end{align*} $$

By expression (24), we know the kernel of equation (34) is

$$ \begin{align*} \bigoplus_{j=0}^{a-1} H^0\left(X, \mathcal{O}_X\left(D_j\right)\right). \end{align*} $$

Since $(p-1)r_1=\deg (\eta )$ , we know from formula (14) that $\deg (D_i)\geq \deg (\eta )-r_0-r_\infty $ . By equation (4) and the Riemann–Roch theorem, we see that $H^0\left (X, \mathcal {O}_{X}\left (D_j\right )\right )$ has dimension $g-1+r_0+r_1+r_\infty - \sum n_{Q,j}$ . Then from equation (15) we know that the kernel of equation (34) has dimension $a\left (g-1+r_0+r_1+r_\infty -\Omega _\rho \right )$ as an $\mathbb {F}_q$ -vector space. To prove the result for V, first note that

$$ \begin{align*} \ker\left(pr:\mathcal{R} \to \mathcal{R}^{trun}\right) \subset \bigoplus_{Q \in W} \bigoplus_{j=0}^{a-1} \mathcal{E}_Q\left(0,r_Q\right], \end{align*} $$

as the kernel consists of functions with finite-order poles. The proposition follows from equation (31).

7.2.2 Choosing a basis

For the remainder of this section, we let $v=U_p \circ C$ , which we view as an operator on V. Then define $J \subset \mathbb {N} \times W\times \{0,\dotsc , a-1\}$ by

(35) $$ \begin{align} J &= \{(n,Q,j) \mid n\geq \mu(Q), \ j \in \{0,\dotsc,a-1\} \}, \end{align} $$

where $\mu $ is the function defined in equation (33). The set $\left \{u_{Q,j}^{-n}\right \}_{\left (n,Q,j\right ) \in J}$ is an orthonormal basis for $\mathcal {R}^{trun}$ over L (recall that $u_{Q,j}$ is the element of $\mathcal {R}$ with $u_Q$ in the $(Q,j)$ -coordinate and zeros in the other coordinates). Let K be a set with $\dim _L(\ker _L(pr\rvert _V))$ elements and set $I=J \sqcup K$ . For $i=(n,Q,j) \in J$ , choose an element $e_i \in V_0$ with $pr(e_i)=u_{Q,j}^{-n}$ . By the first part of Proposition 7.2 we know that such an $e_i$ exists. We also choose an orthonormal basis $\{e_i\}_{i \in K}\subset V_0$ of $\ker _L(pr\rvert _V)$ indexed by K. Then $G=\{e_i\}_{i \in I}$ is an orthonormal basis of $\widehat {V}$ over L. By formula (21) there exists $c_i \in \mathcal {O}_{\mathcal {R}}^{con}$ for each $i \in I$ with

(36) $$ \begin{align} e_i =\begin{cases} u_{Q,j}^{-n} + c_{i}, & i=(n,Q,j) \in J, \\ c_i, & i \in K. \end{cases} \end{align} $$

Define the space

$$ \begin{align*} V^{con}&= \left(\mathcal{O}_{\mathcal{R}}^{con} \cap \widehat{V}\right)\otimes_{\mathbb{Z}_p} \mathbb{Q}_p. \end{align*} $$

Endow $V^{con}$ with a norm so that $V^{con}_0=\mathcal {O}_{\mathcal {R}}^{con} \cap \widehat {V}$ (recall that the subscript $0$ denotes the subset of elements of norm $\leq 1$ ). We now scale each element $e_i \in G$ by an element $x_i$ in $\mathcal {O}_L$ to obtain a formal basis $G^{con}$ of $V^{con}$ . We break up the definition of $x_i$ into four cases: The first case is when $i \in K$ , and the other three cases correspond to the three types of points $Q\in W$ described in Section 5.3.3. We define

(37) $$ \begin{align} x_i &= \begin{cases} 1, & i \in K, \\ \pi_{a \mathfrak{s}_Q}^{q\left(\mathbf{e}_Q,j\right)} \pi_{\mathfrak{s}_Q}^{pn}, & i=(n,Q,j),\ \eta(Q) \in\{0,\infty\} \text{ and } \rho^{wild}_Q \text{ is unramified}, \\ \pi_{a s_Q}^{q\left(\mathbf{e}_Q,j\right)} \pi_{s_Q}^{pn}, & i=(n,Q,j),\ \eta(Q) \in\{0,\infty\} \text{ and } \rho_Q^{wild} \text{ is ramified}, \\ p^{b(n)}, & i=(n,Q,j) \text{ and } \eta(Q)=1, \end{cases} \end{align} $$

where $b(n)$ is the function defined in Section 4.2. From the definition of $\mathcal {O}_{\mathcal {R}}^{con}$ , we see that $G^{con}=\{x_ie_i\}$ is a formal basis of $V^{con}$ . Indeed, we just selected the $x_i$ appropriately for each summand in the definition of $\mathcal {O}_{\mathcal {R}}^{con}$ .

Proposition 7.4. We have

$$ \begin{align*} \det_E\left(1-sU_p\circ C\middle|V\right) &= \det_E\left(1-sU_p\circ C\middle| G^{con}_E\right). \end{align*} $$

Proof. For $Q \in W$ , define a sequence $b_{Q,1},b_{Q,2}, \dotsc \in \mathcal {O}_L$ such that $\left \{\dotsc , u_Q^2,u_Q^1,1,b_{Q,1}u_Q^{-1}, b_{Q,2}u_Q^{-2}, \dotsc \right \}$ is a formal basis of $\mathcal {E}_Q\left (0,r_Q\right ]$ . For $i\in K$ set $y_i=1$ , and for $i=(n,Q,j)$ set $y_i=b_{Q,n}$ . Then $G^{\mathbf {r}}=\{y_ie_i\}$ is an orthonormal basis of V. In particular, we have

$$ \begin{align*} \det_E\left(1-sU_p\circ C\middle| V\right)&=\det_E\left(1-sU_p\circ C\middle| G_E^{\mathbf{r}}\right) \\ &= \det_E\left(1-sU_p\circ C\middle| G^{con}_E\right). \end{align*} $$

The second equality follows from observing that the matrices of $U_p\circ C$ for the bases $G_E^{\mathbf {r}}$ and $G^{con}_E$ are similar.

7.2.3 Estimating the column vectors

To estimate the column vectors we will need the following lemma:

Lemma 7.5. For any $n\geq 0$ , we have $\pi _\circ ^n \mathcal {O}_{\mathcal {R}}^{con} \cap \widehat {V} = \pi _\circ ^nV^{con}_0$ .

Proof. Set $z \in \pi _\circ ^n \mathcal {O}_{\mathcal {R}}^{con} \cap \widehat {V}$ . Then $\pi _\circ ^{-n}z \in \mathcal {O}_{\mathcal {R}}^{con}$ , and since $\widehat {V}$ is a vector space we have $\pi _\circ ^{-n}z \in \widehat {V}$ . It follows that $z \in \pi _\circ ^n\left (\mathcal {O}_{\mathcal {R}}^{con} \cap \widehat {V}\right )$ . The other direction is similar.

We now estimate $\mathbf {col}_{\left (i,1\right )}\left (v,G_E^{con}\right )$ for each $i \in I$ . We break this up into the four cases used for defining $x_i$ .

1. (I) For $i \in K$ , we have $x_ie_i=e_i$ . We know from equation (36) that $e_i \in \mathcal {O}_{\mathcal {R}}^{con}$ . By formula (22) we know $v\left (\mathcal {O}_{\mathcal {R}}^{con}\right ) \subset \mathcal {O}_{\mathcal {R}}^{con}$ , which means $v (e_i) \in V_0^{con}$ . Thus, $\mathbf {col}_{\left (i,1\right )}\left (v,G_E^{con}\right )\geq 0$ and

   $$ \begin{align*} \left\{\mathbf{col}_{\left(i,1\right)}\left(v,G_E^{con}\right)\right\}_{i \in K} \succeq \{\underbrace{0,0,\dotsc,0}_{a\left(g-1+r_0+r_1+r_\infty-\Omega_\rho\right)} \}. \end{align*} $$
   The multiplicity of the zeros follows from Proposition 7.2.

2. (II) Fix Q with $\eta (Q)=1$ and let $i=(n,Q,j)\in J$ . By equation (35), we consider only tuples $(n,Q,j)$ with $n\geq p$ . Recall from equation (37) that $x_i=p^{b(n)}$ and from equation (36) that $e_i=u_{Q,j}^{-n} + c_i$ with $c_i \in \mathcal {O}_{\mathcal {R}}^{con}$ . Write $n=k+pm$ , where $0 \leq k <p$ . By formula (22) we have $v(x_ic_i) \in p^{b(n)}\mathcal {O}_{\mathcal {R}}^{con}$ , and by formula (17) we have $v\left (x_iu_{Q,j}^{-n}\right ) \subset p^m\mathcal {O}_{\mathcal {R}}^{con}$ . From the definition of $b(n)$ in Section 4.2, we know $b(n)\geq m$ , which implies $v (x_ie_i) \in p^m\mathcal {O}_{\mathcal {R}}^{con}$ . Lemma 7.5 tells us that $v (x_ie_i) \in p^m\left (V_0^{con}\right )$ . Thus, we have $\mathbf {col}_{\left (i,1\right )}\left (v,G_E^{con}\right )\geq m$ . This gives

   $$ \begin{align*} P_Q=\left\{\mathbf{col}_{\left(\left(n,Q,j\right),1\right)}\left(v,G_E^{con}\right)\right\}_{\substack{n\geq p \\ 0\leq j < a}} \succeq \{1,2,3,\dotsc \}^{ \times ap}. \end{align*} $$

3. (III) Fix $Q\in W$ such that $\eta (Q)\in \{0,\infty \}$ and $\rho ^{wild}_Q$ is unramified. Consider $i=(n,Q,j)\in J$ . By equation (35) we consider only tuples $(n,Q,j)$ where $n\geq 1$ . Recall from equation (37) that $x_i=\pi _{a \mathfrak {s}_Q}^{q\left (\mathbf {e}_Q,j\right )} \pi _{\mathfrak {s}_Q}^{pn}$ and from equation (36) that $e_i=u_{Q,j}^{-n} + c_i$ with $c_i \in \mathcal {O}_{\mathcal {R}}^{con}$ . Then by formulas (19) and (22), we see that $v(x_ie_i) \in \pi _{\mathfrak {s}_Q}^{n\left (p-1\right )} \pi _{a\mathfrak {s}_Q}^{-\omega _Q} \mathcal {O}_{\mathcal {R}}^{con}$ . Again, by Lemma 7.5 we see that $v (x_ie_i) \in \pi _{\mathfrak {s}_Q}^{n\left (p-1\right )} \pi _{a\mathfrak {s}_Q}^{-\omega _Q} \left (V_0^{con}\right )$ . This gives

   $$ \begin{align*} P_Q=\left\{\mathbf{col}_{\left(\left(n,Q,j\right),1\right)}\left(v,G_E^{con}\right)\right\}_{\substack{n\geq 1 \\ 0\leq j < a}} \succeq \left\{\frac{1}{\mathfrak{s}_Q}-\frac{\omega_Q}{a\mathfrak{s}_Q(p-1)},\frac{2}{\mathfrak{s}_Q}-\frac{\omega_Q}{a\mathfrak{s}_Q(p-1)}, \dotsc \right\}^{\times a}. \end{align*} $$

4. (IV) Finally, fix $Q\in W$ such that $\eta (Q)\in \{0,\infty \}$ and $\rho ^{wild}_Q$ is ramified. Repeating the argument from case III but replacing $\mathfrak {s}_Q$ with $s_Q$ gives

   $$ \begin{align*} P_Q=\left\{\mathbf{col}_{\left(\left(n,Q,j\right),1\right)}\left(v,G_E^{con}\right)\right\}_{\substack{n\geq 1 \\ 0\leq j < a}} \succeq \left\{\frac{1}{s_Q}-\frac{\omega_Q}{as_Q(p-1)},\frac{2}{s_Q}-\frac{\omega_Q}{as_Q(p-1)}, \dotsc \right\}^{\times a}. \end{align*} $$

We put everything together to get

$$ \begin{align*} \left\{\mathbf{col}_{\left(i,1\right)}\left(v,G_E^{con}\right)\right\}_{i \in I} \succeq \{\underbrace{0,0,\dotsc,0}_{g-1+r_0+r_1+r_\infty-\Omega_\rho} \}^{\times a} \bigsqcup \left (\bigsqcup_{Q \in W} P_Q \right ). \end{align*} $$

Then by Lemma 6.12 we see that $\det \left (1-sv,G_E^{con}\right )$ converges and that

$$ \begin{align*} NP_p\left(v\mid G_E^{con}\right) &\succeq \{\underbrace{0,0,\dotsc,0}_{g-1+r_0+r_1+r_\infty-\Omega_\rho} \}^{\times a^2} \bigsqcup \left (\bigsqcup_{Q \in W} P_Q^{\times a} \right ). \end{align*} $$

Then from Proposition 7.4 we have

$$ \begin{align*} \frac{1}{a} NP_p\left(U_p \circ C\mid V\right) &\succeq \{\underbrace{0,0,\dotsc,0}_{g-1+r_0+r_1+r_\infty-\Omega_\rho} \}^{\times a} \bigsqcup \left (\bigsqcup_{Q \in W} P_Q\right ). \end{align*} $$

When Q is from case II, each slope in $P_Q$ is at least $1$ . Also, when Q is from case III, we know from formula (18) that each slope in $P_Q$ is at least $1$ . This gives

$$ \begin{align*} \frac{1}{a} NP_p\left(U_p \circ C\mid V\right)_{< 1} &\succeq \{\underbrace{0,\dotsc,0}_{g-1+r_0+r_1+r_\infty-\Omega_\rho} \}^{\times a} \bigsqcup \left ( \bigsqcup_{i=1}^{\mathbf{m}} S_{\tau_i}^{\times a} \right ). \end{align*} $$

Proposition 7.1 follows from (32).

7.3 Finishing the proof