p-Adic estimates of abelian Artin L-functions on curves

Abstract

The purpose of this article is to prove a ‘Newton over Hodge’ result for finite characters on curves. Let X be a smooth proper curve over a finite field $\mathbb {F}_q$ of characteristic $p\geq 3$ and let $V \subset X$ be an affine curve. Consider a nontrivial finite character $\rho :\pi _1^{et}(V) \to \mathbb {C}^{\times }$ . In this article, we prove a lower bound on the Newton polygon of the L-function $L(\rho ,s)$ . The estimate depends on monodromy invariants of $\rho $ : the Swan conductor and the local exponents. Under certain nondegeneracy assumptions, this lower bound agrees with the irregular Hodge filtration introduced by Deligne. In particular, our result further demonstrates Deligne’s prediction that the irregular Hodge filtration would force p-adic bounds on L-functions. As a corollary, we obtain estimates on the Newton polygon of a curve with a cyclic action in terms of monodromy invariants.

1 Introduction

Let p be a prime with $p\geq 3$ and let $q=p^a$ . Let ${{X}}$ be a smooth proper curve of genus g defined over $\mathbb {F}_q$ with function field $K(X)$ . We define $G_X$ to be the absolute Galois group of $K(X)$ . Let ${ {\rho }}:G_X \to \mathbb {C}^{\times }$ be a nontrivial continuous character. The L-function associated to $\rho $ is defined by

(1) $$ \begin{align} L(\rho,s) =\prod\frac{1}{1 - \rho(Frob_x) s^{\deg(x)}}, \\[-17pt]\nonumber\end{align} $$

with the product taken over all closed points $x \in X$ where $\rho $ is unramified. By the Weil conjectures for curves [31], we know that

$$ \begin{align*} L(\rho,s) = \prod_{i=1}^{d}(1-\alpha_i s) \in \overline{\mathbb{Z}}[s].\\[-17pt] \end{align*} $$

It is then natural to ask what we can say about the algebraic integers $\alpha _i$ . The Riemann hypothesis for curves tell us that $\lvert \alpha _i\rvert _\infty = \sqrt {q}$ for each Archimedean place. Furthermore, we know that the $\alpha _i$ are $\ell $ -adic units for any prime $\ell \neq p$ . This leaves us with the question: What are the p-adic valuations of the $\alpha _i$ ?

The purpose of this article is to study the p-adic properties of $L(\rho ,s)$ . We prove a ‘Newton over Hodge’ result. This is in the vein of a celebrated theorem of Mazur [20], which compares the Newton and Hodge polygons of an algebraic variety over $\mathbb {F}_q$ . Our result differs from Mazur’s in that we study cohomology with coefficients in a local system. Our Hodge bound is defined using two monodromy invariants: the Swan conductor and the tame exponents. The representation $\rho $ is analogous to a rank $1$ differential equation on a Riemann surface with regular singularities twisted by an exponential differential equation (i.e., a weight $0$ twisted Hodge module in the language of Esnault, Sabbah and Yu [9]). In this context one may define an irregular Hodge polygon [8, 9]. The irregular Hodge polygon agrees with the Hodge polygon we define under certain nondegeneracy hypotheses. Our result thus gives further credence to the philosophy that characteristic $0$ Hodge-type phenomena force p-adic bounds on lisse sheaves in characteristic p.

1.1 Statement of main results

To state our main result, we first introduce some monodromy invariants. The character $\rho $ factors uniquely as $\rho =\rho ^{wild} \otimes \chi $ , where $\left \lvert Im\left (\rho ^{wild}\right )\right \rvert =p^n$ and $\lvert Im(\chi )\rvert =N$ with $\gcd (N,p)=1$ .

1. 1. (Local) Let $Q \in X$ be a closed point. After increasing q we may assume that Q is an $\mathbb {F}_q$ -point. Let $u_Q$ be a local parameter at Q. Then $\rho $ restricts to a local representation $\rho _Q:G_Q \to \mathbb {C}^{\times }$ , where $G_Q$ is the absolute Galois group of $\mathbb {F}_q((u_Q))$ . We let $\rho ^{wild}_Q$ (resp., $\chi _Q$ ) denote the restriction of $\rho ^{wild}$ (resp., $\chi $ ) to $G_Q$ .

   1. (a) (Swan conductors) Let $I_Q \subset G_Q$ be the inertia subgroup at Q. There is a decreasing filtration of subgroups $I_Q^s$ on $I_Q$ , indexed by real numbers $s\geq 0$ . The Swan conductor at Q is the infimum of all s such that $I_Q^s \subset \ker (\rho _Q)$ [13, Chapter 1]. We denote the Swan conductor by ${ {s_Q}}$ . Note that $s_Q=0$ if and only if $\rho _Q^{wild}$ is unramified.

   2. (b) (Tame exponents) After increasing q we may assume that $\chi _Q$ is totally ramified at Q. There exists $e_Q \in \frac {1}{q-1}\mathbb {Z}$ such that $G_Q$ acts on $t_Q^{e_Q}$ by $\chi _Q$ . Note that $e_Q$ is unique up to addition by an integer.

      • ○ The exponent of $\chi $ at Q is the equivalence class ${ {\mathbf {e}_Q}}$ of $e_Q$ in $ \frac {1}{q-1}\mathbb {Z}/ \mathbb {Z}$ .

      • ○ We define ${ {\epsilon _Q}}$ to be the unique integer between $0$ and $q-2$ such that $\frac {\epsilon _Q}{q-1} \in \mathbf {e}_Q$ .

      • ○ Write $\epsilon _Q=e_{Q,0} + e_{Q,1}p + \dotsb +e_{Q,a-1} p^{a-1}$ , where $0 \leq e_{Q,i}\leq p-1$ . We define ${ {\omega _Q}}=\sum e_{Q,i}$ , the sum of the p-adic digits of $\epsilon _Q$ . Note that $\omega _Q=0$ if and only if $\chi _Q$ is unramified.

   We refer to the tuple $R_Q=\left (s_Q, \mathbf {e}_Q, \epsilon _Q, \omega _Q\right )$ as a ramification datum and $T_Q=\left (\mathbf {e}_Q,\epsilon _Q,\omega _Q\right )$ as a tame ramification datum. We define the sets

   $$ \begin{align*} S_{Q} &= \begin{cases} \emptyset & s_Q=0, \\ \left\{ \frac{1}{s_{Q}}, \dotsc, \frac{s_{Q}-1}{s_{Q}}\right\}, & s_Q \neq 0 \text{ and } \omega_Q =0, \\ \left\{ \frac{1}{s_{Q}}- \frac{\omega_{Q}}{as_{Q}\left(p-1\right)}, \dotsc, \frac{s_{Q}}{s_{Q}}- \frac{\omega_{Q}}{as_{Q}\left(p-1\right)}\right\}, & s_Q \neq 0 \text{ and } \omega_Q \neq 0. \end{cases} \\[-17pt]\end{align*} $$

2. 2. (Global) Let ${ {\tau _1,\dotsc ,\tau _{\mathbf {m}}}}$ be the points at which $\rho $ ramifies and let $\mathbf {n}\leq { {\mathbf {m}}}$ be such that $\tau _1, \dotsc , \tau _{\mathbf {n}}$ are the points at which $\chi $ ramifies. We define

   $$ \begin{align*} {{\Omega_\rho}} &= \frac{1}{a(p-1)} \sum_{i=1}^{\mathbf{n}} \omega_{\tau_i}.\\[-17pt] \end{align*} $$
   This is a global invariant built up from the p-adic properties of the local exponents. One can show that $\Omega _\rho \in \mathbb {Z}_{\geq 0}$ (see Section 5.3.2).

Using these invariants, we define the Hodge polygon ${ {HP(\rho )}}$ to be the polygon whose slopes are

$$ \begin{align*} \{\underbrace{0,\dotsc,0}_{g-1+\mathbf{m}-\Omega_\rho} \}\sqcup \{\underbrace{1,\dotsc,1}_{g-1+\mathbf{m}-\mathbf{n} + \Omega_\rho} \} \sqcup \left ( \bigsqcup_{i=1}^{\mathbf{m}} S_{\tau_i} \right ), \end{align*} $$

where $\sqcup $ denotes a disjoint union. We can now state our main result:

Theorem 1.1. The q-adic Newton polygon $NP_q(L(\rho ,s))$ lies above the Hodge polygon $HP(\rho )$ .

Remark 1.2. It is worth mentioning that $HP(\rho )$ and $NP_q(L(\rho ,s))$ have the same endpoints. To see this, first note that the x-coordinates of the endpoints of both polygons are $g-1+\mathbf {m} + \sum s_Q$ . For $NP_q(L(\rho ,s))$ this follows from the Euler–Poincaré formula [13, Section 2.3.1], and for $HP(\rho )$ it is clear from the definition. Next let $\left (s_{\tau _i}', \mathbf {e}_{\tau _i}', \epsilon _{\tau _i}', \omega _{\tau _i}'\right )$ be the ramification datum associated to $\rho ^{-1}$ at ${\tau _i}$ . Then we have $s_{\tau _i}'=s_{\tau _i}$ and $\mathbf {e}_{\tau _i}'=-\mathbf {e}_{\tau _i}$ . From this we see that $\omega _{\tau _{i}}'=a(p-1)-\omega _{\tau _{i}}$ for $1\leq i \leq \mathbf {n}$ and $\omega _{\tau _i}'=0$ for $i>\mathbf {n}$ , which implies $\Omega _{\rho ^{-1}}=\mathbf {n}-\Omega _{\rho }$ . Thus, for every slope $\alpha $ of $HP(\rho )$ , there is a corresponding slope $1-\alpha $ of $HP\left (\rho ^{-1}\right )$ . Similarly, by Poincaré duality we know that for every slope $\alpha $ of $NP_q(L(\rho ,s))$ , there is a corresponding slope $1-\alpha $ of $NP_q\left (L\left (\rho ^{-1},s\right )\right )$ . It follows that the y-coordinates of the endpoints of $HP(\rho )\sqcup HP\left (\rho ^{-1}\right )$ and $NP_q(L(\rho ,s))\sqcup NP_q\left (L\left (\rho ^{-1},s\right )\right )$ agree. By applying Theorem 1.1 to $\rho $ and $\rho ^{-1}$ , we see that the endpoints of $HP(\rho )$ and $NP_q(L(\rho ,s))$ are the same.

Remark 1.3. When $\rho $ factors through an Artin–Schreier cover, Theorem 1.1 is due to previous work of the author [15].

Remark 1.4. The only other case where parts of Theorem 1.1 were previously known is when $X=\mathbb {P}^1$ and $\rho $ is unramified outside of $\mathbb {G}_m$ . Work of Adolphson and Sperber [2, 3] studies the case when $\lvert Im(\rho )\rvert =pN$ and $\gcd (p,N)=1$ . We note that the work of Adolphson and Sperber treats the case of higher-dimensional tori as well. These groundbreaking methods were applied to the case when $\rho $ is totally wild by Liu and Wei in [18], introducing ideas from Artin–Schreier–Witt theory. For $\rho $ with arbitrary image there are some results by Liu [17], under strict conditions on the wild part of $\rho $ (this case corresponds to Heilbronn sums).

To the best of our knowledge, Theorem 1.1 was completely unknown outside of the situations described in Remarks 1.3 and 1.4.

Example 1.5. Let $X=\mathbb {P}^1_{\mathbb {F}_q}$ and let $\tau _1,\dotsc ,\tau _{4}$ be the points where $\rho $ ramifies. Assume $\lvert Im(\rho )\rvert =2p^n$ and that $\rho $ is totally ramified at each $\tau _i$ (i.e., the inertia group at $\tau _i$ is equal to $Im(\rho )$ ). Let $f:E \to X$ be the genus $1$ curve over which $\chi $ trivialises and let $\upsilon _i=f^{-1}(\tau _i)$ . Consider the restriction $\rho _E=\rho \rvert _{G_E}$ . Let $(s_i, \mathbf {e}_i, \epsilon _i, \omega _i)$ be the ramification datum of $\rho $ at $\tau _i$ and let $\left (s_i', \mathbf {e}_i', \epsilon _i', \omega _i'\right )$ be the ramification datum of $\rho _E$ at $\upsilon _i$ . By Theorem 1.1 we know that $NP_q(L(\rho _E,s))$ lies above

$$ \begin{align*} HP(\rho_E)&= \{0,0,0,0 \}\sqcup \{1,1,1,1 \} \sqcup \left ( \bigsqcup_{i=1}^4 \left\{\frac{1}{2s_i}, \dotsc, \frac{2s_i-1}{2s_i}\right\} \right ). \end{align*} $$

This follows by recognising that $s_i'=2s_i$ and $\omega _i'=0$ . The factorisation $L(\rho _E,s)=L(\rho ,s)L\left (\rho ^{wild},s\right )$ corresponds to a ‘decomposition’ of $HP(\rho _E)$ into two Hodge polygons, one giving a lower bound for $NP_q(L(\rho ,s))$ and the other for $NP_q\left (L\left (\rho ^{wild},s\right )\right )$ . We have $\omega _i=\frac {a\left (p-1\right )}{2}$ for each i and $\Omega _\rho =2$ . This allows us to compute the Hodge polygons as

$$ \begin{align*} HP(\rho)&= \{0,1\} \sqcup \left ( \bigsqcup_{i=1}^4\left \{\frac{1}{2s_i}, \frac{3}{2s_i}, \dotsc, \frac{2s_i-1}{2s_i}\right\} \right ),\\ HP\left(\rho^{wild}\right) &= \{0,0,0 \}\sqcup \{1,1,1 \} \sqcup \left ( \bigsqcup_{i=1}^4 \left\{\frac{1}{s_i}, \dotsc, \frac{s_i-1}{s_i}\right\} \right ), \end{align*} $$

so that $HP(\rho _E)=HP(\rho )\sqcup HP\left (\rho ^{wild}\right )$ . More generally, we will obtain similar decompositions of the Hodge bounds as long as $Im(\chi )\mid p-1$ .

1.1.1 Newton polygons of abelian covers of curves

Theorem 1.1 also has interesting consequences for Newton polygons of cyclic covers of curves. Let $G=\mathbb {Z}/Np^n\mathbb {Z}$ , where N is coprime to p. Let $f: C \to X$ be a G-cover ramified over $\tau _1,\dotsc ,\tau _{\mathbf {m}}$ . We let $H^1_{cris}(X) \ \left (\text {resp., } H^1_{cris}(C)\right )$ be the crystalline cohomology of X (resp., C). For a character $\rho $ of G, we let $H^1_{cris}(C)^{\rho }$ be the $\rho $ -isotypical subspace for the action of G on $H^1_{cris}(C)$ . Let $NP_C \ \left (\text {resp. } NP_X \text { and } NP_C^{\rho }\right )$ denote the q-adic Newton polygon of $\det \left (1-s\text {F}\mid H^1_{cris}(C)\right ) \ \left (\text {resp., } \det \left (1-s\text {F}\mid H^1_{cris}(X)\right ) \text { and } \det \left (1-s\text {F}\mid H^1_{cris}(C)^{\rho }\right )\right )$ . We are interested in the following question: To what extent can we determine $NP_C$ from $NP_X$ and the ramification invariants of f? The most basic result is the Riemann–Hurwitz formula, which determines the dimension of $H^1_{cris}(C)$ from $H^1_{cris}(X)$ and the ramification invariants. When $N=1$ , there is also the Deuring–Shafarevich formula [7], which determines the number of slope $0$ segments in $NP_C$ . In general, however, a precise formula for the slopes of $NP_C$ seems impossible. Instead, the best we may hope for are estimates. To connect this problem to Theorem 1.1, recall the decomposition

(2) $$ \begin{align} \det\left(1-s\text{F}\mid H^1_{cris}(C)\right)&= \det\left(1-s\text{F}\mid H^1_{cris}(X)\right) \prod_{\rho} \det\left(1-s\text{F}\mid H^1_{cris}(C)^{\rho}\right), \end{align} $$

where $\rho $ varies over the nontrivial characters $\mathbb {Z}/Np^n\mathbb {Z} \to \mathbb {C}^{\times }$ . By the Lefschetz trace formula we know $L(\rho ,s)=\det \left (1-s\text {F}\mid H^1_{cris}(C)^{\rho }\right )$ . Thus, Theorem 1.1 gives lower bounds for $NP_C$ using equation (2).

Consider the case when $N=1$ , so that $G=\mathbb {Z}/p^n\mathbb {Z}$ . Let $r_i$ be the ramification index of a point of C above $\tau _i$ and define

$$ \begin{align*} \Omega = \sum_{i=1}^{\mathbf{m}} p^{n-r_i}\left(p^{r_i}-1\right). \end{align*} $$

For $j=1,\dotsc ,n$ , let $C_j$ be the cover of X corresponding to the subgroup $p^{n-j}\mathbb {Z} \big / p^n\mathbb {Z} \subset G$ . Fix a point $x_i(j)\in C_j$ above $\tau _i$ ; this gives a local field extension of $\mathbb {F}_q\left (\left (t_{\tau _i}\right )\right )$ . We let $s_{\tau _i}(j)$ denote the largest upper numbering ramification break of this extension.

Corollary 1.6. The Newton polygon $NP_C$ lies above the polygon whose slopes are the multiset

$$ \begin{align*} NP_X \sqcup \{\underbrace{0,\dotsc,0}_{(p^n-1)(g-1)+\Omega}, \underbrace{1,\dotsc,1}_{(p^n-1)(g-1)+\Omega}\} \sqcup \left(\bigsqcup_{i=1}^{\mathbf{m}} \bigsqcup_{j=1}^n p^{j-1}(p-1)\times \left\{\frac{1}{s_{\tau_i}(j)}, \dotsc, \frac{s_{\tau_i}(j)-1}{s_{\tau_i}(j)} \right\}\right), \end{align*} $$

where we take $\left \{\frac {1}{s_{\tau _i}\left (j\right )}, \dotsc , \frac {s_{\tau _i}\left (j\right )-1}{s_{\tau _i}\left (j\right )} \right \}$ to be the empty set when $s_{\tau _i}(j)=0$ .

Remark 1.7. When $N>1$ , we can obtain a complicated bound for $NP_C$ from Theorem 1.1 and equation (2). Alternatively, we can replace X with the intermediate curve $X^{tame}$ satisfying $Gal\left (C/X^{tame}\right )=\mathbb {Z}/p^n\mathbb {Z}$ and then apply Corollary 1.6 to the cover $C \to X^{tame}$ to obtain a bound. Both bounds are the same.

1.2 Outline of proof

The classical approaches to studying p-adic properties of exponential sums on tori no longer work when one considers more general curves. Instead, we have to expand on the methods developed in earlier work of the author on exponential sums on curves [15]. We use the Monsky trace formula (see Section 7.1). This trace formula allows us to compute $L(\rho ,s)$ by studying Fredholm determinants of certain operators. More precisely, let $V=X-\{\tau _1,\dotsc ,\tau _{\mathbf {m}} \}$ and let $\overline {B}$ be the coordinate ring of V. Let L be a finite extension of $\mathbb {Q}_p$ whose residue field is $\mathbb {F}_q$ such that the image of $\rho $ is contained in $L^{\times }$ . Let $B^{\dagger }$ be the ring of integral overconvergent functions on a formal lifting of $\overline {B}$ over $\mathcal {O}_L$ (see Section 3). For example, if $V=\mathbb {A}^1$ , then $B^{\dagger }=\mathcal {O}_L\left \langle t \right \rangle ^{\dagger }$ (i.e., $B^{\dagger }$ is the ring of power series with integral coefficients that converge beyond the closed unit disc). Choose an endomorphism $\sigma : B^{\dagger } \to B^{\dagger }$ that lifts the q-power Frobenius of $\overline {B}$ . Using $\sigma $ , we define an operator $U_q: B^{\dagger }\to B^{\dagger }$ , which is the composition of a trace map $Tr:B^{\dagger }\to \sigma \left (B^{\dagger }\right )$ with $\frac {1}{q}\sigma ^{-1}$ .

The Galois representation $\rho $ corresponds to a unit-root overconvergent F-crystal of rank $1$ . This is a $B^{\dagger }$ -module $M=B^{\dagger } e_0$ and a $B^{\dagger }$ -linear isomorphism $\varphi : M \otimes _{\sigma } B^{\dagger } \to M$ . Note that this F-crystal is determined entirely by $\alpha \in B^{\dagger }$ satisfying $\varphi (e_0 \otimes 1) = \alpha e_0$ . We refer to $\alpha $ as the Frobenius structure of M. In our specific setup (see Section 3), the Monsky trace formula can be written as

$$ \begin{align*} L(\rho,s) &= \frac{\det\left(1-sU_q \circ \alpha \mid B^{\dagger}\right)} {\det\left(1-sqU_q \circ \alpha \mid B^{\dagger}\right)}, \end{align*} $$

where we regard $\alpha $ as the ‘multiplication by $\alpha $ ’ map on $B^{\dagger }$ . Thus, we need to understand the operator $U_q \circ \alpha $ . Let us outline how we study this operator.

1.2.1 Lifting the Frobenius endomorphism

Both $U_q$ and $\alpha $ depend on the choice of Frobenius endomorphism $\sigma $ . When $V=\mathbb {G}_m$ , the ring $B^{\dagger }$ is $\mathcal {O}_L\langle t \rangle ^{\dagger }$ , and the natural choice for $\sigma $ sends t to $t^q$ . However, no such natural choice exists for higher-genus curves. Our approach is to pick a convenient mapping $\eta :X \to \mathbb {P}^1$ and then pull back the Frobenius $t \mapsto t^q$ along $\eta $ . We take $\eta $ to be a tamely ramified map that is étale outside of $\{0,1,\infty \}$ . We may further assume that $\eta (\tau _i) \in \{0,\infty \}$ and the ramification index of every point in $\eta ^{-1}(1)$ is $p-1$ (see Lemma 3.1). This leaves us with two types of local Frobenius endomorphisms. For $Q \in X$ with $\eta (Q)\in \{0,\infty \}$ , we may take the local parameter at Q to look like $u_Q=t^{\pm \frac {1}{e_Q}}$ , where $e_Q$ is the ramification index at Q. In particular, the Frobenius endomorphism sends $u_Q \mapsto u_Q^q$ . If $\eta (Q)=1$ , we take the local parameter to look like $u_Q=\sqrt [p-1]{t-1}$ . Thus, the Frobenius endomorphism sends $u_Q \mapsto \sqrt [p-1]{\left (u_Q^{p-1}+1\right )^{p}-1}$ . In Section 4 we study $U_q$ for both types of local Frobenius endomorphisms, and in Section 5.2 we study the local versions of the Frobenius structure $\alpha $ .

1.2.2 The problem of ath roots of $U_q\circ \alpha $

To obtain the correct estimates of $\det \left (1-sU_q \circ \alpha \mid B^{\dagger }\right )$ , it is necessary to work with an ath root of $U_q\circ \alpha $ . That is, we need an element $\alpha _0 \in B^{\dagger }$ and a $U_p$ operator (this is analogous to the $U_q$ operator, but for liftings of the p-power endomorphism) such that $\left (U_p\circ \alpha _0\right )^a=U_q\circ \alpha $ . However, this ath root is only guaranteed to exist if the order of $Im(\chi )$ divides $p-1$ (see Section 5.1). This presents a major technical obstacle. The solution is to consider $ \rho ^{wild}\otimes \bigoplus \limits _{j=0}^{a-1}\chi ^{\otimes p^j}$ , which is a restriction of scalars of $\rho $ . The L-functions of each summand are Galois conjugate, and thus have the same Newton polygon. We can then study an operator $U_p \circ N$ , where N is the Frobenius structure of the F-crystal associated to $ \rho ^{wild}\otimes \bigoplus \limits _{j=0}^{a-1}\chi ^{\otimes p^j}$ . This is similar to the idea used in Adolphson and Sperber’s study of twisted exponential sums on tori [2]. They present it in an ad hoc manner, but the underlying idea is to study $ \rho ^{wild}\otimes \bigoplus \limits _{j=0}^{a-1}\chi ^{\otimes p^j}$ in lieu of $\rho $ .

1.2.3 Global to local computations

When V is $\mathbb {G}_m$ or $\mathbb {A}^1$ , the ring $B^{\dagger }$ is just $\mathcal {O}_L\left \langle t \right \rangle ^{\dagger }$ or $\mathcal {O}_L\left \langle t,t^{-1} \right \rangle ^{\dagger }$ . In both cases, it is relatively easy to study operators on $B^{\dagger }$ . The situation is more complex for higher-genus curves. Our approach to make sense of $B^{\dagger }$ is to ‘expand’ each function around the $\tau _i$ (and some other auxiliary points). Namely, let $t_i\in B^{\dagger }$ be a function whose reduction in $\overline {B}$ has a simple zero at $\tau _i$ . We let $\mathcal {O}_{\mathcal {E}_i^{\dagger }}$ be the ring of formal Laurent series in $t_i$ that converge on an annulus $r<\lvert t_i\rvert _p<1$ (i.e., the bounded Robba ring). Any $f \in B^{\dagger }$ has a Laurent expansion in $t_i$ , and our overconvergence condition implies this expansion lies in $\mathcal {O}_{\mathcal {E}_i^{\dagger }}$ . We obtain an injection

(3) $$ \begin{align} B^{\dagger} \hookrightarrow \bigoplus_{i=1}^{\mathbf{m}} \mathcal{O}_{\mathcal{E}_i^{\dagger}}.\\[-15pt]\nonumber \end{align} $$

The operator $U_p \circ N$ extends to an operator on each summand. By carefully keeping track of the image of $B^{\dagger }$ , we are able to compute on each summand (see Section 7.2). This lets us compute $U_p\circ N$ on the bounded Robba ring, which ostensibly looks like a ring of functions on $\mathbb {G}_m$ . We are thus able to compute $U_p \circ N$ by studying local Frobenius structures and local $U_p$ operators.

1.2.4 Comparing Frobenius structures and $\Omega _\rho $

In Section 5.2 we study the shape of the unit-root F-crystal associated to $\rho $ when we localise at a ramified point $\tau _i$ . We show that the localised unit-root F-crystal has a particularly nice Frobenius structure, which depends on the ramification datum. However, these well-behaved local Frobenius structures do not patch together to give a well-behaved global Frobenius structure. This is a major technical obstacle. When comparing local and global Frobenius structures, we end up having to ‘twist’ the image of formula (3). This process explains the invariant $\Omega _\rho $ occurring in Theorem 1.1 – it arises by ‘averaging’ the local exponents for each $\rho ^{wild} \otimes \chi ^{\otimes p^i}$ . This invariant is essentially absent in the work of Adolphson and Sperber, since $\Omega _\rho =1$ if $V=\mathbb {G}_m$ . It is also absent in the author’s previous work, where the local exponents were all zero.

1.3 Further remarks

Pinning down the exact Newton polygon of a covering of a curve, as well as the Newton polygon of the isotypical constituents, is a fascinating question. A general answer seems impossible, but one can certainly hope for results that hold generically. If the genus of X and the monodromy invariants from Section 1.1 are specified, what is the Newton polygon for a generic character? We believe the bound from Theorem 1.1 should only be generically attained if $N\mid p-1$ and there are some congruence relations between p and the Swan conductors. When $\rho $ factors through an Artin–Schreier cover, this is known by combining work of the author [15] with work of Booher and Pries [5]. The next step would be to study the case arising from a cyclic cover whose degree divides $p(p-1)$ (or even allowing higher powers of p). When $N\nmid p-1$ , the bound from Theorem 1.1 has too many slope $0$ segments. The issue is that a generic tame cyclic cover of degree N is not ordinary, even if X is ordinary [6]. Even when $X=\mathbb {P}^1$ , the study of Newton polygons for tame cyclic covers is already a complicated topic (e.g., [16]). The author plans to return to these questions at a later time. It would also be interesting to prove Hodge bounds for representations with positive weight. In recent work, Fresán, Sabbah and Yu use irregular Hodge theory to study the p-adic slopes of symmetric powers of Kloosterman sums [10]. Not much is known beyond this case.

2 Notation

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 Global setup

We now introduce the global setup, which closely follows [15, Section 3]. We adopt the notation from Section 1.1. Our main goal is to choose a Frobenius endomorphism on a lift of an affine subspace of X. We require two things from this Frobenius endomorphism. First, we want an endomorphism that behaves reasonably with respect to certain local parameters. Second, it should make the Monsky trace formula satisfy a certain form (see Section 7.1). We find this Frobenius endomorphism by bootstrapping from the standard Frobenius endomorphism on the projective line.

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

Lemma 3.1. After increasing q, there exists a tamely ramified morphism $\eta :X \to \mathbb {P}_{\mathbb {F}_q}^1$ , ramified only above $0,1$ , and $\infty $ , such that $\tau _1,\dotsc , \tau _{\mathbf {m}} \in \eta ^{-1}(\{0,\infty \})$ and each $P \in \eta ^{-1}(1)$ has ramification index $p-1$ .

Proof. This is [15, Lemma 3.1].

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 [11, 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 [23, 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 [19, 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 $0<v_p\left (u_Q\right )\leq r_Q$ . 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 [28, 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 Local $U_p$ operators

Let $\nu $ be a p-Frobenius endomorphism of $\mathcal {E}^{\dagger }$ (see Section 2.2). We define $U_p$ to be the map

$$ \begin{align*} \frac{1}{p}\nu^{-1}\circ \text{Tr}_{\mathcal{E}^{\dagger} / \nu\left(\mathcal{E}^{\dagger}\right)}: \mathcal{E}^{\dagger} \to \mathcal{E}^{\dagger}. \end{align*} $$

Note that $U_p$ is $\nu ^{-1}$ -semilinear (i.e., $U_p(y^{\nu } x)=yU_p(x)$ for all $y\in \mathcal {E}^{\dagger }$ ). In this section we will study $U_p$ for the p-Frobenius endomorphisms of $\mathcal {E}^{\dagger }$ appearing in Section 3.4.

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

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

Proof. See [15, Proposition 4.4].

5 Unit-root F-crystals

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.

Definition 5.1. A $\varphi $ -module for $\sigma $ over R is a locally free R-module M equipped with a $\sigma $ -semilinear endomorphism $\varphi : M \to M$ . That is, we have $\varphi (cm)=\sigma (c)\varphi (m)$ for $c\in R$ .

Definition 5.2. A unit-root F-crystal M over $\overline {S}$ is a $\varphi $ -module such that $\sigma ^* \varphi : R \otimes _{\sigma } M \to M$ is an isomorphism. The rank of M is defined as the rank of the underlying R-module.

Theorem 5.3 Katz [12, 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., [27, 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.

5.2.1 Unramified Artin–Schreier–Witt characters

Proposition 5.4. Let $\nu $ be any p-Frobenius endomorphism of $\mathcal {O}_{\mathcal {E}}$ 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 $Im(\psi )\cong \mathbb {Z}/p^n\mathbb {Z}$ and that $\psi $ is unramified. Then there exists a p-Frobenius structure $\alpha _0$ of $M_\psi $ with $\alpha _0 \in 1+\mathfrak m$ (recall that $\mathfrak m$ is the maximal ideal of $\mathcal {O}_L$ ). Furthermore, if $c \in 1+\mathfrak m\mathcal {O}_{\mathcal {E}}$ is another p-Frobenius structure of $M_\psi $ , there exists $b\in 1+\mathfrak m\mathcal {O}_{\mathcal {E}}$ with $\alpha _0=\frac {b^{\nu }}{b}c$ .

Proof. This is essentially the same as [15, Proposition 5.4].

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., [30, Section 4.1] for the exponential-sum situation or [18]). 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 [14, 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] $ .

5.2.3 Tame characters

Let $\psi :G_{\mathbb {F}_q((t))} \to \mathcal {O}_L^{\times }$ be a totally ramified tame character and let $T=(\mathbf {e}, \epsilon , \omega )$ be the corresponding tame ramification datum (see Section 1.1). Write $\epsilon =e_0+\dotsb + e_{a-1}p^{a-1}$ and define $\epsilon _j=\sum _{i=0}^{a-1} e_{i+j}p^i$ .

Proposition 5.6. The following hold:

1. 1. The matrix $C=\mathbf {diag}\left (t^{-\epsilon _0}, \dotsc , t^{-\epsilon _{a-1}}\right )$ (resp., $C_0=\mathbf {tcyc}\left (t^{-e_0}, \dotsc , t^{-e_{a-1}}\right )$ ) is a dual Frobenius structure (resp., dual p-Frobenius structure) of $\bigoplus \limits _{j=0}^{a-1} \psi ^{\otimes p^j}$ and $C=C_0^{\nu ^{a-1}+\dotsb + \nu + 1}$ .

2. 2. Let $A=\mathbf {diag}\left (x_0, \dotsc , x_{a-1}\right )$ (resp., $A_0=\mathbf {tcyc}\left (y_0, \dotsc , y_{a-1}\right )$ ) be another dual Frobenius structure (resp., dual p-Frobenius structure) of $\bigoplus \limits _{j=0}^{a-1} \psi ^{\otimes p^j}$ with $A=A_0^{\nu ^{a-1}+\dotsb + \nu + 1}$ . Then $v_t\left (\overline {x_j}\right )= -\epsilon _j + n_j(q-1)$ for some $n_j \in \mathbb {Z}$ (here $\overline {x_j}$ is the image of $x_j$ in $\mathbb {F}_q((t))$ ). Furthermore, there exists $B=\mathbf {diag}\left (b_0, \dotsc , b_{a-1}\right )$ with $v_t\left (\overline {b_j}\right )=n_j$ such that $B^{\sigma }AB^{-1} = C \ \left (\text {resp., } B^{\nu }A_0B^{-1}=C_0\right )$ .

Proof. Let $G_{\mathbb {F}_q((t))}$ act on $\mathcal {L}=\bigoplus \limits _{j=0}^{a-1} v_j\mathcal {O}_L$ via $ \bigoplus \limits _{j=0}^{a-1} \psi ^{\otimes p^j}$ . Let $u=t^{\frac {1}{q-1}}$ and let $\mathcal {E}'$ be the Amice ring over L with parameter u. The F-crystal associated to $ \bigoplus \limits _{j=0}^{a-1} \psi ^{\otimes p^j}$ is $\left (\mathcal {O}_{\mathcal {E}'} \otimes \mathcal {L}\right )^{G_{\mathbb {F}_q((t))}}$ . In particular, we see that $\left \{u^{-\epsilon _j}\otimes v_j \right \}$ is a basis of $\left (\mathcal {O}_{\mathcal {E}'} \otimes \mathcal {L}\right )^{G_{\mathbb {F}_q((t))}}$ . The first part of the proposition follows from considering the action of $\nu $ and $\sigma $ on this basis. To deduce the second part of the proposition, observe what happens when C and $C_0$ are skew-conjugated by a diagonal matrix.

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., [27, 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 [15, 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 Normed vector spaces and Newton polygons

For the convenience of the reader, we recall some definitions and facts about Newton polygons and normed p-adic vector spaces. Most of what follows is well known (see, e.g., [25] or [22] for many standard facts on p-adic functional analysis). However, we do find it necessary to introduce some notation and definitions that are not standard. In particular, we introduce the notion of a formal basis, which allows us to compute Fredholm determinants by estimating columns (in contrast to estimating rows, which is the approach taken in [1]).

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.

Definition 6.1. Let I be a set. We let $\mathbf {s}(I)$ denote the set of families $x=(x_i)_{i\in I}$ , with $x_i \in L$ such that $\lvert x\rvert =\sup \limits _{i\in I}\lvert x_i\rvert _p < \infty $ . Then $\mathbf {s}(I)$ is a Banach space with the norm $\lvert \cdot \rvert $ . We let $\mathbf {c}(I) \subset \mathbf {s}(I)$ be the subspace of families with $\lim \limits _{i\in I} x_i = 0$ (note that this agrees with $\mathbf {c}(I)$ defined in [25, Section I]).

Definition 6.2. A formal basis of V is a subset $G=\{e_{i}\}_{i \in I} \subset V$ with a norm-preserving embedding $V \to \mathbf {s}(I)$ , where $e_i$ gets mapped to the element in $\mathbf {s}(I)$ with $1$ in the i-coordinate and $0$ otherwise. ² We regard V a subspace of $\mathbf {s}(I)$ .

Definition 6.3. An orthonormal basis of V is a formal basis $G=\{e_{i}\}_{i \in I} \subset V$ such that $V \subset \mathbf {c}(I)$ . This inclusion is an equality if V is a Banach space. By [25, Proposition I], every Banach space over L has an orthonormal basis. Thus, every Banach space is of the form $\mathbf {c}(I)$ .

Example 6.4. Let V be the Banach space $\mathcal {O}_L\left[\!\left[ t\right]\!\right] \otimes \mathbb {Q}_p$ . Then $\{t^n\}_{n \in \mathbb {Z}_{\geq 0}}$ is a formal basis of V and there is an isomorphism $V \cong \mathbf {s}(\mathbb {Z}_{\geq 0})$ . By [25, Lemme I], any orthonormal basis of V reduces to an $\mathbb {F}_q$ -basis of $\overline {V}=\mathbb {F}_q\left[\!\left[ t\right]\!\right] $ and thus must be uncountable. The Tate algebra $L\left \langle t \right \rangle \subset V$ is a Banach space, which we may identify with $\mathbf {c}(\mathbb {Z}_{\geq 0})$ .

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

6.2.1 Completely continuous operators

Let V be a vector space over L with norm $\lvert \cdot \rvert $ . Let $G=\{e_{i}\}_{i \in I}$ be a formal basis of V. We assume I is a countable set. Let $u:V \to V$ (resp., $v: V \to V$ ) be an L-linear (resp., E-linear) operator. Let $\left (n_{i,j}\right )$ be the matrix of u with respect to the basis G.

Definition 6.5. For $i \in I$ , we define $\textbf {row}_i(u,G)=\inf \limits _{j \in I} v_p\left (n_{i,j}\right )$ and $\textbf {col}_i(u,G)=\inf \limits _{j \in I} v_p\left (n_{j,i}\right )$ . That is, $\mathbf {row}_i(u,G)$ (resp., $\mathbf {col}_i(u,G)$ ) is the smallest p-adic valuation that occurs in the ith row (resp., column) of the matrix of u. Note that $\textbf {col}_i(u,G)=\log _p\lvert u(e_i)\rvert $ .

Definition 6.6. Assume that $V=\mathbf {c}(I)$ . We say that u is completely continuous if it is the p-adic limit of L-linear operators with finite-dimensional image. This is equivalent to $\lim \limits _{i \in I} \mathbf {row}_i(u,G) = \infty $ [21, Theorem 6.2]. We make the analogous definition for v.

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 [25, 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

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

6.2.4 Computing Newton polygons using ath roots

When estimating the Newton polygon of an L-linear completely continuous operator u on V, it is convenient to work with an E-linear operator v that is an ath root of u. The reason is that we can translate p-adic bounds on $\det \limits _E(1-sv\mid V)$ to q-adic bounds on $\det (1-su\mid V)$ .

Lemma 6.13. Let V be a Banach space. Let v be a completely continuous E-linear operator on V and let $u=v^a$ . Assume that u is L-linear. We further assume that $\det (1-su\mid V)$ has coefficients in E (a priori, its coefficients could lie in L). Let $\frac {1}{a}NP_p(v\mid V)$ denote the polygon where both the x-coordinates and y-coordinates of the points in $NP_p(v\mid V)$ are scaled by a factor of $\frac {1}{a}$ . Then $NP_q(u\mid V)=\frac {1}{a}NP_p(v\mid V)$ .

Proof. Some version of this lemma is present in most papers proving ‘Hodge bounds’ for exponential sums (see, e.g., [4] or [1]). The proof of [15, Lemma 6.25] is easily adapted to our situation.

7 Finishing the proof of Theorem 1.1

7.1 The Monsky trace formula

Let us recall the Monsky trace formula in the case of curves. For a complete treatment, see [22] or [29, Section 10]. Let $\Omega _{\mathcal {B}^{\dagger }}^i$ denote the space of i-forms of $\mathcal {B}^{\dagger }$ [22, 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 [28, 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 [15, 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

We now finish the proof of Theorem 1.1. From equation (30) and Proposition 7.1, we know

$$ \begin{align*} NP_q(L(\rho,V,s))_{<1} & \succeq \{\underbrace{0,\dotsc,0}_{g-1+r_0+r_1+r_\infty-\Omega_\rho} \} \bigsqcup \left ( \bigsqcup_{i=1}^{\mathbf{m}} S_{\tau_i} \right ). \end{align*} $$

Comparing equation (1) with equation (27) gives

$$ \begin{align*} L(\rho,V,s)&= L(\rho,s)\cdot \prod_{\substack{Q \in W \\ Q \neq \tau_i}} \left(1-\rho\left(Frob_Q\right)s\right). \end{align*} $$

This product has $r_0+r_1+r_\infty -{\mathbf {m}}$ terms, each accounting for a slope $0$ segment. Thus,

$$ \begin{align*} NP_q(L(\rho,s))_{<1} \succeq \{\underbrace{0,\dotsc,0}_{g-1+{\mathbf{m}}-\Omega_\rho} \} \bigsqcup \left ( \bigsqcup_{i=1}^{\mathbf{m}} S_{\tau_i} \right ). \end{align*} $$

From the Euler–Poincaré formula (see, e.g., [24]) we know that $L(\rho ,s)$ has degree $2(g-1+{\mathbf {m}}) + \sum \left (s_{\tau _i} - 1\right )$ . This accounts for the remaining slope $1$ segments. The proof is complete.

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