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

Published online by Cambridge University Press:  10 January 2022

Joe Kramer-Miller*
Department of Mathematics, Lehigh University, 17 Memorial Drive East, Bethlehem, PA18015, United States; E-mail:


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.

Number Theory
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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 [Reference Weil31], 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 [Reference Mazur20], 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 [Reference Esnault, Sabbah and Yu9]). In this context one may define an irregular Hodge polygon [Reference Deligne, Malgrange and Ramis8, Reference Esnault, Sabbah and Yu9]. 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 $I_Q^s \subset \ker (\rho _Q)$ [Reference Katz13, 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 [Reference Katz13, 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 [Reference Kramer-Miller15].

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 [Reference Adolphson and Sperber2, Reference Adolphson and Sperber3] 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 [Reference Liu and Wei18], introducing ideas from Artin–Schreier–Witt theory. For $\rho $ with arbitrary image there are some results by Liu [Reference Liu17], 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 [Reference Crew7], 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 [Reference Kramer-Miller15]. 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 [Reference Adolphson and Sperber2]. 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 [Reference Kramer-Miller15] with work of Booher and Pries [Reference Booher and Pries5]. 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 [Reference Bouw6]. Even when $X=\mathbb {P}^1$ , the study of Newton polygons for tame cyclic covers is already a complicated topic (e.g., [Reference Li, Mantovan, Pries and Tang16]). 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 [Reference Fresán, Sabbah and Yu10]. 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 [Reference Kramer-Miller15, 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 [Reference Kramer-Miller15, 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 [Reference Grothendieck and Murre11, Theorem 4.3.2], there exists a tame cover $\mathbf {X} \to \mathbf {P}^1_{\mathcal {O}_L}$ whose special fibre is $\eta $ , and by formal GAGA [26, Tag 09ZT] there exists a morphism of smooth curves $\mathbb {X} \to \mathbb {P}^1_{\mathcal {O}_L}$ whose formal completion is $\mathbf {X} \to \mathbf {P}^1_{\mathcal {O}_L}$ .

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

3.3 Local parameters and overconvergent rings

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


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:


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

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

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


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

3.4 Global Frobenius and $U_p$ operators

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

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

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

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

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

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

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

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

4 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 [Reference Kramer-Miller15, 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 [Reference Katz12, Section 4]

There is an equivalence of categories

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

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

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

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

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

5.2 Local Frobenius structures

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

5.2.1 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 [Reference Kramer-Miller15, 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., [Reference Wan30, Section 4.1] for the exponential-sum situation or [Reference Liu and Wei18]). However, to the best of our knowledge, the local version presented here does not appear anywhere.

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

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

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

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

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

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

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

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

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

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

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

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

5.2.3 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., [Reference Tsuzuki27, Section 2]). This gives the following proposition:

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

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

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

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

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

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

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

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

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

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

By definition we have

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

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

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

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

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

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

5.3.3 Comparing local and global Frobenius structures

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

There are three cases we need to consider. The first case is when $*=1$ . In this case $\rho _Q^{wild}$ and $\chi _Q$ are both unramified. This is because $\rho $ is only ramified at the points