Exponential Sums of Witt Towers over Affinoids

TL;DR

This paper develops a Dwork theory for exponential sums over affinoids in Witt towers, enabling the computation of L-function degrees and analysis of Hodge and Newton polygons.

Contribution

It introduces a novel Dwork theory framework for exponential sums over affinoids in Witt towers, advancing understanding of their L-functions and polygon relations.

Findings

Computed the degree of the L-function.

Analyzed the Hodge polygon.

Identified conditions for Hodge and Newton polygons to coincide.

Abstract

In this paper we construct a Dwork theory for general exponential sums over affinoids in Witt towers. Using this, we compute the degree of the $L$-function, its Hodge polygon and examine when the Hodge and Newton polygons coincide.

Equations

$$f(x)=\sum _{i=0}^{m-1}\sum _{j=1}^{\ell }\sum _{k=0}^{d_{ij}}{V}^i(a_{ijk}\frac{1}{(x-P_j{)}^k},0,\cdots ,)\in W_m({\mathbb{F}}_q[\frac{1}{x-P_1},\cdots ,\frac{1}{x-P_1}]),$$

$$S_f(k)$$

$$L_f(k)$$

$$C_f(k)$$

$$d=(\sum _{j=1}^{\ell }(\underset{0\le i\le m-1}{\max }{p}^{m-i-1}{d}_{ij}+1))-2.$$

$${\left\{{\left\{\frac{an}{d_{i_j,j}{p}^{m-i_j-1}}\right\}}_{n=0}^{d_{i_j,j}{p}^{m-i_j-1}}\right\}}_{j=1}^{\ell },$$

$$\omega :W_m({\mathbb{F}}_{q^k})$$

$$(x_0,\cdots ,x_{m-1})$$

$$(x_0,x_1,x_2,\cdots )\mapsto \sum _{i=0}^{\infty }\widehat{x_i^{p^{-i}}}{p}^i,$$

$${\chi }_{q^k}(x)=\chi (\omega ({\mathrm{Tr}}_{W_m({\mathbb{F}}_{q^k})/W_m({\mathbb{F}}_p)}(x))).$$

$$S_f(k)$$

$$(x+py{)}^{p^n}$$

$${\mathrm{ord}}_p((\genfrac{}{}{0ex}{}{p^n}{r})x^{n-r}(py{)}^r)\ge (n-{\mathrm{ord}}_p(r))+r=n+(r-{\mathrm{ord}}_p(r))\ge n.$$

$${\mathrm{Tr}}_{{\mathbb{Q}}_{q^k}/{\mathbb{Q}}_p}(x^{p^b})=\sum _{r=0}^{ak-1}{\tau }^r(x^{p^b})=\sum _{r=0}^{ak-1}(x^{p^b}{)}^{p^r}=\sum _{r=b}^{ak-1+b}{x}^{p^r}=\sum _{r=0}^{ak-1}{x}^{p^r}={\mathrm{Tr}}_{{\mathbb{Q}}_{q^k}/{\mathbb{Q}}_p}(x).$$

$$\widehat{x-P_j}=(\widehat{x-P_j}{)}^{q^{bk}}=((\hat{x}-\widehat{P_j})+pg(\hat{x},{\hat{P}}_j){)}^{q^{bk}}=(\hat{x}-{\hat{P}}_j{)}^{q^{bk}}+q^{bk}h(\hat{x},{\hat{P}}_j).$$

$$\widehat{x-P_j}\equiv (\hat{x}-{\hat{P}}_j{)}^{q^{bk}}\mathrm{mod}{p}^m,$$

$${\zeta }_{p^m}^{{\mathrm{Tr}}_{{\mathbb{Q}}_{q^k}/{\mathbb{Q}}_p}({\hat{a}}^{p^{-i}}{(\widehat{x-P_j})}^{-p^{-i}})}$$

$$={\zeta }_{p^m}^{{\mathrm{Tr}}_{{\mathbb{Q}}_{q^k}/{\mathbb{Q}}_p}({\hat{a}}^{p^{-i}}(\hat{x}-{\hat{P}}_j{)}^{-p^{akb-i}})}\cdot {\zeta }_{p^m}^{p^{m}h}$$

$$={\zeta }_{p^m}^{{\mathrm{Tr}}_{{\mathbb{Q}}_{q^k}/{\mathbb{Q}}_p}(\hat{a}(\hat{x}-{\hat{P}}_j{)}^{-1})}.$$

$$\hat{f}=\sum _{i=0}^{m-1}\sum _{j=1}^{\ell }\sum _{k=0}^{d_{ij}}{p}^i{\hat{a}}_{ijk}\frac{1}{(x-{\hat{P}}_j{)}^k}\in {\mathbb{Z}}_q[\frac{1}{x-{\hat{P}}_1},\cdots ,\frac{1}{x-{\hat{P}}_{\ell }}],$$

$$S_f(k)=\sum _{\begin{array}{c}x\in \widehat{{\mathbb{F}}_{q^k}^{\times }},\\ x\ne {\hat{P}}_1,\cdots ,{\hat{P}}_{\ell }\end{array}}{\zeta }_{p^m}^{{\mathrm{Tr}}_{{\mathbb{Q}}_{q^k}/{\mathbb{Q}}_p}(\hat{f}(x))}.$$

$${\mathrm{Tr}}_{W_m({\mathbb{F}}_{q^k})/W_m({\mathbb{F}}_p)}(f(x_0))$$

$${\mathbb{A}}_r=\{x\in {\mathbb{C}}_p:|x{|}_p\le 1/r,|x-\widehat{P_j}{|}_p\ge r\text{ for }2\le j\le \ell \}.$$

$${\mathbb{A}}_1=\{x\in {\mathbb{C}}_p:|x{|}_p=1\text{ and }|x-{\hat{P}}_j{|}_p=1\text{ for }2\le j\le \ell \}=\{x\in \widehat{{\mathbb{F}}_p^{alg}}:\bar{x}\ne P_j\}.$$

$${\mathcal{H}}^{†}=\mathbb{A}⟨\frac{1}{x-{\hat{P}}_1},\cdots ,\frac{1}{x-{\hat{P}}_{\ell }}⟩.$$

$${\mathcal{H}}^{†}\cong \underset{j=1}{\overset{\ell }{⨁}}{\mathcal{H}}_j^{†},$$

$$D$$

$$D_j$$

$$B=\{1,X_1,\cdots ,X_1^{R_1-1},X_2,\cdots ,X_2^{R_2}\}\cup \{X_j^i|j\ge 3,1\le i\le R_j+1\}$$

$$R_j^{\prime }$$

$$r_{j,u}=\frac{1}{R_j{h}_{R_j,j}(-\widehat{P_j}{)}^{\delta (j)}}{X}_j^{u-R_j^{\prime }}\in {\mathcal{H}}_j^{†},$$

$$D_j{r}_{j,u}=(E{H}_j+E)r_{j,u}\equiv X_j^u\mathrm{mod}R.$$

$$E(X_j^i)=\{\begin{array}{ll}i{X}_j^i& \text{ if }j=1,2\\ -i{X}_j^i-i\widehat{P_j}{X}_j^{i+1}& \text{ if }j\ge 3.\end{array}$$

Full text

1. 1 Lifting $f(x)$

2. 2 $p$-adic Banach Spaces

3. 2.1 Quotient Spaces of $\mathcal{H}^{\dagger}$

4. 3 The Degree of the $L$-function

5. 3.1 The truncated Artin-Hasse Exponential

6. 3.2 Dwork Theory via the Truncated Artin-Hasse

7. 4 The Hodge Polygon

8. 4.1 The Artin-Hasse Exponential

9. 4.2 $p$-adic Estimates

10. 4.3 Computing the Hodge

Let $p$ be a prime, $q=p^{a}$ a $p$-power, and let $m\geq 1$ be an integer such that $p>m$. Write $\operatorname{Gal}(\mathbb{Q}_{q}/\mathbb{Q}_{p})=\langle\tau\rangle$. Suppose we have:

[TABLE]

where $P_{1},\cdots,P_{\ell}$ are distinct elements in $\mathbb{F}_{q}\cup\{\infty\}$, $W_{m}(R)$ is the truncated ring of Witt vectors of some ring $R$, and $V$ is the Witt vector shifting operator. Without loss of generality, take $p\nmid d_{ij}$ for all $i,j$ and suppose that for each $1\leq j\leq\ell$, the maximum $\max_{i}d_{ij}p^{m-i-1}$ is uniquely achieved. We will assume that $P_{1}=0$ and $P_{2}=\infty$. If $\zeta_{p^{m}}$ is a primitive $p^{m}$th root of unity, the exponential sum, $L$-function and characteristic function $C_{f}(s)$ attached to $f(x)$ are:

[TABLE]

In this paper we build on the methods developed in [5], where we constructed a “universal” Dwork theory that applies to exponential sums over affinoids. Using the framework from there, we build an alternative dwork cohomology using a truncated Artin-Hasse exponential which allows us to directly compute the degree of the $L$-function. To our knowledge, the general ($m>1$) truncated Artin-Hasse exponential has not been used in this setting before. Furthermore, we compute the Hodge polygon via the traditional Artin-Hasse exponential and use these $p$-adic estimates to generalize Zhu’s result from [6] about when the Newton and Hodge bounds coincide.

Our main results are the following two theorems.

[TABLE]

This result is not new (see Remark 4.7 in [4]), but prior results use the geometry of the Witt tower whereas we utilize $p$-adic methods. Our second result is the computation of the Hodge polygon of $L_{f}(s)$, which is described as follows.

[TABLE]

where $i_{j}$ is such that $p^{m-i_{j}-1}d_{i_{j},j}=\max_{0\leq i\leq m-1}p^{m-i-1}d_{ij}$. Furthermore, if $i_{j}=m-1$ for all $1\leq j\leq\ell$, these polygons coincide if and only if $p\equiv 1\bmod\operatorname{lcm}_{j}d_{i_{j},j}$.

1. Lifting $f(x)$

We will first lift $f(x)$ to a $p$-adic ring in which we can construct a $p$-adic Dwork cohomology. Recall, that if $a\in W(R)$, we denote by $\bar{a}$ the image of $a$ in $R$, that is $a\bmod p$, and for $a\in R$, we define the Teichmüller lift of $a$ in $W(R)$ to be $\hat{a}$.

Lemma 1.1**.**

There is a ring isomorphism:

[TABLE]

Proof.

It’s well known that $\mathbb{Z}_{q^{k}}\cong W(\mathbb{F}_{q^{k}})$ via the isomorphism

[TABLE]

and so the lemma follows from the isomorphism $W_{m}(K)\cong W(K)/V^{m}W(K)$. ∎

Define the additive character $\chi:W_{m}(\mathbb{F}_{p})\to\mathbb{C}_{p}$ by mapping $\chi(x)=\zeta_{p^{m}}^{\omega(x)}$ and extend it to $W_{m}(\mathbb{F}_{q^{k}})$ by composing it with the trace:

[TABLE]

The exponential sum can then be written:

[TABLE]

Using Lemma 1.1 we can lift the exponential sum $S_{f}(k)$ from an object defined over a finite field as in (4), to one defined over a $p$-adic ring. However, before we can choose an appropriate lifting of $f(x)$ some technical lemmas are required.

Lemma 1.2**.**

Let $x,y\in\mathbb{C}_{p}$ such that $\operatorname{ord}_{p}{(}x)\geq 0$ and $\operatorname{ord}_{p}{(}y)\geq 0$. Then for $n\geq 1$, $(x+py)^{p^{n}}=x^{p^{n}}+p^{n}g(x,y)$.

Proof.

Clearly:

[TABLE]

When $r>0$, $\operatorname{ord}_{p}{(}\binom{p^{n}}{r})=n-\operatorname{ord}_{p}{(}r)$ and hence

[TABLE]

Thus $\operatorname{ord}_{p}{(}\sum_{r=1}^{p^{n}}\binom{p^{n}}{r}x^{n-r}(py)^{r})\geq n$ and the lemma follows. ∎

Lemma 1.3**.**

Suppose $x\in\mathbb{C}_{p}$ is such that $\tau(x)=x^{p}$ and $x^{q^{k}}=x$, some $k\geq 1$. Then for any $b\in\mathbb{Z}$, $\operatorname{Tr}_{\mathbb{Q}_{q^{k}}/\mathbb{Q}_{p}}(x^{p^{b}})=\operatorname{Tr}_{\mathbb{Q}_{q^{k}}/\mathbb{Q}_{p}}(x)$.

Proof.

This is just a simple calculation:

[TABLE]

∎

We now make two observations. First, because $\widehat{x-P_{j}}$ is a Teichmüller lift, $\widehat{x-P_{j}}\equiv\widehat{x}-\widehat{P_{j}}\bmod p$ and so $\widehat{x-P_{j}}=(\widehat{x}-\widehat{P_{j}})+pg(\widehat{x},\widehat{P}_{j})$, some $g(X,Y)\in\mathbb{Z}[X,Y]$. Second, because $\widehat{x-P_{j}}$ is again a Teichmüller lift from $\mathbb{F}_{q^{k}}$, $\widehat{x-P_{j}}=(\widehat{x-P_{j}})^{q^{k}}=(\widehat{x-P_{j}})^{q^{bk}}$, any $b\geq 1$. Thus, for any $b\geq 1$, applying Lemma 1.2 yields:

[TABLE]

If we take $b$ sufficiently large so that $q^{b}\geq p^{m-1}$, then

[TABLE]

and so by Lemma 1.3, if $a\in\mathbb{F}_{q}$ then

[TABLE]

The above discussion induces a $p$-adic lifting of $f$ that preserves the exponential sum:

Lemma 1.4**.**

If

[TABLE]

then:

[TABLE]

Proof.

If $x\in{\mathbb{F}_{q^{k}}^{\times}}$, clearly $\omega(f(x))=\hat{f}(\hat{x})$ and hence:

[TABLE]

and the lemma follows. ∎

2. $p$-adic Banach Spaces

Define a $p$-adic affinoid ring as follows.

Definition 2.1**.**

For $0<r\leq 1$, let

[TABLE]

It’s easy to see that

[TABLE]

For convenience, we will often write $\mathbb{A}_{1}=\mathbb{A}$.

Definition 2.2**.**

Consider the affinoid space:

[TABLE]

The ring $\mathcal{H}^{\dagger}$ is the set of overconvergent anaytic elements on $\mathbb{A}$. (That is, any $h(x)\in\mathbb{A}[[\frac{1}{x-\widehat{P}_{1}},\cdots,\frac{1}{x-\widehat{P}_{\ell}}]]$ lies in $\mathcal{H}^{\dagger}$ if and only if it can be evaluated at any $x\in\mathbb{A}$.)

In this section we’ll study some fundamental properties of $\mathcal{H}^{\dagger}$ and look at some special quotient spaces. We start with a basic result, the Mittag-Leffler decomposition

Proposition 2.3**.**

If $\mathcal{H}^{\dagger}_{j}=\mathbb{A}\langle\frac{1}{x-\widehat{P}_{j}}\rangle$, there exists an isomorphism of $\mathbb{A}$-Banach modules

[TABLE]

such that every $g$ in $\mathcal{H}^{\dagger}$ can be written $g=\sum_{j=1}^{\infty}(g)_{j}$, with $(g)_{j}\in\mathcal{H}^{\dagger}_{j}$.

Proof.

See Lemma 2.1 in [6]. ∎

2.1. Quotient Spaces of $\mathcal{H}^{\dagger}$

This section (along with the following Dwork cohomology) is based on the methods used by Lauder and Wan in their papers [1] and [2].

For convenience in this subsection, we will write $X_{j}=\frac{1}{x-\widehat{P}_{j}}$. Fix an arbitrary polynomial $H=\sum_{j=1}^{\ell}\sum_{i=0}^{R_{j}}h_{ij}X_{j}^{i}$ in $\mathbb{A}[X_{1},\cdots,X_{\ell}]$, with $R_{j}\geq 0$ and $h_{R_{j},j}\neq 0$ and say $H_{j}=\sum_{i=0}^{R_{j}}h_{ij}X_{j}^{i}$. Define two operators on $\mathcal{H}^{\dagger}$:

[TABLE]

where $E=x\frac{d}{dx}$.

We wish to understand the quotient space $\mathcal{H}^{\dagger}/D(\mathcal{H}^{\dagger})$. Namely, we are interested in its dimension as a module over $\mathbb{A}$. Our goal will be to show that $\mathcal{H}^{\dagger}/D(\mathcal{H}^{\dagger})$ is isomorphic to the following finite free $\mathbb{A}$-module $R$:

Definition 2.4**.**

Consider the set

[TABLE]

and define the $\mathbb{A}$-module $R=\operatorname{span}_{\mathbb{A}}B\subset\mathcal{H}^{\dagger}$.

For the next lemma we will need some notational sugar:

[TABLE]

Lemma 2.5**.**

Fix $1\leq j\leq\ell$ and take $u\geq R_{j}^{\prime}$. Let

[TABLE]

where $\delta(j)=1$ if $j\geq 3$ and $\delta(j)=0$ if $j=1,2$. (Take $(\infty)^{0}=0^{0}=1$.) There is then a congruence

[TABLE]

Proof.

Observe that the action of $E$ on the terms $X_{j}^{i}$ is nothing but:

[TABLE]

So if $j=1$:

[TABLE]

All the terms except $X_{1}^{u}$ have degree less than or equal to $u-1$. When $u=R_{1}^{\prime}=R_{1}$, then $(EH_{1}+E)r_{1,R_{1}}\equiv X_{1}^{R_{1}}\bmod R$, and so if $u>R_{1}$, by induction $(EH_{1}+E)r_{1,u}\equiv X_{1}^{u}\bmod R$. The proof is similar for $j=2$.

For $j\geq 3$, first compute that

[TABLE]

some $h_{ij}^{\prime}$. Hence:

[TABLE]

Because

[TABLE]

the terms in (2.1) except for $X_{j}^{u}$ have degree strictly less than $u$, and so by the same induction argument with base case $u=R_{j}+1$, the claim follows. ∎

For any $g(x)\in\mathcal{H}_{j}^{\dagger}/D(\mathcal{H}_{j}^{\dagger})$, Lemma 2.5 implies that we can write uniquely

[TABLE]

and so it now remains to decompose $\mathcal{H}^{\dagger}/D(\mathcal{H}^{\dagger})$ in terms of $\mathcal{H}_{j}^{\dagger}/D(\mathcal{H}_{j}^{\dagger})$:

Theorem 2.6**.**

$\dim_{A}\mathcal{H}^{\dagger}/D(\mathcal{H}^{\dagger})=\sum_{j=1}^{\ell}\dim_{A}\mathcal{H}_{j}^{\dagger}/D(\mathcal{H}_{j}^{\dagger})$.

Proof.

For a fixed $j_{0}$ and $i_{0}$, we have by partial fraction expansion

[TABLE]

Hence for appropriate $j$ and $u$, by induction on $u$ (with base case $u=R_{j}^{\prime}$), it follows that

[TABLE]

and so the Theorem follows because every $g(x)\in\mathcal{H}^{\dagger}/D(\mathcal{H}^{\dagger})$ can be written uniquely as

[TABLE]

∎

3. The Degree of the $L$-function

We will now apply the prior section’s results about the space $\mathcal{H}^{\dagger}$ to construct our Dwork theory.

3.1. The truncated Artin-Hasse Exponential

Definition 3.1**.**

Let $1\leq k\leq m$. Define the $k$-truncated Artin-Hasse exponential:

[TABLE]

and take $\pi_{k}\in\mathbb{C}_{p}$ to be a solution to $\sum_{i=0}^{k}\frac{x^{p^{i}}}{p^{i}}=0$ with $\operatorname{ord}_{p}{\pi}_{k}=\frac{1}{p^{k-1}(p-1)}$.

By Theorem 4.1 in [3], the disk of convergence of $E_{k}(x)$ is

[TABLE]

Furthermore, for any $j\leq k$, by Theorem 4.9 also in [3], because $p>m\geq k$, it is guaranteed that $\pi_{k}$ lies inside the disc of convergence of $E_{k}$ and it’s therefore well defined to consider the splitting functions

[TABLE]

Lemma 3.2**.**

If $\theta_{k}(x)=\sum_{i=0}^{\infty}\theta_{ki}x^{i}$, then $\operatorname{ord}_{p}{\theta}_{ki}\geq i\cdot\frac{p-k}{p^{k+1}}$.

Proof.

By Remark 4.5 in [3], $\operatorname{ord}_{p}{u}_{ki}\geq-\frac{i}{p^{k+1}}(\frac{1}{p-1}+k+1)$, and so

[TABLE]

∎

Lemma 3.3**.**

For $k\geq 1$ and $0\leq j\leq k-1$ let $\gamma_{kj}=\sum_{i=0}^{j}\frac{\pi_{k}^{p^{i}}}{p^{i}}$. Then:

[TABLE]

and $\operatorname{ord}\gamma_{kj}=\frac{1}{p^{k-1-j}(p-1)}-j$.

Proof.

A simple computation shows that:

[TABLE]

and so the identity follows by the defining property of $\pi_{k}$. To compute the order of $\gamma_{kj}$, observe that

[TABLE]

which has a unique minimum at $i=j$. ∎

Lemma 3.4**.**

Let $k,v\geq 1$, $t\in\mathbb{C}_{p}$ and suppose $x\in\mathbb{C}_{p}$ such that $x^{p^{v}}=x$. Then

[TABLE]

Proof.

The lemma follows from the following computation:

[TABLE]

∎

3.2. Dwork Theory via the Truncated Artin-Hasse

Definition 3.5**.**

Define:

[TABLE]

The reason for these definitions is clear. For $x\in\widehat{\mathbb{F}_{q^{w}}}$, $w\geq 1$, applying the splitting function $\theta$ to $\hat{f}$ yields:

[TABLE]

Remark*.*

Note that when $x$ is a Teichmüller lift, $(x-\widehat{P}_{j})^{p}\equiv x^{p}-\widehat{P}_{j}^{p}\bmod p$ and using the same argument prior to Lemma 1.4 we see that the last equality above really does hold.

Similar to the above, define the analogous functions:

[TABLE]

so that

[TABLE]

Remark*.*

For each $j$ and $i$, the coefficient of $\frac{1}{(x-\widehat{P}_{j})^{p^{m-i-1}d_{ij}}}$ is equal to $\gamma_{m-i,m-i-1}a_{ij,d_{ij}}$, and since $a_{ij,d_{ij}}\neq 0$ by assumption and $\gamma_{m-i,m-i-1}=\sum_{i=0}^{m-1-1}\frac{\pi_{m-i}^{p^{i}}}{p^{i}}=-\frac{\pi_{m-i}^{m-i}}{p^{m-i}}\neq 0$, the entire coefficient is nonzero. Thus $\deg_{j}H=\max_{i}p^{m-i-1}d_{ij}$.

From these functions define the corresponding Dwork maps $\alpha_{1}=U\circ E_{f}$ and $\alpha_{a}=U^{a}\circ E_{f}^{(a)}$.

Proposition 3.6**.**

The maps $\alpha_{1}$ and $\alpha_{a}$ are $p$-adically completely continuous.

Proof.

This proof is analogous to the corresponding proof in Corollary 6.10 in [5]. Fix $j$, let $i_{j}$ be such that $d_{i_{j},j}=\max_{i}d_{ij}$ and define

[TABLE]

In this case, $\operatorname{ord}_{p}{(}F_{nj})\geq\frac{n}{d_{i_{j},j}}\cdot\frac{p-(m-i_{j})}{p^{m-i_{j}+1}}$ by Lemma 3.2. (Note that we are using here that $\theta_{ki}=u_{ki}\pi_{k}^{i}\neq 0$ because $u_{kn}=\frac{\operatorname{Hom}(C_{p^{k}},S_{n})}{n!}$ and hence $u_{kn}\neq 0$ ) The main observation to make is that all of the proofs following and including Lemma 6.5 in [5] can be generalized to the assumption that $\operatorname{ord}_{p}{(}F_{nj})\geq{\frac{n}{d_{i_{j},j}}}\cdot M_{j}$, where $M_{j}$ is any real number with $0<M_{j}<1$. Here, we see that $M_{j}=\frac{p-(m-i_{j})}{p^{m-i_{j}+1}}$ and the rest of the theory follows accordingly.

∎

Recall from subsection 2.1 that we defined the two operators $E=x\frac{d}{dx}$ and $D=E+E\circ\log\hat{E}_{f}$.

Lemma 3.7**.**

We have the following relations:

1. (1)

$U^{a}\circ E=E\circ(qU^{a})$. 2. (2)

$D=\exp(-H)\circ E\circ\exp(H)$. 3. (3)

$\alpha_{a}=\exp(-H)\circ U^{a}\circ\exp(H)$. 4. (4)

$D\circ(q\alpha_{a})=\alpha_{a}\circ D$.

Proof.

Let $g\in\mathcal{H}^{\dagger}$.

1. (1)

Note that if $z^{q}=x$, then $\frac{dz}{dx}=\frac{1}{q}x^{\frac{1}{q}-1}$ and so $x\frac{dz}{dx}=\frac{1}{q}x^{\frac{1}{q}}=\frac{1}{q}z$. Hence,

[TABLE] 2. (2)

The identity is just the simple calculation:

[TABLE] 3. (3)

The equality is equivalent to the statement $\alpha_{a}=U^{a}(\hat{E}_{f})/\hat{E}_{f}$. By basic properties of $U$ and observing that $\hat{E}_{f}=\prod_{j=0}^{\infty}E_{f}^{\tau^{j}}(x^{p^{j}})$,

[TABLE] 4. (4)

This identity follows from the first three.

∎

Lemma 3.7 yields the commutative diagram:

[TABLE]

where $H_{0}=\mathcal{H}^{\dagger}/D(\mathcal{H}^{\dagger})$.

First, we prove that $D$ is injective.

Lemma 3.8**.**

$D$* is injective.*

Proof.

Suppose that $g\in\ker D$ so that

[TABLE]

which implies that $g\frac{dH}{dx}+\frac{dg}{dx}=0$, or

[TABLE]

some scalar $c$. It remains to show that $\exp(-H)\not\in\mathcal{H}$.

Recall that $\exp(x)$ converges on the disk $\{x\in\mathbb{C}_{p}|\operatorname{ord}_{p}{x}>1/(p-1)\}$. So let $H(x)=\sum_{v,i,j,k}h_{vijk}\frac{1}{(x-\widehat{P_{j}})^{p^{v}k}}$ and

[TABLE]

Hence

[TABLE]

taking $i=0$ and $v=m-i-1$. But this implies that $|H|_{gauss}=p^{-(\frac{1}{p-1}-(m-1))}=p^{m-1}p^{-1/(p-1)}\geq p^{-1/(p-1)}$.

By a well known property of the gauss norm, we can therefore find $x_{0}\in\mathbb{A}_{1}$ such that $|H(x_{0})|_{p}=p^{m-1}p^{-1/(p-1)}$, and so at this $x_{0}$, $\exp(H(x_{0}))$ is not defined, and $\exp(-H)\not\in\mathcal{H}^{\dagger}$. ∎

Proposition 3.9**.**

The degree of the $L$-function is the degree of the first homology space, $\deg L_{f}(s)=\dim_{\mathbb{Z}_{q}[[\pi_{m}]]}H_{0}$.

Proof.

Just as in Theorem 6.13 in [5], because $\alpha_{a}$ is completely continuous by Proposition 3.6, we see that $C_{f}(s)=\det(1-\alpha_{a}s)$. A simple computation then shows that

[TABLE]

Since $\mathcal{H}^{\dagger}=D(\mathcal{H}^{\dagger})\oplus\mathcal{H}^{\dagger}/D(\mathcal{H}^{\dagger})$,

[TABLE]

But by the relation $\alpha_{a}\circ D=D\circ(q\alpha_{a})$ from Lemma 3.7, $D^{-1}\circ\alpha_{a}\circ D=q\alpha_{a}$ (since $D$ is bijective on $D(\mathcal{H}^{\dagger})$). Thus:

[TABLE]

And so (7) becomes:

[TABLE]

that is $\operatorname{Tr}(\alpha_{a}|\mathcal{H}^{\dagger})-\operatorname{Tr}(q\alpha_{a}|\mathcal{H}^{\dagger})=\operatorname{Tr}(\bar{\alpha}_{a}|H_{0}),$ and the same formula holds when $\alpha_{a}$ is replaced with $\alpha_{a}^{k}$ since $D\circ(q^{k}\alpha_{a}^{k})=\alpha_{a}^{k}\circ D$. Thus by the well known formula

[TABLE]

we have the identity

[TABLE]

and the claim follows. ∎

Theorem 3.10**.**

$\dim_{\mathbb{Z}_{q}[[\pi_{m}]]}H_{0}=(\sum_{j=1}^{\ell}(\max_{0\leq i\leq m-1}p^{m-i-1}d_{ij}+1))-2$.

Proof.

We apply the results of subsection 2.1 to the polynomial $H$ in definition 3.5. By Lemma 2.5, it’s clear that:

[TABLE]

where $R_{j}=\max_{0\leq i\leq m-1}p^{m-i-1}d_{ij}$. With this computation, the degree of $L_{f}$ follows from Theorem 2.6. ∎

4. The Hodge Polygon

In this second section we utilize a more classical style Dwork theory from the Artin-Hasse exponential and compute the Hodge polygon of the $L$-function and generalize the main result of [6]. The theory in this section is analogous to the work from [5], but we provide a detailed sketch with a focus on $p$-adic estimates that we will use to compute the Newton polygon.

4.1. The Artin-Hasse Exponential

Definition 4.1**.**

Let $1\leq k\leq m$. Define the Artin-Hasse exponential:

[TABLE]

and take $\pi_{\infty,k}\in\mathbb{C}_{p}$ to be a solution to $\sum_{i=0}^{\infty}\frac{x^{p^{i}}}{p^{i}}=0$ with $\operatorname{ord}_{p}{\pi}_{\infty,k}=\frac{1}{p^{k-1}(p-1)}$.

Define the splitting function:

[TABLE]

When there is no confusion, we will write $u_{i}=u_{\infty,i}$ and $\pi_{\infty,i}=\pi_{i}$.

4.2. $p$-adic Estimates

Definition 4.2**.**

Let $i\geq 0$, $1\leq j\leq\ell$:

[TABLE]

Lemma 4.3**.**

For $i,n\geq 0$ and $1\leq j\leq\ell$,

[TABLE]

with equality if and only if $d_{ij}|n$ and $u_{\frac{n}{d_{ij}}}\in\mathbb{Z}_{p}^{\times}$ (which is satisfied if $\frac{n}{d_{ij}}<p$).

Proof.

We compute that:

[TABLE]

Hence,

[TABLE]

and so the lemma is clear. ∎

Lemma 4.4**.**

For $i,n\geq 0$ and $1\leq j\leq\ell$,

[TABLE]

with equality if and only if $d_{i_{j}j}|n$ and $u_{\frac{n}{d_{i_{j}j}}}\in\mathbb{Z}_{p}^{\times}$ (which is satisfied if $\frac{n}{d_{i_{j}j}}<p$).

Proof.

Observe that

[TABLE]

Then by Lemma 4.3,

[TABLE]

and this minimum is achieved when $n_{i_{j}}=n$ and $n_{k}=0$ otherwise. Furthermore, this minimum is unique, under our assumption that $i_{j}$ is the unique maximum in $d_{i_{j}j}p^{m-i_{j}-1}=\max_{i}d_{ij}p^{m-i-1}$ and obtained if and only if $d_{i_{j}j}|n$ and $u_{\frac{n}{d_{i_{j}j}}}\in\mathbb{Z}_{p}^{\times}$. ∎

Lemma 4.5**.**

For $1\leq j,k\leq\ell$ and $0\leq i,n$:

[TABLE]

and equality holds if and only if $j=k$, $d_{i_{k},k}|(n-i)$ and $u_{\frac{n-i}{d_{i_{k},k}}}\in\mathbb{Z}_{p}^{\times}$.

Proof.

We follow Lemma 4.4.7 in [5]. First note that:

[TABLE]

where $(n_{1},\cdots,n_{\ell})\in\mathbb{Z}_{\geq 0}^{\ell}$ such that $n_{k}-\sum_{\begin{subarray}{c}v=1\\ v\neq k\end{subarray}}n_{v}=n-i$. ∎

Proposition 4.6**.**

For $1\leq j,k\leq\ell$ and $0\leq i,n$:

[TABLE]

with equality if and only if $j=k$, $d_{i_{k},k}|((n-1)p-i+1)$ and $u_{\frac{(n-1)p-i+1}{d_{i_{k},k}}}\in\mathbb{Z}_{p}^{\times}$.

Proof.

Following Proposition 4.4.9 in [5] and using Lemma 4.5,

[TABLE]

where we write $D_{k}=d_{i_{k}k}p^{m-i_{k}-1}$ to ease notation. The minimum is then uniquely achieved when $v=(n-1)p+1$ since $\frac{1}{D_{k}(p-1)}-\frac{1}{p-1}<0$. ∎

Corollary 4.7**.**

For $1\leq j,k\leq\ell$ and $0\leq i,n$:

[TABLE]

with equality if and only if $j=k$, $d_{i_{k},k}|((n-1)(p-1))$ and $u_{\frac{(n-1)(p-1)}{d_{i_{k},k}}}\in\mathbb{Z}_{p}^{\times}$.

4.3. Computing the Hodge

Let $M$ represent the matrix for $\alpha_{1}$ with respect to the weighted basis $\{\pi_{m-i_{j}}^{i/d_{i_{j}j}}\frac{1}{(x-\widehat{P_{j}})^{i}}\}_{ij}$, with the entries of $M$ lying in $\mathcal{O}_{a}$. Write:

[TABLE]

so that

[TABLE]

Following the proof of Theorem 7.2 in [5], line (9) along with Corollary 4.7 yields the Hodge bound with slopes:

[TABLE]

Moreover, noting when equality holds in Corollary 4.7 implies that if this Hodge bound is obtained, then $d_{i_{k}}|(p-1)$ for all $1\leq k\leq\ell$. Oppositely, if $d_{i_{k}}|(p-1)$ for all $1\leq k\leq\ell$, then the Hodge bound is necessarily achieved under the assumption that $d_{i_{j}j}p^{m-i_{j}-1}=d_{i_{j}j}$, because $\frac{n(p-1)}{d_{i_{j}j}}\leq p-1$ and $u_{\frac{n(p-1)}{d_{i_{j}j}}}\in\mathbb{Z}_{p}^{\times}$.

Remark* (The “Truncated” Hodge Polygon).*

If we use the Dwork theory developed in Section 3 utilizing the truncated Artin-Hasse exponential, we can derive lower bounds for the Newton polygon of $L_{f}(s)$, just like we would to compute the Hodge polygon in the traditional case. However, we see that the the resulting lower bound is actually lower than the Hodge bound.

Recall from Proposition 3.6 that

[TABLE]

Applying the same procedure as above yields a lower polygon consisting of slopes

[TABLE]

Interestingly, 10 implies that as $p\to\infty$, this “truncated” Hodge polygon converges upward towards the regular Hodge bound. (This is expected since the coefficients of the truncated Artin-Hasse functions converge upon the coefficients of the classical Artin-Hasse as $p\to\infty$, i.e. $u_{ki}\to u_{i}$ as $p\to\infty$.)