2. Overconvergent Dwork cohomology

After establishing the rationality of the zeta function, Dwork proceeded to pioneer the study of p-adic absolute values of the reciprocal roots and zeros of zeta functions of an algebraic variety Z over $\mathbf{F}_{q}$ . He proved the famous ‘Newton above Hodge’ theorem when Z is a nonsingular hypersurfaces in a projective space [Reference DworkDwo62]. He accomplished this by devising some explicit chain complexes of p-adic Banach spaces, which are equipped with some chain-level representations of the Frobenius operation. A substantial portion of his work was centered around the chain-level investigations. He only delved into the cohomology spaces under special occasions where finite dimensionality can be shown. But even without knowing finiteness of the cohomology spaces, the chain-level approach can still extract numerous properties of the zeta function.

Actually, Dwork did not design just one chain complex, but rather infinitely many different chain complexes, each corresponding to a specific ‘splitting function’ in his terminology. The chain-level Frobenius operators associated with these chain models are all capable of calculating the zeta function of a variety, regardless of whether the variety is nonsingular or not. In some applications, such as the ‘visibility theorem’ 1.2, knowledge of some formal properties of these operators suffices. For more intricate analysis, such as many important results of Adolphson and Sperber [Reference Adolphson and SperberAS87a, Reference Adolphson and SperberAS87b, Reference Adolphson and SperberAS89], and the ‘divisibility theorem’ that relies on them, some specific chain model needed to be used.

Since our first goal is to prove the visibility theorem, in this section we focus exclusively on a most straightforward (overconvergent) Dwork complex. In effect, it is the overconvergent de Rham complex of an exponentially twisted integrable connection on a p-adic polydisk. We state a theorem of Baldassarri and Berthelot that relates the overconvergent Dwork cohomology of an affine variety Z with the rigid cohomology of Z with compact support.

The role of a subtler model, the Artin–Hasse model, is clarified in a subsequent section, before we prove the divisibility theorem.

Definition 2.1. On the structure sheaf of the rigid analytic affine line $\mathbf{A}^{1,\mathrm{an}}_K$ (with coordinate z), we define an integrable connection $\nabla_{\pi}$ by the formula

\begin{equation*}\nabla_{\pi}\xi = \mathrm{d}\xi + \pi \xi \mathrm{d}z.\end{equation*}

The connection is overconvergent. It is equipped with a Frobenius structure

\begin{equation*}\varphi_{\pi}\colon \xi \mapsto \xi^{\tau}(z^p) \cdot \theta(z)^{-1},\end{equation*}

where $\theta(z)$ is the Dwork exponential

\begin{equation*}\theta(z) = \exp(\pi z - \pi z^{p}).\end{equation*}

It is well known that the radius of convergence of $\theta(z)$ is $|p|^{-(({p-1})/{p^2})}$ , and thus overconvergent (cf. [Reference RobertRob00, p. 396, Theorem (b)]). The pair $(\nabla_{\pi}, \varphi_{\pi})$ defines an overconvergent F-isocrystal $\mathcal{L}_{\pi}$ on $\mathbf{A}^{1}_{k}$ , called the Dwork crystal. See [Reference Le StumLS07, § 4.2.1 and § 8.3] for more details. The dual isocrystal of $\mathcal{L}_{\pi}$ is $\mathcal{L}_{-\pi}$ .

As we have said, the benefit of being able to work with the overconvergent Dwork cohomology is that it has a rather explicit chain-level model. We now elaborate on the de Rham complex that is used to compute the overconvergent Dwork cohomology.

Construction 2.3 (Overconvergent Dwork complex). Consider the Monsky–Washnitzer algebra

\begin{align*} B &= K\langle x_{1},\ldots,x_{n+r} \rangle^{\dagger}\\ &= \bigg\{ \sum_{u\in \mathbf{N}^{n+r}} c_{u} x^{u} \in K[\![x_{1},\ldots,x_{n+r}]\!] : \exists \rho > 1, |c_{u}|\rho^{|u|} \xrightarrow{|u| \to \infty} 0 \bigg\},\end{align*}

where $|u| = |u_{1}| + \cdots + |u_{n+r}|$ , $x^{u} = \prod_{i=1}^{n+r} x_{i}^{u_{i}}$ . For a subset $I=\{i_{1},\ldots,i_{m}\}$ of $\{1,\ldots,n+r\}$ , with $i_{1}<\cdots<i_{m}$ , we write $x^{I} = \prod_{i\in I}x_{i}$ , and $dx^{I} = dx_{i_{1}}\wedge\cdots\wedge dx_{i_{m}}$ . We use the notation $\Omega^{m}$ to denote the space of ‘overconvergent m-forms’, that is,

(2.1) \begin{equation}\Omega^{m} = \bigoplus_{\substack{I\subset\{1,\ldots,n+r\}\\|I|=m}} B\cdot dx^{I}= \bigoplus_{\substack{I\subset\{1,\ldots,n+r\}\\|I|=m}} B_{I} \frac{dx^{I}}{x^{I}},\end{equation}

where $B_{I} = x^{I}B$ .

Write $g = \sum_{u\in \mathbf{N}^{n+r}} a_{u}x^{u}$ , with $a_{u}\in \mathbf{F}_{q}$ . Let $A_{u}$ be the Teichmüller lift of $a_{u}$ in $W(\mathbf{F}_{q}) \subset \mathcal{O}_{K}$ , and let $G = \sum_{u\in \mathbf{N}^{n+r}}A_{u}x^{u}$ . Then $G \equiv g \mod \pi$ , and the overconvergent Dwork cohomology is computed by the exponentially twisted de Rham complex

(2.2) \begin{equation}\mathcal{D}^{\bullet}: \quad\Omega^{0} \xrightarrow{d + \pi dG} \Omega^{1} \xrightarrow{d + \pi dG} \cdots \xrightarrow{d + \pi dG} \Omega^{n+r}.\end{equation}

That is,

\begin{equation*}\mathrm{H}^{\ast}_{\mathrm{rig}}(\mathbf{A}^{n+r}_{\mathbf{F}_{q}},g^{\ast}\mathcal{L}_{\pi}) \simeq\mathrm{H}^{\ast}(\mathcal{D}^{\bullet}).\end{equation*}

Remark 2.4. The complex $\mathcal{D}^{\bullet}$ is said to be exponentially twisted because, symbolically,

\[d + \pi dG = \exp(-\pi G) \circ d \circ \exp(\pi G).\]

Construction 2.6 (Frobenius action on the overconvergent Dwork complex). The inverse image isocrystal $g^{\ast}\mathcal{L}_{\pi}$ has a Frobenius structure, which can also be explained using the exponential twist. We denote by $\sigma$ the endomorphism of B defined by

\begin{equation*}\sigma\colon \sum_{u\in \mathbf{N}^{n+r}} a_{u} x^{u} \mapsto \sum_{u\in\mathbf{N}^{n+r}} a_{u}^{\tau} x^{pu}.\end{equation*}

Then the Frobenius structure on $g^{\ast}\mathcal{L}_{\pi}$ with respect to $\sigma$ can be symbolically determined via exponential twist (Remark 2.5) using

\begin{align*} \varphi(\xi) &= \prod_{u} \exp(-\pi A_{u} x^{u}) \cdot \sigma\bigg( \prod_{u} \exp(\pi A_{u} x^{u}) \cdot \xi \bigg) \\ &= \prod_{u} \exp(\pi A_{u}^{\tau} x^{pu} -\pi A_{u} x^{u} ) \cdot \xi^{\sigma} \\ &= \prod_{u} \theta(A_{u} x^{u})^{-1} \cdot \xi^{\sigma},\end{align*}

where u ranges in $\mathbf{N}^{n+r}$ and $\theta$ is the Dwork exponential defined above. Since $\theta$ is overconvergent, it follows that $\varphi$ indeed takes B into B. Recall $\mathbf{F}_{q} = \mathbf{F}_{p^{a}}$ . Let

(2.3) \begin{equation}F_{1}(x) = \prod_{u} \theta(A_{u}x^{u}) \quad \text{and} \quad F_{a}(x) = \prod_{i=0}^{a-1}F^{\tau^{i}}_{1}(x^{p^{i}}).\end{equation}

Then both $F_{1}(x)$ and $F_{a}(x)$ are overconvergent analytic functions in $(x_{1},\ldots,x_{n+r})$ . Hence, $\varphi = F_{1}^{-1} \circ \sigma$ .

The Frobenius $\varphi$ induces an operation on the spaces $\Omega^{m}$ of differential forms.

\begin{equation*}\varphi^{(m)}\Bigg(\sum_{\substack{I\subset\{1,\ldots,n+r\}\\|I|=m}}\xi_{I}(x) \frac{dx_I}{x_{I}}\Bigg)= p^{m}\varphi(\xi_{I}(x))\frac{dx_{I}}{x_{I}},\quad \xi_I \in B_{I}.\end{equation*}

One can check that the above definition turns $(\varphi^{(m)})_{m=0}^{n+r}$ into a chain map $\varphi^{(\bullet)}\colon \mathcal{D}^{\bullet} \to \mathcal{D}^{\bullet}$ .

It induces the semilinear Frobenius map on rigid cohomology

\[\varphi\colon\mathrm{H}^{\ast}_{\mathrm{rig}}(\mathbf{A}^{n+r};g^{\ast}\mathcal{L}_{\pi})\to\mathrm{H}^{\ast}_{\mathrm{rig}}(\mathbf{A}^{n+r};g^{\ast}\mathcal{L}_{\pi}).\]

This analytic issue is common to F-isocrystals in general, and is not limited to the Dwork isocrystal. Consider the pngst example of the standard Frobenius acting on the trivial isocrystal on $\mathbf{A}^{1}$ , when $q = p$ . To take advantage of overconvergence, one is led to examine the Frobenius operation on p-adic disks slightly larger than the unit disk. However, the p-power Frobenius map $a \mapsto a^{p}$ takes the one-dimensional closed ball of radius $|\pi|^{-1/N}$ to the larger closed ball of radius $|\pi|^{-p/N}$ . The Frobenius pullback induces a map on functions:

\[\xi (x) \mapsto \xi(x^{p}) \colon \left\{ \sum a_{n} x^{n} : a_{n} \pi^{-n/N} \to 0 \right\}\to \left\{ \sum b_{n} x^{n} : b_{n} \pi^{-pn/N} \to 0 \right\}.\]

After extending the field K to a larger scalar field by adding Nth roots of $\pi$ , an orthonormal basis of the Banach space $\left\{\sum a_{n} x^{n} : a_{n} \pi^{-n/N} \to 0\right\}$ is given by $1, \pi^{1/N} x, (\pi^{1/N} x)^{2}, \ldots$ , and an orthonormal basis of $\left\{\sum a_{n} x^{n} : a_{n} \pi^{-pn/N} \to 0\right\}$ is given by $1, \pi^{p/N} x, (\pi^{p/N} x)^{2}, \ldots$ . The columns of the matrix representing the linear mapping $\xi(x) \mapsto \xi(x^{p})$ with respect to these bases have larger and larger absolute values. On the other hand, the norm of the column vectors of a completely continuous operator should converge to 0.

Construction 2.8 (Dwork operators). To fix this, following Dwork, we consider the following operator $\psi$ on power series, given by

(2.4) \begin{equation}\psi\bigg( \sum_{u\in \mathbf{N}^{n+r}} a_{u} x^{u} \bigg) = \sum_{u\in \mathbf{N}^{n+r}} a_{pu} x^{u}.\end{equation}

Recall that $q=p^{a}$ and $\tau^{a} = \mathrm{Id}$ . Then

(2.5) \begin{align}(\tau^{-1}\circ \psi)^{a}\left( \sum a_{u}x^{u} \right) &= \sum a_{qu}^{\tau^{-a}} x^{u} = \sum a_{qu}x^{u} \\ &= \psi^{a}\left( \sum a_{u}x^{u} \right)\nonumber,\end{align}

as well as

\begin{equation*}\eta \cdot (\tau^{-1} \circ \psi)(\xi)= (\tau^{-1}\circ\psi)(\eta^{\tau}(x^{p}) \cdot \xi(x)).\end{equation*}

Form the composition

\begin{equation*}\alpha_{1} = \tau^{-1} \circ \psi \circ F_{1} \quad\text{as well as}\quad\alpha_{a} = \alpha_{1}^{a}.\end{equation*}

Then by (2.4) and (2.5),

\begin{align*} \alpha_{a} &= \alpha_{1}^{a} = \underbrace{(\tau^{-1} \circ \psi \circ F_{1}) \circ \cdots \circ(\tau^{-1} \circ \psi \circ F_{1})}_{a\text{ times}}, \\ &= (\tau^{-1} \circ \psi)^{a}\circ \biggl( \prod_{i=0}^{a-1} F_{1}^{\tau^{i}}(x^{p^{i}}) \biggr) \\ &= \psi^{a} \circ F_{a}.\end{align*}

It is clear that $\alpha_{1}$ preserves $B_{I}$ for any $I \subset \{1,2,\ldots,n+r\}$ and is a left inverse to $\varphi$ : $\alpha_{1}\circ\varphi = \mathrm{Id}$ . Similarly, $\alpha_{a}$ also preserves $B_{I}$ and is a left inverse to the ath iteration of $\varphi$ . Note that $\alpha_{1}$ is a $\tau^{-1}$ -semilinear map on the infinite-dimensional K-vector space $B_{I}$ , and $\alpha_{a}$ is a K-linear operator on $B_{I}$ . In the terminology of Monsky [Reference MonskyMon71, Definition 2.1], $\alpha_{1}$ and $\alpha_{a}$ are ‘Dwork operators’ on the spaces $B_{I}$ (the power series $F_{1}(x)$ and $F_{a}(x)$ are integrally defined, i.e., they are elements of $\mathcal{O}_{K}[\![x_{1},\ldots,x_{n+r}]\!] \cap B_{I}$ ). Moreover, both $\alpha_{1}$ and $\alpha_{a}$ are ‘nuclear operators’ on the space B in the sense of Monsky [Reference MonskyMon71, Theorem 2.1]. Therefore, their ‘characteristic power series’ [Reference MonskyMon71, Theorem 1.2] are well defined.

The operators $\alpha_{1}$ and $\alpha_{a}$ extend to endomorphsms of the Dwork complex. We define

\[\alpha_{1}^{(m)} \colon \Omega^{m} \to \Omega^{m},\]

by the formula

\begin{equation*}\alpha_{1}^{(m)}\Bigg(\sum_{\substack{I\subset\{1,\ldots,n+r\}\\|I|=m}}\xi_I(x) \frac{dx_I}{x_{I}}\Bigg)= \alpha_{1}(\xi_I(x)) \frac{dx_I}{x_I}, \quad \xi_I \in B_{I}.\end{equation*}

Then $\alpha_{1}^{(m)} \circ \varphi^{(m)} = p^{m} \mathrm{Id}$ , and the following diagram is commutative.

Similarly, we may define $\alpha_{a}^{(m)}\colon\Omega^{m}\to\Omega^{m}$ , satisfying $\alpha_{a}^{(m)}\circ(\varphi^{(m)})^{a}=q^{m}\mathrm{Id}$ , and the operators $q^{-m}\alpha_{a}^{(m)}$ induce a chain map of $\mathcal{D}^{\bullet}$ . Thus, the operators $(p^{-m}\alpha_{1}^{(m)})_{m=0}^{n+r}$ and $(q^{-m}\alpha_{a}^{(m)})_{m=0}^{n+r}$ induce maps on overconvergent Dwork cohomology,

\begin{equation*}\alpha_{1}, \alpha_{a}\colon \mathrm{H}^{\ast}_{\mathrm{rig}}(\mathbf{A}^{n+r};g^{\ast}\mathcal{L}_{\pi})\to \mathrm{H}^{\ast}_{\mathrm{rig}}(\mathbf{A}^{n+r};g^{\ast}\mathcal{L}_{\pi}).\end{equation*}

Since $\alpha_{a}^{(m)}$ (respectively, $\alpha_{1}^{(m)}$ ) is a left inverse to $\varphi^{(m)}$ (respectively, $(\varphi^{(m)})^{a}$ ) on the chain level, and since the overconvergent Dwork cohomology is finite dimensional, we find on the cohomology level that $q^{-m}\alpha_{a}^{(m)}$ is equal to the inverse to $(\varphi^{(m)})^{a}$ . In particular,

(2.6) \begin{align}&\det(1 - t\cdot q^{-m}\alpha_{a} | \mathrm{H}^{m}_{\mathrm{rig}}(\mathbf{A}^{n+r};g^{\ast}\mathcal{L}_{\pi})) \\=& \det(1 - t\cdot(\varphi^{(m)})^{-a} | \mathrm{H}^{m}_{\mathrm{rig}}(\mathbf{A}^{n+r};g^{\ast}\mathcal{L}_{\pi})).\nonumber\end{align}

Remark 2.9 (Fredholm determinant). For every subset I of $\{1,2,\ldots,n+r\}$ , the collection

(2.7) \begin{equation}\{x^u: u\in \mathbf{N}^{n+r}, u_i \geqslant 1 \text{ if } i \in I\}\end{equation}

is a ‘basis’ of $B_I$ in the sense that every $\xi \in B_I$ can be written uniquely as an infinite linear combination $\xi = \sum a_u x^u$ for some $a_u \in K$ . Furthermore, any continuous linear operator on $B_I$ can be represented by a unique infinite matrix.

According to Monsky’s theory [Reference MonskyMon71, Theorem 1.6], the matrix associated to the Dwork operator $\alpha_a$ has a well-defined Fredholm determinant, which equals the characteristic power series he defined. This is because $B_I$ is a union of certain $\alpha_a$ -invariant Banach subspaces $B_I(b)$ : $B_I=\bigcup_{b>0} B_I(b)$ . Here, $B_I(b)$ is the space of rigid analytic functions converging on the closed polydisk of radius $|p|^{-b}$ . The operators $\alpha_a|_{B_I(b)}$ are all completely continuous in the sense of Serre [Reference SerreSer62]. An appropriately scaled version of (2.7) is an orthonormal basis of the Banach space $B_I(b)$ . For this reason, in the remainder of this paper, we simply refer to the characteristic power series of $\alpha_a|_{B_{I}}$ as the Fredholm determinant of $\alpha_a$ , denoted as $\det(1 - t\alpha_a| B_{I})$ .

On occasion, we will also encounter other overconvergent spaces, such as the spaces C and $C/B$ (which will show up in the proof of Theorem 3.7) among others. In these situations, when we refer to the term ‘basis’, it should be interpreted in a manner analogous to that explained in the preceding paragraph. Consequently, the characteristic power series of Dwork operators on these spaces can all be computed as the Fredholm determinants of the infinite matrices associated to the specified ‘bases’. Therefore, we do not distinguish between the terms ‘characteristic power series’ and ‘Fredholm determinant’.

Now that we have completed the groundwork concerning overconvergent Dwork cohomology, we are ready to present a theorem of Baldassarri and Berthelot that establishes a connection between overconvergent Dwork cohomology and the rigid cohomology of Z. This theorem serves as the cornerstone for our chain-level arguments.

Theorem 2.10 [Reference Baldassarri and BerthelotBB04, Theorem 3.1]. Using notation as in Situation 2.2, we have a natural isomorphism

\begin{equation*}\mathrm{H}^{\ast}_{\mathrm{rig}}\big(\mathbf{A}_{\mathbf{F}_{q}}^{n+r};g^{\ast}\mathcal{L}_{\pi}\big)\simeq \mathrm{H}^{\ast}_{\mathrm{rig},Z}\big(\mathbf{A}_{\mathbf{F}_{q}}^{n}\big),\end{equation*}

which is compatible with Frobenius actions.

In fact, the following local version of the theorem is true.

Here, $\mathcal{O}_{\widehat{\mathbf{P}}^{n},\mathbf{Q}}({}^{\dagger}H)$ is the sheaf of functions on the formal projective space $\widehat{\mathbf{P}}^{n}$ with overconvergent singularities along the infinity hyperplane $H \subset \mathbf{P}^{n}_{\mathbf{F}_{q}}$ . This is what was denoted by $\mathcal{O}_{\widehat{\mathbf{P}}^{n},\mathbf{Q}}(\infty)$ in [Reference Baldassarri and BerthelotBB04]. See Example 5.3 for more details.

However, be aware that Caro’s construction of the local cohomology complex [Reference CaroCar04, 2] differs from Berthelot’s approach when Z is not a divisor. Nevertheless, as noted in [Reference CaroCar04, Remarque 2.2.7], the two constructions coincide when the local cohomology is taken with respect to modules of the form $\mathcal{O}_{\mathcal{P},\mathbf{Q}}({}^{\dagger}H)$ . Also, the proof of [Reference Baldassarri and BerthelotBB04, Theorem 2.14] remains valid using Caro’s version of the local cohomology functor.

We were informed by Steve Sperber that Nobuo Tsuzuki also prepared a proof of Theorem 2.10 in 2011 (unpublished). We appreciate Sperber for sharing Tsuzuki’s manuscript with us.

In addition to the work of Baldassarri and Berthelot, there are many previous works aiming to provide a comparison between rigid cohomology spaces and the cohomology spaces (and their variants) constructed by Dwork and by Adolphson and Sperber, such as Berthelot [Reference BerthelotBer84] (for exponential sums over a one-dimensional torus), P. Bourgeois [Reference BourgeoisBou99] (for Newton nondegenerate toric exponentials sums) and Peigen Li [Reference LiLi22] (for toric exponential sums).

Theorem 2.10 implies a result on cohomology with compact supports by taking Poincaré duality. Before stating it, we recall some standard notation about Tate twists.

For an integer w, let K(w) be the one-dimensional vector space over K equipped with a $\tau$ -semilinear map $x\mapsto p^{-w}\tau(x)$ . For a finite-dimensional K-vector space M equipped with a bijective $\tau$ -semilinear map $\varphi$ , we use M(w) to denote the tensor product $M\otimes_{K}K(w)$ , whose $\tau$ -semilinear map is twisted by $p^{-w}$ . For $(M,\varphi)$ as above, we endow its dual space $M^{\ast}$ the transpose inverse of $\varphi$ . With this convention, we have $(M(w))^{\ast}\simeq M^{\ast}(-w)$ as spaces equipped with $\tau$ -semilinear maps.

Corollary 2.14. Let $f_{1},\ldots,f_{r} \in \mathbf{F}_{q}[x_{1},\ldots,x_{n}]$ be polynomials in n variables. Let Z be the closed subscheme of $\mathbf{A}^{n}$ cut out by $f_{1},\ldots,f_{r}$ . Let $g = \sum_{i=1}^{r}x_{n+i}f_{i} \in \mathbf{F}_{q}[x_{1},\ldots,x_{n+r}]$ . Then we have an isomorphism of rigid cohomology spaces compatible with Frobenius actions:

\begin{equation*}\mathrm{H}^{n-r+j}_{\mathrm{rig},c}(Z)(n) \xrightarrow{\sim} [\mathrm{H}^{n+r-j}_{\mathrm{rig}}(\mathbf{A}^{n+r};g^{\ast}\mathcal{L}_{\pi})]^{\ast}.\end{equation*}

In other words, under the isomorphism, the Frobenius operator $q^{-n}\mathrm{Frob}_{q}|_{\mathrm{H}^{n-r+j}_{\mathrm{rig},c}(Z)}$ corresponds to the transpose of the inverse of the Frobenius operator $(\varphi^{(n+r-j)})^{a}$ on the overconvergent Dwork cohomology. Moreover,

\begin{equation*}{\det}\big(1 - t\mathrm{Frob}_{q}\big|\mathrm{H}^{n-r+j}_{\mathrm{rig},c}(Z)\big)= {\det}\big(1 - q^{j-r}t\alpha^{(n+r-j)}_{a}\big| \mathrm{H}_{\mathrm{rig}}^{n+r-j}(\mathbf{A}^{n+r};g^{\ast}\mathcal{L}_{\pi})\big).\end{equation*}

Proof. Recall the statement of Poincaré duality for rigid cohomology (see [Reference Le StumLS07, Corollary 8.3.14] and [Reference KedlayaKed06, Theorem 1.2.3]; note that the latter article ignores the Tate twist). For a nonsingular, geometrically connected variety X of dimension N, and any closed subvariety Y of X, we have a perfect pairing

\begin{equation*}\mathrm{H}^{i}_{\mathrm{rig},Y}(X) \otimes_K \mathrm{H}^{2N-i}_{\mathrm{rig},c}(Y) \to K(-N).\end{equation*}

Applying Poincaré duality and Theorem 2.10 to $X = \mathbf{A}^{n}$ and $Y=Z$ , we find that the dual space of $\mathrm{H}^{n-r+j}_{\mathrm{rig},c}(Z)(n)$ is isomorphic to $\mathrm{H}^{n+r-j}_{\mathrm{rig},Z}(\mathbf{A}^{n})\simeq\mathrm{H}^{n+r-j}_{\mathrm{rig}}(\mathbf{A}^{n+r};g^{\ast}\mathcal{L}_{\pi})$ , in a way that commutes with semilinear Frobenius actions. This implies that the ath iterations of these semilinear maps, which are linear, also match.

The ath iteration of the semilinear Frobenius operator on $\mathrm{H}^{n-r+j}_{\mathrm{rig},c}(Z)(n)$ is the linear Frobenius operator twisted by $q^{-n}$ , i.e., the map $q^{-n}\mathrm{Frob}_{q}\colon\mathrm{H}^{n-r+j}_{\mathrm{rig},c}(Z)\to\mathrm{H}^{n-r+j}_{\mathrm{rig},c}(Z)$ . By the above discussion, it corresponds to the transpose of the inverse of the ath iteration of the semilinear Frobenius operator $\varphi^{(n+r-j)}$ on the overconvergent Dwork cohomology $\mathrm{H}^{n+r-j}_{\mathrm{rig}}(\mathbf{A}^{n+r};g^{\ast}\mathcal{L}_{\pi})$ . Since the characteristic polynomial of the transpose of a linear operator is identical to that of the operator itself, we conclude that

This completes the proof.

3. Visibility of Frobenius eigenvalues

Let $f_{1},\ldots,f_{r} \in \mathbf{F}_{q}[x_1,\ldots,x_n]$ be a collection of polynomials. For every subset $I \subset \{1,2,\ldots,r\}$ , set

\[Z_{I}= \operatorname{Spec}\mathbf{F}_q[x_1,\ldots,x_n]/(f_i : i\in I),\]

and $Z_{I}^{\ast} = Z_{I} \cap \mathbf{G}_{\mathrm{m}}^{n}$ . Write $Z=Z_{\{1,2,\ldots,r\}}$ .

The purpose of this section is to prove Theorem 1.2; that is, we want to show that the Frobenius eigenvalues of Z are weakly visible in the zeta functions $\zeta_{Z_I^{\ast}}(t)$ .

The idea of the proof of Theorem 1.2 can be explained as follows.

1. (i) By Corollary 2.14, the overconvergent Dwork cohomology associated to Z computes the rigid cohomology of Z. Hence, the Frobenius eigenvalues of Z can be manifestly computed using the operator $\alpha_{a}$ . This is our starting point of the proof of Theorem 1.2.

2. (ii) The chain-level operators $\alpha_{a}|_{B_I}$ are ‘nuclear operators’. Since cohomology spaces are subquotients of $\bigoplus B_{I}$ , Monsky’s spectral theory (see Lemmas 3.1 and 3.2) implies that the cohomological eigenvalues are also ‘eigenvalues’ of $\bigoplus\alpha_{a}|_{B_I}$ . In the latter context, ‘eigenvalue’ should be interpreted as the reciprocal roots of the Fredholm determinant of $\alpha_{a}$ . Thus, Frobenius eigenvalues are visible in the Fredholm determinants $\det(1-t\alpha_{a}| B_I)$ .

3. (iii) The spaces $B_{I}$ are all $\alpha_{a}$ -invariant subspaces of B. Thus, $\det(1-t\alpha_{a}| B)$ witnesses all the ‘chain-level eigenvalues’ of $\alpha_{a}|_{B_I}$ . The Dwork trace formula, applying to $\alpha_{a}|_B$ , equates an alternating product of $\det(1-t\cdot q^{m}\alpha_{a}| B)$ with an alternating product of zeta functions (see Theorem 3.7 and Lemma 3.8). A Möbius inversion (see the formulae in Definition 3.4) then allows us to represent $\det(1-t\alpha_{a}| B)$ as an infinite product of zeta functions. The Frobenius eigenvalues, which are visible in the Fredholm determinant, are thereby weakly visible in the zeta functions.

We carry out the details. To show that the Fredholm determinants $\det(1-t\alpha_{a} | B_I)$ contain all the information of Frobenius eigenvalues, we need the following lemma, extracted from [Reference MonskyMon71, Theorem 1.4].

Lemma 3.2. For each $I \subset \{1,2,\ldots,n+r\}$ and each $i \in \mathbf{Z}$ , $\det(1-t\cdot q^{-i}\alpha_{a} | B_{I})$ is a p-adic entire function. Moreover, the polynomial $\det(1-t\mathrm{Frob}_q| \mathrm{H}^{n-r+j}_{\mathrm{rig},c}(Z))$ is a factor of

\[\prod\limits_{|I|=n+r-j}\det(1-t\cdot q^{j-r}\alpha_{a} | B_{I}).\]

Proof. By Lemma 3.2, Frobenius eigenvalues of $\mathrm{H}^{n-r+j}_{\mathrm{rig},c}(Z)$ are weakly visible in the product

\[\prod\limits_{|I|=j}\det(1-t\alpha_{a}|B_{I}).\]

The corollary follows since $B_{I}$ is a nuclear submodule of B.

Next, we explain how to read off the Fredholm determinants from the zeta functions.

Definition 3.4. Following Dwork, we introduce an operation $\delta$ on the set $1 + t \mathbf{C}_{p}[\![t]\!]$ of formal power series with constant term one given by

\begin{equation*} \delta(\Gamma(t)) \overset{\mathrm{def}}{=\!=} \frac{\Gamma(t)}{\Gamma(qt)}. \end{equation*}

The endomorphism $\delta$ is invertible, and its inverse reads

\begin{equation*} \delta^{-1}(\Gamma(t)) = \prod_{i=0}^{\infty} \Gamma(q^{i}t). \end{equation*}

Proof. Let $\Gamma(t) = \prod_{j=1}^{\infty}(1-\gamma_{i}t)$ be its infinite product expansion, where the reciprocal zero $\gamma_j$ approaches zero as j goes to infinity. Then

\[\delta^{-1}(\Gamma(t)) = \prod_{i=0}^{\infty} \prod_{j=1}^{\infty} (1-q^i\gamma_jt),\]

is also such an infinity product whose reciprocal zero approaches zero.

Proof. Let $Z(t) = \delta(\Gamma)$ . Then $\Gamma(t) = \prod_{i=0}^{\infty}Z(q^{i}t)$ . Write $Z(t) = u(t)/v(t)$ , where u,v are entire functions, without common zeros. Then

\begin{equation*}\Gamma(t) = \prod_{i=0}^{\infty} \frac{u(q^i t)}{v(q^i t)} = \frac{\delta^{-1}(u(t))}{\delta^{-1}(v(t))}.\end{equation*}

By Lemma 3.5, $\delta^{-1}u(t)$ and $\delta^{-1}v(t)$ are entire. If $q^{m}\lambda$ is a reciprocal zero of $\Gamma(t)$ , then $1-q^{m}\lambda t$ must be a factor of the infinite product $\delta^{-1}(u(t))=\prod_{i}u(q^{i}t)$ . Hence, $q^{m}\lambda$ is a reciprocal zero of $u(q^{i}t)$ for some i, i.e., $q^{m-i}\lambda$ is a reciprocal zero of Z(t). Thus, $\lambda$ is weakly visible in Z(t). The polar case is similar.

At this point, we recall the Dwork trace formula [Reference DworkDwo60, p. 637, Lemma 2]. Remember that, for each choice of $\pi$ , the value of the overconvergent function $\theta(t) = \exp(\pi t - \pi t^{p})$ at $t = 1$ is a primitive pth root of unity, and the function

(3.1) \begin{equation}\Psi(t) = \theta(1)^{\mathrm{Tr}_{\mathbf{F}_{q}/\mathbf{F}_{p}}(t)}\end{equation}

is a nontrivial additive character on $\mathbf{F}_{q}$ .

Theorem 3.7 (Dwork trace formula). For each positive integer m, define the mth toric exponential sum

\[S^{\ast}_m(g)=\sum\limits_{x\in\mathbf{G}_{\mathrm{m}}^{n+r}(\mathbf{F}_{q^m})}(\Psi\circ\operatorname{Tr}_{\mathbf{F}_{q^m}/\mathbf{F}_q})(g(x)).\]

Then

\begin{equation*}L^{*}(t)\overset{\mathrm{def}}{=\!=}\exp\bigg\{ \sum_{m=1}^{\infty} S_{m}^{\ast}(g)\frac{t^{m}}{m} \bigg\}= \{ \delta^{n+r}(\det(1 - t\alpha_{a}| B))\}^{(-1)^{n+r-1}}.{}_{\vphantom{\bigg(}}\end{equation*}

Proof. Let $C = K\langle x^{\pm 1}_{1},\ldots,x_{n+r}^{\pm 1} \rangle^{\dagger}$ be the weak completion of $\mathcal{O}_{K}[x_{1}^{\pm 1},\ldots,x_{n+r}^{\pm 1}]$ with $\pi$ inverted. Then the rigid cohomology $\mathrm{H}^{\ast}(\mathbf{G}_{\mathrm{m}}^{n+r};g^{\ast}\mathcal{L}_{\pi})$ is computed by the exponentially twisted de Rham complex

\begin{equation*}C \to \bigoplus_{|I| = 1} C \frac{dx^{I}}{x^{I}} \to \cdots \to \bigoplus_{|I|=m} C \frac{dx^{I}}{x^{I}} \to \cdots,{}_{\vphantom{\bigg(}}\end{equation*}

whose differentials are given by $d+\pi dG$ as in (2.2), and it is equipped with Dwork operators $q^{-m}\alpha_{a}^{(m)}$ as in Construction 2.8. By [Reference Étesse and Le StumELS93, Théorème 6.3II] (noticing that $\theta^{i}$ there is our $q^{-i}\alpha_{a}^{(i)}$ ), the L-function for the isocrystal $g^{\ast}\mathcal{L}_{\pi}$ , which is $L^{\ast}(t)$ in our case, equals

\[\prod_{i=0}^{n+r}\det(1 - t q^{n+r-i}\alpha_{a}^{(i)} |\mathrm{H}^{i}_{\mathrm{rig}}(\mathbf{G}_{\mathrm{m}}^{n+r};g^{\ast}\mathcal{L}_{\pi}))^{(-1)^{i+1}},\]

(note that $\mathrm{H}^{i}_{\mathrm{rig}}(\mathbf{G}_{\mathrm{m}}^{n+r},g^{\ast}\mathcal{L}_{\pi})=0$ for $i > n+r$ ). Since $\alpha_{a}^{(m)}$ are nuclear operators, the above quantity can also be written as

(3.2) \begin{equation}\prod_{i=0}^{n+r}{\det}(1 - tq^{n+r-i}\alpha_{a}| C^{\binom{n+r}{i}})^{(-1)^{i+1}} =\{\delta^{n+r}( \det( 1 - t\alpha_{a} | C) )\}^{(-1)^{n+r-1}}.\end{equation}

If we can substitute C with B in (3.2), we can readily derive the desired result. It suffices to show that $\det(1 - t\alpha_{a} | C/B) = 1$ . Recall the formula $\det(1 - t \gamma) = \exp( -\sum_{m=1}^{\infty}\frac{\mathrm{Tr}(\gamma^m){m}}t^m )$ . We only have to show that $\mathrm{Tr}(\alpha_{a}^{m}| C/B)=0$ for all positive integers m. A ‘basis’ for $C/B$ comprises the images of $x^{u}$ in $C/B$ , where $u \in \mathbf{Z}^{n+r} \mathbin{\rule[0.2em]{0.67em}{0.12em}} \mathbf{N}^{n+r}$ . In particular, these u are never zero. Recall that $\alpha_{a} = \psi^{a} \circ F_{a}$ (see (2.3), Construction 2.8). Because $F_{a} \in B$ , the monomials in its power-series expansion are $x^{v}$ with $v \in \mathbf{N}^{n+r}$ . Thus,

\begin{equation*}\alpha_{a}^{m}(x^{u}) = \sum_{v} b_{v} x^{(u+v)/q^{m}},\end{equation*}

in which $b_{v}=0$ if $q^{m}\nmid u+v$ or if $v \notin \mathbf{N}^{n+r}$ . To compute the trace of $\alpha_{a}^{m}|_{C/B}$ , we need to examine the coefficient of $x^{u}$ in $\alpha_{a}^{m}(x^{u})$ for $u \notin \mathbf{N}^{n+r}$ . If $u = (u+v)/q^{m}$ , then we have $v = (q^{m}-1)u$ in $\mathbf{Z}^{n+r}$ . Since $u \notin \mathbf{N}^{n+r}$ , we have $v \notin \mathbf{N}^{n+r}$ . Therefore, if $u = (u+v)/q^{m}$ , we must have $b_{v}=0$ . From this, we conclude that the diagonal entries of the matrix representation of the operator $\alpha_{a}^{m}|_{C/B}$ on the quotient with respect to the ‘basis’

\begin{equation*}\{\text{The image of }x^{u}\text{ in }C/B : {u\in \mathbf{Z}^{n+r}\mathbin{\rule[0.2em]{0.67em}{0.12em}}\mathbf{N}^{n+r}}\}\end{equation*}

are all zero. The theorem follows.

Combining Corollary 3.3, Lemma 3.6 and Theorem 3.7, we infer that any Frobenius eigenvalue of $\mathrm{H}^{\bullet}_{\mathrm{rig},c}(Z)$ is weakly visible in $L^{\ast}(t)$ .

Our next objective is to relate the function $L^{\ast}(t)$ to the zeta functions of $Z_{I}$ .

Lemma 3.8 We have

\begin{equation*} L^*(t) = \prod_{J\subset \{1,2,\ldots,r\}} \zeta_{Z^*_{J}}(q^{|J|}t)^{(-1)^{r-|J|}}. \end{equation*}

Proof. For any subset J of $\{1,2,\ldots,r\}$ , let $\mathbf{A}^{J}$ denote the affine space with coordinates $(y_{j})_{j\in J}$ , and let $g_{J}$ denote the regular function $\sum_{j\in J}y_{j}f_{j}(x)$ . The orthogonality of characters implies that

\[ S_{J} \overset{\text{def}}{=\!=} \sum\limits_{x\in \mathbf{G}_{\mathrm{m}}^{n}(\mathbf{F}_q)}\sum_{y\in\mathbf{A}^{J}(\mathbf{F}_q)} \Psi(g_{J}(x,y)) =q^{|J|}|Z^*_{J}(\mathbf{F}_q)|. \]

Since $\mathbf{A}^{n} = \mathbf{G}_{\mathrm{m}}^{n} \sqcup D$ , where D is the union of all coordinate hyperplanes, and since D admits the standard semisimplicial resolution

we deduce from the inclusion-exclusion that

\begin{align*} \sum_{x\in \mathbf{G}_{\mathrm{m}}^{n+r}(\mathbf{F}_q)}\Psi(g(x)) &= \sum_{J\subset \{1,2,\ldots,r\}}(-1)^{r-|J|} S_J\\ &= \sum_{J\subset \{1,2,\ldots,r\}} (-1)^{r-|J|}q^{|J|} |Z_{J}^*(\mathbf{F}_q)|. \end{align*}

Replacing $\mathbf{F}_q$ by $\mathbf{F}_{q^m}$ and $\Psi$ by $\Psi\circ \mathrm{Tr}_{\mathbf{F}_{q^m}/\mathbf{F}_q}$ in the above calculation, and exponentiating, we get the desired result.

4. Reciprocal roots of Fredholm determinants

To establish Theorem 1.3 and Theorem 1.5, we provide a more robust result at the chain level. In this section, we apply Adolphson–Sperber’s theory to the specific context of our interest and deduce a lower bound of the q-orders of ‘eigenvalues’ of the operator $\alpha_{a}\colon B_{I} \to B_{I}$ introduced in Construction 2.8. Given that rigid cohomologies are subquotients of the entries in the overconvergent Dwork complex (2.2), divisibility at the chain level will imply divisibility at the cohomological level, as per Lemma 3.2.

For a given subset I of $\{1,2,\ldots,n+r\}$ , we define $I^{\prime}$ as $I \cap\{1,2,\ldots,n\}$ and $I^{\prime\prime}$ as $I \cap \{n+1,\ldots,n+r\}$ .

Recall that $B_{I}$ represents the subspace $x^{I}B$ within the Monsky–Washnitzer algebra B (refer to Construction 2.3), and that $\alpha_{1}$ and $\alpha_{a}$ are the Dwork operators acting on $B_{I}$ (Construction 2.8).

The main result of this section is the following lemma, which is a straightforward consequence of Adolphson–Sperber’s result (Theorem 4.10) once we have explained the relations between two different representations of Frobenius structures.

Lemma 4.2. Let the notation be as in Construction 2.8. Let $\lambda$ be a reciprocal root of the p-adic entire function $\det(1-t\alpha_{a}| B_I)$ . Then

\begin{equation*}\operatorname{ord}_q \lambda \geqslant \frac{1}{d_1}\bigg( |I^{\prime}| + \sum_{i\in I^{\prime\prime}} (d_1-d_i)\bigg)= \frac{1}{d_1}\bigg( |I| + \sum_{i\in I^{\prime\prime}} (d_1-d_i-1) \bigg),\end{equation*}

where $\operatorname{ord}_q$ is the p-adic valuation normalized so that $\operatorname{ord}_q(q)=1$ .

Our objective is to provide sharp lower bounds for the q-orders, or, equivalently, upper bounds for the p-adic absolute values of the reciprocal roots of the Fredholm determinant $\det(1-t\alpha_{1} | B_{I}) = \sum_{m} t^{m} \cdot\mathrm{Tr}(\wedge^{m}\alpha_{1})$ . By the theory of Newton polygons, we need to give upper bounds for $|\mathrm{Tr}(\wedge^{m}\alpha_{1})|$ .

We can view $\alpha_{1}$ as an infinite matrix by fixing the standard ‘basis’ of the infinite-dimensional K-vector space $B_{I}$ that comprises the monomials

(4.1) \begin{equation}\{x_{1}^{u_{1}}\cdots x_{n+r}^{u_{n+r}}:u_{j}\geqslant 0\quad\text{and}\quad u_{i}\geqslant 1\text{ if }i\in I\}.\end{equation}

A straightforward computation shows that the infinite matrix representing $\alpha_{1}$ is $[F_{pu-v}]$ ( $u_{j}, v_{j}\geqslant 0, u_{i},v_{i}\geqslant 1$ if $i \in I$ ), where $F_{1}(x) = \sum_{u\in\mathbf{N}^{n+r}} F_{u} x^{u}$ is the power-series expansion of the function $F_{1}$ (2.3) that defines the Frobenius structure. The number $\mathrm{Tr}(\wedge^{m}\alpha_{1})$ is just the sum of $m \times m$ principal minors of the infinite matrix $[F_{pu-v}]$ . Therefore, ultimately, we need a good upper bound for $|F_{u}|$ .

Since the function $F_{1}$ is derived from the Dwork exponential $\theta(t) = \exp(\pi t - \pi t^{p})$ , to get an upper bound for $|F_{u}|$ , we need an upper bound for the coefficient $a_{j}$ of $t^{j}$ in the power-series expansion of $\theta$ . The most optimal bound is

(4.2) \begin{equation}|a_{j}| \leqslant |p|^{\frac{j(p-1)}{p^{2}}}\end{equation}

(cf. [Reference DworkDwo62, Equation (4.7) and p. 57, line 1]). But this estimate falls short of meeting our need (see (4.4) below).

Therefore, the matrix of the Dwork operator with respect to the naive ‘basis’ (4.1) does not yield an ideal estimate. From an analytic perspective, the key to improving the estimate lies in finding a suitable ‘basis’ for B, such that the matrix of the Dwork operator under the new ‘basis’ has smaller entries. Equivalently, we would like to construct an isomorphism $\rho\colon B \to B$ , inducing the following commutative diagram.

We hope that the matrix of $\beta_{1}$ under the ‘basis’ (4.1) should have smaller entries than those of $\alpha_{1}$ . In addition, we expect that $\beta_{1}$ should map $B_{I}$ to $B_{I}$ .

For every choice of $s=1,2,\ldots,\infty$ , Dwork was able to construct a ‘splitting function’ of level s [Reference DworkDwo62, Equation (4.1)]. These splitting functions then give rise to Dwork operators $\alpha_{1,s}$ on B, $B_{I}$ , and hence $\Omega^{m}$ . All of them can play the role of $\beta_{1}$ above. For $s=1$ , we are reduced to the exponentially twisted $\alpha_{1}$ that we have been using so far. It is the choice $s = \infty$ , which corresponds to what we refer to as the Artin–Hasse representation of the Dwork crystal, that has the most optimal convergence property (cf. (4.3) below) among all splitting functions, and this was extensively employed in the work of Adolphson and Sperber. Below, we introduce Dwork’s construction for $s=\infty$ .

Construction 4.5. Let $E(t) = {\exp}\big\{ \sum_{j=0}^{\infty} t^{p^{j}}/p^{j} \big\}$ be the Artin–Hasse exponential. It is well known that $E(t) \in \mathbf{Z}_{(p)}[\![t]\!]$ . By the theory of Newton polygons, the series $\sum_{j=0}^{\infty}t^{p^{j}}/p^{j}$ has a root $\gamma$ satisfying $|\gamma| = |\pi| = |p|^{{1}/{p-1}}$ . We define $\theta_{\infty}(t) = E(\gamma\cdot t)$ . Write

\begin{equation*}\theta_{\infty}(t) = \sum_{j=0}^{\infty} c_{j}t^{j}.\end{equation*}

Then (cf. (4.2))

(4.3) \begin{equation}|c_{j}| \leqslant |\pi|^{j} = |p|^{\frac{j}{p-1}}.\end{equation}

The twisting function $\exp(\pi t)$ appearing in the overconvergent Dwork complex may be explained as the infinite product

\begin{equation*}\widehat{\theta}(t) = \exp(\pi t) = \prod_{j=0}^{\infty} \theta(t^{p^{j}}).\end{equation*}

Dwork introduced the twisting factor associated with $\theta_{\infty}$ as

\begin{equation*}\widehat{\theta}_{\infty}(t) = \prod_{j=0}^{\infty} \theta_{\infty}(t^{p^{j}})= \exp\bigg\{ \sum_{j=0}^{\infty} \gamma_{j} t^{p^{j}} \bigg\},\quad\text{so we have }\theta_{\infty}(t) = \widehat{\theta}_{\infty}(t)/\widehat{\theta}_{\infty}(t^{p}).\end{equation*}

Here, $\gamma_{j}=\sum_{i=0}^{j}\gamma^{p^{i}}/p^{i}$ . It is easy to show that $\widehat{\theta}_{\infty}(t)$ is a rigid analytic function on the open unit disk bounded by 1 (see [Reference DworkDwo62, Equation (4.13)]). For a conceptual explanation of this unusual exponential function in connection with Frobenii liftings on the formal multiplicative group, we recommend reading Pulita’s article [Reference PulitaPul07].

The vertical arrows send $\xi$ to $\xi\cdot\prod_{u}\widehat{\theta}(A_{u}x^{u})$ .

Define the Dwork operator associated to the ‘Artin–Hasse representation’ as a different twist using the commutativity of the following diagram.

The vertical arrows are now given by $\xi \mapsto \xi \cdot \prod_{u}\widehat{\theta}_{\infty}(A_{u}x^{u})$ . Using the relation between $\theta_{\infty}$ and $\widehat{\theta}_{\infty}$ ,

\begin{equation*}\beta_{1}(\xi) = (\psi(\xi\cdot \Phi))^{\tau^{-1}} ,\end{equation*}

where $\Phi(x) = \prod_{u} \theta_{\infty}(A_{u}x^{u})$ . Since $B_{I}$ are ideals of B and $\Phi$ is overconvergent, by definition, it is clear that $\beta_{1}\colon B_{I} \to B_{I}$ is well defined for any subset I of $\{1,2,\ldots,n+r\}$ . The operator $\beta_{1}$ is a Dwork operator in the sense of Monsky [Reference MonskyMon71] since $\Phi \in \mathcal{O}_{K}[\![x_{1},\ldots,x_{n+r}]\!] \cap B$ . We define the operator $\beta_{a}$ as the ath iteration of $\beta_{1}$ : $\beta_{a} = \beta_{1}^{a}$ . Then $\beta_{1} \colon B_{I} \to B_{I}$ is $\tau^{-1}$ -semilinear and $\beta_{a}$ is K-linear.

The following lemma tells us that the we can use the operator $\beta_{a}$ to study the Frobenius eigenvalues of the rigid cohomology.

The hook arrows are the natural inclusions and, for a power series $\xi \in K[\![x_1,\ldots,x_{n+r}]\!]$ ,

\begin{align*} \widetilde{\alpha}_1(\xi) &= \bigg\{ \prod_{u}\widehat{\theta}(A_u x^u) \bigg\}^{-1}\cdot \tau^{-1}\bigg(\psi\bigg(\prod_{u}\widehat{\theta}(A_u x^u)\cdot \xi\bigg)\bigg),\\ \widetilde{\beta}_1(\xi) &= \bigg\{ \prod_{u}\widehat{\theta}_{\infty}(A_u x^u) \bigg\}^{-1}\cdot \tau^{-1}\bigg(\psi\bigg(\prod_{u}\widehat{\theta}_{\infty}(A_u x^u)\cdot \xi\bigg)\bigg),\\ \rho(\xi) &= \xi \cdot \frac{\prod_u \widehat{\theta}(A_u x^u)}{\prod_u \widehat{\theta}_{\infty}(A_u x^u)}.\end{align*}

It suffices to show that the ratio of twisting factors $\prod_{u}\widehat{\theta}(A_{u}x^{u})/\widehat{\theta}_{\infty}(A_{u}x^{u})$ , and its reciprocal, are overconvergent, i.e., are elements of B. This will imply that $\rho$ and $\rho^{-1}$ take $B_I$ into $B_I$ , as $B_I$ is an ideal of B. By transporting the structures, the ath iterations $\alpha_a$ and $\beta_{a}$ of $\alpha_1$ and $\beta_1$ will correspond under $\rho\colon B_I \xrightarrow{\sim} B_I$ , and they will have the same Fredholm determinant.

To prove the overconvergence of $\prod_{u}\widehat{\theta}(A_{u}x^{u})/\widehat{\theta}_{\infty}(A_{u}x^{u})$ , it, in turn, suffices to prove that

\begin{equation*}\widehat{\theta}(t)/\widehat{\theta}_{\infty}(t)\quad\text{and} \quad\widehat{\theta}_{\infty}(t)/\widehat{\theta}(t)\end{equation*}

are overconvergent, i.e., are elements of $K\langle t \rangle^{\dagger}$ . For a documented proof of this basic fact, we refer the reader to Peigen Li’s article [Reference LiLi22, Proposition 2.1(i)], where Li also works out the radius of convergence of the ratio.

Remark 4.8. The mappings

\begin{equation*}\varphi_{\infty}\colon \xi \mapsto \xi^{\sigma} \cdot \bigg(\prod_{u} \widehat{\theta}_{\infty}(A_{u}x^{u}) \bigg)^{-1}\quad\text{and}\quad\varphi\colon \xi \mapsto \xi^{\sigma} \cdot \bigg( \prod_{u} \widehat{\theta}(A_{u}x^{u}) \bigg)^{-1},\end{equation*}

(see Construction 2.6) together with the connections

\begin{equation*}\nabla = \widehat{\theta}^{-1} \circ d \circ \widehat{\theta}\quad \text{and}\quad\nabla_{\infty} = \widehat{\theta}_{\infty}^{-1} \circ d \circ \widehat{\theta}_{\infty},\end{equation*}

define two overconvergent unit-root F-isocrystal structures on

\begin{equation*}B = K\langle x_{1},\ldots,x_{n+r} \rangle^{\dagger}.\end{equation*}

The above argument shows that the mapping $\rho$ induces an isomorphism between $(B,\nabla,\varphi) = g^{\ast}\mathcal{L}_{\pi}$ and $(B,\nabla_{\infty}, \varphi_{\infty})$ . What we need is the slightly stronger result, namely, that $\beta_{1}$ also preserves the subspaces $B_{I}$ of B: this is to facilitate the chain-level argument.

The plan now is to examine traces and Fredholm determinants of the operator $\beta_{a}\colon B_{I} \to B_{I}$ for subsets I of $\{1,2,\ldots,n+r\}$ , under the ‘basis’

\[\{x_{1}^{u_{1}}\cdots x_{n+r}^{u_{n+r}}:u_{j}\geqslant 0\quad \text{and}\quad u_{i}\geqslant 1\text{ if }i\in I\}\]

of $B_{I}$ . Adolphson and Sperber’s idea is to use some combinatorial quantity to measure the absolute value of the entries of the matrix representation of $\beta_{a}$ , which we introduce below.

Define the weight function $w\colon \mathbf{R}^{n+r}\to \mathbf{R}$ by

\[w(y_1,\ldots, y_{n+r}) =y_{n+1}+\cdots + y_{n+r},\]

and define

\begin{equation*}w_I = \min\{w(y) : y \in C(\Delta) \cap \mathbf{N}^{n+r}, y_i > 0, \forall i \in I\}.\end{equation*}

By construction, if $g(x) = \sum_{u\in \mathbf{N}^{n+r}} a_{u}x^{u}$ , then $a_{u} \neq 0$ implies that $u \in \Delta \cap \mathbf{N}^{n+r}$ . Since $\Phi(x) = \prod_{u} \theta_{\infty}(A_{u}x^{u})$ (see Construction 4.6), the subscripts of the nonzero coefficients of $\Phi$ in the power-series expansion are all non-negative integral linear combinations of $u \in \Delta$ . Hence, if we write $\Phi(x) = \sum_{u\in \mathbf{N}^{n+r}} \Phi_{u} x^{u}$ , then $\Phi_{u} \neq 0$ implies that $u \in C(\Delta)$ .

The following theorem is due to Adolphson–Sperber. We have formulated only a weaker version on estimating the first slope, which is sufficient for the purpose of this paper.

Theorem 4.10 [Reference Adolphson and SperberAS87b, Proposition 4.2]. For any reciprocal root $\lambda$ of the Fredholm determinant $\det(1-t\beta_{a}| B_{I})$ ,

\begin{equation*}\mathrm{ord}_{q}(\lambda) \geqslant w_{I}.\end{equation*}

Recall that $\beta_{1} = \tau^{-1}\circ\psi\circ \Phi$ . We write $\Phi(x) = \sum_{u\in \mathbf{N}^{n+r}}\Phi_{u} x^{u}$ . Then, by [Reference Adolphson and SperberAS87a, Equation (2.12)],

(4.4) \begin{equation}\operatorname{ord}_{p}(\Phi_{u}) \geqslant \frac{1}{p-1}w(u).\end{equation}

(If we use F instead of $\Phi$ , we would get a worse bound. So it is crucial to work with the Artin–Hasse representation here.) A $\mathbf{Q}_{p}(\zeta_{p})$ -‘basis’ of $B_{I}$ is of the form

\begin{equation*}\lambda_{l} \cdot x^{u}, \quad l=1,\ldots,a, u \in \mathbf{N}^{n+r}, u_{i} \geqslant 1 \text{ if } i\in I,\end{equation*}

where $\{\lambda_{l}:l=1,\ldots,a\}$ is an integral basis of $\mathcal{O}_{K}$ over $\mathbf{Z}_{p}[\zeta_{p}]$ . When $I = \emptyset$ , the operator $\beta_{1}$ is represented by an infinite matrix

\begin{equation*}C=\begin{bmatrix} C_{00} & \quad C_{01} & \quad \cdots & \quad C_{0i} & \quad \cdots \\ C_{10} & \quad C_{11} & \quad \cdots & \quad C_{1i} & \quad \cdots \\ \vdots & \quad \vdots & \quad & \quad \vdots \\ C_{i0} & \quad C_{i1} & \quad \cdots & \quad C_{ii} & \quad \cdots \\ \vdots & \quad \vdots & \quad & \quad \vdots\end{bmatrix},\end{equation*}

where each $C_{ij}$ is a finite block matrix, corresponding to the components of $\beta_{1}(\lambda_{l}x^{v})$ with respect to $\lambda_{m}x^{u}$ , with $w(v) = i$ , $w(u)=j$ . Any entry in this block matrix is of the form $\lambda_{m}\lambda_{l}^{-1}\Phi_{pu-v}$ . When $I\neq \emptyset$ , the matrix representation of $\alpha_{1}|_{B_{I}}$ is a suitable submatrix $C_{I}$ of C.

An entry of $C_{I}$ on the diagonal is of the form $\Phi_{(p-1)u}$ , where $u_{i}\geqslant 1$ , for all $i\in I$ . By (4.4), we find that the trace of $\beta_{1}|_{B_{I}}$ satisfies the estimate

\begin{equation*}\operatorname{ord}_{p}\operatorname{Tr}(\beta_{1}| B_{I}) \geqslant w_{I}.\end{equation*}

This shows that, in the power-series expansion of $\det(1-t\beta_{1}| B_{I})$ , the p-order of the coefficient of t is at least $w_{I}$ . When it comes to the coefficient of $t^{m}$ in the Fredholm determinant, where $m\geqslant 2$ , these are computed as sums of $m\times m$ principal minors of the matrix $C_{I}$ . By employing the determinant formula in terms of the matrix entries, one can readily observe that this coefficient has p-order $\geqslant mw_{I}$ . The theorem can be deduced by examining the Newton polygon of $\det(1-t\beta_{1}| B_{I})$ .

End of proof of Lemma 4.2. Since the problem only concerns the Fredholm determinant, we can replace $\alpha_{a}$ by $\beta_{a}$ by virtue of Lemma 4.7. By Theorem 4.10, it suffices to show that

\[w_I \geqslant \frac{1}{d}\bigg( |I^{\prime}|+ \sum_{i\in I^{\prime\prime}} (d-d_i) \bigg).\]

This is a problem of linear programming. We are dealing with non-negative integers $y_1,\ldots,y_{n+r}$ subject to the constraints

(4.5) \begin{equation}\begin{cases} y_i \geqslant 1, \forall i \in I, \\ y_1+\cdots + y_n\leqslant d_1y_{n+1}+\cdots +d_r y_{n+r},\end{cases}\end{equation}

and we want to control the minimum of $y_{n+1} + \cdots + y_{n+r}$ . Write

\[\xi_i =\begin{cases} y_i - 1, & i \in I; \\ y_i, & i \notin I.\end{cases}\]

Then (4.5) is equivalent to

(4.5′)

It follows that

This completes the proof.

5. Two lemmas

The goal of this section is to prove two lemmas, Lemma 5.2 and Lemma 5.4 which will be used in the proofs of Theorem 1.3 and Theorem 1.5.

5.1 An algebraic lemma

Throughout this subsection, we assume that k is an algebraically closed field. The common zero locus, in $\mathbf{A}_{k}^n$ , of a collection of polynomials $f_1,\ldots,f_r \in k[x_1,\ldots,x_n]$ is be denoted by $Z(f_1,\ldots,f_r)$ .

Lemma 5.1. Let $g_{1},\ldots,g_{m}$ and $f_{m+1},\ldots,f_{r}$ be elements of $k[x_{1},\ldots,x_{n}]$ . Assume that

(5.1) \begin{equation}\dim Z(g_1,\ldots,g_m) = n-m, dim Z(g_1,\ldots, g_m) > \dim Z(g_1,\ldots, g_m, f_{m+1}, \ldots, f_r).\end{equation}

Then there are constants $c_{m+1}, \ldots, c_r$ in k, not all zero, such that:

• – if $f_{m+1}$ does not vanish on $Z(g_1,\ldots,g_m)$ , $c_{m+1}=1$ ;

• – if $f_{m+1}$ vanishes identically on $Z(g_1,\ldots,g_m)$ , $c_{m+1}=0$ ; and

• – $\dim Z(g_1,\ldots, g_m, c_{m+1}f_{m+1}+ \cdots + c_rf_r) = n - m -1$ .

Proof. Denote the irreducible components of the variety $Z(g_1, ..., g_m)$ by $D_1, \ldots, D_h$ . These components have the same dimension by the unmixedness theorem. If $f_{m+1}$ vanishes (identically) on all components $D_1, \ldots, D_h$ , then $\dim Z(g_1,..., g_m, f_{m+1}) = \dim Z(g_1,..., g_m)$ . In this case, we set $c_{m+1}=0$ , drop $f_{m+1}$ from our list and consider the shorter list $\{g_1,\ldots, g_m, f_{m+2}, \ldots, f_r\}$ , which still satisfies the condition (5.1).

Hence, without loss of generality, we may assume that $f_{m+1}$ does not identically vanish on $D_1, \ldots, D_{h_1}$ with $h_1>0$ but vanishes on $D_{h_1+1}, \ldots, D_h$ . If $h_1=h$ , then

\[ \dim Z(g_1,\ldots, g_m, f_{m+1}) < \dim Z(g_1,\ldots, g_m). \]

Since the variety $Z(g_1,\ldots,,g_{m},f_{m+1})$ is nonempty, this forces that

\[ \dim Z(g_1,\ldots, g_m, f_{m+1}) = \dim Z(g_1,\ldots, g_m) - 1, \]

that is, $\dim Z(g_1, ..., g_m, f_{m+1}) = n-m-1$ .

Now assume that $h_1 < h$ . By condition (5.1), there is another polynomial among $\{f_{m+2}, \ldots, f_r\}$ , say, $f_{m+2}$ , which does not vanish identically on all of $D_{h_1+1}, \ldots, D_h$ . Without loss of generality, we may assume that $f_{m+2}$ does not vanish on $D_{h_1+1}$ , $\ldots$ , $D_{h_1+h_2}$ with $h_2>0$ but vanishes on $D_{h_1+h_2+1}, \ldots, D_h$ .

Claim. There is a nonzero constant c in k such that $f_{m+1}+cf_{m+2}$ does not vanish on the irreducible components $D_1, \ldots, D_{h_1}, \ldots, D_{h_1+h_2}$ .

Proof of Claim. Because $f_{m+1}$ vanishes on $D_{h_1+1},\ldots, D_{h_1+h_2}$ and $f_{m+2}$ does not, for any nonzero constant c in k, the polynomial $f_{m+1} + cf_{m+2}$ does not vanish on $D_{h_1+1},\ldots, D_{h_1+h_2}$ . For each $i =1, \ldots, h_1$ , we can choose $x_i$ in $D_i$ such that $f_{m+1}(x_i)$ is nonzero as $f_{m+1}$ is not identically zero on $D_i$ . Choose nonzero constant c in k such that none of the $h_1$ numbers

\begin{equation*} f_{m+1}(x_i) + c f_{m+2}(x_i) , i=1,\ldots, h_1, \end{equation*}

is zero: one simply chooses any nonzero c in k such that c is not among the $h_1$ numbers

\[\{-f_{m+1}(x_i)/f_{m+2}(x_i), i=1,\ldots, h_1\},\]

which is possible since k is an infinite field. The claim is proved.

Repeating the above procedure, we see that there are constants $c_{m+1},\ldots, c_r$ in k such that the linear combination

\[ g_{m+1} = c_{m+1}f_{m+1}+ c_{m+2}f_{m+2}+ \cdots + c_r f_r \]

does not vanish identically on the component $D_i$ for $i=1,\ldots, h$ . It follows that

\begin{align*}\;\qquad\qquad\qquad\qquad\qquad\qquad \dim Z(g_1,\ldots,g_m,g_{m+1}) = n-m-1. \end{align*}

Lemma 5.2. Let $f_1,\ldots, f_r \in k[x_1,\ldots,x_n]$ be a collection of polynomials. Set $d_i = \deg f_i$ . Assume that $d_1 \geqslant d_2\geqslant \cdots \geqslant d_r$ . Let $Z = Z(f_1,\ldots,f_r)$ . Then there exists a new sequence of polynomials $g_1,\ldots,g_r \in k[x_1,\ldots,x_n]$ such that:

1. (1) $Z(g_1,\ldots,g_r) = Z$ ;

2. (2) $\deg g_i \leqslant d_i$ ; and

3. (3) $\dim Z(g_1,\ldots,g_{n-\dim Z}) = \dim Z$ .

Proof. Applying Lemma 5.1 repeatedly gives rise to a new sequence of polynomials $g_1,\ldots,g_r \in k[x_1,\ldots,x_n]$ , which satisfies the following.

• – $g_1 = f_1$ , $g_m = f_m$ if $m > n-\dim Z$ .

• – There exists an upper-triangular square matrix $B = (b_{\alpha\beta})_{1\leqslant \alpha,\beta\leqslant r}$ with entries in k, whose diagonal entries are either 0 or 1, such that

  (5.2) \begin{equation} \begin{bmatrix} g_1 \\ \vdots \\ g_r \end{bmatrix} = B \cdot \begin{bmatrix} f_1 \\ \vdots \\ f_r \end{bmatrix}. \end{equation}

• – $\dim Z(g_1,\ldots,g_{n-\dim Z})=\dim Z$ .

Thus, the condition (3) is ensured. By construction, $\deg g_i \leqslant d_i$ for any $i=1,2,\ldots,r$ . The condition (2) is checked.

Proof of (1). Since $g_1,\ldots,g_r$ are k-linear combinations of $f_1,\ldots,f_r$ , Z is contained in the variety $Z(g_1,\ldots,g_r)$ . We prove that $Z(g_1,\ldots,g_r) \subset Z$ .

If the jth diagonal entry of B is zero, we say that j is a ‘jumping’ index. By Lemma 5.1, for each jumping j, $f_{j}$ vanishes identically on $Z(g_1,\ldots,g_{j-1})$ ; hence, $f_j$ vanishes identically on $Z(g_1,\ldots,g_r)$ as well.

It remains to show that if $\beta$ is not a jumping index, then $f_{\beta}(Q)=0$ for any $Q \in Z(g_1,\ldots,g_r)$ . For each jumping j, remove the jth row and jth column from the matrix B. The resulting matrix C is upper triangular, and its diagonal entries are all 1. In particular, C is invertible. Evaluating (5.2) at Q, using the vanishing of jumping $f_j$ at Q, we see that, for any nonjumping index $\alpha$ ,

\begin{equation*} 0 = g_{\alpha}(Q) = \sum_{\beta \text{ nonjumping}} b_{\alpha\beta}f_{\beta}(Q), \end{equation*}

The matrix associated with the above system of linear equations is the invertible matrix C. Thus, $f_{\beta}(Q)=0$ for any nonjumping $\beta$ . This concludes the proof.

5.2 A lemma on local cohomology

In this subsection, we prove a lemma (Lemma 5.4) on local cohomology in the theory of arithmetic $\mathscr{D}$ -modules. It is the analogue of the fact that, in $\ell$ -adic sheaf theory, the Verdier dual of the shifted constant sheaf $\mathbf{Q}_{\ell}[\dim Z]$ (hence, $\mathbf{Q}_{\ell}[\dim Z]$ itself) is a perverse sheaf if Z is a local complete intersection (cf. [Reference Kiehl and WeissauerKW01, Lemma III 6.5]).

We review some basic concepts about arithmetic $\mathscr{D}$ -modules. The reader is referred to Abe and Caro [Reference Abe and CaroAC18, § 1] or Abe [Reference AbeAbe18, § 1] for up-to-date surveys.

Let k be a perfect field (for us, $k = \mathbf{F}_{q}$ ). Let $\mathcal{O}_{K}$ be a complete discrete valuation ring with residue field k and field of fractions K. Assume that K has characteristic 0. Let $S = \operatorname{Spf}(\mathcal{O}_{K})$ .

Let $\mathcal{P}$ be a smooth formal scheme over S. Berthelot [Reference BerthelotBer96, 2.4] introduced a sheaf $\mathscr{D}^{\dagger}_{\mathcal{P},\mathbf{Q}}$ on $\mathcal{P}$ whose sections are infinite-order differential operators on $\mathcal{P}$ of finite level. Caro introduced several finiteness conditions on coherent $\mathscr{D}^{\dagger}_{\mathcal{P},\mathbf{Q}}$ -modules: overcoherence [Reference CaroCar04, Définition 3.1.1], overholonomicity [Reference CaroCar09, Définition 3.1] and ‘devissability’ by overconvergent F-isocrystals [Reference CaroCar07, Définition 3.2.5 and Reference Caro and TsuzukiCT12, Definition 2.3.1]. Finally, in [Reference Caro and TsuzukiCT12, Theorem 2.3.16], Caro and Tsuzuki proved that with the presence of Frobenius structures, these finiteness notions are equivalent.

Let $\mathrm{Hol}(\mathcal{P})$ denote the strictly full, thick subcategory of $\mathscr{D}^{\dagger}_{\mathcal{P},\mathbf{Q}}$ -modules generated by overholonomic $\mathscr{D}^{\dagger}_{\mathcal{P},\mathbf{Q}}$ -modules that can be endowed with $q^{s}$ -power Frobenius structures for some s, although the Frobenius structure is not part of the defining data. Let $D^{b}_{\mathrm{hol}}(\mathcal{P})$ be the strictly full subcategory of $D^{b}(\mathscr{D}^{\dagger}_{\mathcal{P},\mathbf{Q}})$ consisting of complexes with cohomology lying in $\mathrm{Hol}(\mathcal{P})$ .

Since $D^{b}_{\mathrm{hol}}(\mathcal{P})$ is stable under the usual and extraordinary direct image functors, the usual and extraordinary inverse image functors, and duality functors, as shown in [Reference CaroCar09], one can use these categories to canonically associate to each realizable k-variety (defined below) V a coefficient category $D^{b}_{\mathrm{hol}}(V/K)$ for p-adic cohomology theory, amenable to the Grothendieck six-functor formalism (see [Reference CaroCar12, Reference Abe and CaroAC18]).

A realizable variety V over k is a variety that admits an immersion $V \to \mathcal{P}$ , where $\mathcal{P}$ is a smooth, proper formal scheme over S. For each realizable variety V over k, and any immersion $V \to \mathcal{P}$ as above, we have a functor

(5.3) \begin{equation}D^{b}_{\mathrm{hol}}(V/K) \to D^{b}_{\mathrm{hol}}(\mathcal{P}),\end{equation}

and this functor induces an equivalence between $D^{b}_{\mathrm{hol}}(V/K)$ and the strictly full subcategory of $D^{b}_{\mathrm{hol}}(\mathcal{P})$ consisting of objects that are supported on V, in the sense that the following natural arrows are isomorphisms: i.e.,

\begin{equation*}\mathbf{R}\underline{\Gamma}_{\overline{V}}^{\dagger} (\mathcal{M})\xrightarrow{\sim} \mathcal{M} \xrightarrow{\sim} \mathcal{M}({}^{\dagger}(\overline{V}\mathbin{\rule[0.2em]{0.67em}{0.12em}} V)),\end{equation*}

where $\overline{V}$ is the Zariski closure of V in $\mathcal{P} \otimes_{\mathcal{O}_{K}}k$ . For the definition of the local cohomology functor $\mathbf{R}\underline{\Gamma}^{\dagger}_{Z}$ , see [Reference CaroCar04, 2.1.3 (divsior case) and Définition 2.2.6 (general case)]. For the definition of the functor $\mathcal{M} \mapsto \mathcal{M}({}^{\dagger}Z)$ , see [Reference CaroCar04, Définition 2.2.6].

For any realizable variety V over k, $D^{b}_{\mathrm{hol}}(V/K)$ has a standard t-structure [Reference Abe and CaroAC18, § 1.2]. The objects in the heart of $D^{b}_{\mathrm{hol}}(V/K)$ are analogues to perverse sheaves on V in the $\ell$ -adic theory.

For a morphism $f\colon V \to V^{\prime}$ of realizable varieties, we have the ordinary and extraordinary inverse image functors

\begin{equation*}f^{+}, f^{!} \colon D^{b}_{\mathrm{hol}}(V^{\prime}/K) \to D^{b}_{\mathrm{hol}}(V/K),\end{equation*}

and the ordinary and extraordinary direct image functors

\begin{equation*}f_{+}, f_{!} \colon D^{b}_{\mathrm{hol}}(V/K) \to D^{b}_{\mathrm{hol}}(V^{\prime}/K).\end{equation*}

They satisfy the usual adjunction properties. For each V, there is a duality functor

\[\mathbf{D}_{V}\colon D^{b}_{\mathrm{hol}}(V/K)^{\mathrm{op}} \to D^{b}_{\mathrm{hol}}(V/K),\]

which is a t-exact anti-equivalence satisfying $\mathbf{D}_{V}^{2} = \mathrm{Id}$ . The duality functors swap ordinary and extraordinary direct and inverse images: i.e.,

\begin{align*}\mathbf{D}_{V^{\prime}} f_{!} = f_{+}\mathbf{D}_{V}, \quad&\mathbf{D}_{V^{\prime}} f_{+} = f_{!}\mathbf{D}_{V}, \\f^{!}\mathbf{D}_{V^{\prime}} = \mathbf{D}_{V}f^{+}, \quad&f^{+}\mathbf{D}_{V^{\prime}} = \mathbf{D}_{V}f^{!}.\end{align*}

Example 5.3. Suppose that $\mathcal{P}$ is a purely n-dimensional, proper smooth formal scheme over S. Let H be a divisor of $P = \mathcal{P}\otimes_{\mathcal{O}_{K}}k$ and $U = P \mathbin{\rule[0.2em]{0.67em}{0.12em}} H$ . Then we have an equivalence

\begin{equation*}D^{b}_{\mathrm{hol}}(U/K) \simeq \{\mathcal{M} \in D^{b}_{\mathrm{hol}}(\mathcal{P}) : \mathcal{M} \xrightarrow{\sim} \mathcal{M}({}^{\dagger}H)\}.\end{equation*}

Under this equivalence, the complex $K_{U}[n] = a^{+}K[n]$ , where $a\colon U \to \operatorname{Spec}k$ is the canonical morphism, is represented by $\mathcal{O}_{\mathcal{P},\mathbf{Q}}({}^{\dagger}H)$ , the (specialization of the) sheaf of function on the rigid analytic space $\mathcal{P}_{K}$ overconvergent along H. Moreover, $\mathcal{O}_{\mathcal{P},\mathbf{Q}}({}^{\dagger}H)$ is self-dual, that is, $\mathbf{D}_{U}(K_{U}[n]) \simeq K_{U}[n]$ (see [Reference AbeAbe18, § 1.5.6]). Lastly, we mention that the objects in the heart of the t-structure of $D^{b}_{\mathrm{hol}}(U/K)$ are represented by actual overholonomic modules with overconvergent singularities along H: i.e.,

\begin{equation*}\{ \mathcal{M} \in \mathrm{Hol}(\mathcal{P}) : \mathcal{M} \xrightarrow{\sim} \mathcal{M}({}^{\dagger}H)\}.\end{equation*}

Lemma 5.4. Regard $\mathbf{A}^{N}_{k}$ as a locally closed subscheme of the formal projective space $\widehat{\mathbf{P}}^{N}$ over $\operatorname{Spf}(\mathcal{O}_{K})$ . Set $H = \mathbf{P}^{N}_{k}\mathbin{\rule[0.2em]{0.67em}{0.12em}} \mathbf{A}^{N}_{k}$ to be the hyperplane at infinity. Let $f_{1},\ldots, f_{r}$ be regular functions on $\mathbf{A}^{N}_{k}$ , defining a closed subscheme Z of $\mathbf{A}^{N}_{k}$ . Then

\[\mathcal{H}^{m}\{\mathbf{R}\underline{\Gamma}_{Z}^{\dagger}(\mathcal{O}_{\widehat{\mathbf{P}}^{N},\mathbf{Q}}({}^{\dagger}H))[r]\} = 0\text{ unless } -[\dim Z - (N-r)]\leqslant m \leqslant 0.\]

In particular, if $\dim Z = N-r$ , then $\mathcal{H}^{m}\{\mathbf{R}\underline{\Gamma}_{Z}^{\dagger}(\mathcal{O}_{\widehat{\mathbf{P}}^{N},\mathbf{Q}}({}^{\dagger}H))[r]\}$ is zero unless $m = 0$ .

Proof. First, we prove that

(5.4) \begin{equation}\mathbf{R}\underline{\Gamma}^{\dagger}_{Z}(\mathcal{O}_{\widehat{\mathbf{P}}^{N},\mathbf{Q}}({}^{\dagger}H))[r]\in D_{\mathrm{hol}}^{\leqslant 0}(\widehat{\mathbf{P}}^{N}).\end{equation}

Consider the function $g\colon \mathbf{A}^{N}_{k}\times \mathbf{A}_{k}^{r} \to \mathbf{A}_{k}^{1}$ defined by $g = \sum x_{N+i}f_{i}$ . Let $\mathcal{L}$ be the $\mathscr{D}^{\dagger}$ -module on $\mathbf{A}_{k}^{N+r}$ obtained by regarding the Dwork isocrystal as a $\mathscr{D}^{\dagger}$ -module: i.e., $\mathcal{L}=\mathrm{sp}_{+}(g^{\ast}\mathcal{L}_{\pi})$ (cf. [Reference Abe and CaroAC18, § 1.2.14]). Let $\varpi\colon \mathbf{A}_{k}^{N+r} \to \mathbf{A}^{N}_{k}$ be the projection. Then, by the theorem of Baldassarri and Berthelot (Theorem 2.11),

\begin{equation*}\varpi_{+}(\mathcal{L}) \simeq \mathbf{R}\underline{\Gamma}^{\dagger}_{Z}(\mathcal{O}_{\widehat{\mathbf{P}}^{N},\mathbf{Q}}({}^{\dagger}H))[r].\end{equation*}

The inclusion (5.4) follows from the Artin vanishing theorem for arithmetic $\mathscr{D}$ -modules (see [Reference Abe and CaroAC18, Proposition 1.3.3]).

Next, we show that

(5.5) \begin{equation}\mathbf{R}\underline{\Gamma}^{\dagger}_{Z}(\mathcal{O}_{\widehat{\mathbf{P}}^{N},\mathbf{Q}}({}^{\dagger}H))[r]\in D^{\geqslant N-r-\dim Z}_{\mathrm{hol}}(\widehat{\mathbf{P}}^{N}).\end{equation}

Let $i\colon Z \to \mathbf{A}_{k}^{N}$ be the inclusion morphism. Then, by [Reference Abe and CaroAC18, § 1.1.7], the local cohomology complex $\mathbf{R}\underline{\Gamma}^{\dagger}_{Z}(\mathcal{O}_{\widehat{\mathbf{P}}^{N},\mathbf{Q}}({}^{\dagger}H)) \in D^{b}_{\mathrm{hol}}(\widehat{\mathbf{P}}^{N})$ represents $i_{+}i^{!}(K_{\mathbf{A}_{k}^N}[N]) \in D^{b}_{\mathrm{hol}}(\mathbf{A}^{N}_{k}/K)$ via the functor (5.3). Since $\mathbf{D}_{\mathbf{A}_{k}^{N}}(K_{\mathbf{A}_{k}^N}[N])=K_{\mathbf{A}_{k}^N}[N]$ , (5.5) is equivalent to $i_{+}i^{+}(K_{\mathbf{A}_{k}^N}[N-r]) \in D^{\leqslant \dim Z-(N-r)}_{\mathrm{hol}}(\mathbf{A}_{k}^{N}/K)$ by duality. Since $i_{+}$ is an exact functor [Reference Abe and CaroAC18, Proposition 1.3.2(iii)], it remains to show that

(5.5bis)

To prove this last inclusion, we use some formal properties of the ‘constructible t-structure’ that introduced in [Reference AbeAbe18, § 1.3]. This is a t-structure $({}^{c}D^{\leqslant 0}_{\mathrm{hol}}(V/K),{}^{c}D^{\geqslant 0}_{\mathrm{hol}}(V/K))$ on $D^{b}_{\mathrm{hol}}(V/K)$ for any variety V. The properties of this t-structure that are relevant to us are the following.

1. (a) For any closed subvariety W of V, let $\iota_{W}$ denote the inclusion map. Then ${}^{c}D^{\leqslant 0}_{\mathrm{hol}}(V/K)$ is the full subcategory of $D^{b}_{\mathrm{hol}}(V/K)$ consisting of $\mathcal{M}$ satisfying the property that, for any closed subvariety W,

   \begin{equation*}\mathcal{H}^{m}\iota_{W}^{+}\mathcal{M} = 0 \text{ for any }m > \dim W\end{equation*}
   (see [Reference AbeAbe18, § 1.3.1, second bullet]).

2. (b) If V is nonsingular of pure dimension n, then any overconvergent isocrystal E which admits some $q^{s}$ -Frobenius structure on V determines a $\mathscr{D}^{\dagger}$ -module $\mathcal{E}$ . Then $\mathcal{E}[-n]$ (an object in $D^{b}_{\mathrm{hol}}(V/K)$ of this form is called a smooth object in [Reference AbeAbe18]) is in the heart of the constructible t-structure. This is simply because the constructible t-structure is obtained by gluing smooth objects on smooth locally closed subvarieties (see [Reference AbeAbe18, Proposition 1.3.3]).

We apply Property (5.2) with $V=W=Z$ . This gives ${}^{c}D^{\leqslant 0}_{\mathrm{hol}}(Z/K) \subset D^{\leqslant \dim Z}_{\mathrm{hol}}(Z/K)$ . So, to prove (5.5bis), we just need to show that $i^{+}K_{\mathbf{A}_{k}^N}$ belongs to ${}^{c}D^{\leqslant 0}_{\mathrm{hol}}(Z/K)$ .

We know that $K_{\mathbf{A}_{k}^N}[N]$ comes from the constant overconvergent isocrystal on $\mathbf{A}_{k}^{N}$ . By Property (5.2), it lies in the heart of the constructible t-structure, and, in particular, we have that $K_{\mathbf{A}_{k}^N} \in {}^{c}D^{\leqslant 0}_{\mathrm{hol}}(\mathbf{A}_{k}^{N}/K)$ . Since the functor $i^{+}$ is exact with respect to the constructible t-structure by [Reference AbeAbe18, Lemma 1.3.2], we have $i^{+}K_{\mathbf{A}^{N}_{k}} \in {}^{c}D^{\leqslant 0}_{\mathrm{hol}}(Z/K)$ . This proves (5.5bis) and completes the proof of the lemma.

6. Divisibility of Frobenius eigenvalues

We return to the following situation.

Notation 6.1. We are given a collection of polynomials $f_1,\ldots,f_r \in \mathbf{F}_{q}[x_1,\ldots,x_n]$ , and we denote by

\[ Z = \operatorname{Spec}\mathbf{F}_q[x_1,\ldots,x_n]/(f_1,\ldots,f_r)\]

the vanishing scheme of $f_1,\ldots,f_r$ . By rearranging the order, we assume that $d_1\geqslant \cdots \geqslant d_r$ , where $d_i=\deg f_{i}$ . The codimension $n - \dim Z$ of Z is denoted by c.

The following easy lemma should be well known. It shows that the vanishing of compactly supported cohomology of Z can be controlled by the number of defining equations of Z. So Theorems 1.3 and 1.5 cover all nontrivial cohomology degrees. In its statement, $\mathrm{H}^{i}_{c}(Z)$ could either be $\mathrm{H}^{i}_{\mathrm{rig},c}(Z)$ or $\mathrm{H}^{i}_c(Z_{\overline{\mathbf{F}}_q},\mathbf{Q}_{\ell})$ .

Proof. We have a long exact sequence

\begin{equation*} \cdots \to \mathrm{H}^{i}_c(Y) \to \mathrm{H}^{i}_c(Z) \to \mathrm{H}^{i+1}_c(Y\mathbin{\rule[0.2em]{0.67em}{0.12em}} Z) \to \mathrm{H}^{i+1}_{c}(Y) \to \cdots. \end{equation*}

If $i \leqslant n-1$ , then $\mathrm{H}^{i}_{c}(Y)=0$ by smoothness of Y, Poincaré duality and Artin vanishing. Thus, it suffices to prove that $\mathrm{H}^{i}_{c}(Y\mathbin{\rule[0.2em]{0.67em}{0.12em}} Z)=0$ for $i<n-r+1$ .

Write $Y \mathbin{\rule[0.2em]{0.67em}{0.12em}} Z = \bigcup_{i=r}^{r} U_{i}$ , where $U_i = Y\mathbin{\rule[0.2em]{0.67em}{0.12em}} \{f_i=0\}$ . Then, for $I\subset \{1,2,\ldots,r\}$ , $U_{I}=\bigcap_{i\in I}U_{i}$ equals $Y\mathbin{\rule[0.2em]{0.67em}{0.12em}}\{\prod_{i\in I}f_{i}=0\}$ . We have a Mayer–Vietoris spectral sequence

\begin{equation*} E_{1}^{-a,b} = \bigoplus_{|I|=a+1}\mathrm{H}_{c}^{b}(U_{I}) \Rightarrow \mathrm{H}^{b-a}_c(Y\mathbin{\rule[0.2em]{0.67em}{0.12em}} Z). \end{equation*}

Since each $U_{I}$ is a smooth affine variety of dimension n, by Poincaré duality and Artin vanishing again, $\mathrm{H}^{i+1}_c(U_{I}) = 0$ if $i < n-1$ . It follows that

\begin{equation*} E^{-a,b}_{1} \neq 0 \Longrightarrow \begin{cases} b \geqslant n \text{ and }\\ a \leqslant r-1, \end{cases} \Longrightarrow b-a\geqslant n-r+1. \end{equation*}

Therefore, $\mathrm{H}^{i}_{c}(Y\mathbin{\rule[0.2em]{0.67em}{0.12em}} Z)=0$ if $i < n-r+1$ .

The remainder of this section is devoted to the proofs of Theorems 1.3 and 1.5.

6.1 Step 1: Reduction

Recall that $n - \dim Z$ is denoted by c. By Lemma 5.2, there exists a finite extension $k^{\prime}$ of $\mathbf{F}_q$ , and a collection of polynomials $g_1,\ldots,g_r$ , such that:

• – $\deg g_1=d_1$ , $\deg g_i \leqslant d_i$ ;

• – Z is the common zero locus of $g_1,\ldots,g_r$ ; and

• – $\operatorname{Spec}k^{\prime}[x_1,\ldots,x_n]/(g_1,\ldots,g_c)$ has dimension equal to $\dim Z$ .

Since the conclusion of Theorem 1.3 is not sensitive to the base field, and since we have

\begin{align*} \nu_j(n;\deg g_1,\ldots,\deg g_r) & \geqslant \nu_j(n;d_1,\ldots,d_r)\\ \epsilon_{m}(n;\deg g_1,\ldots,\deg g_r) &\geqslant \epsilon_{m}(n;d_1,\ldots,d_r),\end{align*}

it suffices to prove the theorems with $\mathbf{F}_q$ replaced by $k^{\prime}$ and $f_i$ replaced by $g_i$ . Thus, it suffices to prove Theorems 1.3 and 1.5 under the following additional hypothesis.

(6.1) \begin{equation} \textit{The scheme $\operatorname{Spec}\mathbf{F}_q[x_1,\ldots,x_n]/(f_1,\ldots,f_c)$ has dimension equal to $\dim Z$.}\end{equation}

6.2 Step 2: Slope estimates

First, we prove that the numbers $\nu_{j}(n; d_1,\ldots,d_r)$ and $\epsilon_{m}(n;d_1,\ldots,d_r)$ provide lower bounds of the q-order of the Frobenius eigenvalues of $\mathrm{H}^{\ast}_{\mathrm{rig},c}(Z)$ . Later, we will bootstrap this bound to a bound of q-divisibility of algebraic numbers.

Recall the meaning of g (2.2), G (Construction 2.3), and our convention of the sets I, $I^{\prime}$ and $I^{\prime\prime}$ made in Notation 4.1, as well as the spaces $B_{I}$ (2.1). Rewrite the overconvergent Dwork complex (2.2) as the total complex of the following double complex (in order to save ink, we have omitted the monomials $\mathrm{d}x^I/x^I$ in the expression).

(6.2)

In the diagram, the horizontal differentials are induced by $D^{\prime}_{n+1},\ldots,D^{\prime}_{n+r}$ , and the vertical ones are induced by $D^{\prime}_1,\ldots,D^{\prime}_{n}$ , where

(6.3) \begin{equation}D^{\prime}_{i} = x_{i}\frac{\partial}{\partial x_i} + \pi x_{i}\frac{\partial G}{\partial x_{i}}=\exp(-\pi G) \circ x_{i}\frac{\partial}{\partial x_i}\circ \exp(\pi G), \quad i=1,2,\ldots,n+r.\end{equation}

The following lemma shows that the 0th, 1st, $\ldots$ and $(c-1)$ st columns of the $E_1$ -page of the spectral sequence associated to the double complex (6.2) are all zero.

Lemma 6.4. For each $0\leqslant i\leqslant n$ , the ith row of (6.2) is exact in cohomology degree $0,1,\ldots,c-1$ .

Proof. The complex (6.2) uses ‘toric’ conventions, and it is more suitable for later chain-level manipulations. In the following proof, we use an equivalent ‘affine’ convention. Write $B = K\langle x_{1},\ldots,x_{n+r} \rangle^{\dagger}$ , and let

\begin{equation*}D_{j} = \frac{\partial}{\partial x_{j}} + \pi \frac{\partial G}{\partial x_{j}}, \quad j=1,\ldots,n+r.\end{equation*}

Then (6.2) can be written as follows.

(6.2′)

In this double complex, the horizontal arrows are induced by $D_{n+1},\ldots,D_{n+r}$ , and the vertical arrows are induced by $D_{1},\ldots,D_{n}$ .

First, we show that the zeroth row of (6.2ʹ) is exact in degrees $0,1,\ldots,c-1$ . Since the ith row of (6.2ʹ) is an $\binom{n}{i}$ -fold direct sum of the zeroth row, the desired exactness, in general, will follow.

The zeroth row of (6.2ʹ) is itself the total complex of a double complex.

(6.4)

In the diagram, the horizontal arrows are induced by

\begin{equation*}D_{n+1},\ldots, D_{n+c}\end{equation*}

and the vertical arrows are induced by

\begin{equation*}D_{n+c+1},\ldots, D_{n+r}.\end{equation*}

Thus, the ith row is the $\binom{r-c}{i}$ -fold direct sum of the zeroth row. If we prove that the zeroth row of (6.4) is acyclic except in top cohomology degree, then every row of (6.4) will be exact except in top cohomology degree. Thus, the total complex of (6.4), which is the zeroth row of the original (6.2ʹ), will have vanishing cohomology in degrees $0,1,\ldots,c-1$ .

To prove acyclicity, we make the following auxiliary construction. Consider the projection

\[\varpi^{\prime}\colon \mathbf{A}^{n+r} \to \mathbf{A}^{n+r-c}, \quad (x_{1}, \ldots, x_{n+r}) \mapsto (x_{1}, \ldots, x_{n}, x_{n+c+1}, \ldots, x_{n+r}).\]

Define $\mathcal{P} = \widehat{\mathbf{P}}^{n+r-c}_{\mathcal{O}_{K}} \times \widehat{\mathbf{P}}^{c}_{\mathcal{O}_{K}}$ and $\mathcal{P}^{\prime} = \widehat{\mathbf{P}}^{n+r-c}_{\mathcal{O}_{K}}$ . The special fiber of $\mathcal{P}$ (respectively, $\mathcal{P}^{\prime}$ ) contains $\mathbf{A}^{n+r}$ (respectively, $\mathbf{A}^{n+r-c}$ ) as a Zariski open subset, with complement H (respectively, $H^{\prime}$ ). Let $\mathcal{L}$ be the overholonomic $\mathscr{D}^{\dagger}_{\mathcal{P}, \mathbf{Q}}$ -module associated with the Dwork crystal $g^{\prime\ast}\mathcal{L}_{\pi}$ , where

\[g^{\prime} = x_{n+1} f_{1} + \cdots + x_{n+c} f_{c}.\]

By construction, $\mathcal{L}$ has an overconvergent singularity along H, i.e., $\mathcal{L} = \mathcal{L}({}^{\dagger}H)$ , and is naturally a $\mathscr{D}^{\dagger}_{\mathcal{P},\mathbf{Q}}({}^{\dagger}H)$ -module, where $\mathscr{D}^{\dagger}_{\mathcal{P},\mathbf{Q}}({}^{\dagger}H)$ is the sheaf of differential operators on $\mathcal{P}$ of finite level with overconvergent singularities along H (see [Reference BerthelotBer96, § 4.2.5] or [Reference Baldassarri and BerthelotBB04, § 2.5]).

Thus, $\mathcal{L}$ represents an object of $D^{b}_{\mathrm{hol}}(\mathbf{A}^{n+r}/K)$ . By Theorem 2.11,

\begin{equation*}\varpi^{\prime}_{+} \mathcal{L} \simeq \mathbf{R}\underline{\Gamma}^{\dagger}_{Z^{\prime}}(\mathcal{O}_{\mathcal{P}^{\prime}, \mathbf{Q}}({}^{\dagger}H^{\prime}))[c],\end{equation*}

where $Z^{\prime}$ is the zero locus of $f_{1}, \ldots, f_{c}$ in the affine space $\mathbf{A}^{n} \times \mathbf{A}^{r-c}$ : i.e.,

\[Z^{\prime} = \{x \in \mathbf{A}^{n} : f_{1}(x) = \cdots = f_{c}(x) = 0\} \times \mathbf{A}^{r-c}.\]

Given Hypothesis (6.1), we have $\dim Z^{\prime} = n + r - 2c$ , and it follows from Lemma 5.4 that $\mathbf{R}\underline{\Gamma}^{\dagger}_{Z^{\prime}}(\mathcal{O}_{\mathcal{P}^{\prime}, \mathbf{Q}}({}^{\dagger}H^{\prime}))[c]$ , which is initially a complex of $\mathscr{D}^{\dagger}_{\mathcal{P}^{\prime},\mathbf{Q}}({}^{\dagger}H^{\prime})$ -modules, is concentrated in degree 0 only. Consequently, by [Reference Baldassarri and BerthelotBB04, Theorem 2.3] (which asserts that the category of $\mathscr{D}^{\dagger}_{\mathcal{P}^{\prime},\mathbf{Q}}({}^{\dagger}H)$ -modules is equivalent to the category of modules over the overconvergent Weyl algebra, the equivalence being induced by the functor $\mathrm{H}^{0}(\mathcal{P}^{\prime},-)$ ), we find that

\[\mathrm{H}^{0}(\mathcal{P}^{\prime}; \mathbf{R}\underline{\Gamma}^{\dagger}_{Z^{\prime}}(\mathcal{O}_{\mathcal{P}^{\prime}, \mathbf{Q}}({}^{\dagger}H^{\prime}))[c])\]

is a module over the overconvergent Weyl algebra $\mathrm{H}^{0}(\mathcal{P}^{\prime};\mathscr{D}^{\dagger}_{\mathcal{P}^{\prime},\mathbf{Q}}({}^{\dagger}H^{\prime}))$ , and is thus concentrated in degree 0 only.

Now, the zero row of (6.4), shifted to the left by r, is

\[{B} \to B^{\oplus c} \to B^{\oplus \binom{c}{2}} \to \cdots \to B^{\oplus \binom{c}{c-1}} \to \underset{\bullet}{B^{\oplus \binom{c}{c}}}\]

(where the bullet indicates the degree zero item of the chain complex). We want to show that it is acyclic except in degree 0. Since this is a relative de Rham complex with respect to $\varpi^{\prime}$ , according to [Reference Baldassarri and BerthelotBB04, p. 197, Remark], it represents the complex $\mathrm{H}^{0}(\mathcal{P}^{\prime}; \varpi^{\prime}_{+}\mathcal{L})$ , which is isomorphic to

\[\mathrm{H}^{0}(\mathcal{P}^{\prime}; \mathbf{R}\underline{\Gamma}^{\dagger}_{Z^{\prime}} (\mathcal{O}_{\mathcal{P}^{\prime}, \mathbf{Q}} ({}^{\dagger} H^{\prime})) [c]),\]

by Theorem 2.11. As discussed in the preceding paragraph, this latter complex is indeed concentrated in degree 0. This completes the proof.

Since Lemma 6.4 implies that the spectral sequence associated to the double complex (6.2) satisfies

\begin{equation*} E_{1}^{i,j} = 0, \quad \forall i < c,\end{equation*}

in Figure 1, only the shaded part contributes to the final abutment of the spectral sequence.

Figure 1 The double complex.

For this reason, $\mathrm{H}^{n+r-j}_{\mathrm{rig}}(\mathbf{A}_{\mathbf{F}_q}^{n+r},\mathcal{L}_{\pi})$ is in fact a subquotient of

(6.5) \begin{equation} \bigoplus_{\substack{|I|=n+r-j \\ |I^{\prime\prime}| \geqslant c}} B_{I},\end{equation}

and Lemma 3.2 can be refined.

Lemma 6.5. In Situation 6.1, under Hypothesis 6.1, we have that the Fredholm determinant $\det(1-t\cdot F| \mathrm{H}^{n-r+j}_{\mathrm{rig},c}(Z))$ is a factor of

\[ \prod_{\substack{|I|=n+r-j \\ |I^{\prime\prime}| \geqslant c}}\det(1-t\cdot q^{j-r}\alpha_{a}| B_{I}).\]

Hence, every Frobenius eigenvalue of $\mathrm{H}^{n-r+j}_{\mathrm{rig},c}(Z)$ is a reciprocal root of $\det(1-t\cdot q^{j-r}\alpha_{a} | B_{I})$ for some I with $|I|=n+r-j$ , $|I^{\prime\prime}| \geqslant c$ .

To proceed, there are two cases.

6.2.1 First case

$j\geqslant r-c$ . This case corresponds to Theorem 1.3. All the relevant spaces $B_{I}$ lie on the lower slant line of Figure 1. Let $\gamma$ be one reciprocal root of $\det(1-t\cdot q^{j-r}\alpha_{a} | B_{I})$ . By Lemma 4.2, the q-order of any reciprocal root $\gamma$ of $\det(1-t\cdot \alpha_{a}| B_I)$ is at least

\[\frac{1}{d_1}\bigg( n+r-j+ \sum_{i\in I^{\prime\prime}} (d_1-d_i-1) \bigg).\]

Since $d_1\geqslant \cdots \geqslant d_r$ and $|I^{\prime\prime}|\geqslant c$ ,

\begin{align*} \sum_{i\in I^{\prime\prime}}(d-d_i-1) &\geqslant \sum_{i=1}^{c}(d-d_i-1)-\sum_{i=c+1}^{|I^{\prime\prime}|-c}d_{i}^{*} \\ &\geqslant \sum_{i=1}^{c}(d-d_i-1)-\sum_{i=c+1}^{r}d_{i}^*,\end{align*}

where recall that

\begin{equation*} d_i^* = \begin{cases} d_i & \text{ if }1\leqslant i \leqslant c, \\ 1 & \text{ if }i>c \text{ and }d_i = d_1, \\ 0 & \text{ if }i> c \text{ and }d_i < d_1. \end{cases}\end{equation*}

Moreover, by [Reference WanWan00, Lemma 3.1], the q-order of every reciprocal root of $\det(1-t\cdot \alpha_{a}| B_I)$ is at least $|I^{\prime\prime}| \geqslant c$ .

Remark 6.6. It should be noted that the cited lemma was stated for a certain Banach space denoted by $B^{J_1,J_2}$ in [Reference WanWan00]. In our context, its role is subsumed by the overconvergent space $B_{I}$ , where $J_1=I^{\prime}$ , $J_2=I^{\prime\prime}$ . The proof of the cited lemma uses only Dwork trace formula, which is applicable to $B_{I}$ as well.

In fact, the cited lemma actually says that the reciprocal roots of $\det(1-t\cdot \alpha_{a} | B_I)$ are algebraic integers and are divisible by $q^{|I^{\prime\prime}|}$ in the ring of algebraic integers.

Hence, Lemma 6.5 implies that the q-order of every Frobenius eigenvalue of $\mathrm{H}^{n-r+j}_{\mathrm{rig},c}(Z)$ is at least

(6.6) \begin{equation} j - (r - c) + \max\left\{ 0, \left\lceil \frac{n-j+(r-c)-\sum_{i=1}^{r}d^*_i}{d_1} \right\rceil \right\}.\end{equation}

Making the change of variable $j -(r-c) \to j$ , the above argument implies that the q-order of Frobenius eigenvalues of $\mathrm{H}^{\dim Z+j}_{\mathrm{rig},c}(Z)$ are at least $\nu_{j}(n;d_1,\ldots,d_r)$ .

6.2.2 Second case

$0\leqslant j < r-c$ , which corresponds to Theorem 1.5. In this case, the spaces $B_I$ appearing in (6.5) all lie on the upper slant line of Figure 1. Thus, $r\geqslant |I^{\prime\prime}| \geqslant r-j > c$ . It follows from Lemma 4.2 that every reciprocal root $\gamma$ of $\det(1-t\cdot \alpha_{a}| B_I)$ satisfies $\operatorname{ord}_q\gamma \geqslant ({1}/{d_1})( n+r-j + \sum_{i\in I^{\prime\prime}} (d_1-d_i-1))$ . Since $|I^{\prime\prime}| \geqslant r-j$ , arguing as in the first case,

(6.7) \begin{align} &\frac{1}{d_1}\bigg( n+r-j + \sum_{i\in I^{\prime\prime}} (d_1-d_i-1)\bigg) \nonumber\\ \geqslant &\frac{1}{d_1} \bigg( n+r-j + \sum_{i=1}^{r-j} (d_1-d_i-1) - \sum_{i=r-j+1}^{r}d_{i}^* \bigg)\nonumber\\ = &\frac{1}{d_1} \bigg( n - \sum_{i=1}^{r-j} d_i - \sum_{i=r-j+1}^{r}d_{i}^{*}\bigg) + r-j. \end{align}

Again, by [Reference WanWan00, Lemma 3.1], we have $\operatorname{ord}_q\gamma \geqslant |I^{\prime\prime}|\geqslant r-j$ . Hence, by Lemma 6.5, the q-order of every Frobenius eigenvalue of $\mathrm{H}^{n-r+j}_{\mathrm{rig},c}(Z)$ is at least

\begin{equation*} \max\left\{ 0, \left\lceil \frac{n-\sum_{i=1}^{r-j}d_i-\sum_{i=r-j+1}^{r}d_i^*}{d_1} \right\rceil \right\} = \epsilon_{j}(n;d_1,\ldots,d_r).\end{equation*}

6.3 Step 3. Bootstrap

To finish the proof of Theorems 1.3 and 1.5, it remains to explain why the a priori weaker q-order estimate given above implies the integrality as well as divisibility in the ring of algebraic integers. All we need is the following lemma (see Lemma 6.8 below).

Lemma 6.7. If $\gamma$ is a reciprocal root of the Fredholm determinant $\det(1 - t \alpha_{a} | B_I)$ , then $\gamma$ is an algebraic integer, and any Galois conjugate of it is still a reciprocal root of $\det(1-t\alpha_{a}| B_I)$ .

That $\gamma$ is an algebraic integer had been shown by the first author in [Reference WanWan00, Lemma 3.1], but it will naturally come up again in the argument below.

Before proving Lemma 6.7 let us take a tour through the Dwork theory of exponential sums. For $J \subset \{1,2,\ldots,n+r\}$ , let $X_{(J)}$ be the linear subspace of $\mathbf{A}^{n+r}$ defined by the vanishing of the variables $(x_{j})_{j\in J}$ , and let $X_{(J)}^*$ be its standard embedded torus. Let $g_{J}$ be the restriction of g to $X_{(J)}$ . For the nontrivial additive character $\Psi$ (see (3.1)), consider the exponential sum

\begin{equation*} S_{(J),m}^* = \sum_{x\in X_{(J)}^*(\mathbf{F}_{q^m})} (\Psi\circ \mathrm{Tr}_{\mathbf{F}_{q^m}/\mathbf{F}_q})(g_J(x)).\end{equation*}

Let $B^J = B/\sum_{j\in J}x_{j}B$ . Applying Dwork trace formula (Theorem 3.7) to $g_{J}$ gives

(6.8) \begin{equation} (q^m-1)^{n+r-|J|} \mathrm{Tr}(\alpha_{a}^{m}|{B^J}) = S_{(J),m}^*,\end{equation}

where, as before, $\alpha_{a}$ is the nuclear operator defined in § 2.

For $I \subset \{1,2,\ldots,n+r\}$ , let $B_{I} = (\prod_{i\in I}x_{i})\cdot B$ . Then $B/B_I$ should be thought of as a dagger algebra lifting the divisor $D_{I}=\{\prod_{i\in I}x_{i}=0\}$ . The divisor $D_I$ , being of strict normal crossings, has a standard semisimplicial resolution

By inclusion-exclusion, $\mathrm{Tr}(\alpha_{a}^m|{B_I})=\sum_{J\subset I} (-1)^{|J|}\mathrm{Tr}(\alpha_{a}^m|{B^J})$ . Exponentiating, we get

(6.9) \begin{equation} \det(1 - t\alpha_{a}|{B_I}) = \prod_{J\subset I} \det(1 - t\alpha_{a}|{B^J})^{(-1)^{|J|}}.\end{equation}

Proof of Lemma 6.7. The proof of the assertion is based on a similar, but more precise, argument used in the visibility proof.

For each $J \subset \{1,2,\ldots,n+r\}$ , by (6.8),

\begin{equation*} \det(1-t\alpha_{a}|{B^J})^{\delta^{n+r-|J|}} = L^*_{J}(t)^{(-1)^{n+r-|J|-1}}, \end{equation*}

where

\begin{equation*} L_{J}^*(t) = \exp\biggl\{ \sum_{m=1}^{\infty}S_{(J),m}^{*} \frac{t^m}{m}\biggr\}. \end{equation*}

But, by Lemma 3.8 (applying to the lower dimensional affine space $X_{(J)}$ ), $L_{J}^{*}$ is an alternating product of zeta functions of $\zeta_{Z^*\cap X_{(E^{\prime})}}(q^{|E^{\prime}|}t)$ , where $E^{\prime}$ is the intersection of a subset E of J with $\{1,2,\ldots,r\}$ . Thus, by the second equation in Definition 3.4, as well as (6.9), $\det(1-t\alpha_{a}|{B_I})$ is an infinite alternating product of ‘shifted’ zeta functions $\zeta_{Z^*\cap X_{(E^{\prime})}}(q^{M}t)$ , where $M \in \mathbf{N}$ .

Therefore, any reciprocal root of $\det(1-t\alpha_{a}| B_I)$ , say, $\gamma$ , is a reciprocal zero or reciprocal pole of some zeta function $\zeta_{Z^*\cap X_{(J)}}(q^{M}t)$ , not being canceled in the infinite product, for some natural number M. Therefore, $\gamma$ is an algebraic integer. Since such a shifted zeta function is a ratio of integral polynomials of constant term one, the (reciprocal) minimal polynomial of $\gamma$ must not be canceled either.

By Lemma 6.7, all the conjugates of $\gamma$ are still reciprocal roots of $\det(1-t\alpha_{a}| B_I)$ . Thus, the q-orders of the conjugates are bounded by (6.6) or (6.7), depending on $|I|$ . The proof of Theorems 1.3 and 1.5 is then concluded thanks to the following elementary lemma.

Lemma 6.8. Fix an algebraic closure $\overline{\mathbf{Q}}_p$ of $\mathbf{Q}_p$ . Let $\gamma \in \overline{\mathbf{Q}}_{p}$ be an algebraic integer. Suppose that, for any automorphism $\sigma$ of $\overline{\mathbf{Q}}_{p}$ , $\operatorname{ord}_q(\sigma(\gamma))\geqslant m$ . Then $q^{m} \mid \gamma$ in the ring of algebraic integers.

Proof. Let $P(T) = T^{e} - a_{1}T^{e-1} + \cdots + (-1)^{e}a_{e}$ be the minimal polynomial of $\gamma$ . Then, for every i, $a_{i} \in \mathbf{Z}$ . Since $a_i$ is an elementary symmetric polynomial of $\sigma(\gamma)$ , the hypothesis implies that $\operatorname{ord}_q(a_i) \geqslant m$ . Hence, we can write $a_i = q^{im} \cdot b_i$ for some $b_i \in \mathbf{Z}$ . The numbers $\sigma(\gamma)\cdot q^{-m}$ all satisfy the polynomial equation $T^e-b_1T^{e-1} + \cdots + (-1)^{e}b_{e} = 0$ , and thus are all algebraic integers. The lemma is proved.