Abstract
We provide an effective version of Katz’ criterion for finiteness of the monodromy group of a lisse, pure of weight zero, ℓadic sheaf on a normal variety over a finite field, depending on the numerical complexity of the sheaf.
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1 Introduction
Let X be a smooth, geometrically irreducible variety over a field \(k=\mathbb {F}_{q}\) of characteristic p. Fix a prime ℓ≠p, and consider the category \(\mathcal S(X,\bar {\mathbb {Q}}_{\ell })\) of lisse \(\bar {\mathbb {Q}}_{\ell }\) sheaves on X. A sheaf will be said to be pure (of a certain weight) if it is so for every embedding \(\bar {\mathbb {Q}}_{\ell }\to \mathbb {C}\).
Fix a geometric generic point \(\bar \eta \) of X. Every sheaf \(\mathcal {F}\in \mathcal S(X,\bar {\mathbb {Q}}_{\ell })\) of rank r corresponds to a continuous representation \(\pi _{1}(X,\bar \eta )\to \text {GL}(r,\bar {\mathbb {Q}}_{\ell })\). The Zariski closure of its image (respectively of the image of the subgroup \(\pi _{1}(\overline X,\bar \eta )\), where \(\overline X=X\otimes _{\mathbb {F}_{q}}\overline {\mathbb {F}_{q}}\)) is called the arithmetic monodromy group (resp. the geometric monodromy group) of \(\mathcal {F}\). It is well known that, under certain conditions, these groups govern the distribution of the Frobenius traces of the sheaf \(\mathcal {F}\) on the set of rational points of X over larger and larger extensions of \(\mathbb {F}_{q}\).
Several methods have been used in the literature to determine these groups, such as using local monodromies to deduce enough properties of it so that there is only one possibility [3, Chapter 11], or computing their moments and applying Larsen’s alternative [6, Chapter 2]. In some cases, one can determine that there are only two options: either the group is finite of it is one specific group [11]. In any case, the question of determining whether the monodromy groups are finite is an interesting one.
In general, we have the following criterion, proven in [4, 8.14] for curves and in [7, Proposition 2.1] for higher dimensional varieties:
Proposition 1
Suppose that \(\mathcal {F}\) is geometrically irreducible and pure of weight 0 and its determinant is arithmetically of finite order. Then the following conditions are equivalent:

1.
The geometric monodromy group G_{geom} of \(\mathcal F\) is finite.

2.
The arithmetic monodromy group G_{arith} of \(\mathcal F\) is finite.

3.
For every finite extension \(\mathbb {F}_{q}\subseteq \mathbb {F}_{q^{r}}\) and every \(t\in X(\mathbb {F}_{q^{r}})\), the Frobenius trace of \(\mathcal F\) at t is an algebraic integer.

4.
For every finite extension \(\mathbb {F}_{q}\subseteq \mathbb {F}_{q^{r}}\) and every \(t\in X(\mathbb {F}_{q^{r}})\), the Frobenius eigenvalues of \(\mathcal F\) at t are roots of unity.
However, this criterion does not provide an effective algorithm, since one needs to check that condition (3) or (4) holds for every finite extension of \(\mathbb {F}_{q}\). In practice, one usually needs to resort to adhoc methods to check that these conditions hold for a particular \(\mathcal F\) (see e.g. [7, Theorem 3.1]). In this article, we prove the following effective version of the criterion, relying on the complexity of \(\mathcal F\) as defined by Sawin [10, Definition 6.4].
Theorem 2
Suppose that X is given with a projective embedding \(u:X\hookrightarrow {\mathbb P}^{n}_{k}\). Then there exist explicit constants N_{1} = N_{1}(u, d, r, C) and N_{2} = N_{2}(u, d, r, C) such that for every geometrically irreducible \(\mathcal F\in {\mathcal S}(X,\bar {\mathbb {Q}}_{\ell })\) of rank r, pure of weight 0, whose determinant is arithmetically of finite order, with complexity \(c_{u}(\mathcal F)\leq C\) and Edefined, where \(\mathbb Q\subseteq E\) is a finite extension of degree ≤ d, the following conditions are equivalent:

1.
The geometric monodromy group G_{geom} of \(\mathcal F\) is finite.

2.
The arithmetic monodromy group G_{arith} of \(\mathcal F\) is finite.

3.
For every finite extension \(\mathbb {F}_{q}\subseteq \mathbb {F}_{q^{r}}\) of degree ≤ N_{1} and every \(t\in X(\mathbb {F}_{q^{r}})\), the Frobenius trace of \(\mathcal F\) at t is an algebraic integer.

4.
For every finite extension \(\mathbb {F}_{q}\subseteq \mathbb {F}_{q^{r}}\) of degree ≤ N_{2} and every \(t\in X(\mathbb {F}_{q^{r}})\), the Frobenius eigenvalues of \(\mathcal F\) at t are roots of unity.
If X is a smooth curve, we have a more explicit version which does not make use of the general concept of complexity:
Theorem 3
Suppose that X is a smooth curve, Y its smooth projective closure and D = Y ∖X. Then there exist explicit constants N_{1} = N_{1}(X, d, r, e) and N_{2} = N_{2}(X, d, r, e) such that for every geometrically irreducible \(\mathcal F\in {\mathcal S}(X,\bar {\mathbb {Q}}_{\ell })\) of rank r, pure of weight 0, whose determinant is arithmetically of finite order, all whose breaks at every \(t\in D(\overline {\mathbb F}_{q})\) are ≤ e, and Edefined, where \(\mathbb Q\subseteq E\) is a finite extension of degree ≤ d, the following conditions are equivalent:

1.
The geometric monodromy group G_{geom} of \(\mathcal F\) is finite.

2.
The arithmetic monodromy group G_{arith} of \(\mathcal F\) is finite.

3.
For every finite extension \(\mathbb {F}_{q}\subseteq \mathbb {F}_{q^{r}}\) of degree ≤ N_{1} and every \(t\in X(\mathbb {F}_{q^{r}})\), the Frobenius trace of \(\mathcal F\) at t is an algebraic integer.

4.
For every finite extension \(\mathbb {F}_{q}\subseteq \mathbb {F}_{q^{r}}\) of degree ≤ N_{2} and every \(t\in X(\mathbb {F}_{q^{r}})\), the Frobenius eigenvalues of \(\mathcal F\) at t are roots of unity.
Unfortunately, the explicit constants N_{1} and N_{2} we obtain in these theorems are still too large to be useful in practice, so at the moment these results are mainly of theoretical interest. See Section 4 for some numerical examples and some potential ways to optimize them.
2 ℓAdic Sheaves and Trace Functions
Let X be as in the previous section. Every sheaf \(\mathcal F\in \mathcal S(X,\bar {\mathbb {Q}}_{\ell })\) defines a trace function \({\varPhi }_{\mathcal F}:{\coprod }_{m\geq 1}X(k_{m})\to \bar {\mathbb {Q}}_{\ell }\) (where k_{m} denotes the degree m extension of k) given by
where t ∈ X(k_{m}) and \(\bar t\) is a geometric point over t. By Chevotarev’s density theorem, two semisimple sheaves \(\mathcal F,~\mathcal {G}\in \mathcal S(X,\bar {\mathbb {Q}}_{\ell })\) are isomorphic if and only if \({\varPhi }_{\mathcal F}={\varPhi }_{\mathcal G}\).
For every m ≥ 1, denote by \({\varPhi }_{\mathcal F,m}:X(k_{m})\to \bar {\mathbb {Q}}_{\ell }\) the restriction of \({\varPhi }_{\mathcal F}\) to X(k_{m}). For a smooth curve X, Deligne proved [1] the following bounded version of the previous statement. Let Y be a smooth compactification of X, and D = Y ∖X. For every \(s\in D(\bar k)\) and \(\mathcal F\in \mathcal S(X,\bar {\mathbb {Q}}_{\ell })\), let \(\alpha _{s}(\mathcal F)\) be the largest break of \(\mathcal F\) at s. Then we have
Theorem 4
[1, Proposition 2.5] Let X be a smooth curve and \(\mathcal F,\mathcal G\in \mathcal S(X,\bar {\mathbb {Q}}_{\ell })\) two lisse semisimple sheaves of rank r. Then \(\mathcal F\) and \(\mathcal G\) are isomorphic if and only if \({\varPhi }_{\mathcal F,m}={\varPhi }_{\mathcal G,m}\) for every m ≤ N, where
\(\log _{q}^{+}=\max \limits \{0,\log _{q}\}\) and \(b_{1}(X)=\dim \mathrm {H}^{1}_{c}(X,\bar {\mathbb {Q}}_{\ell })\) is the first Betti number with compact supports of X.
We will generalize this to higher dimensional varieties, using Sawin’s complexity theory [10] as a replacement for the ramification data. Assume that X is geometrically irreducible, smooth quasiprojective of dimension d over k, given with an embedding \(u:X\hookrightarrow \mathbb {P}^{n}_{k}\). See [10, Definitions 3.2 and 6.4] for the definition of the complexity \(c_{u}(\mathcal F)\in \mathbb N\) of a lisse sheaf \(\mathcal F\in \mathcal S(X,\bar {\mathbb {Q}}_{\ell })\) (or, more generally, an object of \({D^{b}_{c}}(X,\bar {\mathbb {Q}}_{\ell })\)). By [10, Theorem 5.2] there is an explicit constant A_{n} such that \(c_{u}(\mathcal F\otimes \mathcal G)\leq A_{n} c_{u}(\mathcal F)c_{u}(\mathcal G)\) for any \(\mathcal F, \mathcal G\in \mathcal S(X,\bar {\mathbb {Q}}_{\ell })\). In fact, by [10, Theorem 8.1], one may take
We then have the following generalization of Theorem 4:
Theorem 5
Let \(\mathcal F,\mathcal G\in \mathcal S(X,\bar {\mathbb {Q}}_{\ell })\) be two lisse semisimple sheaves of rank r of complexity ≤ C. Then \(\mathcal F\) and \(\mathcal G\) are isomorphic if and only if \({\varPhi }_{\mathcal F,m}={\varPhi }_{\mathcal G,m}\) for every m ≤ N, where
The proof follows closely that of [1, Proposition 2.5]. As there, we can decompose \(\mathcal F\) and \(\mathcal G\) as direct sums
where, for every i, there is some n_{i} ≥ 1 such that \({\mathcal H}_{i}\) is a geometrically irreducible lisse sheaf on \(X_{n_{i}}:=X\otimes _{k} k_{n_{i}}\) with determinant of finite order, \(\mathcal W_{i}\) and \(\mathcal W^{\prime }_{i}\) are geometrically constant on \(X_{n_{i}}\) (and at least one of them is nonzero), \(p_{i}:X_{n_{i}}\to X\) is the natural projection, and the \({\mathcal H}_{i}\) and their Galois conjugates are pairwise geometrically nonisomorphic.
Let \(\mathcal H_{i,j}\) for 0 ≤ j < n_{i} denote the Frobenius conjugates of \(\mathcal H_{i}\). For every m ≥ 1, let I_{m} be the set {i ∈ I : n_{i}m}. As in [1, Lemme 2.6], we can show:
Lemma 6
Let \(C=\max \limits \{c_{u}(\mathcal F),c_{u}(\mathcal G)\}\) and \(N_{0}=2\log _{q}^{+}(2A_{n} C^{2})\). For m > N_{0}, the functions \({\varPhi }_{\mathcal H_{i,j},m}:X(k_{m})\to \bar {\mathbb {Q}}_{\ell }\) (i ∈ I_{m}, 0 ≤ j < n_{i}) are linearly independent.
Let \({\sum }_{i,j}\lambda _{i,j}{\varPhi }_{\mathcal H_{i,j},m}=0\) be a nontrivial linear combination, and assume without loss of generality that \(\lambda _{i_{0},j_{0}}=1\) and λ_{i, j}≤ 1 for every i, j. We may also assume that \(\mathcal H_{i_{0},j_{0}}\) is a direct summand of \(\mathcal F\), by interchanging \(\mathcal F\) and \(\mathcal G\) if necessary. Then
Since the \({\mathscr{H}}_{i,j}\) are geometrically irreducible and pairwise nonisomorphic, we have \(\mathrm {H}^{2d}_{c}(X\otimes \bar k,\mathcal H_{i,j}\otimes \widehat {\mathcal H_{i_{0},j_{0}}})=0\) if (i, j) ≠ (i_{0},j_{0}) and \(\bar {\mathbb {Q}}_{\ell }(d)\) if (i, j) = (i_{0},j_{0}), so
and
since \(\oplus _{i,j}\mathcal H_{i,j}\) and \({\mathcal H_{i_{0},j_{0}}}\) are direct summands of \(\mathcal F\oplus \mathcal G\) and \(\mathcal F\) respectively.
By the properties of the complexity [10, Theorem 5.1, Lemma 6.12, Theorem 8.1], we have
So, from (1), we deduce
or
The proof of Theorem 5 now concludes exactly as in [1, 2.8]: for every m > N_{0}, we have
and
so, by the linear independence of the \({\varPhi }_{\mathcal H_{i,j},m}\), \({\varPhi }_{\mathcal F,m}={\varPhi }_{\mathcal G,m}\) if and only if \(\text {Tr}(F_{k_{m}}\mathcal W_{i})=\text {Tr}(F_{k_{m}}\mathcal W^{\prime }_{i})\) for every i ∈ I_{m}. For every i ∈ I, \(\mathcal W_{i}\) and \(\mathcal W^{\prime }_{i}\) have dimension at most ⌊r/n_{i}⌋ so, in order to show they are isomorphic (that is, that they have the same eigenvalues for the action of \(F_{k_{n_{i}}}\)), it suffices to show that the traces of the action of \(F_{k_{n_{i}}}^{l}=F_{k_{ln_{i}}}\) on them coincide for ⌊2r/n_{i}⌋ consecutive values of l [1, Lemme 2.9]. But, under the hypotheses of Theorem 5, these traces coincide for N_{0} < ln_{i} ≤ N_{0} + 2r, in which there are at least ⌊2r/n_{i}⌋ possible values of l.
3 An Effective Criterion for Finite Monodromy
For a finite extension E of \(\mathbb Q\) and a positive integer r, let M(E, r) denote the least common multiple of the n ≥ 1 such that [E(ζ_{n}) : E] ≤ r, where ζ_{n} is a primitive nth root of unity. For instance, we have
and, in general, \(M(E,r)\leq M(\mathbb Q,r\cdot [E:\mathbb Q])\), since
Given an \(\mathcal H\in \mathcal S(X,\bar {\mathbb {Q}}_{\ell })\), we say that \(\mathcal H\) is Evalued if, for every m ≥ 1 and \(x\in U(\mathbb {F}_{q^{m}})\), the characteristic polynomial of the Frobenius action on \(\mathcal H\) at x has coefficients in E. By [8, Théorème VII.6], if \(\mathcal H\) is irreducible and its determinant is arithmetically of finite order then it is Evalued for some finite extension \(\mathbb Q\subseteq E\).
Proposition 7
Let X be a smooth variety given with a projective embedding \(u:X\hookrightarrow \mathbb {P}^{n}_{k}\), \(\mathcal H\) a geometrically irreducible lisse ℓadic sheaf on X of rank r, pure of weight 0, of complexity \(c_{u}(\mathcal H)=C\), and whose determinant is arithmetically of finite order, let \(\mathbb Q\subseteq E\) be a finite extension such that \(\mathcal H\) is Evalued, and M = M(E, r). Let
where
Then \(\mathcal H\) has finite (arithmetic and geometric) monodromy if and only if all Frobenius eigenvalues of \(\mathcal H\) at x are roots of unity for every m ≤ N and every \(x\in X(\mathbb {F}_{q^{m}})\).
Proof
Suppose that the Frobenius eigenvalues of \(\mathcal H\) at x are roots of unity for every m ≤ N and every \(x\in X(\mathbb {F}_{q^{m}})\). Since these eigenvalues are roots of a polynomial of degree r over E, their order divides M by definition. That is, the Mth power of Frobenius acts trivially on \({\mathscr{H}}_{\bar x}\) for every \(x\in X(\mathbb {F}_{q^{m}})\), m ≤ N.
Let \({\mathscr{H}}^{[M]}:={\sum }_{i=1}^{M}(1)^{i1}i[\text {Sym}^{Mi}{\mathscr{H}}\otimes \wedge ^{i}{\mathscr{H}}]\) be the Mth Adams power of \(\mathcal H\). It is an element of the Grothendieck group of the category of constructible sheaves on X and, by [2, 1], its trace function is given by \({\varPhi }_{\mathcal H^{[M]}}(m,x)={\varPhi }_{\mathcal H}(Mm,x)\). By [9, Proposition 3.4], we have the optimized expression \({\mathscr{H}}^{[M]}={\sum }_{i=0}^{M1}(1)^{i}[{\mathscr{H}}_{i}]\), where \({\mathscr{H}}_{i}\in \mathcal S(X,\bar {\mathbb {Q}}_{\ell })\) denotes the sheaf defined by \({\mathscr{H}}om_{\mathfrak S_{M}}(\wedge ^{i} V,{\mathscr{H}}^{\otimes M})\), where V is a constant sheaf of rank M − 1 on X, \(\mathfrak S_{M}\) acts on V via its standard (M − 1)dimensional representation, and on \({\mathscr{H}}^{\otimes M}\) by permutation of the factors. The relationship between both expressions for the Adams power is given by \(\text {Sym}^{Mi}{\mathscr{H}}\otimes \wedge ^{i}{\mathscr{H}}\cong {\mathscr{H}}_{i}\oplus {\mathscr{H}}_{i1}\). By the previous paragraph, we have
for every m ≤ N and \(x\in X(\mathbb {F}_{q^{m}})\). This can be rewritten in terms of “real” sheaves by splitting the positive and negative components of \({\mathscr{H}}^{[M]}\): let \(\mathcal {F}=\oplus _{i\text { even}}{\mathscr{H}}_{i}\) and \(\mathcal G=\oplus _{i\text { odd}}{\mathscr{H}}_{i}\), then
for every m ≤ N and \(x\in X(\mathbb {F}_{q^{m}})\).
Since \(\mathcal F\) and \(\mathcal G\) are subsheaves of \({\mathscr{H}}^{\otimes M}\), their complexity is bounded by that of \({\mathscr{H}}^{\otimes M}\), which in turn, applying [10, Theorem 5.2] repeatedly, is bounded by \(A_{n}^{M1} C^{M}\). Then the complexity of \(\mathcal G\oplus \bar {\mathbb {Q}}_{\ell }^{r}\) is bounded by \(A_{n}^{M1} C^{M}+r\cdot c_{u}(X)\).
Since \(\mathcal F\) and \(\mathcal G\oplus \bar {\mathbb {Q}}_{\ell }^{r}\) have rank \({\sum }_{i\text { even}}{{r+Mi1}\choose {M}}{{M1}\choose i}\) [9, Remark 3.6] and are pure of weight 0, by Theorem 5 we conclude that \(\mathcal F\cong \mathcal G\oplus \bar {\mathbb {Q}}_{\ell }^{n}\) or, equivalently, \({\mathscr{H}}^{[M]}=[\mathcal F][\mathcal G]=[\bar {\mathbb {Q}}_{\ell }^{n}]\) in the Grothendieck group. That is, the Mth power of Frobenius acts trivially on \({\mathscr{H}}_{\bar x}\) for every \(x\in X(\mathbb {F}_{q^{m}})\) and every m ≥ 1. By Proposition 1, we conclude that \({\mathscr{H}}\) has finite monodromy. □
If X is a smooth curve, then we can improve the bound by using Theorem 4 instead of Theorem 5. Let Y be the smooth projective closure of X and D := Y ∖X. We then get
Proposition 8
Let \({\mathscr{H}}\) be a geometrically irreducible lisse ℓadic sheaf on X of rank r, pure of weight 0 and whose determinant is arithmetically of finite order, let \(\mathbb Q\subseteq E\) be a finite extension such that \({\mathscr{H}}\) is Evalued, and M = M(E, r). For every \(x\in D(\overline {\mathbb {F}_{q}})\), assume that the breaks of \({\mathscr{H}}\) at x are ≤ e_{x}, and let \(e:={\sum }_{x\in D(\overline {\mathbb {F}_{q}})}e_{x}\) and
where
Then \({\mathscr{H}}\) has finite (arithmetic and geometric) monodromy if and only if all Frobenius eigenvalues of \({\mathscr{H}}\) at t are roots of unity for every m ≤ N and every \(x\in X(\mathbb {F}_{q^{m}})\).
Proof
The proof goes exactly as in Proposition 7, using Theorem 4 and the fact that all breaks of \({\mathcal H}^{\otimes r}\) at \(x\in D(\overline {\mathbb {F}_{q}})\) are ≤ e_{x}. □
Next, we give similar results based on the integrality of the Frobenius traces instead of its Frobenius eigenvalues, which are generally easier to compute. We start by showing
Lemma 9
Let \(\mathbb Q_{p}\subseteq E_{\pi }\) be a finite extension with ramification index e, and let α_{1},…,α_{r} ∈ E_{π}. Let \(a=\lfloor \log _{p} r\rfloor \) and
Then α_{1},…,α_{r} are integral if and only if \({\sum }_{i=1}^{r}{\alpha _{i}^{k}}\) is integral for every 1 ≤ k ≤ N.
Proof
Let \(M=1+\left \lfloor \frac {e}{p1}\left (1\frac {1}{p^{a}}\right )\right \rfloor \); \(p_{k}:={\sum }_{i=1}^{n}\alpha _{i}^{Mk}\) for k ≥ 1 and s_{k} be the kth elementary symmetric function on \({\alpha ^{M}_{1}},\ldots ,{\alpha ^{M}_{r}}\) for k = 0,…,r. By Newton’s identities, \(ks_{k}={\sum }_{i=1}^{k}(1)^{i1}s_{ki}p_{i}\), so k!s_{k} is integral for every k = 0,…,r.
If ν denotes the valuation on E_{π}, normalized so that ν(p) = 1, we get
so the Newton polygon of the polynomial \({\prod }_{i=1}^{r}(1{\alpha _{i}^{M}}T)\) is bounded below by the polygon with vertices
and, in particular, its largest slope (in absolute value) is bounded by
so \(\nu ({\alpha _{i}^{M}})\geq \frac {1}{p1}\left (1\frac {1}{p^{a}}\right )\) for every i, and
But \(\nu (E_{\pi })=\frac {1}{e}\mathbb Z\), so we conclude that ν(α_{i}) ≥ 0 for every i = 1,…,n. □
Theorem 10
Let \({\mathscr{H}}\) be a geometrically irreducible lisse ℓadic sheaf on X of rank r, pure of weight 0, of complexity \(c_{u}(\mathcal H)=C\), and whose determinant is arithmetically of finite order, let \(\mathbb Q\subseteq E\) be finite extension such that \({\mathscr{H}}\) is Evalued, f be the maximum among the ramification indices of the primes of E above p, and \(a=\lfloor \log _{p} n\rfloor \). Let
where
Then \({\mathscr{H}}\) has finite (arithmetic and geometric) monodromy if and only if \({\varPhi }_{{\mathscr{H}}}(m,x)\) is integral at all places of E over p for every m ≤ N and every \(x\in X(\mathbb {F}_{q^{m}})\).
Proof
Assume that all Frobenius traces of \({\mathscr{H}}\) at x are integral at all places of E over p for every m ≤ N and every \(x\in X(\mathbb {F}_{q^{m}})\). For every m ≥ 1 and every \(x\in X(\mathbb {F}_{q^{m}})\), the Frobenius eigenvalues of \({\mathscr{H}}\) at x are contained in some extension of E of degree ≤ r, in which all primes above p have ramification index ≤ rf. By Lemma 9, all Frobenius eigenvalues of \({\mathscr{H}}\) at x are integral for every \(m\leq N^{\prime }\) and every \(x\in X(\mathbb {F}_{q^{m}})\) at all places over p, where
By [8, Théorème VII.6], they are also integral at all other nonarchimedean places, so they are algebraic integers. Since their product (which is the Frobenius trace at x of \(\det ({\mathscr{H}})\), which is arithmetically of finite order by hypothesis) is a root of unity, they must all be roots of unity, and we conclude by Proposition 7. □
For X a smooth curve we get the following, more optimized, result by using Proposition 8 instead of Proposition 7:
Theorem 11
Let \({\mathscr{H}}\) be a geometrically irreducible lisse ℓadic sheaf on X of rank r, pure of weight 0 and whose determinant is arithmetically of finite order, let \(\mathbb Q\subseteq E\) be a finite extension such that \({\mathscr{H}}\) is Evalued, f be the maximum among the ramification indices of the primes of E above p, and \(a=\lfloor \log _{p} r\rfloor \). For every \(x\in D(\overline {\mathbb {F}_{q}})\), assume that the breaks of \({\mathscr{H}}\) at x are ≤ e_{x}, and let \(e:={\sum }_{x\in D(\overline {\mathbb {F}_{q}})}e_{x}\) and
where
Then \({\mathscr{H}}\) has finite (arithmetic and geometric) monodromy if and only if \({\varPhi }_{{\mathscr{H}}}(m,x)\) is integral at all places of E over p for every m ≤ N and every \(x\in X(\mathbb {F}_{q^{m}})\).
4 Examples
Let q = p be a prime, \(X=\mathbb A^{1}_{\mathbb {F}_{p}}\) the affine line, fix an additive character \(\psi :\mathbb {F}_{p}\to \mathbb {C}\) and an integer n ≥ 2, and consider the sheaf \(\mathcal F\in {\mathcal S}(X,\bar {\mathbb {Q}}_{\ell })\) whose trace function at \(t\in X(\mathbb {F}_{p^{r}})\) is given by
where \(G={\sum }_{x\in \mathbb {F}_{p}}\psi (x^{2})\) is the Gauss sum if p≠ 2 and 1 + i if p = 2. That is, the (normalized) Fourier transform of the pullback of the Artin–Schreier sheaf \(\mathcal L_{\psi }\) by the nth power map. The sheaf \(\mathcal F\) is lisse of rank n − 1, pure of weight 0 and \(\mathbb {Q}(\zeta _{p})\)valued, where ζ_{p} is a pth root of unity (respectively \(\mathbb {Q}(i)\)valued) if p≠ 2 (resp. p = 2). We have
if p≠ 2 and
if p = 2, b_{1}(X) = 0, \(e=\frac {1}{n1}\) and \(f=\max \limits \{2,p1\}\) (as p is totally ramified in \(\mathbb {Q}(\zeta _{p})\) if p≠ 2), so in this case we get finite monodromy iff all Frobenius eigenvalues are roots of unity at all points defined over extensions of \(\mathbb {F}_{p}\) of degree up to
or if all Frobenius traces are integral at all points over extensions of \(\mathbb {F}_{p}\) of degree up to
if p≠ 2 and
if p = 2. Even for p = 2, this gives degrees up to 61, 1178 and 18432701 respectively for n = 3,4,5 for the eigenvalues criterion, far beyond what we can compute in practice.
Let now \(X={\mathbb {G}_{m}}_{,\mathbb {F}_{q}}\) be the onedimensional torus over \(\mathbb {F}_{q}\) where q = p^{r}, fix an additive character \(\psi :\mathbb {F}_{p}\to \mathbb {C}\) and two disjoint sets of multiplicative characters χ = {χ_{1},…,χ_{a}} and ρ = {ρ_{1},…,ρ_{b}}, and consider the (normalized) hypergeometric sheaf
[4, Chapter 8]. Assume a > b and, if p = 2, also a + b odd, then \({\mathscr{H}}\) is lisse of rank a on X, pure of weight 0 [4, Theorem 8.4.2], and \(\mathbb {Q}(\zeta _{pm})\)valued, where m = lcm(ord(χ_{i}),ord(ρ_{j})). Here
b_{1}(X) = 1, \(e=\frac {1}{ab}\) and f = p − 1 (as p is totally ramified in \(\mathbb {Q}(\zeta _{p})\) and unramified in \(\mathbb {Q}(\zeta _{m})\)), so in this case we get finite monodromy iff all Frobenius eigenvalues are roots of unity at all points defined over extensions of \(\mathbb {F}_{q}\) of degree up to
or if all Frobenius traces are integral at all points over extensions of \(\mathbb {F}_{p}\) of degree up to
For p = 2 and a = 2, this gives degrees up to 54 and 124 respectively for m = 3,5 for the eigenvalues criterion, which is again beyond what we can compute in practice.
Proposition 7 can be improved in particular cases if we can get better estimates for the dimensions of the cohomology groups of the tensor powers of \(\mathcal F\) (for instance, if the trace functions of these tensor powers can be written in terms of sums of additive or multiplicative characters, one may use the estimates for the sum of the Betti numbers given in the last section of [5]).
The main obstacle that makes the bounds for N so large in these theorems is the big rank of the components of \(\mathcal F^{[M]}\). It is easy to see that the N in Theorem 5 can not be made smaller than O(r)  even in dimension 0, one could take for instance the pushforward of the trivial sheaf from \(\text {Spec} \mathbb {F}_{q^{r}}\) to \(\text {Spec} \mathbb {F}_{q}\), which has rank r and trace 0 over \(\mathbb F_{q^{i}}\) for i < r. However, in the proof of Proposition 7 we have equality not only of the Frobenius traces of \(\mathcal F\) and \(\mathcal G\oplus \bar {\mathbb {Q}}_{\ell }^{r}\), but also of their Frobenius characteristic polynomials. Any improvement of Theorem 5 in the case where we have equality of Frobenius characteristic polynomials, and not just of traces, would lead to similar improvements in the size of the N’s in the theorems, which could hopefully make them usable in practice.
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Acknowledgements
The author would like to thank N. Katz and P.H. Tiep for their useful comments on earlier versions of the manuscript, H. Esnault for her remarks about Deligne’s Theorem 4, and the referees for their corrections and suggestions.
The author was partially supported by PID2020114613GBI00 (Ministerio de Ciencia e Innovación), P2001056 and US1262169 (Consejería de Economía, Conocimiento, Empresas y Universidad, Junta de Andalucía and FEDER)
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Dedicated to Professor Pham Huu Tiep on the occasion of his 60th birthday.
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RojasLeón, A. An Effective Criterion for Finite Monodromy of ℓAdic Sheaves. Vietnam J. Math. 51, 703–713 (2023). https://doi.org/10.1007/s10013022006031
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DOI: https://doi.org/10.1007/s10013022006031