Density of Arithmetic Representations of Function Fields

We propose a conjecture on the density of arithmetic points in the deformation space of representations of the \'etale fundamental group in positive characteristic. This? conjecture has applications to \'etale cohomology theory, for example it implies a Hard Lefschetz conjecture. We prove the density conjecture in tame degree two for the curve $\mathbb{P}^1\setminus \{0,1,\infty\}$. v2: very small typos corrected.v3: final. Publication in Epiga.


Introduction
Let X 0 be a smooth geometrically connected variety defined over a finite field k = F q of characteristic p. We fix an algebraic closure k ⊂k and a geometric point x ∈ X 0 (k). In this note we study representations of the geometric étale fundamental group G = πé t 1 (X, x), where X = X 0 ⊗ kk , and the action of the Frobenius on the set of representations.
For a given prime p, we fix a finite field F of characteristic , and a continuous semi-simple representationρ : G → GL r (F ). We define the set Sρ of isomorphism classes of continuous semi-simple representations ρ : πé t 1 (X, x) → GL r (Q ) with the property that the associated semi-simple residual representation is isomorphic toρ. We endow Sρ with a Noetherian Zariski topology in Section 2.
There is a canonical Frobenius action Φ : Sρ ∼ − → Sρ. A point [ρ] ∈ Sρ is fixed by Φ n for some integer n > 0 if and only if the representation ρ extends to a continuous representation πé t 1 (X 0 ⊗ k k , x) → GL r (Q ), for some finite extension k ⊂ k . We call such a point in Sρ arithmetic and we let Aρ ⊂ Sρ be the subset of arithmetic points.
The aim of our note is to propose and to study (two variants of) a conjecture about the density of arithmetic points, see Section 3.
Weak Conjecture. The arithmetic points Aρ are dense in Sρ.
Strong Conjecture. For a Zariski closed subset Z ⊂ Sρ with Φ n (Z) = Z for some integer n > 0 the subset of arithmetic points Z ∩ Aρ is dense in Z.
One application of the Strong Conjecture is that it implies a Hard Lefschetz isomorphism for semi-simple perverseQ -sheaves in characteristic p, see Section 9. This application is motivated by the corresponding work of Drinfeld for complex varieties [Dri01].
For degree r = 1 and X either proper or a torus the Strong Conjecture is shown in [EK21, Theorem 1.7 and Lemma 3.1].
In Section 6 we prove the following reductions for the Strong Conjecture, see also Proposition 3.6. Here the algebraically closed fieldk is fixed.
• If the Strong Conjecture holds for given degree r for all smooth curves X overk then it holds in degree r for all smooth varieties X overk. • If the Strong Conjecture holds in any degree r for X = P 1 k \ {0, 1, ∞} andρ tame then it holds in general overk.
These reductions motivate our two main theorems, see Section 3.
Theorem A. The Weak Conjecture holds when X is a curve, > 2 andρ is absolutely irreducible.
Theorem B. The Strong Conjecture holds for X = P 1 k \ {0, 1, ∞} whenρ is tame of degree two. We now explain the ideas of our proofs. The main ingredient in the proof of Theorem A is de Jong's conjecture [deJ01] proven in [Gai07,§ 1.4] under the assumption > 2, using the geometric Langlands program. Indeed, ifρ is absolutely irreducible, then Sρ is the set ofQ -points of Mazur's deformation space which is smooth if X is a curve, and on which we can apply de Jong's technique [deJ01, § 3.14].
The proof of Theorem B is very different. We embed Sρ in the completion of the affine space of dimension 6 at the closed point which corresponds to the characteristic polynomials of three well chosen elements of the geometric fundamental group G on which Φ acts by raising to the q th power. We can then apply our main density theorem in [EK21] on the cover which separates the roots of those polynomials. In particular, this also shows that the arithmetic points are precisely those which have quasi-unipotent monodromy at infinity. We remark in Section 9.2 that our method yields a proof de Jong's conjecture in this particular case, which does not use automorphic forms.

The Zariski topology on the set of semi-simple representations
Let be a prime number, O be the ring of integers of a finite extension of Q with residue field F , O →Q be an embedding of O into an algebraic closure ofQ defining an embedding of F into an algebraic closureF . Let G be a pro-finite group which satisfies Mazur's -finiteness property, i.e. for any open subgroup U ⊂ G the set Hom cont (U , Z/ Z) is finite. Let be a continuous representation. We define Sρ to be the set of isomorphism classes of continuous semi-simple representations ρ : G → GL r (Q ) with semi-simple reduction isomorphic toρ ss : G → GL r (F ) ⊂ GL r (F ). In this section we define a Zariski topology on Sρ. In Section 4 we relate Sρ to the deformation space of pseudorepresentations.
We endow the maximal spectrum with the usual Zariski topology, which is thus Noetherian.
Proposition 2.1. There exists an integerm > 0 and a familyg ∈ Gm such that for any finite family g ∈ G m which containsg we have: (1) char g is injective with Zariski closed image.
(2) The induced topologies on Sρ via the embeddings char g and charg are the same.
Proposition 2.1 is an immediate consequence of Lemma 5.1. From now on we endow Sρ with the induced Zariski topology from Proposition 2.1.
Remark 2.2. By the same procedure we can define the -adic topology on Sρ, which we do not consider in this note, compare [Che14,Theorem D], and [Lit21] where it is used in an essential way.

The density conjectures
In this section we formulate a strong conjecture and a weak one on the density of arithmetic representations in the Zariski space of all semi-simple representations Sρ defined in Section 2. Then we formulate our main results concerning them.
Let X 0 be a smooth geometrically connected variety defined over a finite field k = F q of characteristic wherek is an algebraic closure of k. Fix a geometric point x ∈ X 0 (k) and let G be the geometric fundamental group πé t 1 (X, x). Fix a lift Φ ∈ πé t 1 (X 0 , x) of the arithmetic Frobenius. Then Φ acts by conjugation on G. This action depends on the lift up to an inner automorphism, so it canonically acts on isomorphism classes of representations of G. We assume that Φ(ρ) is isomorphic toρ which is always fulfilled after replacing Φ by a power, or equivalently X 0 by X 0 ⊗ k k for a finite extension k of k. Thus the action of Φ on G induces a well defined automorphism Φ of Sρ. By the construction of the Zariski topology on Sρ via Proposition 2.1 the automorphism Φ is a homeomorphism.
We define the arithmetic points of Sρ as the fixed points of powers of Φ Remark 3.1. The arithmetic points in Sρ correspond to those continuous semi-simple representations ρ : G → GL r (Q ) which can be extended to a continuous representation  Note that the formulation of the conjectures depends only on X and not on the choice of X 0 or the base point x.
Remark 3.4. If r = 1 and X is projective or X is a torus, then the strong conjecture is true by virtue of [EK21, Theorem 1.7 and Lemma 3.1].
Remark 3.5. If we endow Sρ with the -adic topology as in Remark 2.2, then the subset of arithmetic points Aρ is discrete and closed, see [Lit21, Theorem 1.1.3].
Using the Lefschetz theorem on fundamental groups and the Belyi principle we reduce in Section 6 the Strong Conjecture to the case where X is a curve.
Proposition 3.6. For varieties over the fixed fieldk we have the implications: (1) If for fixed r the Strong Conjecture holds for dim(X) = 1, then it holds for any X and the given degree r.
(2) If the Strong conjecture holds for all r > 0 for tame representationsρ on the variety X = P 1 k \ {0, 1, ∞}, then it holds in general.
The main results of our note are the following.
Theorem 3.7 (Theorem A). Assume that > 2. Ifρ is absolutely irreducible and X is a curve, then the weak conjecture holds. The only reason why we assume > 2 in Theorem 3.7 is that de Jong's conjecture [deJ01, Conjecture 2.3] is known only under this assumption at the moment, see [Gai07,§ 1.4]. In fact our proof of Theorem 3.8 yields a geometric proof of de Jong's conjecture on P 1 k \ {0, 1, ∞} in rank 2 for any , without any use of the Langlands program, see Section 9.2.

The deformation space of pseudorepresentations
In this section we recall some properties of the deformation space of pseudorepresentations PDρ following [Che14]. The reason why we work with pseudorepresentations is that they naturally give rise to a parametrization of the semi-simple representations Sρ defined in Section 2. As in Section 2, G is a profinite group satisfying Mazur's -finiteness property.
Let C be the category of complete local O-algebras (A, m A ) such that O → A/m A identifies the residue fields of A and O. Following [Che14, Section 3] we define the functor of pseudodeformations ofρ PDρ : C → Sets which assigns to A the set of continuous r-dimensional A-valued determinants D : Recall that a determinant is given by a compatible collection of maps We define the coefficients of the characteristic polynomial of D as the maps Λ i : G → A determined by the formula With a slight abuse of notation we define PDρ(Z ) to be the set of r-dimensionalZ -valued determinants  (1) The functor PDρ is representable by R Pρ ∈ C with universal determinant D R Pρ : (2) The complete local ring R Pρ is Noetherian and topologically generated as an O-algebra by the finitely many elements Λ j (g i ) where 1 ≤ j ≤ r and g 1 , . . . , g m ∈ G is a suitable family. (

3) Ifρ is absolutely irreducible, R Pρ coincides with Mazur's universal deformation ring and D R Pρ is the determinant of the universal deformation.
We refer to [Che14,Proposition 7.59] for part (1), to [Che14,Remark 7.61] for part (2), to [Che14, Example 7.60] for part (3). Recall that Mazur's deformation functor C → Sets assigns to A ∈ C the set of isomorphism classes of continuous representations ρ : G → GL r (A) such that ρ ⊗ A F is isomorphic toρ, see for example [Til96,Section 3]. Note that for anyρ we have by definition  Proof. By [Che14, Theorem A] there is a bijection between the isomorphism classes of not necessarily continuous semi-simple representations ρ : G → GL r (Q ) and the not necessarily continuous determinants D :Q [G] →Q . We combine this with the simple fact from representation theory that a semi-simple representation ρ : G → GL r (Q ) is continuous precisely when its character tr • ρ has image in a finite extension of Frac(O) insideQ and is continuous.

Combining Remark 4.2 and Proposition 4.3 we obtain a canonical identification
We shall see in the next section that (1) induces the same Zariski topology on Sρ as the one defined in Section 2.

Characteristic polynomials
In this section G is a profinite group satisfying Mazur's -finiteness property. Recall that for a family p = (p 1 , . . . , p m ) of m monic polynomials p i of degree r over the finite field F , we introduced the complete local deformation ring R p in Section 2. When p i is the characteristic polynomial of the matrixρ(g i ) for a representationρ : G → GL r (F ) and for i = 1, . . . , m, we also write Rρ (g) for R p . Let be the corresponding formal scheme over O. We obtain a canonical morphism of formal schemes which sends a pseudorepresentation to the family of associated characteristic polynomials of g 1 , . . . , g m . In view of the identification (1) this induces the map char g : Sρ → Spm(Rρ (g) ⊗ OQ ) from Section 2.
Lemma 5.1. There is a finite familyg ∈ Gm such that for any finite family g ∈ G m containingg, the morphism Proof. This is an immediate consequence of Proposition 4.1(2).
Proof of Proposition 2.1. Takeg as in Lemma 5.1 and use the identification (1).
For an integer n we write µ n for ((µ induces a finite morphism of O-formal schemes There exists a unique lower horizontal morphism [n] of O-formal schemes making the square p has the obvious meaning, that is its j th component is defined to be We now study the compatibility of the universal deformation space of pseudodeformations with restriction. Let U ⊂ G be an open subgroup,ρ| U : U → GL r (F ) be the restriction of ρ to U . It induces a morphism Lemma 5.2. The morphism rest is finite.
Proof. Fix a family g in G as in Lemma 5.1. Choose an integer n > 0 such that g n i ∈ U for all the g i of the family and denote by g n the family (g n 1 , . . . , g n m ). We have a commutative diagram in which the upper horizontal arrow char g is a closed immersion and the right vertical arrow [n] is finite. This implies that the left vertical arrow rest is finite as well.
It is likely that the notion of induction for pseudorepresentations with respect to an open subgroup U ⊂ G can be defined and induces a finite morphism of universal deformation spaces of pseudorepresentations. Unfortunately, this is not documented in the literature.
We now describe a weak form of induction which is sufficient for our purpose. For simplicity assume that U ⊂ G is a normal subgroup of index n. Letρ U : U → GL r (F ) be a continuous representation and set Similarly, sending a pseudorepresentation D : (2) We expect that (2) can be extended to a commutative diagram by the indicated dashed induction arrow. As ξ is finite by Lemma 5.3, the dashed arrow, if it exists, is automatically finite. rn letters. In other words, it is the quotient by the product j Σ rn . The embedding a Σ r ⊂ Σ rn induces the embedding a,j Σ r ⊂ j Σ rn , and thus defines the requested dashed arrow which is finite. This finishes the proof.
Remark 5.4. When evaluated onZ the diagram (2) becomes commutative if we define the dashed arrow onZ -points as the induction on representations. Here the isomorphisms are coming from (1) and induction is understood up to semisimplification.

Compatibility with restriction and induction
In this section we prove some reductions and compatibilities which enable us to prove Proposition 3.6. So as there G is a profinite group satisfying Mazur's -finiteness property. As we are interested in the density of the fixed points of powers of an automorphism on a topological space, we formulate the simple Lemma 6.1 in this context. For a topological space S and a homeomorphism Φ : S → S, we define and study the following density property.
(D) S,Φ : For any closed subset Z ⊂ S with Φ n (Z) = Z for some integer n > 0 the intersection If Φ is clear from the context we omit it in our notation.
(1) If ψ is surjective then (2) If the ring homomorphism ψ : R 2 → R 1 is finite then Proof. Part (1) is obvious. To show part (2) consider Z ⊂ S 1 closed with Φ n 1 (Z) = Z for some n > 0. Then, replacing n by m for some m > 0 the latter is true for each irreducible component of Z, so in order to show that Z ∩ S Φ ∞ 1 1 is dense in Z we can assume without loss of generality that Z is irreducible.
We assume that the closure Z of Z ∩ S Φ ∞ 1 1 is not equal to Z and we are going to deduce a contradiction. Incomparability, see [Bou98, Section V.2.1, Corollary 1], tells us that we get a proper inclusion ψ(Z ) ψ(Z) of closed subsets of S 2 . As the fibres of ψ are finite we have ψ −1 (S . But then (D) S 2 ,Φ 2 applied to the closed subset ψ(Z) says that ψ(Z ) = ψ(Z), which is a contradiction.
Let U ⊂ G be an open subgroup and letρ : G → GL r (F ) be a continuous representation. Let Φ : G → G be an automorphism with Φ(U ) = U and with Φ(ρ) ρ. We can then deduce compatibility of our density property with restriction and induction.

Proposition 6.2.
(1) We have the implication (D) Sρ | U ⇒ (D) Sρ . ( Proof. For part (1) we observe that the restriction map Sρ → Sρ | U is induced via the identification (1) by the finite homomorphism of Noetherian Jacobson rings The Noetherian Jacobson property of these rings follows from the general fact that for a Noetherian complete local O-algebra R with finite residue field, the ring R ⊗ OQ is Noetherian and Jacobson, see [ (1) follows from Lemma 6.1(2) with S 1 = Sρ, S 2 = Sρ | U and ψ = rest.
We prove part (2). In view of Remark 5.4 takingZ -points in the diagram (2) we get the commutative diagram By Lemma 5.2 the image S = rest(Sρ) ⊂ Sρ | U is closed, so it is naturally a maximal spectrum. By Lemma 6.1(1) our assumption (D) Sρ implies that (D) S holds. As by Lemma 5.3 the map ξ : Sρ U → S is induced by a finite ring homomorphism on maximal spectra, we can apply Lemma 6.1(2) and deduce that (D) Sρ U holds.
Let X be a smooth connected variety over an algebraically closed field k. Fix a geometric point x ∈ X(k). We apply the results from the previous sections to the case G = πé t 1 (X, x). As in Section 3, we consider a continuous representationρ : G → GL r (F ).

Proposition 6.3.
There is a smooth connected one-dimensional C and a locally closed immersion ι : C → X such that the induced morphism ι * : PDρ → PD ι * ρ is a closed immersion.
Proof. Let m ⊂ R Pρ be the maximal ideal. By [Che14,Lemma 7.52] there exists an open normal subgroup U ⊂ G such that the composed determinant It is sufficient to choose ι such that ι * : R P ι * ρ → R Pρ /m 2 is surjective. Recall from Proposition 4.1(2) that the O-algebra R Pρ /m 2 is generated by the coefficients of the characteristic polynomials Λ j (ḡ) with 1 ≤ j ≤ r andḡ ∈ G/U . So any closed immersion ι : Y → X such that the composition To find such a ι we can assume without loss of generality that X → A N k is affine and use Bertini's theorem [Jou83, Theorem 6.3] applied to the étale covering X of X corresponding to U ⊂ G and the unramified map X → A N k . In fact Bertini tells us that a generic affine line L ⊂ A N k has the property that X × A N k L is smooth connected and one-dimensional, so one can take C = X × A N k L.
Remark 6.4. The above argument shows in addition that if X • ι − → X is an open embedding, then the induced morphism ι * : PDρ → PD ι * ρ is a closed immersion. Indeed πé t 1 (X • , y) and one argues as above.
We prove part (2). By part (1) we may assume that X 0 is a smooth geometrically connected curve. Let π : Y 0 → X 0 be a finite étale cover trivializingρ. Let1 be the trivial rank r representation on Y with value in GL r (F ). By Proposition 6.2(1), we may assume thatρ =1 on the one-dimensional X 0 .

Proof of Theorem B
The aim of this section is to prove Theorem B. Let X be the scheme P 1 W \ {0, 1, ∞} over the ring of Witt vectors W = W (k). Set K = Frac(W ) and fix an algebraic closureK of K together with an embeddinḡ K → C and an isomorphism of the residue field ofK withk. We also fix a lift x ∈ X (K) of our base point x ∈ X 0 (k).
Fix an orientation for C and let γ 0 , γ 1 , γ ∞ ∈ π top 1 (X (C), x ) be suitable "simple" loops around 0, 1 and ∞ such that Proof. Up to inner automorphisms one can replace the base point x in the étale fundamental group by the base point y a : Spec (K a ) → X K , whereK a is an algebraic closure of the fraction field K a of O h P 1 C ,a . As explained in [Del73, § 1.1.10] there is a corresponding generalized topological base point of X (C), which we simply write asD * a . Here D * a is a small punctured disk around the point a ∈ P 1 C (C) andD * a is its universal covering. Then by loc. cit. we have a commutative diagram where the vertical maps are pro-finite completions.
By [Gro71, Exposé XIII, § 2.10 and 4.7] the specialization homomorphism sp : πé t 1 (XK , x ) → π t 1 (X, x) is surjective and compatible with the action of the Frobenius lift Φ. Here the codomain is the tame fundamental group which we also write G t in the following. We set g a = sp(γ a ) for a ∈ {0, 1, ∞}, so we have In addition, G t is topologically generated by g 0 , g 1 . By Lemma 5.5 the map is a closed immersion, where g = (g 0 , g 1 , g −1 ∞ ) and where PDρ classifies pseudorepresentations of the group G t . From Claim 7.1 we conclude that for a = 0, 1, ∞. In particular, asρ is fixed by Φ, one has char(ρ(g q )) = char(ρ(g)).
We assume without loss of generality, by replacing k by a finite extension and thus Φ by a power, that the family of roots µ of the polynomials char(ρ(g)) satisfy µ q = µ, thus the isomorphism D µ

[q]
− − → D µ is well defined. This implies that with the notation as in Section 5 we obtain a commutative diagram As char g is a closed immersion we have the implication

Proof of Theorem A
The aim of this section is to prove Theorem A. Soρ is supposed to be absolutely irreducible. The arguments rely on [deJ01], and are partly similar to [BK06] and [BHKT19, Section 5].
As recalled in Proposition 4.1(3) R Pρ is then Mazur's universal deformation ring, which parametrizes isomorphism classes of continuous representations ρ : G → GL r (A) for A ∈ C such that ρ ⊗ A F is isomorphic toρ. In this case we simply write Rρ for R Pρ and Dρ for PDρ.
Proof. Let D detρ = Spf R detρ be the universal deformation space of the degree one representation Let 0 → I → B → A → 0 be an extension in C such that I · m B = 0 (so the B-module structure on I factors through F ). By [Maz89, § 1.6], there exists a canonical commutative obstruction diagram with exact rows Here F is the lisse étale sheaf on X corresponding toρ and we use that the canonical map induced by the Hochschild-Serre spectral sequence of the universal coveringX → X is injective. The latter injectivity is due to the fact that F is trivialized onX and that its first cohomology onX vanishes.
First case: X is affine. By [Art73, Corollary 3.5] we have H 2 (X, End(F )) = 0, so the first exact row in (4) tells us that Dρ is formally smooth.
Second case: X is projective. This case follows from the following two claims and a chase in the diagram (4). Proof of Claim 8.2. We know that if detρ =1 is trivial, then D1 = O G ab, , where G ab, is the abelian, -adic étale fundamental group of X. In general, D detρ is isomorphic to D1 by translating with the Teichmüller lift of detρ, see [Maz89,§ 1.4]. Thus D detρ is formally smooth, since G ab, is torsion free.
Proof of Claim 8.3. As the trace map of étale sheaves End(F ) → F is surjective and X has dimension one, the map tr in (4) is surjective as well. So it suffices to show that both F -vector spaces have dimension one. For H 2 (X, F ) this is immediate from Poincaré duality. In order to apply Poincaré duality to H 2 (X, End(F )) we recall that the trace pairing induces an isomorphism End(F ) ∨ End(F ). So we obtain from duality an isomorphism H 2 (X, End(F )) End G (ρ) ∨ = F . The equality comes from the absolute irreducibility ofρ and Schur's lemma.
For an integer n > 0 we consider the quotient ring (Rρ) Φ n = Rρ/I n , where I n is the ideal generated by Φ n (α) − α for all α ∈ Rρ. Based on the presentation of Lemma 8.1, we see that I n is generated by the b elements By definition We use the following two propositions.
Proposition 8.4. The ring (Rρ) Φ n is finite, flat and a complete intersection over O for any n > 0.
Proposition 8.5 is the same as Proposition 5.12 in [BHKT19].
Sketch of proof of Proposition 8.4. As in [deJ01, § 3.14], we have to show that the images of the elements (5) form a regular sequence in Rρ ⊗ O F . The latter is equivalent to (Rρ) Φ ⊗ O F being zero-dimensional. This is deduced by verbatim the same argument as loc. cit. in view of the fact that de Jong's conjecture is known for > 2 by [Gai07].
Proof of Proposition 8.5. Consider a continuous representation ρ : where O is a discrete valuation ring which is a finite extension of O. Then up to replacing k by a finite extension, ρ can be extended to a continuous representation ρ 0 : πé t 1 (X 0 , x) → GL r (O ), see Remark 3.1. Asρ is absolutely irreducible, ρ 0 ⊗ O Q is irreducible. After a suitable twist we can assume without loss of generality that det(ρ 0 ) is finite, see [Del80, Proposition 1.3.4]. Then by the Langlands correspondence [Laf02, Theorem VII.6] the lisse sheaf F 0 corresponding to ρ 0 is pure of weight zero. The tangent space to ρ in (Rρ) Φ ⊗ OQ is given by where the last equality follows from the fact that H 1 (X, End(F )) has weight one as End(F 0 ) has weight zero. This finishes the proof.
Proof of Theorem A. We have to show that an element α ∈ Rρ ⊗ OQ which vanishes on the points Aρ is zero. After replacing O by a finite extension we may assume without loss of generality that α ∈ Rρ. The vanishing condition means that α is contained in all the maximal ideals corresponding to the points of Aρ, i.e. that the image of α in the ring (Rρ) Φ n ⊗ OQ is contained in its nilpotent radical for all n > 0. As (Rρ) Φ n ⊗ OQ is reduced by Proposition 8.5, this means that the image of α vanishes in (Rρ) Φ n ⊗ OQ for all n > 0. By the flatness in Proposition 8.4 it actually vanishes in (Rρ) Φ n for any n > 0. By Claim 8.6 this implies that α = 0.
Proof. Let m be the maximal ideal of Rρ. For any integer m > 0, there is an integer n > 0 such that Φ n acts trivially on Rρ/m m as the latter ring is finite, so I n ⊂ m m . Thus ∩ n>0 I n ⊂ ∩ m>0 m m = {0}.

Some Applications
In this section we make two remarks concerning applications.

The Hard Lefschetz theorem in positive characteristic.
This application of our Strong Conjecture is motivated by [Dri01]. Let f : X → Y be a projective morphism of separated schemes of finite type over an algebraically closed fieldk. Let η ∈ H 2 (X, Q ) be the Chern class of a relative ample line bundle. Here we omit Tate twists for simplicity of notation.
One conjectures (see [EK21,Remark 1.4]) that if F ∈ D b c (X,Q ) is a semi-simple perverse sheaf, then the Hard Lefschetz property holds, i.e. the cup-product is an isomorphism for all i ≥ 0. It is known that this holds if (i) F is of geometric origin in the sense of [BBD82, § 6.2.4-6.2.5] (ii)k is the algebraic closure of a finite field k and f , η and F descend to schemes X 0 , Y 0 over the field k. Sketch of proof. Similar to [BBD82, Lemma 6.1.9] one uses a spreading argument in order to reduce to the case in whichk is the algebraic closure of a finite field k 0 and f and η are defined over k 0 . Then F corresponds to an irreducible representation ρ F : πé t 1 (U ) → GL r (Q ), where U ⊂ X is a smooth locally closed geometrically irreducible subvariety (over which F is a shifted smooth sheaf).
Letρ : πé t 1 (U ) → GL r (F ) be the semi-simple reduction of ρ F . In fact each representation ρ ∈ Sρ gives rise via the intermediate extension of the associated smooth sheaf to aQ -perverse sheaf F ρ on X.
Similarly to [EK21,Corollary 4.3] one shows Claim 9.2. The subset Z • ⊂ Sρ of those ρ for which the Hard Lefschetz property for the perverse sheaf F ρ fails to hold, is constructible.
In order to give a complete proof of the claim one would need a theory of perverse étale adic sheaves over fields more general thanQ . Unfortunately, such a theory does not exist in the literature at the moment, but should be rather formal in terms of the pro-étale topology.
As Z • is also stabilized by the Frobenius Φ, we can apply the Strong Conjecture to the Zariski closure Z of Z • . This implies that Z • contains an arithmetic point, which contradicts (ii) above.
Remark 9.3. Using Proposition 3.6, it would be enough in Proposition 9.1 to prove the Strong Conjecture in rank ≤ r on all curves or in any rank on P 1 \ {0, 1, ∞} for a tameρ.  We observe that our proof avoids the use of the geometric Langlands correspondence, which is used in [deJ01, Theorem 1.2] to establish the degree two case of de Jong's conjecture.