MONODROMY FILTRATION AND MOTIVES

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HAL Id: hal-01068065

https://hal.archives-ouvertes.fr/hal-01068065v2

Preprint submitted on 28 Mar 2015

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Abhijit Laskar

To cite this version:

Abhijit Laskar. MONODROMY FILTRATION AND MOTIVES. 2014. �hal-01068065v2�

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ABHIJIT LASKAR

Abstract. Given a (pure) motive M over a number fieldF and a non-archimedean valuation v on F, we state an analogue of the weight monodromy conjecture forM, via a local weight filtrationW• on the`-adic realizationH`(M). We prove special cases of this conjecture. In some of these cases, we are able to show that the localL-factor atv, of the L-function ofM, is well-defined i.e., rational and independent of`.

0. Introduction and notation

LetF be a number field with an embedding τ :F ,→C; ¯F is a fixed algebraic closure of F; ¯τ : ¯F ,→Cis an extension ofτ. Let vbe a discrete valuation on F;Fv is the completion of F at v; ¯v is an extension of v to ¯F; ¯F_v is the localization of ¯F at ¯v. The residue fields of Fv and ¯Fv are denoted by kv and ¯kv, respectively. The characteristic of kv is p >0. We write Γ_v := Gal( ¯F_v/F_v) ⊂Γ_F := Gal( ¯F /F) andI_v ⊂ Γ_v is the inertia group. A geometric Frobenius Ψv ∈Γv is an element which induces the inverse of the Frobenius automorphism φ_vof ¯k_v. We denote byW_v theWeil groupofF_v, i.e., the dense subgroup formed by elements w∈Γ_v which induce on ¯k_v an integral power φ_v^α(w). The mapα :W_v →Z thus defined is a group homomorphism andker(α) = Iv. We will suppose all our algebraic varieties to be geometrically irreducible.

Consider a (pure) motive M over F. For any prime number `, the `-adic realization H_`(M) gives us a representation ρ_`: Γ_F →GL(H_`(M)) of Γ_F. Let v be a non-archimedean valuation ofF, by restricting ρ_` to Γ_v and using Grothendieck’s`-adic monodromy theorem, we obtain a nilpotent morphism N_` : H_`(M)(1) → H_`(M) of Γv-modules. This morphism induces the local monodromy filtration M_• on H_`(M). In case if H_`(M)∼= H^r_´_et(XF¯,Q`) ( as ΓF-modules) for some smooth projective algebraic varietyX over F, then Deligne’s weight monodromy conjecture ( cf. [3]) predicts thatfor any element w∈W_v, the eigenvalues of the induced autormphisms ρ¯`,j(w) on thej-th graded of M•, are algebraic integers with identical complex absolute valueq(r+j)α(w)/2

v . We generalize (see Con.3.2) this conjecture to the case of any arbitrary motiveM, via a local (w.r.t. v) weight filtrationW_•onH_`(M). We verify this conjecture (see Cor.3.7) for the category of motives which is⊗-generated by Artin motives;

abelian varieties; surfaces; and smooth complete intersections in any projective space, as a consequence of some more general results.

Next we prove (see Thm.4.4) that for any motive M and w ∈ Wv, the characteristic polynomial ofρ_M,`(w) has coefficients inQand is independent of`. This fact combined with the above mentioned results, allows us to verify in a large number of new cases ( see Thm.4.2 and Cor.4.5), that the local L-factors in the expression for the L-function of motives, are well-defined, i.e., rational and independent of `. In particular, we deduce that ifX is finite product of hyperk¨ahler varieties of K3^[n] type; unirational varieties of dim ≤ 4; uniruled varieties of dim≤ 3; cubic 4-folds; moduli spaces of stable vector bundles of coprime rank and degree over smooth projective curves and Fermat hypersurfaces; then H^r_et_´(XF¯,Q`) has a well-definedL-function for 0≤r≤2 dimX, which is independent of the choice of`.

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1. Monodromy filtration

1.1. Generalities. Let A be an abelian category and A ∈ Ob(A). Let N ∈ End(A) be a nilpotent endomorphism. Then, we define the kernel filtration associated to N, as the increasing filtration F_• on A satisfying

• F_pA= Ker (N^p+1 :A→A) for p≥0 and

• F_pA= 0 for p <0.

We define the image filtrationG^•associated toN, as the decreasing filtration onA, satisfying

• G^qA= Im (N^q:A→A) for q >0 and

• G^qA=A forq≤0.

The convolutionM=F∗Gis the increasing filtration onA satisfying M_rA:= X

p−q=r

F_pA∩G^qAforr ∈Z,

where + stands for product in A. Now, by the universal property of coproducts, the natural “inclusion” morphisms F_pA∩G^qA (0,···1,···0)

−−−−−−→M_rA induces a morphism

Π : M

p−q=r

F_pA∩G^qA→M_rA,

whereL

stands for coproduct inA.

Now, note that the filtrationGalso induces a filtration G^qGr^F_pA:= Im (G^qA∩F_pA→Gr^F_pA)

on the graded pieces Gr^F_pA := F_pA/Fp−1A. The following is an elementary, but useful characterization of graded pieces ofM_•, in terms of the above filtration.

Lemma 1.2. There is a natural isomorphism L

p−q=r

Gr_G^qGr^F_pA→Gr_r^MA

Proof. First note that the morphism Π :⊕p−q=rF_pA∩G^qA→M_rA defined above is surjective, i.e.,cokerΠ = 0. Now, as (G^qA∩F_pA)∩(G^q+1A∩Fp−1A) = (G^qA∩Fp−1A) + (G^q+1A∩ F_pA), it follows that

kerΠ = M

p−q=r

((G^qA∩F_p−1A) + (G^q+1A∩F_pA)).

Now, as

Gr^q_GGr_p^FA=G^qA∩Fp−1A/((G^qA∩Fp−1A) + (G^q+1A∩F_pA)), the assertion follows.

The next lemma is well-known

Lemma 1.3. The convolutionM=F∗Gis the unique filtration onAsatisfying the following properties :

(i) is an increasing filtration · · ·Mi−1A⊂M_iA⊂ M_i+1A· · · of Γ_v representations, such that M_iA= 0 for sufficiently small i and M_iA=A for sufficiently large i.

(ii) N_`(M_iA)⊆M_i−2A for all i.

(iii) Using the second condition we can define an induced mapN¯ : Gr^M_i A→Gr^M_i−2A, where Gr^M_i A=M_iA/Mi−1A. Then N¯^r : Gr^M_r A→Gr^M_−rA is an isomorphism for eachr ≥0.

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1.4. Local monodromy filtration for motives. Let the notation be as§0. We denote by V_F the category of smooth projective varieties over F. In this article, we will be interested in the following three different categoriesM_α(F) of pure motives over F :

• the category of (Grothendeick) motives,M_hom(F) defined in terms of algebraic cycles modulo homological equivalence.

• the category of motives M_mot(F), defined in terms of motivated correspondences;

• the category of motivesM_AH(F) defined in terms of absolute Hodge correspondences.

In each of these categories, there is a functorh_α :V_F → M_α(F), such that classical Weil- cohomology theories (with Tate twists)H_B (Betti cohomology),H_` ( `-adic cohomology, for any prime number`) andH_dR(De Rham cohomology) onV_F, factors throughM_α(F). This allows to define variousrealization functors onM_α(F), corresponding to these cohomology theories. In particular, if M := (X, p, n) ∈ObM_α(F), then its`-adic realization is defined as

H_`(M) =

2 dimX

M

w=0

p^∗H^w+2n_et (XF¯,Q`)(n),

wherep^∗ denotes the image ofp∈Corr_α⁰(X×X) under the ΓF-invariant isomorphism H^{2 dimX}_et (XF¯ ×XF¯,Q`)(dimX)∼=

2 dimX

M

w=0

End(H_et^w(XF¯,Q`)).

It is also clear that the Q`-vector space V_` := H_`(M) has a natural action of Γ_F. This gives rise to the `-adic representations ρ_` : ΓF → GL(V_`), which in turn gives us the local representations

ρ`,v: Γ_v →GL(V_`).

For any prime number `, we denote by µ`ⁿ the group of `ⁿ-th roots of unity in ¯kv and Z`(1) := lim←−

n

µ_`ⁿ. For any Q`-vector space U, we writeU(1) :=U ⊗

Q`

Q`(1), where Q`(1) = Q`⊗_Z_`Z`(1).

The inertia group Iv ⊂Γv fits into the following exact sequence 1→P →I_v −→^t Z(p⁰)(1)→1, where P is a pro-p-group and Z(p⁰)(1) = Y

`6=p

Z`(1). Let ` 6= p be a prime number. We denote by t_` :I_K → Z`(1), the `-component of t. Explicitly, the surjective map t_` is given as x 7→

x(π^`n¹ )/π^`n¹

n, where π ∈ O_v is an uniformizer. The map t_` is unique upto multiplication by an element ofZ^×_` .

By Grothendieck’s monodromy theorem there exists a unique nilpotent morphism N_` : V_`(1)→V_` of Γ_v-representations such that for a sufficiently small open subgroup J of I_v we have

(1) ρ

`,v(x) = exp(t_`(x)N_`), for allx∈J.

The morphism N_` is called the monodromy operator associated to ρ

`,v. Let M_• be the unique filtration onV` induced by N` and satisfying the properties of Lemma 1.3.

We leave it to the reader to verify the validity of the next lemma.

Lemma 1.5. Let M and M⁰ be motives inM_α(F).

(a) IfM is a sub-motive of M⁰, thenN_M,` =N_M⁰_,`|

H`(M)andM_iH_`(M) =M_iH_`(M⁰)|

H`(M)

for all i.

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(b) The monodromy operator associated to the direct sum of motives M ⊕M⁰, is given by NM⊕M⁰,`=NM,`⊕NM⁰,` and M_iH`(M⊕M⁰) =M_iH`(M)⊕M_iH`(M⁰).

(c) The monodromy operator associated to the tensor product of motives M ⊗M⁰ is given by N_M⊗M⁰_,`=N_M,`⊗Id + Id⊗N_M⁰_,` and

M_iH_`(M ⊗M⁰) =X

f+g

M_fH_`(M)⊗M_gH_`(M).

(d) The monodromy operator associated to the dual M^∨ is given by NM^∨,` = −^tNM,` and M_iH_`(M^∨) = (M−i−1H_`(M))^⊥.

Theorem 1.6. LetM be any motive overF andva non-archimedean valuation onF. Then there exists an open subgroupJ of Iv such thatρ_M,`|_J is unipotent for every `6=p.

Proof. LetM be a direct summand of a motiveh(X)(n) for some smooth projective algebraic variety X and n ∈ Z. Then it follows that the `-adic realization H_`(M) of M, is a Γ_F- submodule of the`-adic realization H_`(h(X)(n)) of h(x)(n). Let us denote by ρ_X,` : ΓF → GL(H_`(h(X))) the `-adic representation associated to the motive h(X). Let ξ_n,` : Γ_F → GL(Q`(n)) be the `-adic representation associated to the motive Q(n). Write ρX,`(n) :=

ρ_X,` ⊗ξ_n,`, then it is clear that ρ_X,`(n) is the `-representation associated to the motive h(X)(n). Assume that there exists a JX ⊂Iv such that ρX,`|_J_X is unipotent for all `6=p, i.e., there exists a r ∈Nsuch that (ρ_X,`(σ)−Id)^r = 0 for all σ ∈J_X. Asξ_1,` is unramified at all non-archimedean places v ofF, the same holds forξn,` = (ξ1,`)^⊗n. Therefore,

(ρ_X,`(n)(σ)−Id)^r = (ρ_X,`(σ)−Id)^r⊗Id = 0, for allσ ∈JX. This implies thatρ_M,`|_J_X is unipotent for all `6=p.

It remains to show that that there exists aJ_X ⊂I_v such that ρ_X,`|_J_X is unipotent for all

`6=p. In order to see this, first note thath(X) has a direct sum decomposition ash(X) =

⊕^{2 dimX}_i=0 hⁱ(X). Let ρⁱ_X,` : ΓF → GL(H`(hⁱ(X))) be the `-adic representation associated to the motive hⁱ(X). Then it follows that

ρ_X,`=

2 dimX

M

i=0

ρⁱ_X,`.

Assume that for every i, there exists aJ_Xⁱ ⊂Iv such thatρⁱ_X,`|_Ji

X is unipotent for all`6=p.

By settingJX =∩^{2 dim}_i=0 ^XJ_Xⁱ , we see thatρ_X,`|_J_X is unipotent for all `6=p.

So we are reduced to showing that for every 0≤i≤2 dimX, there exists aJ_Xⁱ ⊂I_v such thatρⁱ_X,`|_Ji

X is unipotent for all `6=p.

First assume thatXFv has a proper strictly semi-stable modelX over the ring of integers O_v of F_v. LetX_¯_k

v =X ⊗_O_v ¯k_v be the geometric special fiber of X. Now it is a well-known fact that Iv acts trivially on the sheaf of vanishing cycles R^qΨ(Q`) on X¯kv. It now follows from the Γ_v-invariant spectral sequence of vanishing cycles

E₂^p,q =H_et^p(X¯kv, R^qΨ(Q`))⇒H_etⁿ(XF¯v,Q`),

that for allσ ∈Iv, (σ−Id)ⁱ⁺¹ acts trivially onH_etⁱ (XF¯v,Q`) for every`6=p.

Now in general the existence of a semi-stable model for X_F_v is unknown. But we know from [2, Thm. 6.5]that, there is a finite extensionF_v⁰0 ofF_v and a strictly semi-stable scheme Y⁰over the ring of integersO_v⁰ ofF_v⁰0, with generic fiberX⁰and aFv-alterationf :X⁰ →XFv. The induced morphism f⁰ : X⁰ → X_F⁰

v0 is again an alteration. Since f⁰ is finite of degree degf⁰ on some open dense subscheme ofX⁰, the pull backf^0∗ :H_etⁱ(XF¯v,Q`)→H_etⁱ(X_F⁰_¯

v,Q`) and the pushforward f_∗⁰ :H_etⁱ (X_F⁰_¯

v,Q`) → H_etⁱ (XF¯v,Q`) satisfy f_∗⁰ ◦f^0∗ = degf⁰·id. As f^0∗

and f_∗⁰ commute with the action of Γ_v⁰, it follows that H_etⁱ(XF¯v,Q`) is a direct summand 4

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of H_etⁱ(X_F⁰_¯

v,Q`), as Γ_v⁰-module and hence as I_v⁰ (inertia subgroup of Γ_v⁰) module. As X⁰ has a semi-stable model, it follows from our previous discussion that I_v⁰ acts trivially on H_etⁱ (X_F⁰_¯

v,Q`) and hence on H_etⁱ (XF¯v,Q`), for every `6=p.

Finally, as the`-adic realizationH_`(hⁱ(X)) of the motivehⁱ(X) is isomorphic toH_etⁱ (XF¯v,Q`) as Γ_F-modules, the proposition follows.

2. Weight Filtration

Definition 2.1. A Γv- mdouleU is said to have a (unique) weight filtrationW_• if it verifies the following properties :

(i) W_• is an increasing filtration · · ·W_i−1U ⊂ W_iU ⊂ W_i+1U· · · of Γv-representations such thatW_iU = 0 for sufficiently smalli andW_i+1U =U for sufficiently largei;

(ii) the action of Iv on Gr^w_i =W_iU/Wi−1U is through a finite quotient;

(iii) the eigenvalues of the action of Ψ_v on Gr^w_i =W_iU/Wi−1U are all algebraic integers of complex absolute valueq^i/2, where q=|k_v|.

Example 2.2. Q`(1) is a one-dimensional Γv-representation which is unramified for every v ( such that p 6= `). Moreover, we know that the action of any arithmetic Frobenius on Q`(1) is through multiplication by q, hence the action of the geometric Frobenius is through multiplication by q⁻¹. This allows, us to define the weight filtration as following

W_jQ`(1) =Q`(1) forj≥ −2 andW_jQ`(1) = 0 forj <−2.

Lemma 2.3. Let U, U⁰ be Γv- modules.

(a) IfU has weight filtration and U⁰ is a sub-module ofU, then U⁰ has weight filtration.

(b) If bothU andU⁰ has weight filtrations, then the Γ_v-modules U⊕U⁰ andU⊗U⁰ has weight filtrations.

(c) IfU has weight filtration then the dual U₁^∨ has weight filtration.

Proof. (a) We verify that W_iU⁰ :=W_iU∩U⁰ defines the weight filtration onU⁰.

(b) We verify thatW_i(U⊕U⁰) :=W_iU⊕W_iU⁰ defines the weight filtration onU ⊕U⁰ and W_i(U ⊗U⁰) = X

f+g=i

W_fU ⊗W_gU⁰, defines the weight filtration on U⊗U⁰.

(c) we verify that the dual filtration W_iU^∨ := (W−i−1U)^⊥ defines the weight filtration on U^∨.

Proposition 2.4. Let M be any motive overF. Then for every prime number `6=p, there exists a weight filtration on the Γ_v-moduleH_`(M).

Proof. Let M = (X, p, n) be a motive over F, where X is a smooth projective algebraic variety,p∈Corr⁰(X×X) and n∈Z. By definition, the`-adic realization of M is

H_`(M) =

2 dimX

M

w=0

p^∗H^w+2n_et (XF¯,Q`)(n).

Now, p^∗H^w+2n_et (XF¯,Q`) is a Γv-sub-representation of H^w+2n_et (XF¯,Q`) ∼= H^w+2n_et (XF¯v,Q`).

Now suppose that H_et^w(XF¯v,Q`) has a weight filtration. Then by 2.3(a) we get a weight filtration on p^∗H^w+2n_et (XF¯,Q`). Now, it follows from 2.2(i) and 2.3(b), that there exists a weight filtration on p^∗H^w+2n_et (XF¯,Q`)⊗Q`(1)^⊗n = p^∗H^w+2n_et (XF¯,Q`)(n). Finally, using

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2.3[(b)] we conclude that there exists a weight filtration on ⊕^{2 dim}_w=0 ^Xp^∗H^w+2n_et (XF¯,Q`)(n) = H`(M).

It remains to exhibit thatH_et^w(XF¯v,Q`) has a weight filtration. To start with let us assume that X_F_v has a proper strictly semi-stable model X over the ring of integers O_v of F_v. Let X1,· · ·, Xm be the irreducible components of the special fiber ofX, and set

X^(k):= a

1≤i1<···<i_k≤m

Xi1 ∩ · · · ∩Xi_k.

Then X^(k) is a proper smooth variety of dimensionn−k+ 1 over the residue field kv of F_v. Now, by [6], there is a Γ_v- invariant spectral sequence

(2) E₁^−r,w+r= M

k≥max{0,−r}

H_et^w−r−2k(X_k_¯^(2k+r+1)

v ,Q`(−r−k))⇒H_et^w(XF¯v,Q`).

Now the inertia group I_v acts trivially on each E₁^i,j and hence there is an action of Gal(¯k_v/k_v). By the Weil conjectures, we know that H_et^w−r−2k(X_¯_k^(2k+r+1)

v ,Q`(−r −k)) has weight (w−r−2k)−2(r−k) =w+r. Thus, E₁^i,j has weightj and the filtration induced by the spectral sequence (2), is the required weight filtration defined in 2.1.

We now treat the general case. As in the proof of Thm.1.6, there is a finite extension F_v⁰0 of F_v, such that there is a strictly semi-stable schemeY⁰ over the ring of integers O_v⁰ of F_v⁰0, with generic fiberX⁰ and aFv-alterationf :X⁰ →XFv. It follows that H_et^w(XF¯v,Q`) is a direct summand of H_et^w(X_F⁰_¯

v,Q`), as Γ_v⁰-modules. As X⁰ has a semi-stable model, hence by the above observation, there is a weight filtration on H_et^w(X_F⁰_¯

v,Q`). It now follows from Lemma 2.3, that H_et^w(XF¯v,Q`) has a weight filtration.

3. On the purity of local monodromy filtration

Definition 3.1. Let M be a motive overF. The weight wt(M) of M is defined to be the weight of the underlying Hodge structure of the Betti realizationHτ(M). It is independent of the choice of the embeddingτ.

The weight monodromy conjecture (WMC) predicts that the monodromy filtration on the absolute`-adic cohomology of any proper smooth variety overFv, coincides with the weight filtration but upto a shift. This conjecture has the following analogue for motives.

Conjecture 3.2. Let M be any motive over F of weight w. Then, M_iH_`(M) =W_i+wt(M)H_`(M)∀i.

Proposition 3.3. Let M be an Artin motive over F. Then, conjecture 3.2 holds for M.

Proof. We know that the category of Artin motives M⁰_F ∼= Rep_Q(Γ_F). Let ρ: Γ_F →GL(V) be the image ofM under this isomorphism. Then we have,

Hτ(M) =V andH_`(M) =V ⊗Q`, as a Γ_F −module.

Thus, the`-adic representationρ

`: Γ_F →GL(H_`(M)) associated to M, is simply ρ⊗Q`. Now, as ρis continuous for the Krull topology on ΓF and the discrete topology on V, there exists a finite extensionF⁰/F, such that Gal( ¯F /F⁰) acts trivially on V. In particular, ifv⁰ is an extension ofvtoF⁰, then Gal( ¯Fv/F_v⁰0) and hence the inertia group corresponding to this extension, acts trivially on V. Thus, the `-adic representation ρ

` is potentially unramified at every v such that p 6= `. This implies that the monodromy operators corresponding to

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the local representations ρ

`,v are all trivial. In particular, the monodromy filtration M_• of H`(M) is trivial, i.e.

M_jH_`(M) =H_`(M) forj≥0 andM_jH_`(M) = 0 forj <0.

Now, as the weight of an Artin motive is 0, in order to prove the proposition, we only need to show that the weight filtration on H`(M) coincides with the monodromy filtration on H_`(M). It is obvious that M_•H_`(M), verifies property (i) of 2.1. As, ρ

` is potentially unramified at every v, so M_•H_`(M) also verifies property (ii) of 2.1. Finally, note that as ρ` = ρ ⊗Q`, the eigenvalues of the action of Ψv on Gr^M₀ H`(M) = H`(M) = V ⊗Q`

are algebraic integers ( independent of ` ). Now, as Gal( ¯F /F⁰) acts trivially on V, the eigenvalues of the action of Ψ^[k⁰^v⁰^:k^v^]onV are all 1, wherek_v⁰0 denotes the residue field of F_v⁰0. This implies that the eigenvalue of the action of Ψ_v on V is a root of unity and hence of complex absolute valueq^0/2= 1. This establishes property (iii) of 2.1. By the uniqueness of the weight filtration, we conclude that on H_`(M), the monodromy filtration coincides with

the weight filtration.

Definition 3.4. Forβ = motorAH, we denote byM^pmf_β (F) the Tannakian subcategory of M_β(F), which is ⊗-generated by the familyC := (Mi|i∈I) of motives, such that eachMi

verifies Conjecture 3.2.

Proposition 3.5. Conjecture 3.2 holds for every M ∈Ob(M^pmf_β (F)).

Proof. Any motive M ∈Ob(M^pmf_β (F)) is isomorphic to a sub-quotient of an object of the form P(M_j⁰) ( a polynomial expression), where {M_j⁰}_j∈J⊂I consists of objects of C or their duals and P(t_j) ∈ N[t_j]j∈J; multiplication in P(M_j⁰) is interpreted as ⊗ and addition as

⊕. As M_β(F) is a semi-simple Tannakian category, it follows that every quotient object of P(M_j⁰), can be identified with a subobject. Thus, in view of 3.5, it suffices to verify the conjecture for the motiveP(M_j⁰). Now, by combining Lemma1.5 and Lemma2.3, we see that Conjecture 3.2 is stable under sub-objects, ⊕, ⊗ and taking duals of motives. A repeated application of this fact and the definition of C, implies the proposition.

Definition 3.6. Define M^acs_AH(F) ( resp. M^acs_mot(F)) to be the Tannakian subcategory of M_AH(F) ( resp. M_mot(F) ), which is ⊗-generated by the following motives

(i) Artin motives;

(ii) h¹(A)’s of abelian varieties;

(iii) h²(S)’s of surfaces;

(iv) for any m ∈ N, h^m(Y)’s of m-dimensional (smooth) complete intersections in any projective space.

We also define M^acs_hom(F) to be the abelian ⊗-subcategory of M_hom(F), generated by (i) the Tate motive Q(1) (ii) abelian varieties (iii) 0 dimensional varieties (iv) surfaces and (v) complete intersections in projective spaces.

Corollary 3.7. Forα=hom, motorAH, Conjecture 3.2 holds for everyM ∈Ob(M^acs_α (F)).

Proof. First note that it suffices to show the corollary for M^acs_AH(F)), as M_hom(F) and M_mot(F) are subcategories ( a priori not full) ofM_AH(F) and the the`-adic realization of a motive is unchanged by passing from one category to the other.

Now, by Prop. 3.3 the Artin motives verifies Conjecture 3.2, by [1] we know thath¹(A)’s for A any abelian variety, verifies the conjecture, by [6] the conjecture is true for h²(S) for any surface S, and finally by [8], the conjecture holds for h^m(Y)’s (for any m ∈ N ) of m-dimensional (smooth) complete intersections in any projective space. It follows that M^acs_AH(F) is a subcategory of M^pmf_AH(F) and hence verifies Conjecture 3.2.

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Remark 3.8. Denote by M_num(F) the category of (Grothendieck) motives for numerical equivalence on algebraic cycles. LetM^av_num(F) be the abelian ⊗-subcategory of M_num(F), generated by (i) the Tate motiveQ(1) (i) abelian varieties and (iii) 0 dimensional varieties.

By [4], we know thatMâv_num(F) is aF-linear semi-simple Tannakian category. As homological equivalence coincides with numerical equivalence for zero dimensional varieties and abelian varieties [5], we can identifyMâv_num(F) to a subcategory ofMâcs_AH(F). Hence it makes sense to speak of`-adic realizations of motives in Mâv_num(F). The previous corollary then implies that Conjecture 3.2 holds for objects inMâv_num(F).

Corollary 3.9. Let X be a smooth projective variety over F which is a finite product of (i) hyperk¨ahler varieties of K3^[n] type;

(ii) unirational varieties of dim≤4;

(iii) uniruled varieties of dim≤3;

(iv) cubic 4-folds;

(v) moduli spaces of stable vector bundles of coprime rank and degree over smooth projective curves;

(vi) Fermat hypersurfaces;

(vii) curves;

(viii) surfaces;

(ix) abelian varieties;

then the weight monodromy conjecture holds forHⁱ(XF¯,Q`) for any i≥0.

Proof. In view of 3.5, it suffices to establish the result for each of the varieties in the list.

If X is either a hyperk¨ahler variety of K3^[n] type, a cubic 4-fold, a unirational variety of dim≤3, a Fermat hypersurface, moduli space of stable vector bundles of coprime rank and degree over a smooth projective curve, then the motiveh(X)∈Ob(M^av_F ). So, in these cases the weight monodromy conjecture follows from 3.7.

The case of curves [1] and surfaces [6], are well known. The case of unirational varieties of dimension 4, follows from the fact that the motive of any such variety is an object of the Tannakian category generated by the motive of variety of dimension at most 2. Similarly, the motive of a uniruled 3-fold belongs to the Tannakian category of motives generated by a surface, and hence the WMC holds in this case as well.

In the case of any abelian variety A the only new thing that we obtain is that WMC holds for the higher cohomology groups Hⁱ(A_/F¯v,Q`) (i ≥ 2) as well, since the motive

hⁱ(A) =∧ⁱh¹(A).

4. L-functions

LetM be any motive in M_α(F), for α= hom, motorAH. For any prime number `, we denote byV` :=H`(M). For any non-archemdean valuation vonF of residual characteristic p, consider the characteristic polynomial

Z(V_`, T) := det_Q_`(1−T ρ_M,`(Ψ_v)|V_`^I^v)), (l6=p)

for the action of a geometric Frobenius element Ψ_v ∈ Γ_v on the inertia invariant part V_`^I^v part ofV`. By definition, Z(V`, T)∈Q`[T].

Definition 4.1. A motiveM as above is said to have a well-definedL-function, ifZ(V`, T)∈ Q[T] and is independent of`. In this case, theL-function ofM is defined as the Euler product

L(M, s) =Y

v

Lv(M, s), 8

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where the localL-factors L_v(M, s) is defined as Lv(M, s) = 1

Z(V_`, q^−sv ), for`6=p.

Theorem 4.2. Let M ∈ObM^pmf_AH(F) be a motive, such that for every non-archemdean val- uationvonF andw∈Wv, the characteristic polynomialP_`(w, T)ofρ_M,`(w)has coefficients in Qand is independent of `. Then M has a well-defined L-function.

Proof. We need to verify that Z(V_`, T) has coefficients in Qand is independent of `.

First consider the non-archemdean valuationsv, whereM has good reduction, i.e., V_` is unramified at v for every`6=p. In this case

Z(V_`, T) =P_`(Ψ_v, T),

hence by hypothesis,Z(V`, T) has coefficients in Q and is independent of`.

Now consider the non-archemdean valuations v, whereM has bad reduction. AsV_`^I^v is a subspace ofV_`, it follows from the hypothesis, that for every w∈W_v, the coefficients of the characteristic polynomial det_Q_`(1−T ρ_M,`(w)|V_`^I^v)) are algebraic numbers independent of`.

Thus, in order to prove our claim, it suffices to show that Tr(w^∗:V_`^I^v)∈Qand independent of` and then use Newton’s lemma.

Let ¯ρ_`,j : Γv → GL(Gr^M_j V_`) denote the representation induced by ρ_M,` on the graded parts of the local monodromy filtrationM_• on V_`. For every w ∈W_v, let ¯P_`,j(w, T) be the characteristic polynomial of ¯ρ`,j(w).

Lemma 4.3. Fix a w∈W_v, then for everyj, P¯_`,j(w, T)∈Q[T]and is independent of `.

Proof. Let L be the sub-field of an algebraic closure ¯Q` of Q`, generated by the roots of P_`(w, T). By hypothesis,P_`(w, T)∈Q[T] and has coefficients independent of`. This implies that if P_`(w, β) = 0, then P_`(w, σ(β)) = 0, for every σ ∈ Gal(L/Q). Now, from linear algebra, we know that P_`(w, T) = Q

jP¯_`,j(w, T). Let η be an eigenvalue ¯ρ_`,j(w) and hence ofρ_`(w). By Prop. 3.5, the complex absolute value

(3) |η|_C=q^(wt(M_v ^)+j)α(w)/2 for everyj.

As|β|_C =|σ(β)|_C, it follows from (3), that β and σ(β) occurs as the roots of a same factor (say) ¯P_`,j₀(w, T) of P_`(w, T). Now suppose ¯P_`,j₀(w, T) = T^m +Pm

r=1am−rT^m−r. As the coefficients ar’s are symmetric polynomials in the roots of ¯P_`,j₀(w, T), it follows from the previous observation that σ(a_r) = a_r, for every σ ∈ Gal(L/Q). In other words, a_r ∈ Q, i.e., ¯P`,j0(w, T)∈Q[T] and is independent of `. By varying β over all roots of P`(w, T), we conclude the ¯P`,j(w, T)∈Q[T] and is independent of `, for everyj.

Now, let us denote byF•andG^•the kernel filtration and image filtration onV`. By lemma 1.2, we have an equality of Γv-modules

Gr^M_−j+1V_` =Gr_G^jGr^F₁V_`⊕Gr^M_−j−1V_`(1), for everyj≥0.

As, Gr_G^jGr₁^FV_`=Gr_G^jF₀V_` (this is the graded for the filtration induced by G^• on F₀V_`), we get that

Gr_−j+1^M V_`=Gr^j_GF₀V_`⊕Gr^M_−j−1V_`(1)

for everyj≥0. This implies that the traces Tr(w^∗ :Gr_G^jF₀V_`)∈Qand is independent of `.

It now follows from linear algebra that Tr(w^∗:F₀V_`)∈Q and is independent of`.

Now, the action ofI_v onF₀V_`factors through a finite quotientI_v/J andV_`^I^v = (F₀V_`)^I^v^/J. 9

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Hence for a complete set ofT ⊂I_v of representatives of I_v/J, we have [Iv :J]·Tr(w^∗:V_`^I^v) =X

τ∈T

Tr((w◦τ)^∗ :F₀V`).

Now by our previous discussion, Tr((w◦τ)^∗ :F₀V`) ∈Q and is independent of ` and by Thm.1.6, [I_v :J] is independent of`. It follows that Tr(w^∗ :V_`^I^v)∈Qand independent of `.

This completes the proof of the theorem.

Theorem 4.4. Assume that the K¨unneth Standard conjecture holds for (absolute) `-adic cohomology (with Tate twists) of varieties defined over F. Then, for every motive M ∈ M_hom(F), and non-archimedean valuation v onF, the characteristic polynomialP_M,`(w, T) for the action of any element w∈W_v, has coefficients in Qand is independent of `.

Proof. First note that it suffices to show that for every w ∈ Wv, the traces Tr (ρ_M,`(w)) are in Q and is independent of `, for then one can resolve the Newton’s identities relating symmetric polynomials and power sums, to conclude that the polynomialPM,`(w, T)∈Q[T] and is independent of`.

LetM := (X, p, n) and M⁰ := (X, p,0). Then it is easy to see that Tr (ρ_M,`(w)) =χⁿ_`(w)·Tr ρ_M⁰_,`(w)

,

whereχ_` : Γ_F → Q^×_` is the `-adic cyclotomic character of Γ_F. As χ_`(λ) = 1,for every λin the inertia subgroupI_v, it follows that if w= Ψ^m_v ·λ(for some m∈Z), then

χⁿ_`(w) =q_v^mn.

Asq^mn_v ∈Q and is independent of `, thus it suffices to show the proposition for motives of the form M⁰ = (X, p,0).

First, we make an observation. Lete, e⁰ ∈CH^d(X×X) be any two algebraic correspondences. Then, by definition

e⁰◦e=pr13∗(pr₁₂^∗ e·pr^∗₂₃e⁰),

where pr_αβ denotes the projection maps from X ×X×X to X×X, for α, β ∈ {1,2,3}.

Writed:= dimX andγ_X×X^d :CH^d(X×X)→H^2d_et(XF¯×XF¯,Q`)(d) be the cycle class map to`-adic cohomology of X×X. As cycle class maps are compatible with push-forward , we getγ^d_X×X(e⁰◦e) =pr13∗(γ^d_X×X×X(pr₁₂^∗ e·pr^∗₂₃e⁰)). Since cycle class maps are compatible with intersection products inCH^∗(−) and cup products in cohomology, we have γ_X×X^d (e⁰◦e) = pr13∗(γ_X×X^d _×X(pr^∗₁₂e)·γ_X×X×X^d (pr₂₃^∗ e⁰)). Finally, as cycle class maps are compatible with pull-backs we get γ_X×X^d (e⁰ ◦ e) = pr13∗(pr₁₂^∗ (γ_X^d_×X(e))·pr^∗₂₃(γ_X^d_×X(e⁰))). But this is by definition the composition of correspondneces γ_X×X^d (e⁰)◦γ_X×X^d (e) under the identification (4) H^2d_et(XF¯ ×XF¯,Q`)(d)∼=⊕_r≥0End_Q_`(H^r_et(XF¯,Q`)).

Hence, writinge:=γ_X×X^d (e), when viewing it as a Q`-linear endomorphism of the (graded) vector space V :=⊕^2d_r≥0H^r_et(XF¯,Q`) (under the identification (4)), we conclude that

e⁰◦e=e⁰◦e.

Now consider the motiveM⁰ = (X, p,0), by definitionpis an idempotent inCH^d(X×X), i.e.,p◦p=p. It follows from the above discussion thatp is an idempotent linear map. This implies that

V =Ker(p)⊕Im(p),

as Γ_F-modules. Let w∗ be the automorphism of V, induced by the action of w. It now follows by linear algebra that

Tr ρ_M⁰_,`(w)

= Tr

w∗|_Im(p)

= Tr w∗◦p . 10

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Thus in order to prove the proposition it suffices to show that Tr w∗◦p

∈Q and is independent of`.

For brevity, let us denote by Tr^alt(w∗◦p) the alternating sum

2d

X

r=0

(−1)^rTr(w∗◦p: H^r_et(XF¯,Q`)).

By hypothesis the K¨unneth standard conjecture holds forX, i.e, for every 0≤r≤2d, there exists an algebraic cycleπ_r such thatπ_r is the canonical projection

V H^r_et(XF¯,Q`),→V,

followed by inclusion. It now follows from the above discussion that Tr w∗◦(p◦π_r)

= Tr w∗◦p: H^r_et(XF¯,Q`)

= Tr^alt w∗◦(p◦π_r) . By [7, Thm 0.1], Tr^alt w∗◦(p◦πr)

∈Qand is independent of`. It follows that

2d

X

r=0

Tr w∗◦p: H^r_et(XF¯,Q`)

=

2d

X

r=0

Tr w∗◦(p◦π_r)

= Tr

w∗◦(p◦∆X×X)

= Tr w∗◦p , is inQ and is independent of`.

Corollary 4.5. Every motiveM ∈ M^acs_hom(F) has a well-defined L-function.

Proof. As M^acs_hom(F) is a subcategory of M^pmf_AH(F), thus in view of Thm.4.2, we only need to verify that for any non-archemdean valuation v on F and w ∈ W_v, the characteristic polynomialPM,`(w, T) ofρM,`(w) has coefficients in Qand is independent of `.

WriteM = (X, p, n), then X is a product of abelian varieties, zero dimensional varieties, surfaces or complete intersection in projective spaces. As each of these varieties satisfies K¨unneth standard conjecture, it follows that X satisfies K¨unneth standard conjecture. It now follows from the proof of Thm.4.4, thatP_M,`(w, T)∈Q[T] and is independent of`.

Corollary 4.6. Let X be as in Cor. 3.9. Then, for every i ∈ N, the cohomology groups H_etⁱ (X_/F¯,Q`) has well defined L-function, which is independent of the choice of`.

References

[1] Groupes de monodromie en géométrie algébrique. I. Lecture Notes in Mathematics, Vol. 288. Springer- Verlag, Berlin, 1972. Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I), Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim.

[2] A. J. de Jong. Smoothness, semi-stability and alterations.Inst. Hautes ´Etudes Sci. Publ. Math., (83):51–93, 1996.

[3] Pierre Deligne. Théorie de Hodge. I. InActes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, pages 425–430. Gauthier-Villars, Paris, 1971.

[4] Uwe Jannsen. Motives, numerical equivalence, and semi-simplicity.Invent. Math., 107(3):447–452, 1992.

[5] David I. Lieberman. Numerical and homological equivalence of algebraic cycles on Hodge manifolds.Amer.

J. Math., 90:366–374, 1968.

[6] M. Rapoport and Th. Zink. ¨Uber die lokale Zetafunktion von Shimuravariet¨aten. Monodromiefiltration und verschwindende Zyklen in ungleicher Charakteristik.Invent. Math., 68(1):21–101, 1982.

[7] Takeshi Saito. Weight spectral sequences and independence of l. J. Inst. Math. Jussieu, 2(4):583–634, 2003.

[8] Peter Scholze. Perfectoid spaces.Inst. Hautes ´Etudes Sci. Publ. Math., (116):245–313, 2012.

Department of Mathematics, VU University - Faculty of Sciences, Amsterdam, Netherlands.

E-mail address: a.laskar@vu.nl

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