Derivatives of Frobenius and Derivatives of Hodge weights 6 1 0 Bingyong Xie 2 n Department of Mathematics, East China Normal University, Shanghai, China a J [email protected] 9 1 ] Abstract T N In this paper we study the derivatives of Frobenius and the derivatives . of Hodge weights for families of Galois representations with triangulations. h t We generalize the Fontaine-Mazur -invariant and use it to build a formula a L m which is a generalization of the Greenberg-Stevens-Colmez formula. For the purposeof provingthis formulaweshow two auxiliary resultscalled projection [ vanishingpropertyand “projection vanishingimplying -invariants” property. 1 L v 4 0 Introduction 9 4 0 It is well known that Galois Representation is one of the most fundamental ob- 1. jects in number theory. In this paper we concentrate on the p-adic representations 0 of the absolute Galois group of Q , where p is a fixed prime number. Among p 6 them semistable representations are special but important. To such representa- 1 : tions Fontaine [18] attached linear algebra objects called filtered (ϕ,N)-modules. v i Colmez and Fontaine [14] proved that there is an equivalence of categories between X the category of semistable representations and the category of admissible filtered r a (ϕ,N)-modules. Using the associated filtered (ϕ,N)-module we can attach to each semistable representation two kinds of invariants, i.e. Hodge weights and the eigen- values of Frobenius ϕ. A famous fact is that the Newton polygon is always above the Hodge polygon, which is the main significance of admissibility. Recently there are a lot of papers studying families of Galois representations. For example, see [6, 21, 22, 27]. A natural question on families of Galois representations is the following 1 Question 0.1. For a family of p-adic representations of G , what is the relation Qp of derivatives of Hodge weights and derivatives of eigenvalues of Frobenius? However, Hodgeweights andeigenvalues ofFrobeniusarenotdefinedforageneral representation of G . Therefore, we need to specify certain conditions so that Qp the two kinds of derivatives in Question 0.1 can be reasonably explained. A good choice is the families with triangulations. The significance of triangulations has been confirmed by many works. See [23, 11, 13, 24, 8] for example. To explain what a triangulation is, we need the theory of (ϕ,Γ)-modules. The (ϕ,Γ)-modules are modules over various rings of power series (denoted by E, E† and R). See [16, 9, 20] for precise constructions of these rings and definitions of (ϕ,Γ)-modules. Theorem 0.2. ([16, 9, 20]) Thereis an equivalenceof categories between the category of p-adic representations of G and the category of ´etale (ϕ,Γ)-modules over either Qp E, E† or R. What we need is a version of Theorem 0.2 with p-adic representations (i.e. Q - p representations) replaced by E-representations where E is a finite extension of Q . p Such a variant version follows directly from Theorem 0.2 itself. Let E be a finite extension of Q . For a (not necessarily ´etale) (ϕ,Γ)-module M p over R , by a triangulation of M we mean a filtration Fil M on M consisting of E • saturated (ϕ,Γ)-submodules of M with rank Fil M = i such that Fil M/Fil M RE i i i−1 (1 i rank M) is of rank 1, i.e. of the form R (δ ) where δ is an E×-valued ≤ ≤ RE E i i character of Q×. We call (δ , ,δ ) the triangulation data for M. p 1 ··· n When M comes from a semistable representation V, w , , w coincide − δ1 ··· − δn with the Hodge weights of V, and δ1(p)pwδ1, ,δn(p)pwδn coincide with eigenvalues ··· of Frobenius ofV. Here for a character δ of Q×, w is the weight of δ whose definition p δ is given in Section 3. Hence, for a family of representations of G with triangulation data (δ , ,δ ) Qp 1 ··· n we can regard dw (i = 1, ,n) as the derivatives of Hodge weights, and regard δi ··· dδi(p) + log(p)dw (i = 1, ,n) formally as the derivatives of “logarithmic of δi(p) δi ··· Frobenius eigenvalues”. The value of log(p) depends on which component of the logarithmic we take. Now specifying the families of representations of G with triangulations, Ques- Qp tion 0.1 becomes the following Question 0.3. For an S-representation of G with triangulation date (δ , ,δ ), Qp 1 ··· n what is the relation among dδ1(p), , dδn(p),dw , ,dw ? We will always take S δi(p) ··· δn(p) δ1 ··· δn to be an affinoid E-algebra. 2 When n = 2, Question 0.3 has been researched by Greenberg-Stevens [19] (for ordinary semistable point) and Colmez [12] (for general semistable point). The pre- cise statement of Colmez’s theorem will be recalled below. Later Colmez’s theorem was generalized by Zhang [28] (again for n = 2 but the base field Q is replaced by p any finite extension of Q ). p Let S be an affiniod E-algebra, a 2-dimensional S-representation of G . V Qp Without loss of generality we may assume that is free, and let v ,v be a basis 1 2 V { } of over S. Let σ A be the matrix of σ G with respect to this basis. Then V 7→ σ ∈ Qp there exist δ,κ S such that ∈ log(detA ) = δψ (σ)+κψ (σ) σ 1 2 for any σ G . Here, ψ : G E is the unramified additive character of G ∈ Qp 1 Qp → Qp such that ψ (σ) = 1 if σ induces the Frobenius x xp on F ; ψ : G E is the 1 7→ p 2 Qp → additive character that is the logarithmic of the cyclotomic character χ . cyc Theorem 0.4. ([12]) Suppose that admits a fixed Hodge weight 0 and there exists α ∈ S such that (Bϕcr=isα,S⊗SV)GQp isVlocally free of rank 1 over S. Suppose z0 is a closed point of Max(S) such that is semistable with Hodge weights 0 and k 1. b Vz0 ≥ Then the differential dα 1 1 dκ+ dδ α − 2L 2 is zero at z , where is the Fontaine-Mazur -invariant of . 0 L L Vz0 Theorem 0.4 hints that Question 0.3 should be closely related to the following Question 0.5. What is the generalization of Fontaine-Mazur -invariants? L Let (D,ϕ,N,Fil•) be an admissible filtered E-(ϕ,N)-module with a refinement . Throughout this paper we assume that ϕ is semisimple on D.1 The monodromy F N induces an operator N on the grading module F dimED grFD = D/ D. • Fi Fi−1 Mi=1 1This isnotanessentialcondition. However,toinclude the resultforthe generalcase(ϕmaybe not semisimple) we need much more knowledge and technique from the theory of (ϕ,Γ)-modules, whichdoesnotfitwiththestyleofthepresentpaper. Thegeneralcasewillbeconsideredinasequel paper, where we study the families of (not necessarily ´etale) (ϕ,Γ)-modules instead of families of Galois representations. 3 If s,t 1, ,dim D satisfy s < t and N (grFD) = grFD, then we say that s is ∈ { ··· E } F t s critical for and write t = t (s). The criticality does not depend on ϕ and Fil•. We F F will introduce another notion “strong criticality” (see Definition 4.8) which depends not only on N and but also on ϕ and Fil•. If s is strongly critical, we can attach F to s an invariant denoted by . For the solution to Question 0.5 we regard the set F,s L : s is strongly critical for F,s {L F} as the generalization of the Fontaine-Mazur -invariant. 2 L Now we can state our main theorem as follows. Theorem 0.6. Let S be an affinoid E-algebra. Let be an S-representation of G V Qp with a triangulation and the associated triangulation date (δ , ,δ ). Let z be a 1 n 0 ··· closed point of Max(S), E the residue field of S at z . Suppose that is semistable z0 0 Vz0 and ϕ is semisimple on D, where D is the filtered E -(ϕ,N)-module attached to . z0 Vz0 Let be the refinement on D corresponding to the triangulation of . Suppose that F Vz0 s 1, ,n 1 is strongly critical for , t = t (s). Then F ∈ { ··· − } F dδ (p) dδ (p) t s + (dw dw ) δ (p) − δ (p) LF,s δt − δs t s is zero at z . 0 We remark that, when s is critical for and t (s) = s+1, s is strongly critical F F for if and only if w > w . F δs,z0 δs+1,z0 An especially interesting case is when the rank of the monodromy N of D is equal to dim D 1. Let e be an element not in N(D) such that ϕ(e ) Ee . E n n n − ∈ For i = 1, ,n 1 put e = Nn−ie . Then D admits a unique triangulation and i n ··· − F D = Ee Ee for all i = 1, ,n. Write k = w . Then k , ,k are Fi 1 ⊕···⊕ i ··· i − δi,z0 1 ··· n Hodge weights of . There always exists an upper-triangular matrix (ℓ ) such Vz0 j,i n×n that e + ℓ e : i = 1, ,n is an E-basis of D compatible with the Hodge i j,i j { ··· } 1≤Pj<i filtration. 2 The readermaybe mystifiedthatour -invariantisdefinedforGaloisrepresentationswithtri- L angulationsinsteadofGaloisrepresentationsthemselves. Ononehand,suchadefinitionissuitable for Question 0.3, which can be seen in Theorem 0.6. On the other hands, Galois representations with triangulations play the fundamental rolein many aspects,for example the definition of p-adic L-functions for modular forms [26] and the construction of eigenvarieties [1, 7]. In Fontaine and Mazur’s definition of -invariants,the information of triangulationis hidden. Indeed, a semistable L (but non-crystalline) 2-dimensional Galois representation admits a unique triangulation. 4 Theorem 0.7. With the above notations suppose that k < k < < k . Then 1 2 n ··· dδ (p) dδ (p) s+1 s +ℓ (dw dw ) δ (p) − δ (p) s,s+1 δs+1 − δs s+1 s is zero at z . 0 When n = 2, the condition k < k automatically holds. So Theorem 0.7 1 2 covers Theorem 0.4. Indeed, under the condition of Theorem 0.4 we have dw = 0, δ1 dα = dδ1(p), dδ = dδ1(p) dδ2(p) and dκ = dw . α δ1(p) − δ1(p) − δ2(p) δ2 We sketch the proof of Theorem 0.6. From we obtain an infinitesimal deformation of and attach to this infinites- V Vz0 imal deformation a 1-cocycle c : G ∗ . Let e , ,e be a basis of Qp → Vz0 ⊗Ez0 Vz0 { 1 ··· n} D that is s-perfect for , e∗, e∗ the dual basis of e , ,e . (See Definition F { 1 ··· n} { 1 ··· n} 4.10 for the precise meaning of s-perfect basis.) Let π be the composition of the h,ℓ inclusion ∗ ֒ B ( ∗ ) Vz0 ⊗Ez0 Vz0 → st,Ez0 ⊗Ez0 Vz0 ⊗Ez0 Vz0 and the projection B ( ∗ ) B , b e∗ e b . st,Ez0 ⊗Ez0 Vz0 ⊗Ez0 Vz0 → st,Ez0 ij j ⊗ i 7→ ℓh Xi,j We have the following projection vanishing property (Theorem 0.8) and “projec- tion vanishing implying -invariant” property (Theorem 0.9). L Theorem 0.8. Suppose that ϕ is semisimple on D. Let c : G ∗ be a Qp → Vz0 ⊗Ez0 Vz0 1-cocycle coming from an infinitesimal deformation of . If h < ℓ, then π ([c]) = 0 Vz0 h,ℓ in H1(B ). st,Ez0 Theorem 0.9. Suppose that ϕ is semisimple on D. Let c be a 1-cocycle G Qp → ∗ satisfying the projection vanishing property. If s is strongly critical for Vz0 ⊗Ez0 Vz0 and t = t (s), then there exist γ ,γ ,γ ,γ E and x ,x Bϕ=1 such F F s,1 s,2 t,1 t,2 ∈ z0 s t ∈ st,Ez0 that π (c ) = γ ψ +γ ψ +(σ 1)x , i = s,t. i,i σ i,1 1 i,2 2 i − Furthermore γ γ = (γ γ ). s,1 t,1 F,s s,2 t,2 − L − Theorem 0.6 follows from Theorem 0.8, Theorem 0.9 and a computation relating γ , γ to dδi(p) and dw . i,1 i,2 δi(p) δi Our paper is organized as follows. In Section 1 we provide preliminary results on Galois cohomology. The proof of the “projection vanishing implying -invariant” L 5 property needs the functors X and X used in [14] where they are denoted by V0 st dR st and V1 respectively. In Section 2 we give a systematic study on these two functors. st The relation between triangulations and refinements is reviewed in Section 3. In Section 4 we introduce the concepts of criticality and strong criticality and define -invariants. The “projection vanishing implying -invariant” property is proved in L L Section 5, and the projection vanishing property is proved in Section 6. Finally in section 7 we combine results in Section 5 and Section 6 to prove Theorem 0.6. There are two directions to generalize Theorem 0.6. One is to consider families of (not necessarily ´etale) (ϕ,Γ)-modules instead of families of Galois representations. The other is that the base field Q is replaced by a finite extension of Q . These are p p in progress. There may be two possible applications of Theorem 0.6. One is to the Excep- tional Zero phenomenon, and the other is to the local-global compatibility in p-adic Langlands program. In the case of n = 2 the former is done in [19] and the latter is done in [15]. Notation For a G -module M we write Hi(M) for the cohomology group Hi(G ,M). For Qp Qp a 1-cocycle c : G M let [c] denote the class of c in H1(M). For a G -module Qp → Qp M let M(i) denote the twist of M by χi , where χ is the cyclotomic character. cyc cyc Let E be a finite extension of Q considered as a base field with trivial action p of G . Let ψ : G E be the unramified additive character of G such that Qp 1 Qp → Qp ψ (σ) = 1 if σ induces the Frobenius x xp on F . Let ψ : G E be the 1 7→ p 2 Qp → additive character that is the logarithmic of χ . Then [ψ ] and [ψ ] form a basis of cyc 1 2 H1(E) = Hom(G ,E) over E. Qp If δ is a multiplicative character of Q×, the character of Q× whose restriction to p p Z×p coincides with δ|Z×p and whose value at p is 1, is again denoted by δ|Z×p by abuse of notation. For an affinoid E-algebra S and a closed point z Max(S), let E denote the z ∈ residue field of S at z. For an S-module we put = E . z S z M M M⊗ Let N, Z and Q denote the set of natural numbers, integers and rational numbers respectively. 6 Acknowledgement This paper is partly supported by the National Natural Science Foundation of China (grant 11371144). Part of this work was done while the author was a visitor at Shanghai Center forMathematical Sciences. Theauthorisgratefultothisinstitution for its hospitality. I would like to dedicate this paper to my teacher Professor Chunlai Zhao for his 70th birthday. 1 Fontaine period rings and Galois cohomology Let B , B and B be Fontaine’s period rings [17]. Put cris st dR B = B E, B = B E, B = B E. cris,E cris ⊗Qp st,E st ⊗Qp dR,E dR ⊗Qp We extend the actions of G on B , B and B E-linearly to B , B Qp cris st dR cris,E st,E and B . We also extend the operators ϕ and N on B E-linearly to B . dR,E st st,E Then B is stable under ϕ and B = BN=0. Let t be Fontaine’s p-adic cris,E cris,E st,E cyc “2π√ 1” [17]. We have ϕ(t ) = pt , Nt = 0 and g(t ) = χ (g)t for cyc cyc cyc cyc cyc cyc − g G . Let Fil be the filtration on B such that FiliB = FiliB E. ∈ Qp dR,E dR,E dR ⊗Qp Put B+ = Fil0B = B+ E. Then we have the following short exact dR,E dR,E dR ⊗Qp sequence, the so called fundamental exact sequence [14, Proposition 1.3 v)] 0 // E //Bϕ=1 //B /B+ // 0. cris,E dR,E dR,E The following lemma is well known. See [12, Proposition 1.1]. Lemma 1.1. Let a b be in Z ,+ . If either a > 0 or b 0, then ≤ ∪{−∞ ∞} ≤ H0(FilaB /FilbB ) = H1(FilaB /FilbB ) = 0 dR,E dR,E dR,E dR,E with the convention Fil−∞B = B and Fil+∞B = 0. dR,E dR,E dR,E For i N and j Z put U = BNi+1=0,ϕ=pj. Note that U coincides with the ∈ ∈ i,j st,E i,i−1 notation U in [12]. i Lemma 1.2. For any i 1 we have the following short exact sequence ≥ 0 //Bϕ=pj //U N //U // 0. cris,E i,j i−1,j−1 7 Proof. We only need to prove the surjectivity of N : U U . Let u be the i,j i−1,j−1 → element in B , considered as an element in B , that is denoted by log[π] in [14, st st,E §1.5]. Then B = B [u] and ϕ(u) = pu, N(u) = 1. For x U write st,E cris,E i−1,j−1 − ∈ x = i−1 a uℓ with a B . Then a is in Bϕ=pi−1−ℓ. So y = i−1 a uℓ+1 is in ℓ=0 ℓ ℓ ∈ cris,E ℓ cris,E − ℓ=0 ℓ ℓ+1 Ui,j aPnd N(y) = x. P Proposition 1.3. If i 1, then the inclusion E U induces an isomorphism i,0 ≥ ⊂ H1(E) ∼ ker(H1(U ) N H1(B )). i,0 st,E −→ −→ Proof. We prove the assertion by induction on i. For i = 1, the assertion is [12, Proposition 1.2]. By definition U = Bϕ=p−i. From the fundamental exact sequence we obtain 0,−i cris,E the following exact sequence 0 // Et−i //U // B /Fil−iB //0. 0,−i dR,E dR,E So we have an isomorphism H1(E( i)) = H1(Et−i) H1(U ) since by Lemme 0,−i − → 1.1 H0(B /Fil−iB ) = H1(B /Fil−iB ) = 0 dR,E dR,E dR,E dR,E for i 0. When i 1, each nontrivial extension of E by E( i) is not semistable. ≥ ≥ − (This is a well known fact; it also follows from Proposition 2.6 below.) Thus H1(E( i)) H1(B ) induced by the natural inclusion E( i) B is injective. st,E st,E − → − ⊂ AsH1(E( i)) H1(U )isanisomorphism, itfollowsthatH1(U ) H1(B ) 0,−i 0,−i st,E − → → is also injective. Note that ker(H1(U ) N H1(B )) ker(H1(U ) Ni H1(B )). i,0 st,E i,0 st,E −→ ⊂ −→ We consider the exact sequence 0 // U //U Ni //U // 0. i−1,0 i,0 0,−i As H0(U ) = 0, from this short exact sequence we derive an isomorphism 0,−i H1(U ) ∼ ker(H1(U ) Ni H1(U )). i−1,0 i,0 0,−i −→ −→ In particular the natural map H1(U ) H1(U ) is injective. As H1(U ) i−1,0 i,0 0,−i → injects into H1(B ), we have st,E ker(H1(U ) Ni H1(B )) = ker(H1(U ) Ni H1(U )). i,0 st,E i,0 0,−i −→ −→ 8 It followsthatker(H1(U ) N H1(B ))lies intheimageofH1(U ) H1(U ). i,0 st,E i−1,0 i,0 −→ → Since H1(U ) injects into H1(U ), we have an isomorphism i−1,0 i,0 ker(H1(U ) N H1(B )) ∼ ker(H1(U ) N H1(B )). i−1,0 st,E i,0 st,E −→ −→ −→ This completes the inductive proof. Corollary 1.4. The inclusion E Bϕ=1 induces an isomorphism ⊂ st,E H1(E) ∼ ker(H1(Bϕ=1) N H1(B )). −→ st,E −→ st,E Proof. First we prove that H1(E) H1(Bϕ=1) is injective. Let c bea 1-cocycle with → st,E values in E. If the image of [c] in H1(Bϕ=1) is zero, then there exists some y Bϕ=1 st,E ∈ st,E such that c = (σ 1)y for all σ G . As Bϕ=1 = U , y is in U for some σ − ∈ Qp st,E ∪i≥1 i,0 i,0 i 1, which implies that the image of [c] in H1(U ) is zero. But by Proposition i,0 ≥ 1.3, H1(E) injects to H1(U ), so [c] = 0 (in H1(E)). i,0 Now, let c be a 1-cocycle with values in Bϕ=1 such that the image of [c] by st,E N : H1(Bϕ=1) H1(B ) is zero. Then there exists some z B such that st,E → st,E ∈ st,E N(c ) = (σ 1)z for all σ G . Let i be a positive integer such that Ni(z) = 0. σ − ∈ Qp Then c U for all σ G . In other words, [c] comes from an element in σ ∈ i,0 ∈ Qp ker(H1(U ) N H1(B )) by the map H1(U ) H1(Bϕ=1). So by Proposition i,0 −→ st,E i,0 → st,E 1.3, [c] comes from an element in H1(E) by the map H1(E) H1(Bϕ=1). → st,E 2 Some facts on Galois representations Throughout this section a filtration on an E-vector space D means an exhaustive descending Z-indexed filtration. 2.1 X and X st dR We will use the functors X and X defined in [12]. These functors were already st dR used in[14] to show that every admissible filtered (ϕ,N)-module comes fromaGalois representation. In [14] X and X are denoted by V0 and V1 respectively. st dR st st We refer the reader to [12] for the notions of E-(ϕ,N)-modules, filtered E- modules, filtered E-(ϕ,N)-modules and admissible filtered E-(ϕ,N)-modules. Note that, if D, D and D are filtered E-(ϕ,N)-modules, then there exist natural filtered 1 2 E-(ϕ,N)-module structures on D∗ and D D . 1 E 2 ⊗ 9 If V is a finite-dimensional E-representation of G , then D (V) = (B Qp st st,E ⊗E V)GQp is a filtered E-(ϕ,N)-module induced from the natural filtered E-(ϕ,N)- module structure on B V. We always have dim D (V) dim V, and say st,E E E st E ⊗ ≤ that V is semistable if dim D (V) = dim V. E st E If D is a finite-dimensional E-(ϕ,N)-module, let X (D) be the Bϕ=1 -module st cris,E defined by X (D) = (B D)ϕ=1,N=0. st st,E E ⊗ If Fil = (Filj) is a filtration on a finite-dimensional E-vector space D, let j∈Z X (D,Fil) or just X (D) if there is no confusion, be the B+ -module dR dR dR,E X (D,Fil) = (B D)/Fil0(B D). dR dR,E E dR,E E ⊗ ⊗ By [14, Proposition 5.1, Proposition 5.2] X and X are exact. st dR If (D,Fil) is a filtered E-(ϕ,N)-module, then there is a natural E-linear map X (D) X (D,Fil) induced by the inclusion B D B D. Let st dR st,E E dR,E E → ⊗ → ⊗ V (D,Fil) be the kernel of this map, which is an E-vector space. st By [12, Theorem 2.1] V is an equivalence of categories from the category of st admissible filtered E-(ϕ,N)-modules to the category of semistable E-representations of G , with quasi-inverse D . Furthermore, V and D respect tensor products Qp st st st and duals. If (D,Fil) is an admissible filtered E-(ϕ,N)-module, then the sequence 0 // V (D,Fil) //X (D) //X (D,Fil) //0 (2.1) st st dR is exact, and the natural map B V (D,Fil) B D st,E E st st,E E ⊗ → ⊗ is an isomorphism respecting the actions of G , ϕ, N and the filtrations. Qp If D is an E-(ϕ,N)-module, and e∗ is an element in the dual E-(ϕ,N)-module D∗, we have a G -equivariant map Qp πe∗ : Xst(D) Bst,E, x < e∗,x > . → 7→ Here < , > denotes the B -bilinear pairing st,E · · (B D∗) (B D) B st,E E st,E E st,E ⊗ × ⊗ → induced by the canonical E-bilinear pairing D∗ D E. × → Lemma 2.1. (a) We have N πe∗ = πNe∗. ◦ 10