F -crystalline representation and Kisin module. October 4th, 2015

Transcription

1 Bryden Cais University of Arizona Purdue University October 4th, 2015

2 Basic Settings Let k be a perfect field of characteristic p with ring of Witt vectors W := W (k), write K 0 := W [1/p] and let K /K 0 be a finite and totally ramified extension. We fix an algebraic closure K of K and set G K := Gal(K /K ). If K is a p-adic field (i.e., K is a finite extension of Q p ) then K 0 is just the maximal unramified subfield inside K and W (k) = O K0. If A is a W (k)-algebra then we reserve ϕ for arithmetic Frobenius so that ϕ(x) = x p over A/pA. Fix a uniformizer π O K and its Eisenstein polynomial E(u) W (k)[u].

3 Basic Settings Let k be a perfect field of characteristic p with ring of Witt vectors W := W (k), write K 0 := W [1/p] and let K /K 0 be a finite and totally ramified extension. We fix an algebraic closure K of K and set G K := Gal(K /K ). If K is a p-adic field (i.e., K is a finite extension of Q p ) then K 0 is just the maximal unramified subfield inside K and W (k) = O K0. If A is a W (k)-algebra then we reserve ϕ for arithmetic Frobenius so that ϕ(x) = x p over A/pA. Fix a uniformizer π O K and its Eisenstein polynomial E(u) W (k)[u].

4 Basic Settings Let k be a perfect field of characteristic p with ring of Witt vectors W := W (k), write K 0 := W [1/p] and let K /K 0 be a finite and totally ramified extension. We fix an algebraic closure K of K and set G K := Gal(K /K ). If K is a p-adic field (i.e., K is a finite extension of Q p ) then K 0 is just the maximal unramified subfield inside K and W (k) = O K0. If A is a W (k)-algebra then we reserve ϕ for arithmetic Frobenius so that ϕ(x) = x p over A/pA. Fix a uniformizer π O K and its Eisenstein polynomial E(u) W (k)[u].

5 Crystalline Representation Let V be a finite dimensional Q p -vector space with continuous Q p -linear G K -action, and V the Q p -dual of V. Note V always contains a G K -stable Z p -lattice T. Recall V is crystalline if dim K0 D cris (V ) = dim Qp V, where D cris (V ) := (V Qp B cris ) G K. Here B cris is a huge W (k)-algebra with many structures: G K -action, Frobenius action ϕ and filtration. Caution: D cris is defined slightly differently from the classical one but will be convenient for integral theory.

6 Crystalline Representation Let V be a finite dimensional Q p -vector space with continuous Q p -linear G K -action, and V the Q p -dual of V. Note V always contains a G K -stable Z p -lattice T. Recall V is crystalline if dim K0 D cris (V ) = dim Qp V, where D cris (V ) := (V Qp B cris ) G K. Here B cris is a huge W (k)-algebra with many structures: G K -action, Frobenius action ϕ and filtration. Caution: D cris is defined slightly differently from the classical one but will be convenient for integral theory.

7 D cris and Hodge-Tate weights By Colmez and Fontiane, D cris : induces (anti)-equivalence {crystalline rep. V with HT(V ) [0,..., r] } D=D cris(v ) { w. a. (D, Fil i D K, ϕ D ), Fil 0 D K = D K, Fil r+1 D K = {0} } Here w.a. = weakly admissible, D K = K K0 D and Hodge-Tate weights HT(V ) of V can be read by HT(V ) = {i Z Fil i D K /Fil i+1 D K {0}}. Example Let A be an abelian variety over K with good reduction. Then the p-adic Tate module T p (A) is crystalline with HT(T p (A)) = {0, 1}.

8 D cris and Hodge-Tate weights By Colmez and Fontiane, D cris : induces (anti)-equivalence {crystalline rep. V with HT(V ) [0,..., r] } D=D cris(v ) { w. a. (D, Fil i D K, ϕ D ), Fil 0 D K = D K, Fil r+1 D K = {0} } Here w.a. = weakly admissible, D K = K K0 D and Hodge-Tate weights HT(V ) of V can be read by HT(V ) = {i Z Fil i D K /Fil i+1 D K {0}}. Example Let A be an abelian variety over K with good reduction. Then the p-adic Tate module T p (A) is crystalline with HT(T p (A)) = {0, 1}.

9 Classification for lattices Question How to classify G K -stable Z p -lattices T V? One of partial answers: Kisin modules. Define S := W (k)[[u]] and extends ϕ W (k) to S via ϕ S (u) = u p. Choose π n K such that π p n+1 = π n and π 0 = π. Write K = n 0 K (π n) and G = Gal(K /K ). Definition A Kisin module of height r is a finite free S-module M endowed with ϕ-semilinear endomorphism ϕ M : M M whose linearization 1 ϕ : S ϕ,s M M has cokernel killed by E(u) r.

10 Results of Kisin For a Kisin module M, define T S (M) := Hom S,ϕ (M, W (R)). One can show that T S (M) is a finite free Z p -representation of G and rank Zp T S (M) = rank S M. Let Rep cris,r Q p (G K ) denote the category of crystalline representations V with HT(V ) [0,..., r] and Rep Qp (G ) denote the category of representations of G.

11 Results of Kisin (Cont.) Theorem (Kisin) 1 Let V be an crystalline representation with Hodge-Tate weights in [0,..., r] and T V a G K -stable Z p -lattice. Then there exists a unique Kisin module M of E(u)-height r satisfying T S (M) T G. 2 The restriction functor Rep cris,r Q p (G K ) Rep Qp (G ) induced by V V G is fully faithfull. 3 The category of Kisin modules of height 1 is equivalent to the category of Barsotti-Tate groups over O K Remark: Theorem (1) also works for semi-stable representations.

12 Results of Kisin (Cont.) Theorem (Kisin) 1 Let V be an crystalline representation with Hodge-Tate weights in [0,..., r] and T V a G K -stable Z p -lattice. Then there exists a unique Kisin module M of E(u)-height r satisfying T S (M) T G. 2 The restriction functor Rep cris,r Q p (G K ) Rep Qp (G ) induced by V V G is fully faithfull. 3 The category of Kisin modules of height 1 is equivalent to the category of Barsotti-Tate groups over O K Remark: Theorem (1) also works for semi-stable representations.

13 Variation of Frobnius Question Note there are many lift of Frobnius on S. How about try ϕ(u) = u p + p 2 u?

14 F -crystalline setup Let F K be a subfield which is finite totally ramified over Q p and ϖ a uniformizer of O F. Fix a polynomial f (u) := a 1 u + a 2 u a p 1 u p 1 + u p O F [u] with f (u) u p mod ϖ. Set π 0 := π and we choose π := {π n } n 1 with π n K satisfying f (π n ) = π n 1 for n 1. Set K π := n 0 K (π n) and G π := Gal(K /K π ). Example 1 F = Q p, f (u) = u p. This is the classical situation discussed before. 2 F = Q p, f (u) = (1 + u) p 1 and K = Q p (ζ p ). Set π = ζ p 1. Then K π = n 1 Q p(ζ p n).

15 Observation Observation: Theorem (2) fails for Example 2 if V is the p-adic cyclotomic character. So to extend Kisin s result, we need distinguish the situation between Example 1 and Example 2. Also need to make restriction on category of crystalline representations.

16 F -crystalline representations Let V be a finite dimensional F-vector space with continuous F -linear G K -action. Let D = D cris (V ). Then D K is naturally a module over the semilocal ring K F := K Qp F, so we have a decomposition D K = m D K,m with the product running over all maximal ideals of K F. Let m 0 denote the maximal ideal corresponds to F K. Definition We say that V is F-crystalline if it is crystalline (as a Q p -linear G K -representation) and the filtration on D K,m is trivial (Fil j D K,m = 0 if j > 0 and Fil 0 D K,m = D K,m ) unless m = m 0.

17 Kisin modules in the F -crystalline setting Let F 0 := F K 0 K and S F := O F Zp S = O F0 [[u]]. Extend ϕ from W (k) to S F so that ϕ acts on O F -trivially and ϕ(u) = f (u). Denote E(u) O F0 [u] the minimal polynomial of π over F 0. One can embed S F to W (R) F := O F Zp W (R) so that the embedding is compatible with Frobenius and G π acts on S F trivially. One defines Kisin module of height r similarly, simply replace S by S F, and defines T SF (M) := Hom SF,ϕ(M, W (R) F ), which can be proved to be finite free O F -representation of G π.

18 Kisin modules in the F -crystalline setting Let F 0 := F K 0 K and S F := O F Zp S = O F0 [[u]]. Extend ϕ from W (k) to S F so that ϕ acts on O F -trivially and ϕ(u) = f (u). Denote E(u) O F0 [u] the minimal polynomial of π over F 0. One can embed S F to W (R) F := O F Zp W (R) so that the embedding is compatible with Frobenius and G π acts on S F trivially. One defines Kisin module of height r similarly, simply replace S by S F, and defines T SF (M) := Hom SF,ϕ(M, W (R) F ), which can be proved to be finite free O F -representation of G π.

19 Main results I Theorem Let V be an F-crystalline representation with Hodge-Tate weights in {0,..., r} and T V a G π -stable O F -lattice. Then there exists a Kisin module M of height r satisfying T SF (M) T. Remark: Partial results have been known, e.g. Kisin-Ren M may not be unique in general. For example, In Example 2, V is p-adic cyclotomic character.

20 Main Results II Let Rep F cris,r F (G K ) denote the category of F-crystalline representations V with HT(V ) [0,..., r] and Rep F (G π ) denote the category of F-representations of G π. Theorem Assume that ϕ n (f (u)/u) is not a power of E(u) for all n 0 and that ϖ r+1 a 1, with a 1 = f (0) the linear coefficient of f (u). Then the restriction functor Rep F cris,r F (G K ) Rep F (G π ) induced by V V Gπ is fully faithfull. Remark: The condition on ϕ n (f (u)/u) is a technical condition, while the condition ϖ r+1 a 1 can not be removed. See Example 2, where a 1 = p. There are infinitely many different K π so that the above theorem holds.

21 Main Result III An F-Barsotti-Tate group over O K is a Barsotti-Tate group H over O K so that the p-adic Tate module T p (H) is F-crystalline Theorem Assume ϖ 2 a 1. Then the category of Kisin modules of height 1 is equivalent to the category of F -Barsotti-Tate groups over O K.

22 Thank you!