Higher Hasse Witt matrices. Masha Vlasenko. June 1, 2016 UAM Poznań

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1 Higher Hasse Witt matrices Masha Vlasenko June 1, 2016 UAM Poznań

2 Motivation: zeta functions and periods X/F q smooth projective variety, q = p a zeta function of X: ( Z(X/F q ; T ) = exp m=1 #X(F q m) T m) = m 2 dim X i=0 P i (T ) ( 1)i+1 P i (T ) = det(1 T F rob q H i crys(x)) Z[T ] X/Q[t] family of varieties X t ω t an algebraic differential i-form on X t, ξ t H i (X t ; Z) a cycle period p(t) = ξ t ω t satisfies the Picard Fuchs diff. equation X t : y 2 = x(x 1)(x t) The elliptic integral p(t) = 1 satisfies t(1 t)p (t) + (1 2t)p (t) 1 4p(t) = 0. dx x(x 1)(x t) B. Dwork Deformation theory for zeta functions (1962): can one use solutions to the Picard Fuchs differential equations to give a p-adic analytic formula for Z(X t /F p ; T )?

3 Zeta function modulo p X/F q smooth projective variety, q = p a Katz s congruence formula: Z(X/F q ; T ) dim X i=0 det(1 T F a H i (X, O X )) ( 1)i+1 mod p F : H i (X, O X ) H i (X, O X ) Frobenius map p-linear map induced by h h p on O X

4 Hasse Witt operation definition and some properties X projective hypersurface (or complete intersection) dim X = n F : H n (X, O X ) H n (X, O X ) the Hasse Witt operation # p-adic unit eigenvalues of F a = stable rank r(f ) generically r(f ) = dim H n (X, O X ) (Miller, Koblitz 1975)

5 Hasse Witt operation computation for hypersurfaces X = {f(x 0,..., x n+1 ) = 0} P n+1, deg f = d > n O P n+1( d) f O P n+1 O X 0 f p 1 F F F 0 O P n+1( d) f O P n+1 O X 0 H n (X, O X ) H n+1 (P n+1, O P n+1( d)) basis: x u = x u x un+1 n+1, u U = { n+1 i=0 u i = d, u i Z 1 } Hasse Witt matrix: ( F )u,v U = the coefficient of xpv u in f(x) p 1

6 Main result R commutative ring f = u a ux u R[x ±1 1,..., x±1 N ] (f) R N Newton polytope of f = convex hull {u : a u 0} J = (f) Z N internal integral points, g = #J (β m ) u,v J = the coefficient of x (m+1)v u in f(x) m β m Mat g g (R), m 0 p prime α s := β p s 1, s 0 α 0 = β 0 = Id g g α 1 = β p 1 ( mod p = the Hasse Witt matrix )

7 Main result Theorem 1. Assume there exists σ End(R), a lift of Frobenius on R/pR (i.e. σ(a) a p mod p for any a R). Then: (i) α s α 1 σ(α 1 )... σ s 1 (α 1 ) mod p If all α s are invertible over R = lim R/p s R then: (ii) α s+1 σ(α s ) 1 α s σ(α s 1 ) 1 mod p s (iii) for any derivation D : R R D(α s+1 ) αs+1 1 D(α s) αs 1 mod p s

8 Idea of proof R ring with a pth power Frobenius lift σ End(R): σ(a) a p mod p for any a R For a R define recursively δ s (a) R: a ps 1 = δ 1 (a) σ(a ps 1 1 ) + δ 2 (a) σ 2 (a ps 2 1 ) δ s (a). δ 1 (a) = a p 1 δ 2 (a) = a p2 1 a p 1 σ(a p 1 ) = a p 1 (a (p 1)p σ(a p 1 )) pr Lemma. δ s (a) p s 1 R apply in R = R[x ±1 1,..., x±1 ], define matrices N (γ s ) u,v J = the coefficient of x ps v u in δ s (f) p s 1 Mat g g (R) Lemma. α s = γ 1 σ(α s 1 ) + γ 2 σ 2 (α s 2 ) γ s

9 Corollary Theorem 1 when the Hasse Witt matrix α 1 mod p is invertible, F = lim s α s+1 σ(α s ) 1 Mat g g ( R) (D) = lim D(α s ) α 1 s s Mat g g ( R) for every derivation D Der(R) Observations: F α 1 mod p F is invertible The rest of the talk: meaning of the limiting matrices?

10 Example (elliptic curve) c m = 1 dx 2 y E : y 2 = x 3 + ax + b, a, b R ( ) = 1 + 2a T 4 + 3b T 6 + 6a 2 T ab T dt ( = c m T m) dt T, T = x/y m=1 { the coefficient of x m 1 in (x 3 + ax + b) (m 1)/2, m odd, 0, m even. R = Z or Z p Atkin and Swinnerton-Dyer congruences c m + a p c m/p + p c m/p 2 0 a p = p + 1 #E(F p ) mod p ordp(m)

11 Example (elliptic curve) (ASD) c m + a p c m/p + p c m/p 2 0 mod p ordp(m) a p = p + 1 #E(F p ) β m = the coefficient of x m y m in (x 3 + ax + b y 2 ) m ( ) m = ( 1) m/2 c m+1 ; α s = β p m/2 s 1 Theorem 1 if p α 1 ( p a p ordinary curve) α ( ) s lim = λ Z p = lim c p s/c p s 1 s s α s 1 ASD c p s + a p c p s 1 + p c p s 2 0 mod p s λ 2 + a p λ + p = 0 λ = the p-adic unit eigenvalue of Frobenius

12 Proposition. When R/pR = F p and X 0 = X Spec(R) Spec(R/pR) = {f(x) = 0} is a smooth projective variety over F p, then the eigenvalues of the limiting matrix F = lim s α s+1 σ(α s ) 1 are p-adic unit eigenvalues of the Frobenius operator on H n crys(x 0 ). F p F q F F σ(f )... σ a 1 (F ) = lim α s+a α 1 s s q = p a Strategy of proof: formal group laws & generalized ASD due to Stienstra

13 Formal group laws A commutative formal group law of dimension g over a ring R is a tuple of g power series G = (G 1,..., G g ), G i = G i (x, y) R x, y satisfying G(x, 0) = G(0, x) = x G(x, G(y, z)) = G(G(x, y), z) G(x, y) = G(y, x) x = (x 1,..., x g ), y = (y 1,..., y g ) If R R Q (characteristic 0 ring), there exists the logarithm: log G (x) := l(x) = x +... (R Q) x G(x, y) = l 1 (l(x) + l(y))

14 Main result 2: formal group laws from a polynomial R characteristic 0 ring; f R[x ±1 1,..., x±1 N ] J = (f) Z N or (f) Z N g = #J > 0 (β m ) u,v J = the coefficient of x (m+1)v u in f(x) m l(x) = (l u (x)) u J := m=1 1 m β m 1x m G f (x, y) := l 1 (l(x) + l(y)) (R Q) x, y x = (x v ) v J Theorem 2. If R can be endowed with a pth power Frobenius endomorphism σ End(R), then G f actually has coefficients in R Z (p). { m } Z (p) = k : p k Q.

15 Artin-Mazur formal groups and generalized ASD J. Stienstra (1987) X = {f(x) = 0} P n+1 R smooth projective R ring of characteristic 0 G f (x, y) is a coordinalization of the Artin-Mazur formal group H n (X, Ĝm,X) R = Z det(1 T F H n crys(x 0 )) = 1 + c 1 T c k T k Z[T ] (ASD) α s + c 1 α s 1 + c 2 α s c k α s k 0 mod p s F k + c 1 F k c k 1 F + c k = 0

16 Example: hyperelliptic curve y 2 = x 5 + 2x 2 + x + 1 β n = the coefficients of p = 11: ( ) x n y n x 2n+1 y n x n 1 y n x 2n y n in (y 2 x 5 2x 2 x 1) n s 0 ( ) 1 ( ) α s = β p s tr(α s α 1 s 1 ) mod ps 8 + O(11) O(11 2 ) O det(α s α 1 s 1 ) mod ps 7 + O(11) O(11 2 ) ( ) det 1 T F Hcrys(C 1 0 ) = 1 + 3T + 18T T T 4 = (1 + 4T + 11T 2 )(1 T + 11T 2 ) λ 1,2 = 2 ± 7, λ 3,4 = 1 ± λ 1 = O(11 3 ) λ 3 = O(11 3 ) λ 1 + λ 3 = O(11 3 ) λ 1 λ 3 = O(11 3 )

17 Conjecture Assume: X = {f(x 0,..., x n+1 ) = 0} P n+1 R R smooth algebra over Z p smooth for every point of Spec(R/pR), the Hasse-Witt operation for the respective fibre of X 0 is invertible Then matrices F = lim α s s+1 σ(α s ) 1, (D) = lim D(α s ) α 1 s s for D Der(R) describe respectively the Frobenius operator and the Gauss Manin connection on the unit-root F-crystal of X in the basis [ω u ] H 0 (X, Ω n X/S ), u J = (f) Z n+2, where. ω u = res X ( x u f(x) n+1 ( 1) i dx 0 x 0 i=0... dx i... dx ) n+1. x i x n+1

18 Crystals R smooth algebra over Z p, R = lim An F-crystal (H,, F ) over R is locally free R-module H R/p s R integrable connection : H H Ω R/Zp (D) : H H for each D Der( R) for every lift of Frobenius σ : R R, a horizontal morphism F (σ) : σ H H X/Spec(R) smooth projective; X 0 = X Spec(R) Spec(R/pR) H dr (X/Spec(R)) R R H crys(x 0 ) is an F-crystal

19 Example: variation of hypersurfaces f Λ (x 0,..., x n+1 ) = M k=1 Λ k x a k, R = Z[Λ 1,..., Λ M ] a k = (a 0k,..., a (n+1)k ) Z n+2 with n+1 i=0 a ik = d X = {f Λ (x) = 0} P n+1 R ω u = res X ( x u f(x) n+1 ( 1) i dx 0 x 0 i=0... dx i... dx ) n+1 x i x n+1 ( Λ i ) : H n (X) H n (X) Question: construct differential operators that anihilate classes [ω u ] H 0 (X, Ω n X )

20 Example: variation of hypersurfaces A-hypergeometric system of PDEs A = {a 1,..., a M } Z n+2, a + k = (a 0k,..., a (n+1)k, 1) Z n+3 L = { l = (l 1,..., l M ) Z M M } l k a + k = 0 k=1 lattice of relations The A-hypergeometric system of partial differential equations attached to the set A = {a + k }M k=1 with the set of parameters µ = (µ 0,..., µ n+2 ) C n+3 consists of box operators l = ( ) lk ( ) lk, Λ k Λ k l k >0 l k <0 and Euler (or homogeneity) operators l L Z i = M a ik Λ k µ i, i = 0,..., n + 3. Λ k k=1

21 Example: variation of hypersurfaces f Λ (x) = M k=1 Λ k x a k, deg f Λ = d = span{a 1,..., a M } R n+2 ; J = Z n+2 Proposition. (A. Adolphson & S. Sperber) Assume that J = {u = (u 0,..., u n+1 ) : u i Z 1, n+1 i=0 u i = d}. Then for each u J the class [ω u ] H n (X) is anihilated by the A-hypergeometric system of PDEs attached to the set A = {a + k }M k=1 with the parameter µ = u+. Proposition. For any u, v J the matrix entry (α s ) u,v (Λ) = the coefficient of x ps v u in f Λ (x) ps 1 is anihilated modulo p s by the same A-hypergeometric system with µ = u +.

22 Example: the Legendre family y 2 = x(x 1)(x t) the period integral ω(t) = 1 π 1 dx x(x 1)(x t) = t t = k=0 ( ) 2k 2( t ) k k 16 is anihilated by the hypergeometric differential operator t(1 t) d2 dt 2 + (1 2t) d dt 1 4

23 Example: the Legendre family y 2 = x(x 1)(x t) R = Z[t], σ(h(t)) = h(t p ), R = Z p t β m = β m (t) = the coefficient of x m y m in (y 2 x(x 1)(x t)) m { 0, m odd, = ) m/2 ) 2 t k, m even. ( m m/2 k=0 ( m/2 k ( ) t(1 t) d2 + (1 2t) d dt 2 dt 1 4 α s (t) 0 mod p s α 1 (t) = β p 1 (t) = ( (p 1)/2) + O(t) is invertible in R when p 2, and the limits from Theorem 1 are given by F = ( 1) p 1 2 ω(t) ω(t p ), ( d dt ) = ω(t) ω(t)

24 Thank you!