HASSE-WITT MATRICES, UNIT ROOTS AND PERIOD INTEGRALS

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1 HASSE-WITT MATRICES, UNIT ROOTS AND PERIOD INTEGRALS AN HUANG, BONG LIAN, SHING-TUNG YAU, CHENGLONG YU Abstract. Motivated by the work of Candelas, de la Ossa and Rodriguez-Villegas [6], we study the relations between Hasse-Witt matrices and period integrals of Calabi-Yau hypersurfaces in both toric varieties and partial flag varieties. We prove a conjecture by Vlasenko [23] on higher Hasse-Witt matrices for toric hypersurfaces following Katz s method of local expansion [14, 15]. The higher Hasse-Witt matrices also have close relation with period integrals. The proof gives a way to pass from Katz s congruence relations in terms of expansion coefficients [15] to Dwork s congruence relations [8] about periods. Contents 1. Introduction Period Integral Hasse-Witt matrix Statement of the theorem Acknowledgment 6 2. Local expansions and Hasse-Witt matrices 6 3. Generalized flag vareities Complete intersections Frobenius matrices of toric hypersurfaces Unit root of toric Calabi-Yau hypersurfaces and periods Frobenius matrices for Calabi-Yau hypersurfaces Unit root of Calabi-Yau hypersurfaces in G/P 25 References Introduction The relations among Hasse-Witt matrices, unit roots of zeta-functions and period integrals were pioneered in Dwork s work on the variation of zeta-functions of hypersurfaces [7], [8], [9]. Some well-known examples are Legendre family (1.1 y 2 = x(x 1(x λ, see Example 8 in [12]; and the Dwork family (1.2 X n X n+1 n (n + 1tX 0 X n = 0, 1

2 2 AN HUANG, BONG LIAN, SHING-TUNG YAU, CHENGLONG YU see and in [13] and also [24]. There is a canonical choice of holomorphic n- forms ω λ for these Calabi-Yau families since they are hypersurfaces in P n. These families both have maximal unipotent monodromy at λ = t (n+1 = 0. The period integral I γ of ω λ over the invariant cycle γ near λ = 0 is the unique holomorphic solutions to the corresponding Picard-Fuchs equation. On the other hand, these families are defined over Z. We can consider the p-reductions of these families and the Hasse-Witt matrices associated to ω λ. According to a theorem of Igusa-Manin-Katz, they are solutions to Picard-Fuchs equations mod p. We first state the relations for Dwork family, see [24]. The period is given by hypergeometric series (1.3 ( 1 I γ = F (λ = n F n+1, 2 n+1,, n n 1 = r=0 ( 1 n+1 r r=0 1, 1,, 1 ( 2 n+1 n+1 ; λ ( n r n+1 r (r! n λ r. The Hasse-Witt matrix H-W p is given by the truncation of F (λ ( ( ( p (1.4 H-W p (λ = (p 1 n+1 r n+1 n r n+1 r F (λ = (r! n λ r. Here (k F (λ means the truncation of F (λ with terms of λ of degree less or equal than k. Let (1.5 g(λ = F (λ F (λ p Z p[[λ]] Then g is an element in lim s Z p [λ, ( (p 1 F (λ 1 ]/p s Z p [λ, ( (p 1 F (λ 1 ] and it satisfies Dwork congruences (p s 1 (F (λ (1.6 g(λ (p s 1 1 (F (λ p Especially it is related to Hasse-Witt matrix by (1.7 mod p s. F (λ F (λ p (p 1 F (λ mod p. Let q = p r and t F q. Assume p n + 1 and t n+1 0, 1 H-W p (λ 0. Then there exists exactly one p-adic unit root in the factor of zeta function of Dwork family corresponding to Frobenius action on middle crystalline cohomology. It is given by (1.8 g(ˆλg(ˆλ p g(ˆλ pr 1 with ˆλ being the Teichmüller lifting under λ λ p. In this paper, we generalize the above relation to hypersurfaces in toric varieties and partial flag varieties. We first prove the mod p results. Complete intersections are treated in Section 4. The key algorithm of Hasse-Witt matrix is a generalization of the result on

3 HASSE-WITT, UNIT ROOTS AND PERIODS 3 hypersurfaces in P n. See Katz s algorithm 2.7 in [13]. For general hypersurfaces in a Fano variety X, we use the Cartier operator on ambient space to localize the calculation in terms of local expansion similar to [15]. When X is toric variety, the algorithm depends on the toric data associated to X. The algorithm implies generic invertibility of Hasse- Witt matrices for toric hypersurfaces, generalizing Adolphson and Sperber s result for P n in [2], see remark 2.7 and corollary 2.8. For generalized flag varieties, Bott-Samelson desingularization is used to reduce the calculation to a similar situation as toric varieties. The affine charts on Bott-Samelson varieties also give an explicit algorithm to calculate the power series expansions of period integrals of hypersurfaces in G/P. The second part of the paper applies Katz s local expansion method [14, 15] to prove a conjecture in [23]. The crystalline cohomology of the hypersurface family has an F - crystal structure. When the Hasse-Witt matrix is invertible, there exists a unit root part of the F -crystal. We consider the p-adic approximation of the Frobenius matrix on the unit root part. Especially, the Hasse-Witt matrix is the Frobenius matrix mod p. In [15], Katz gives a p-adic approximation of the Frobenius matrix in terms of the local expansions of top forms on a formal chart along a section of the family. In [23], Vlasenko constructed a sequence of matrices related to a Laurent polynomial f and proved congruence relations similar to Katz s algorithm in [15]. The p-adic limit is conjectured to be the Frobenius matrix for hypersurfaces in P n when f is a homogenous polynomial. According to Corollary 2.3 in section 2, the first matrix α 1 mod p appeared in [23] is the Hasse-Witt matrix for toric hypersurfaces. So it is natural to generalize Vlasenko s conjecture to toric hypersurfaces. We give a proof of the conjecture in section 5. We first recall some notations for period integral 1.1. Period Integral. (1 Let X be a smooth semi-fano variety of dimension n over C. In this paper X is toric variety or partial flag variety G/P with P parabolic subgroup in a semisimple algebraic group G. (2 We denote V = H 0 (X, ω 1 X to be the space of anticanonical sections. (3 For any nonzero section s V, the zero locus Y s is a Calabi-Yau hypersurface in X. (4 Let B be the set of s V such that Y s is smooth. Then B is Zariski open subset of V and there is a family of smooth Calabi-Yau varieties π : Y B with Y s as fibers.

4 4 AN HUANG, BONG LIAN, SHING-TUNG YAU, CHENGLONG YU (5 The section s induces the adjunction formula ω Ys = ωx ω 1 X Y s. The constant function 1 on the right hand side corresponds to a canonical section ω s H 0 (Y s, ω Ys. In other words, the section ω s is the residue of rational form 1 s. Putting ω s together, we get a canonical section of R 0 π (ω Y/B, denoted by ω Period integral. Consider the local system L on B formed by H n 1 (Y s, Q, which is the dual of R n 1 π Q Y. For any flat section γ of L on U B an open subset, the period integral I γ is defined by γ ω Picard-Fuchs system. Let D V be the sheaf of linear differential operators generated by Der(V. Then any section D of D V acts on the de-rham coholomogy sheaf R n 1 π (Ω (Y/B via restriction to B and Gauss-Manin connection. Define the sheaf of Picard-Fuchs system for ω to be P F (U = {D D V (U (D U B ω = 0}. Period integrals are solutions to Picard-Fuchs system. In other words, we have DI γ = 0 for any D P F (U Solution-rank-1 points. Consider the classical solution sheaf Sol = Hom DV (D V /P F, O V. The solution rank at a point s V is defined to be the dimension of the stalk Sol s. We will consider the points in V having solution rank Special point s 0. There exist special solution-rank-1 points when X is toric or G/P. The theorem we will state is for those special solution-rank-1 points. We characterize s 0 up to scaling in terms of its zero locus Y s0 as follows. If X is toric variety, we consider the stratification of X by the torus action. Then Y s0 is the union of toric invariant divisors. If X = G/P, we consider the stratification of X by projected Richardson varieties, see [16]. Then Y s0 is the union of projected Richardson divisors. Especially, if P is a Borel subgroup, the divisor Y s0 is the union of Schubert divisors and opposite Schubert divisors. They are proven to have solution rank 1 from GKZ systems and tautological systems by Huang-Lian-Zhu [11]. The unique solution at s 0 is realized as a period integral I γ0 invariant cycle γ 0. The special point s 0 is known as the large complex structure limit in the moduli of toric hypersurfaces. We expect the same result holds for flag varieties. Example 1.1. Let X = P n with homogenous coordinate [x 0,, x n ]. Then V is identified with space of homogenous polynomials of degree n + 1. In this case, the special solutionrank-1 point s 0 = x 0 x n. over

5 HASSE-WITT, UNIT ROOTS AND PERIODS Hasse-Witt matrix. Next we define the Hasse-Witt matrix. Let k be a perfect field of characteristic p. Assume π : Y S is a smooth family of Calabi-Yau variety over k with relative dimension n 1. Let ω be a trivializing section of R 0 π (ω Y/S. Let ω be the dual section of R n 1 π (O Y. The p-th power endomorphism of O Y induces a p-semilinear map R n 1 π (O Y R n 1 π (O Y sending ω to aω. Then H-W p = a as a section of O S is the Hasse-Witt matrix under the basis ω. The choice of ω for Calabi-Yau hypersurfaces is made by adjunction formula similar to period integral Statement of the theorem. Now we state our main theorem. When X is toric or G/P, it has an integral model over Z. Let s 0 H 0 (X, K 1 X be the special solution-rank-1 point chosen in section Then s 0 can be extended as a basis s 0 s N of H 0 (X, K 1 X. Let a 0 a N be the dual basis for s 0, s N. Suitable choices of s 0 s N are still basis considering the p-reduction of X. See section 2 and section 3 for the details of choice of s i and the integration cycle γ 0. There exists the following truncation relation between Hasse-Witt invariants of hypersurfaces over F p and period integrals. It can also be viewed as a relation between mod p solutions to Picard-Fuchs systems and solution over C. Theorem 1.2. (1 The Hasse-Witt matrix H-W p defined above are polynomials of a I of degree p 1. (2 The period integral I γ0 defined above can be extended as holomorphic functions at s 0 and has the form 1 a 0 P ( a I a 0, where P ( a I a 0 is a Taylor series of a 1 a 0,, a N a0 with integer coefficients. (3 They satisfy the following truncation relation 1 (1.9 a p 1 H-W p = (p 1 P ( a I mod p a 0 0 where (p 1 P ( a I a 0 is the truncation of P at degree p 1. Remark 1.3. In characteristic p, the conjugate spectral sequence provides a horizontal filtration for relative de-rham cohomology. In particular, the Hasse-Witt matrix gives part of the coordinate for the projection of R n 1 π (O Y to the horizontal subbundle in de-rham cohomology. This is how Katz [13] proved elements in Hasse-Witt matrix satisfy Picard-Fuchs equations. On the other hand, period integrals give the coordinate of horizontal sections in relative de-rham cohomology over C. The above relations suggest that the horizontal subbundle provided by conjugate spectral sequence can approximate the horizontal section in characteristic zero near some degeneration point when p. Remark 1.4. The Hasse-Witt matrices count rational point on Calabi-Yau hypersurfaces mod p by Fulton s fixed point formula [10]. In the case of Calabi-Yau hypersurfaces in toric varieties, the relation between point counting and period integrals has been studied by Candelas, de la Ossa and Rodriguez-Villegas [6].

6 6 AN HUANG, BONG LIAN, SHING-TUNG YAU, CHENGLONG YU Next we state the relation between period integrals and unit roots of zeta-function of toric hypersurfaces. Let a 0 = 1. The formal power series (1.10 g(a I = P (a I P (a p I lies in lim s Z p [a I1 a IN, ( (p 1 P (a I 1 ]/p s Z p [a I1 a IN, ( (p 1 P (a I 1 ] and satisfies Dwork congruences (p s 1 (P (a I (1.11 g(a I (p s 1 1 (P ((a I p mod p s. Theorem 1.5. Let a I F q. Assume the hypersurface Y defined by a I s I is smooth and H-W p (a I 0. Then there exists exactly one p-adic unit root in the factor of zeta function of Y corresponding to Frobenius action on Hcris n 1 (Y. It is given by the formula (1.12 g(â I g(â p I g(âpr 1 I with â I being the Teichmüller lifting under a I a p I. Similar unit root formulas for general-type toric hypersurfaces and Calabi-Yau hypersurfaces in G/P is given in 5 and Acknowledgment. The authors are grateful to Mao Sheng, Zijian Yao, Dingxin Zhang, Jie Zhou for their interests and helpful discussions. 2. Local expansions and Hasse-Witt matrices Now we prove a algorithm of calculating Hasse-Witt matrices of hypersurfaces of X in terms of local expansions of the sections. The key ingredient is to related the Hasse-Witt operator of Calabi-Yau or general type families Y to the Cartier operator on X. Then we apply the algorithm to toric and generalized flag varieties. Especially this recovers the algorithm for P n. We make the following assumptions for this section. (1 Let X n be a smooth projective variety defined over k and satisfies H n (X, O = H n 1 (X, O = 0 (2 Let L be an base point free line bundle on X and V = H 0 (X, L 0 and W = H 0 (X, L K X 0. Let a 1, a 2 a N and e 1 e r be basis of V and W.

7 HASSE-WITT, UNIT ROOTS AND PERIODS 7 (3 Consider the smooth hypersurfaces Y over S V {0}. Let X be X S and i: Y X be the embedding. The projections to S are denoted by π (4 Let F S be the absolute Frobenius on S and X (p = X FS S the fiber product. Then we have absolute Frobenius F X the relative Frobenius F X/S : X X (p. Denote W : X (p X and π (p : X (p S to be the projections. The corresponding diagram for family Y is defined in a similar way. Consider the following diagram (2.1 f 0 W L 1 O X (p i O Y (p 0 f p 1 F F f 0 F X/S L 1 F X/S O X i F Y/S O Y 0 The map f p 1 : W L 1 F X/S L 1 is induced by F X/S W L 1 = F X L 1 = L p multiplied by f p 1. This induces the diagram (2.2 R n 1 π (p (O Y (p R n π (p (W L 1 F f p 1 R n 1 π (F Y/S O Y R n π (L 1 The two horizontal maps are isomorphism. The left vertical map is the Hasse-Witt operator H-W : FS (Rn 1 π (O Y = R n 1 π (p (O Y (p R n 1 π (p (F Y/S O Y = R n 1 π O Y. Definition 2.1. The basis e 1 e r of H 0 (X, K X L induces a basis of R 0 π (ω Y/S by residue map and dual basis e 1 e r of R n 1 π (O Y under Serre duality. The Hasse-Witt matrix a ij is defined by H-W(F S (e i = j a ije j. Let C X/S : ω X/S ω X (p /S X, the Grothendieck duality be the top Cartier operator. For any coherent sheaf M on (2.3 F X/S Hom(M, ω X/S = Hom(F X/S M, ω X (p /S is related to C X/S by the natural pairing (2.4 F X/S Hom(M, ω X/S F X/S M ω X (p /S sending g m to C X/S (g(m. Consider M = L p. Since F X/S L p = W L 1, we have (2.5 F X/S Hom(L p, ω X/S = Hom(W L 1, ω X (p /S Then we have a morphism induced by multiplication by f p 1 (2.6 F X/S Hom(L 1, ω X/S F X/S Hom(L p, ω X/S = Hom(W L 1, ω X (p /S.

8 8 AN HUANG, BONG LIAN, SHING-TUNG YAU, CHENGLONG YU After taking R 0 π (p on both sides, we have an morphism (2.7 R 0 π Hom(L 1, ω X/S R 0 π Hom(W L 1, ω X (p /S. This the dual of (2.8 R n π (p (W L 1 R n π (L 1 Now we can conclude the dual of Hasse-Witt matrix is given by the following algorithm. Let (t 1,, t n be local coordinate of X at a point x. Denote g(t = a I t I to be a formal power series. Then define τ(g = J a It J with I = (p 1,, p 1 + pj. Fix a local trivialization section ξ of L on an Zariski open section U containing x. Then any section e i ξ is a section of ω X U and has the form h i (tdt 1 dt 2 dt n. Under the same trivilization, the section e i f p 1 ξ has the form as g p i (tdt 1 dt 2 dt n. Then we claim τ(g i has the form τ(g i = j a jih j. Theorem 2.2. The matrix a ij defined above is the Hasse-Witt matrix under the basis e 1 e r. Notice that τ is p 1 -semilinear τ(h p g = hτ(g. So the same algorithm works if we use a rational section ξ. Now we specialize this algorithm to toric hypersurfaces or Calabi-Yau hypersurfaces. Let X be a smooth complete toric variety defined by a fan σ. The 1-dimensional primitive vectors v 1, v N correspond to toric divisors D i. Assume L = O( a i D i with a i 1. Let = {v R n v, v i a i } and the interior of. Then H 0 (X, L has a basis corresponding to u I Z n and H 0 (X, L K X has basis e i identified with u i Z n. Let f = a I t u I be the Laurent series representing the universal section of H 0 (X, L and f p 1 = A u t u. Then we have Corollary 2.3. The Hasse-Witt matrix of hypersurface family over L under the basis e i Z n is given by a ij = A puj u i. Proof. After an action of SL(n, Z, we can assume v 1 v n is the standard basis of Z n. Then t 1,, t n is an affine chart on X. We choose a section of L K X to be s 0 corresponding to origin in Z n and a meromorphic section of K X to be θ = dt 1 dt 2 dt n t 1 t n. Let s = s 0 θ be a meromorphic section of L. Then (2.9 e i s = tui t 1 t n dt 1 dt 2 dt n. If we view f as a section in H 0 (X, L, then f (2.10 s = t 1 t n a I t u I I

9 HASSE-WITT, UNIT ROOTS AND PERIODS 9 Hence e i f p 1 s p = g i dt 1 dt 2 dt n with (2.11 g i = tu i ( I a It u I p 1 Here 1 = (1,, 1. So t 1 t n (2.12 τ(g i = v = u A u t v A u t u+u i 1. with u + u i 1 = pv + (p 11. On the other hand, we have (2.13 τ(g i = j a ji t u j 1. So a ij = A puj u i. Remark 2.4. When X is P n, Corollary 2.3 gives the same algorithm as Katz [13]. In [23], Vlasenko defines the higher Hasse-Witt matrices for Laurent polynomial f. When f is a homogenous polynomial of degree d, the first matrix α 1 in [23] mod p is the Hasse- Witt matrix for hypersurface Y in P n. The p-adic limit of the matrices is conjectured to give the Frobenius matrix of the unit root part of Hcris n 1 (Y, which is a dual analogue of matrices by Katz [12]. Corollary 2.3 proves that α 1 mod p is also the Hasse-Witt matrix for toric hypersurfaces. Hence it is natural to generalize Vlasenko s conjecture to toric hypersurfaces. If X is any smooth variety satisfying the assumptions in this section and L = K 1 X, then we have a Calabi-Yau family. In this case, the algorithm coincides with the criterion for Frobenius splitting of X respect to Y. The basis of H 0 (X, L K X is chosen to be constant function 1. The Hasse-Witt matrix is a function a on S. We can choose the trivializing section of L to be (dt 1 dt 2 dt n 1. The local algorithm in this case it the following. Corollary 2.5. Let f = g(t(dt 1 dt 2 dt n 1. Then the Hasse-Witt matrix a is given by τ(g p 1. More explicitly a is the coefficient of (t 1 t n p 1 in local expansion of (g(t p 1. Remark 2.6. For any closed point s S(k, the corresponding section fs p 1 H 0 (X, ω 1 p X determines a Frobenius splitting of X compatibly with Y s if and only if a(s 0. It is also equivalent to Y s being Frobenius split. Especially, Corollary 2.3 implies the well-known fact that toric variety X is Frobenius split compatibly with torus invariant divisors. See Chapter 1 of [4].

10 10 AN HUANG, BONG LIAN, SHING-TUNG YAU, CHENGLONG YU Proof of Theorem 1.2 for toric X. Following the previous notations, let X be smooth complete toric variety and L = K 1 X = O X( i D i. Then the basis of H 0 (X, L is identified with the integral points u I in the polytope = {v R n v, v i 1}. The universal section f(t = a I t u I with u 0 = (0,, 0. Then H-W p is the coefficient of constant term in f p 1 according to Corollary 2.3 or 2.5. On the other hand, the period integral 1 (2.14 I γ = (2π 1 n γ dt 1 dt n t 1 t n f(t along the cycle γ : t 1 = t 2 = t n = 1 is the coefficient of constant term in the Laurent expansion of f 1. So (2.15 and I γ = 1 a 0 (1 + ( 1 k k=1 (2.16 p 1 H-W p = a p 0 (1+ k 1 u I1 + +k l u Il =0, k j =k,i j 0 k=1 k 1 u I1 + +k l u Il =0, k j =k,i j 0 ( ( k k 1, k 2,, k l p 1 k 1, k 2,, k l, p 1 k Then apply the congruence relation ( ( p 1 (2.17 ( 1 k k k 1, k 2,, k l, p 1 k k 1, k 2,, k l in the two expansions to get the conclusion. ( a I 1 a 0 k1 ( a I l a 0 k l mod p ( a I 1 a 0 k1 ( a I l a 0 k l Remark 2.7 (General toric hypersurfaces. The same argument also applies to generaltype toric hypersurfaces. The entries in Hasse-Witt matrix are truncations of period integrals. The results for hypersurfaces in P n are proved by Adolphson and Sperber in [2]. We follow the notations in Corollary 2.3. The sections s H 0 (X, L K X determines a section of R 0 π (Y, ω Y/B via residue map and we can define period integral of s in a similar way as Calabi-Yau hypersurfaces. Let s 0 H 0 (X, K 1 X be the large complex structure limit point with zero locus equal to the union of D i. Let e i be the basis of H 0 (X, L K X corresponding to u i Z n and denote s i = s 0 e i H 0 (X, L. Let f = a I t u I be the universal section of L. In the Laurent series expression of f, the section s i defined above is identified with multi-index t u i. The period integral of e i the cycle γ : t 1 = t 2 = t n = 1 near s j is given by 1 (2.18 I γ,i = (2π t u i dt 1 dt n 1 n t 1 t n f(t and it is equal to the coefficient of t u i γ along in the Laurent expansion of f 1. On the other hand, the ijth entry a ij of the Hasse-Witt matrix under the basis e 1 e r is given by the coefficient of t pu j u i in the Laurent expansion of f p 1. So we have the following

11 (2.19 HASSE-WITT, UNIT ROOTS AND PERIODS 11 (1 The function a ij on S are polynomials of a I of degree p 1. (2 The period integral I γ,i is a holomorphic functions at s j and has the form 1 a j P i ( a I a j, where P i ( a I a j is a Taylor series of a I a j with integer coefficients. (3 They satisfies the following truncation relation 1 a p 1 j a ij = (p 1 (P i ( a I a j mod p where (p 1 (P i ( a I a j is the truncation of P at degree p 1. Since the period integral of ω i = tu i dt 1 dt n t 1 t n f(t satisfies the corresponding Gel fand-kapranov-zelevinski hypergeometric differential system, the entries a ij of Hasse-Witt matrices are mod p solutions to the same differential system. See [1] for mod p solutions to general hypergeometric systems. In [2], Adolphson and Sperber also proved the generic invertibility of Hasse-Witt matrices for hypersurfaces in P n. Similar idea gives the same result for toric hypersurfaces. Corollary 2.8. The Hasse-Witt matrices for generic smooth toric hypersurface are not degenerate. In other words, the determinant det(a ij 0. Proof. Consider the determinant of matrix (B ij = ( (p 1 (P i ( a I a j = ( 1 (p 1 (P i ( a I a j has the form (2.20 p 1 ( 1 k k=0 ( a I1 ( a I k. a j a j u I1 + +u Ik =(k+1u j u i a p 1 j a ij. The entry The indices I l are not required to be distinct. The constant term in B ij = δ ij. Now we prove the constant term in det B is 1. Let ɛ be a permutation of r-elements. Assume a I 1 a I 1 a ( k1 ai2 I 2 1 k2 air 1 air kr a 1 a 1 a 2 a 2 a r a r be a constant term appearing in the product B ɛ(11 B ɛ(rr. Then all indices I m appearing in the numerator correspond to interior integer points u i Z n and satiesfy (2.22 u I i u I i ki + u ɛ(i = (k i + 1u i. Consider the vertex u l of the convex polytope generated by all u i Z n. Since the convex expression for such u l is unique, the indices I l 1 = = k l l = ɛ(l = l. Hence other terms in the product does not involve u l. We can delete the vertices and consider the convex polytope generated by the remaining u i and get ɛ(i = i inductively. Then the only constant term is 1.

12 12 AN HUANG, BONG LIAN, SHING-TUNG YAU, CHENGLONG YU 3. Generalized flag vareities Now we prove similar proposition for generalized flag variety X = G/P using Corollary 2.5. There is a natural candidate for large complex structure limit in Calabi-Yau hypersurfaces family of G/P, which is the union of codimension one stratum of projections of Richardson varieties, denoted by Y 0. See [16] for the definition of Y 0 and [11] for indentification of Y 0 as solution rank 1 point of Picard-Fuchs system. In the proof of toric Calabi- Yau families, we only used the following fact. There is an affine chart (t 1 t n A n Z on X Z with s 0 = t 1 t n (dt 1 dt n 1. So we expect that Y 0 = Y Y n + W,where Y 1,, Y n has complete intersections at some point x and Z is an effective divisor outside x. But this only happens in some special cases, for example, projective spaces and Grassmannian G(2, 4. In general, the projections of Richardson varieties are not intersecting transversely to one point. We need the Bott-Samelson-Demazure-Hansen type resolution of projections of Richardson varieties to lift the anticanonical sections to rational anticanonical sections. This construction is also used in the proof of Frobenius splitting for projections of Richardson varieties, see [16]. Now let ψ : Z X be a proper birational morphism between smooth varieties Z and X over k. Let ω Z = ψ ω X + E, where E is a Weil divisor supported on exceptional divisor. Then we have ψ ω 1 X = ω 1 Z + E inducing the isomorphism (3.1 ψ (ω 1 Z + E = ψ (ψ ω 1 X = ω 1 X. This isomorphism is induced by pulling back anticanonical sections on X to anticanonical sections on Z with poles along E and hence fits in the commutative diagram of sheaves (3.2 F (ω 1 p X ψ F (ω 1 p Z ((p 1E ˆτ O X ˆτ ψ (O Z (E Here ˆτ is the same trace map induced by Cartier operator as follows. If σ = I f It I (dt 1 dt n 1 p is a local section of ω 1 p X, then ˆτ(F (σ = J f It J with I = (p 1,, p 1 + pj. The map ˆτ : F (ω 1 p Z ((p 1E O Z(E is defined as follows ˆτ(F (σ = ˆτ(F ( 1 η η p σ = ˆτ( 1 p η F (η p σ = 1 η ˆτ(F (η p σ. Here η is local defining section of E. Then η p σ is a holomorphic section of ω 1 p Z and ˆτ(F (η p σ is defined the same as X. After taking global sections, we reduce the calculation of Hasse-Witt matrix to Z. If we have a section s 0 H 0 (X, ω 1 Z (E with the desired property as toric case, then similar conclusion follows. Note that we need to take into account meromorphic sections. When E is union of some coordinate hypersurfaces at a point x, the same formula for ˆτ in terms of Laurent expansion of σ in local coordinates still applies.

13 HASSE-WITT, UNIT ROOTS AND PERIODS 13 Now we apply the discussion above to Bott-Samelson-Demazure-Hansen varieties. They arise as resolutions of singularities of Schubert varieties and projections of Richardson varieties. See [4], section 2, [3], or [16]. First we fix some notations. Let G be simple complex Lie group with Lie algebra g. Fix upper Borel subgroup B = B + and lower Borel subgroup B of G. Denote the simple roots by α 1 α l. Let W be the Weyl group and s i W the simple reflection generated by α i. Let w = s i1 s in be a reduced expression for w W and we denote it by w = (s i1,, s in. Let P ij be the minimal parabolic subgroup corresponding to simple root α ij. Then the Bott-Samelson variety Z w is defined to be P i1 P il /B n. Here the right action by B n is defined by (p 1,, p n (b 1,, b n = (p 1 b 1, b 1 1 p 2b 2,, b 1 n 1 p nb n. The image of (p 1,, p n under the quotient map is denoted by [p 1,, p n ]. We now recall some basic properties of Bott-Samelson variety. The map ψ w : Z w G/B defined by [p 1,, p n ] p 1 p n is a birational map to the Schubert variety X w = BwB/B. Let Z w(j = P i1 ˆP ij P il /B n 1 be a divisor of Z w via the embedding [p 1,, ˆp j,, p n ] [p 1,, 1,, p n ]. The boundary of Z w is defined to be Z w = Z w(1 + + Z w(n. These components have normal crossing intersection at [1,, 1]. Let L(λ = G B k λ be the equivariant line bundle on G/B associated to character λ and L w (λ = ψwl(λ. Then ω Zw = OZw ( Z w L w ( ρ with ρ being the sum of fundamental weights. From the previous discussion for toric case, we are looking for a special section s 0 H 0 (X, ω 1 X and suitable affine chart on Z w. Since the Picard group of G/B is generated by opposite Schubert divisors, we have a section σ of L(ρ vanishing exactly along all opposite Schubert divisors. Let s 0 be the tensor product of ψ wσ and canonical section of O Zw ( Z w. Then s 0 vanishes along Z w and preimage of opposite Schubert divisors. Let U i j be the negative unipotent root subgroup P ij U. The natural map U i 1 U i n Z w gives an affine neighborhood of [1,, 1] which is isomorphic to A n with coordinate (t 1,, t n. Then Z w(j on this affine chart is defined by t j = 0. The image of this chart under ψ is inside the opposite Schubert cell C id = B B/B. So s 0 vanishes with simple zero along coordinate hyperplanes on this chart. After rescaling, we can take s 0 = t 1 t n (dt 1 dt n 1. Let W P be the set of minimal representatives in cosets W/W P and w P be the longest element in W P with reduced expression w P. Then ψ : Z wp G/P is birational and it is an isomorphism restricted to Z wp Z wp Bw P B/B Bw P P/P.The exceptional divisor is supported on Z wp. Next we identify s 0 with a special anticanonical form defined on X = G/P. Let X v w = X v X w be the intersection of Schubert variety X w = BwB/B with opposite Schubert variety X v = B vb/b. The image in X = G/P is denoted by Π v w. This forms a stratification of X. The codimension one strata form an anticanonical divisor Π Π s. See [16],

14 14 AN HUANG, BONG LIAN, SHING-TUNG YAU, CHENGLONG YU section 3. (Note that our notations for Schubert variety and opposite Schubert variety are different from [16]. So there is an anticanonical section s 0 vanishing to the first order along Y 0 = Π 1 Π s. Lemma 3.1. The two anticanonical sections are related by s 0 = ψ (s 0 up to a rescaling. Proof. We compare the two divisors ( s 0 and (ψ (s 0. Since σ vanishes along opposite Schubert divisors on G/B, then s 0 vanishes along the preimage of opposite Schubert divisors under ψ w and Z w. Let C wp be the Schubert cell and C id the opposite Schubert cell. The restriction of ψ wp : Z wp Z wp C wp is an isomorphism. Let D i be divisors supported on C wp C id. Then ( s 0 = i ψ 1 w P (D i + j Z w P (j. The restriction of projection C wp G/P is also isomorphism on its image. The divisors Π j are exactly the complement of the image of C wp C id. So we have (ψ (s 0 ψ 1 w P (C wp = i ψ 1 w P (D i. The exceptional locus of ψ is supported on ZwP. So (ψ (s 0 = i ψ 1 w P (D i + j n jz wp (j as a meromorphic anticanonical section. Since ( s 0 and (ψ (s 0 are linear equivalent, then j Z w P (j and j n jz wp (j are linear equivalent. On the other hand, the divisors Z wp (1 Z wp (n form a basis for Pic(Z wp, see [4] Excercise 3.1.E (3. So n j = 1. So the Hasse-Witt invariants have similar expansion algorithm as toric case according to the discussion above. On the other hand, the period integral near s 0 can also be calculated by pulling-back to Z ω. The cycle γ : t 1 = t 2 = t n = 1 has nontrivial image in H n (X Y s0 since the integral of γ 0. This is the unique invariant cycle near s 0 since dim Hc n (X Y 0 = 1. According to Theorem 1.4 in [11], the period integral 1 f = γ ψ ( 1 f is the unique holomorphic solution to the Picard-Fuchs system near ψ γ s 0. So we proved Theorem 1.2 for generalized flag variety X = G/P. Note that the basis of H 0 (X, ω 1 X including s 0 can also be written down explicitly in terms of standard monomials, see [5]. This method also gives a way to calculate the power series expansions of period integrals of hypersurfaces in G/P. Remark 3.2. The anticanonical form s 0 appears in [19] and [16]. In [19], the form s 0 is constructed on torus chart of the open Richardson cell R id w = C id C w and glued together by coordinate transformations. We use the construction in [16] that the complement of R id w is an anticanonical divisor. Lemma 3.1 proves that ψ (s 0 = t 1 t n (dt 1 dt n 1 on the affine coordinate of Z ω, which is the local formula on torus chart appeared in [19]. This gives an explanation of the footnote in section 7 of [19]. The cycle γ appears in [20] 7.1 for complete flag variety G/B, in [18] Theorem 4.2 for Grassmannians and in [17] 12.4 for general G/P. 1 s 0

15 HASSE-WITT, UNIT ROOTS AND PERIODS 15 Now we give some explicit examples of the resolution and the anticanonical form s 0 under the resolution. Example 3.3. Let X be Grassmannian G(2, 4. Then X = G/P with G = SL(4 and P = { 0 0 }. The Weyl group is S 4 and W P = S 2 S 2. The element 0 0 w p = (13(24 = (23(34(12(23 = s 2 s 3 s 1 s 2. So Z w = P 1 P 2 P 3 P 4 /B 4 with P 1 = { 0 0 t 1 }, P 2 = { }, P 3 = { t } and P 4 = t { 0 0 t }. The largest Schubert cell is { a b 1 0 c d (a, b, c, d. The affine coordinate (t 1,, t 4 A 4 on Z wp is {[ t 1 1 0, , t , t So the map ψ : Z wp X under these local charts is given by }P/P with coordinates t ]}. (3.3 a = 1, b = t 1 + t 4, c = 1, d = 1 t 1 t 3 t 1 t 2 t 3 t 4 t 1 t 1 t 2 Recall the( anticanonical section s 0 in [11] is given in terms of standard monomials as follows. Let 12 a 13 a 14 a11 a be the basis of any two plane. The Plücker coordinates a 21 a 22 a 23 a 24 x ij are the determinant of i, j columns. The section s 0 = x 12 x 23 x 34 x 14. In coordinate of Schubert cell, we have s 0 = ad(ad bc(da db dc dd 1. A direct calculation using (3.3 shows that ψ s 0 = t 1 t 2 t 3 t 4 (dt 1 dt 2 dt 3 dt 4 1. The other sections of H 0 (X, L can also be written as homogenous polynomials of x ij of degree 4. Remark 3.4. The proof for both toric and flag varieties only depends on the the following fact. There is a torus chart (t 1,, t n on the complement of Y s0 with s 0 = t 1 t n (dt 1 dt n 1 on the chart. So Theorem main can hold for more general ambient spaces X. This also implies the Frobenius splitting of X compatibly with Y Complete intersections We further discuss the algorithm for Hasse-Witt matrix for complete intersections.

16 16 AN HUANG, BONG LIAN, SHING-TUNG YAU, CHENGLONG YU (1 Let X n be a smooth projective variety defined over k. Let L 1, L s be line bundles on X and E = i L i. Assume the following vanishing conditions H i (X, K X s i E = H i 1 (X, K X s i E = 0 for i = 1,, s. (2 Let Vi = H 0 (X, L i 0 and W = H 0 (X, det E K X 0. We further assume the zero locus of a generic element of V = H 0 (X, E is smooth with codimension s. Let e 1 e r be a basis of W. Let f i be the universal section of L i and f = (f 1,, f s be universal section of E. (3 Consider the family of complete intersections defined by f over the smooth locus S V {0}. Let X be X S and i: Y X is the embedding of universal family. The projections to S are denoted by π. (4 Let F S be the absolute Frobenius on S and X (p = X FS S the fiber product. Then we have absolute Frobenius F X the relative Frobenius F X/S : X X (p. Denote W : X (p X and π (p : X (p S to be the projections. The corresponding diagram for family Y is defined in a similar way. We repeat the argument in the hypersurfaces using the Koszul resolution (4.1 0 s E s 1 E E O X i O Y 0. Standard spectral sequence argument together with the vanishing assumptions gives an isomorphism R n s π (O Y R n π ((det E 1. The maps in the Koszul resolution is given as follows. Identify the section of k E with the sections (fj 1,,j k of j1,,j k L j 1 L j k with ordered set j 1,, j k, such that j 1,, j k are distinct and fj 1,,j k = ±fj 1, if,j k j 1,, j k is a permutation of j 1,, j k with signature ±1. Then (f j 1,,j k+1 is mapped to (fj 1,,j k = ( j f j 1,,j k,j f j. We have similar commutative diagram as (2.1. (4.2 f 0 W det E O X (p i O Y (p 0 f p 1 F F f 0 F X/S det E F X/S O X i F Y/S O Y 0 The map f p 1 : W k E ((f j 1,,j k p f p 1 j 1 F X/S k E is induced by multiplication (f j 1,,j k f p 1 j k. So we have commutative diagram (4.3 R n s π (p (O Y (p R n π (p (W det E F f p 1 R n s π (F Y/S O Y R n π (det E with horizontal maps being isomorphisms. The left vertical map is the Hasse-Witt operator H-W : FS (Rn s π (O Y = R n s π (p (O Y (p R n s π (p (F Y/S O Y = R n s π O Y. So we have similar definition of Hasse-Witt matrix under basis e 1 e r.

17 HASSE-WITT, UNIT ROOTS AND PERIODS 17 Definition 4.1. The basis e 1 e r of H 0 (X, K X det E induces a basis of R 0 π (ω Y/S by residue map and dual basis e 1 e r of R n s π (O Y under Serre duality. The Hasse- Witt matrix a ij is defined by H-W(FS (e i = j a ije j. The same argument in hypersurfaces case gives us the algorithm of computing Hasse- Witt matrix in terms of local expansion of f. Now f p 1 is replaced by (f 1 f s p 1. Under the trivialization ξ of det E under local coordinates (t 1,, t n, the section e i ξ has the form h i (tdt 1 dt 2 dt n and e i (f 1 f s p 1 ξ has the form as g p i (tdt 1 dt 2 dt n. Then τ(g i has the form τ(g i = j a jih j. On the other hand, the period integral has the form γ Res Ω f 1 f s = Ω γ f 1 f s where γ is a cycle in the complement of {f 1 f s = 0}. So it is the same form as hypersurfaces with f replaced by f 1 f s. The section f 1 f s also defines a subfamily of hypersurfces in the linear system det E. Both Hasse-Witt matrices and period integrals can be calculated with the same algorithm applied to this subfamily. So the truncation relation still holds for complete intersections in both toric variety and flag variety. For example, the statement for toric Calabi-Yau hypersurfaces is as follows. Let X be a smooth toric variety and = D D s be a partition of toric invariant divisors. Let L i = O(D i. Let f ij be a basis of H 0 (X, L i consisting of monomials with f i0 the defining section of D i. The K 1 X universal section is f = ( j b ijf ij i. Then the period integral of the unique invariant cycle near f 0 = (f i0 has the form 1 b 10 b s0 P ( b 1j 1 b sjs of b 1j 1 b sjs b 10 b s0 b 10 b s0, in which P ( b 1j 1 b sjs b 10 b s0 is a Taylor series b i0 multiplied with integer coefficients. The degree-(p 1 truncation (p 1 P ( b ij by (b 10 b s0 p 1 is a degree-(p 1 polynomial of b 1j1 b sjs and gives the Hasse-Witt matrix for the Calabi-Yau complete intersection family. 5. Frobenius matrices of toric hypersurfaces Now we give a proof of the conjecture in [23] for toric hypersurfaces. First we state the conjecture. The notations follow [15]. Let k be a perfect field of characteristic p. Let W (k be the ring of Witt vectors of k. Denote σ : W W be the absolute Frobenius automorphism of W. For any W -scheme Z, let Z 0 = Z W k be the reduction mod p. Let S = Spec(R be an affine W -scheme. Let R = lim R/p s R and S = Spf(R. We fix a Frobenius lifting on R and also denote it by σ, which is a ring endomorphism σ : R R such that σ(a = a p mod pr. Let X be a smooth complete toric variety defined by a fan. The 1-dimensional primitive vectors v 1, v N correspond to toric divisors D i. Assume L = O( k i D i with k i 1. Let = {v R n v, v i k i } and the interior of. Then H 0 (X, L has a basis corresponding to u I Z n and H 0 (X, L K X has basis

18 18 AN HUANG, BONG LIAN, SHING-TUNG YAU, CHENGLONG YU e i identified with u i Z n. Let f = a I t u I, a I R be a Laurent series representing a section of H 0 (X, L. Let (α s i,j be a matrix with ij-th entry equal to the coefficient of t ps u j u i in (f(t ps 1. The endmorphism σ is also extended entry-wisely to matrices. It is proved in [23] that α s satisfies the following congruence relations Theorem 5.1 (Theorem 1 in [23]. (1 For s 1, α s α 1 σ(α 1 σ s 1 (α 1 mod p. (2 Assume α 1 is invertible in R. Then α s+1 σ(α s 1 α s σ(α s 1 1 mod p s. (3 Under the condition of (2, for any derivation D : R R, we have D(σ m (α s+1 σ m (α s+1 1 D(σ m (α s σ m (α s 1 mod p s+m. Suppose that f defines a smooth hypersurface π : Y S. We assume Y satisfies the condition (HLF in [15], the Hodge cohomology groups H j (Y, Ω i Y/S are locally free R-modules for i + j = n 1. We also assume the pair (X, Y satisfies (HLF, the Hodge cohomology groups H j (X, Ω i X/S (log Y are locally free R-modules for i + j = n. Consider the F -crystal Hcris n 1 (Y 0/S = H n 1 DR (Y/S R R. We further assume the family Y/S satisfy condition HW (n 1 in [15], which says for any s 0 : R 0 K with K perfect field, the Hasse-Witt operator H n 1 (Y s (p 0, O (p Y H n 1 (Y s0, O Ys0 is an s 0 automorphism. Notice that α 1 mod p is the Hasse-Witt matrix under the dual basis of ω i = Res tu idt 1 dt n t 1 t nf(t H 0 (Y, Ω n 1 Y/S according to Corollary 2.3. The condition HW (n 1 can be checked using α 1. In particular, this condition implies that α 1 mod p is invertible. The unit-root F -crystal U 0 Hcris n 1 (Y 0/S and slope n 2 sub-crytal U n 2 are defined under the assumption. The quotient Q n 1 = Hcris n 1 /U n 2 is isomorphic to the p n 1 -twist of the dual U0 to U 0. The projection of ω i to Q n 1 gives a dual basis of U 0. In [23], the Frobenius matrix and connection matrix of U 0 are conjectured to be the limits of matrices in Theorem 5.1. Conjecture 5.2 ([23]. The Frobenius matrix is the p-adic limit (5.1 F = lim s α s+1 σ(α s 1. The connection matrix is given by (5.2 D = lim s D(α s (α s 1. Now we give the proof of this conjecture under an additional assumption on (X, L.

19 HASSE-WITT, UNIT ROOTS AND PERIODS 19 Theorem 5.3. Let (X, L be a smooth toric variety with line bundle L = O( k i D i. Let p i be toric invariant points corresponding to top dimensional cone in the fan decomposition. If a generic section of L does not vanish at some p i, then Conjecture 5.2 is true. The assumption in the theorem can be checked from toric data, or replaced by the equivalent assumption on the polytope of L. Let p i be the intersection of D 1 D n. Under a transformation of SL(n, Z, we can assume the corresponding cone is generated by standard basis of R n. Let f = I a It u I as before. Then under a trivialization of L, the universal section f is t k 1 1 tkn n ( I a It u I. So a generic section f does not vanish at (t 1,, t n = (0,, 0 means ( k 1,, k n is a vertex of. The assumption the theorem is equivalent to that at least one of the vertices of is the intersection of hyperplanes v, v i = k i, 1 i n with v i v n generating a cone of X. This is satisfied by X = P n with L = O(d, d n + 1. Proof. The proof follows the ideas in Katz s proof of Theorem 6.2 [15]. Consider the F - crystal constructed by logarithmic crystalline cohomology H n cris (X 0, Y 0 = H n DR (X, Y R. From the long exact sequence (5.3 HDR(X n HDR(X, n Y H n 1 DR (Y ( 1 and H k DR (X is concentrated in H k 2, k 2, the corresponding subcrystal U n 1 and quotient Q n are also defined on H n cris (X 0, Y 0 by taking the inverse image of U n 2 subcrystal in H n 1 cris (Y 0 and Q n (H n cris(x 0, Y 0 = Q n 1 (H n 1 cris (Y 0( 1. Here H( 1 means the Frobenius action is multiplied by p. We also have isomorphism H n cris (X 0, Y 0 = (H 0 (X, Ω n X/S (Y R U n 1. So we only need to consider the Frobenius matrix acting on projections of log n-forms ω i = tu idt 1 dt n t 1 t nf(t onto Q n. We can assume the primitive vectors v 1, v n are the standard basis of R n. The cone generated by v 1 v n defines an affine coordinate (t 1,, t n on X which is isomorphic to A n. First we assume Y is away from (t 1,, t n = 0 and consider the formal expansion map along (t 1,, t n = 0 (5.4 P : H n DR(X, Y R H n DR(R [[t 1,, t n ]]/R. Similar as Katz s proof of Theorem 6.2 [15], we have the following conjecture Conjecture 5.4. U n 1 is the kernel of formal expansion map.

20 20 AN HUANG, BONG LIAN, SHING-TUNG YAU, CHENGLONG YU Actually a weaker statement that U n 1 is contained in the the kernel can imply that U n 1 is the kernel, see remark 5.5. The conjecture might be proved by log version of the theory of de Rham-Witt following Katz s proof. We will first give the proof of Theorem 5.3 assuming the conjecture and state the method to get around the conjecture at the end. Assume the local expansion of 1 f exist in R[[t 1,, t n ]][ 1 t 1,, 1 t n ] and has the form (5.5 1 f = u A u t u. Notice that f may not have an inverse in R[[t 1,, t n ]][ 1 t 1,, 1 t n ]. We can consider the localization of R by inverting the coefficient a u0 of the vertex u 0 = ( k 1,, k n. Let a u0 = 1, then (5.6 1 f = t u0 1 + u I u 0 a I t u I u 0 = t u 0 (1 + k So the local expansion of ω i has the form (5.7 ω i = tu i dt 1 dt n t 1 t n f(t = dt 1 dt n t 1 t n ( 1 k ( a I t ui u0 k = A u t u. u I u 0 u A u t u+u i with all u + u i > 0. Assume the Frobenius action on ω i is in the form (5.8 F (ω (σ i j and the connection of D has the form f ij ω j mod U n 1 u (5.9 (D(ω i j (D ij ω j mod U n 1 Assume Conjecture 5.4 is true, then (5.10 F (ω (σ i = j f ij ω j in H n DR(R [[t 1,, t n ]]/R and (5.11 (D(ω i = j (D ij ω j in H n DR(R [[t 1 t n ]]/R. According to the Frobenius action on H n DR (R [[t 1,, t n ]]/R, we compare the coefficient of t pkv for multi-index v Z n and v > 0 (5.12 p n σ(a u i j f ij A u j mod p k with p(u i + u i = p k v = u j + u j. On the other hand, we compare the expansions of f ps 1 = u Ãs ut u and 1 f (5.13 Ã s ut u = f 1 ps f = ( But s u ( u u A u t u.

21 HASSE-WITT, UNIT ROOTS AND PERIODS 21 So Ãs u = u +u =u Bs u A u. We can extend σ to any Laurent series with coefficients in R by σ(t u = t pu. Since σ(f = f p + pg, then (5.14 σ(f ps 1 = σ(f ps 1 = (f p + pg ps 1 f ps mod p s. Let f ps 1 1 = u Ãs 1 u t u, then (5.15 σ(ãs 1 u t pu f ps σ( 1 f = ( But s u ( u u u σ(a u t pu mod p s. So σ(ãs 1 u u +pu =pu Bs u σ(a u mod p s. Now we compute σ(α s 1 im = σ(ãs 1 p s 1 u m u i in terms of A u and B u. It is the sum of Bu s σ(a u with u + pu = p(p s 1 u m u i. The factor Bu s is the sum of the terms ( p s (5.16 a k 1 I k 1, k 2,, k 1 a k l I l l with k 1 u I1 + +k l u Il = u. Denote ν p to be the p-adic valuation. Let k = min{ν p (k 1 ν p (k l } and p k v = p s u m u = p(u + u i in (5.12, then (5.17 σ(a u j f ij A u j mod p k with u + u j = ps u m u j. Since the p-adic valuation of multinomial has estimate ( p s (5.18 ν p s min{ν p (k 1 ν p (k l }, k 1, k 2,, k l then (5.19 ( p s k 1, k 2,, k l a k 1 I 1 a k l I l σ(a u j f ij ( p s k 1, k 2,, k l a k 1 I 1 a k l I l A u j mod p s. So we have (5.20 B u σ(a u j f ij B u A u j mod p s with u + u = p s 1 u m u i and u + u j = ps u m u j. Summing all such terms implies (5.21 p n σ(α s 1 (f ij α s mod p s and p n (f ij 1 α s σ(α s 1 1 mod p s n. Similar calculation as [15] and congruence relation D(B u 0 mod p s imply (5.22 D(α s ( (D ij α s mod p s. If the coefficient of the vertex u 0 is zero, we regard a I as formal variables and the universal hypersurface family. Then we can prove the result on an open subset of S and the p-adic limit formulas holds on the open subset. Since Vlasenko proved the congruence relations in Theorem 5.1 without any constraints on the coefficients, the p-adic limits

22 22 AN HUANG, BONG LIAN, SHING-TUNG YAU, CHENGLONG YU always exist. So the limits coincide with Frobenius matrices and connection matrices because they are equal restricted to an open subset of S. Now we state the proof without assuming Conjecture 5.4. We claim p l(n 1 P (U n 1 p ln HDR n (R [[t 1,, t n ]]/R. Applying Katz s argument of extension of scalars, we only need to prove this when R is the Witt vectors of a perfect field. The Frobenius action on U n 1 divides p n 1. So there exists σ 1 -linear map F on U n 1 such that F F = F F = p n 1. On the other hand, the Frobenius action on each element in HDR n (R [[t 1 t n ]]/R has a factor p n. So p n 1 P (U n 1 = P (p n 1 U n 1 = P (F F U n 1 p n HDR n (R [[t 1,, t n ]]/R and l iterations give p l(n 1 P (U n 1 p ln HDR n (R [[t 1,, t n ]]/R. Multiplying both (5.10 and (5.11 by p l(n 1, we get (5.23 p l(n 1 F (ω (σ i = p l(n 1 j and f ij ω j mod p ln in H n DR(R [[t 1,, t n ]]/R (5.24 p l(n 1 (D(ω i = p l(n 1 j (D ij ω j mod p ln in H n DR(R [[t 1 t n ]]/R. Let s = nl, similar congruence relation as (5.12 still holds for k s (5.25 p n+l(n 1 σ(a u i p l(n 1 j f ij A u j mod p k with p(u i + u i = p k v = u j + u j and v > 0. The same argument shows (5.26 p n+l(n 1 σ(α s 1 p l(n 1 (f ij α s mod p s and (5.27 p l(n 1 D(α s p l(n 1 ( (D ij α s mod p s. Dividing both sides by p l(n 1 and letting l, we see that the subsquence α s σ(α s 1 1 and D(α s (α s 1 converges to the Frobenius matrix and connection matrix. Remark 5.5. If U n 1 is contained in the the kernel of formal expansion map, then it is exactly the kernel. We only need to show the restriction of expansion map on H 0 (X, Ω n X/S (Y R HDR n (R [[t 1 t n ]]/R is injective. This can be proved by similar argument in the proof and invertibility of α s. Remark 5.6. The proof also gives a weaker version of the second and third congruence relations in Vlasenko s Theorem 5.1. The first congruence relation α s α 1 σ(α s 1 mod p can also be proved geometrically using the argument in Theorem 2.2 and Corollary 2.3. We can consider the s-iterated Hasse-Witt operation H n 1 (Y (ps 0, O (p Y s H n 1 (Y 0, O Y0. 0 Using similar commutative diagram 2.1 with the first vertical map L 1 L 1 being replaced by the composition L 1 L ps L 1 with ξ ξ ps f ps 1, we can see the matrix for s-iterated Hasse-Witt operation is given by α s mod p. Hence α s α 1 σ(α s 1 mod p.

23 HASSE-WITT, UNIT ROOTS AND PERIODS Unit root of toric Calabi-Yau hypersurfaces and periods. Now we discuss the relation between unit roots of zeta functions and period integrals for toric Calabi-Yau hypersurfaces. Let f be a Laurent series defining toric Calabi-Yau hypersurfaces. For the sake of simplicity, let a 0 = 1 be the constant term (or the coefficient of interior point of. The unique holomorphic period integral at the special solution-1 point or large complex structure limit is I γ = the constant term of the expansion ( f = u I 0 a It u I = 1 + k ( 1 k ( u I 0 a I t u I k. It can be written as formal power series of a I with constant term being 1 and denoted by P (a I. Then (5.29 α s (ps 1 (P (a I mod p s because of the congruence ( p s ( 1 (5.30 k 1, k 2,, k l, p s ( 1 k k 1 k k 1, k 2,, k l Then (5.31 α s σ(α s 1 1 (p s 1 (P (a I (p s 1 1 mod p s (P (σ(a I mod p s. according to Vlasenko s congruences without any geometric constraints. So the p-adic P (a limit I P (σ(a I exists in R and equal to the Frobenius matrix. We can fix σ(a I = a p I. Then the formal power series (5.32 g(a I = P (a I P (a p I has p-adic limit in lim Z s p [a I1 a IN, α1 1 ]/ps Z p [a I1 a IN, α1 1 ] and it satisfies Dwork congruences (p s 1 (P (a I (5.33 g(a I (p s 1 1 (P (σ(a I Especially it is related to Hasse-Witt matrix by (5.34 mod p s. P (a I P (a p I (p 1 (P (a I mod p. Let q = p r and a I F q defining a smooth Calabi-Yau variety Y 0 over F q. Assume the Hasse-Witt matrix (p 1 (P (a I mod p is not zero. Then there exist exactly one p- adic unit root in the factor of zeta function of Y 0 corresponding to Frobenius action on H n 1 cris (Y 0. It is given by (5.35 g(â I g(â p I g(âpr 1 I with â I being the Teichmüller lifting under σ. For example, if a I has lifting as an integer, then â I = lim s a ps I.