Zeta Functions of Toric Calabi-Yau Hypersurfaces

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1 Zeta Functions of Toric Calabi-Yau Hypersurfaces Daqing Wan March 5, 2004 Contents 1 Toric Geometry n-torus Basic problem Projective toric hypersurfaces regularity Homological formulation of -regularity Hodge numbers Reflexive and Calabi-Yau hypersurface A basic example Zeta Functions L-functions of exponential sums Dwork s p-adic analytic character Analytic representation of S k (x 0 f) Frobenius endmorphism Rationality of L(x 0 f, T ) and Z(U f /F q, T ) p-adic Cohomological formula for L(x 0 f, T ) Newton polygon for L(x 0 f, T ) ( 1)n Variation of NP (f) with p Variation of NP (f) with f (p fixed) Generically ordinary primes Generically ordinary Calabi-Yau hypersurfaces Basic example Zeta functions of affine toric hypersurfaces MCM workshop lectures (Beijing, Dec , 2003), notes taken by Guohua Peng. Expanded by Doug. Haessig for the Arizona Winter School (March 14-17, 2004). 1

2 3 p-adic Variation p-adic Analytic formula for the Frobenius matrix Deformation theory and Picard-Fuch equation Hodge-Newton decomposition and unit root formula Unit root L-function and p-adic Galois representation Dwork s unit root conjecture p-adic Monodromy group Toric Geometry 1.1 n-torus Denote by G n m the algebraic n-torus over F q. Notice that its F q k-rational points are G(F q k) = (F q k ) n and so #G n m(f q k) = (q k 1) n. It follows that its zeta function is rational: 1.2 Basic problem Z(G n #G n m(f q k) m/f q, T ) : = exp( T k ) k k=1 n = (1 q i T ) ( 1)n i 1 ( n i). Given a Laurant polynomial f(x 1,, x n ) F q [x ± 1,, x± n ], we may define an affine toric hypersurface U f := {x G n m f(x) = 0} G n m. Wanting to understand the sequence of integers obtained by counting the F q k-rational points of U f leads to its zeta function: Z(U f /F q, T ) = exp( #U f (F q k) T k ) 1 + T Z[[T ]]. k k=1 As with the n-torus, we wonder whether this too will be a rational function. Indeed, Dwork has shown this to be true. Theorem 1.1 (Dwork). Z(U f /F q, T ) Q(T ). A consequence of this theorem is the existence of a formula for the numbers #U f (F q k) in terms of the zeros and poles of the zeta function. However, how well do we know these zeros and poles? Knowing their precise values seems to be too difficult in general, so, we may ask for weaker results concerning their absolute values (p-adic and over C). Also, we may wonder how these zeros and poles vary in a family. 2

3 1.3 Projective toric hypersurfaces Let K be a field. With our Laurant polynomial f, write f(x) = J j=1 a jx V j where a j K and V j = (v 1j,, v nj ) Z n. Associated to f is its Newton polytope: (f) := := the closed convex hull of the {V j } s in R n. We will assume dim = n. The Newton polytope will be used to define a graded algebra S as follows. First, define the the polytope R n+1, which is one dimension higher than, as the closed convex hull of the origin in R n+1 and the points (1, V j ) R n+1. Next, define the cone C( ) as the cone generated by. Observe that C( ) = k=1 k. Next, define the monoid L( ) as the lattice points in the cone C( ). It may be shown that L( ) is a finitely generated monoid. Finally, define the K-algebra S := K[L( )]. This means S consists of all finite sums of a u x u where a u K and u L( ). Since L( ) is a finitely generated monoid, S is a finitely generated K-algebra. Further, we may define a grading on S by deg(x u ) := u 0 where u = (u 0,..., u n ). Therefore, S = (S ) d d=0 where (S ) d is the K-submodule of S consisting of all elements of S of degree d. Since S is a finitely generated graded K-algebra, we may define a K- scheme P := Proj S. This is the toric variety associated to. Observe that this toric variety only depends on those terms of f that lie on the vertices of. So, we may think of P as an analogue of projective space. That is, since x 0 f (S ) 1, we may define U f := Proj S /(x 0 f). Notice that Ū f embeds in P by construction. Thus, we call Ūf a toric hypersurface in P. It follows that we have the diagram: U f projective closure P compactification w.r.t. U f G n m. This raises the new questions: what is Z(U f, T ) and how is it related to Z(U f, T )? 3

4 1.4 -regularity In this section, we define the notion of a -regular polynomial f. Let τ be a face of the polytope of any dimension ranging between zero and n. Define the restriction of f to τ as f τ = V j τ a jx V j. Using the operator E i := x i x i, define f i := E i f for each i = 1,, n. Definition 1.2. f is called -regular if for each face τ of any dimension, the system f τ = f τ 1 = = f τ n = 0 has no common solutions in G n m(k alg. clos. ). We may reformulate the definition of -regularity as follows. F := x 0 f 1 S. Notice that { F x 0 f, i = 0 F i := E i F = x i = x i x 0 f i i = 1,, n and F i (S ) 1. For each i, define U Fi = ProjS /(F i ). Proposition 1.3. f is -regular if and only if n U F i =. 1.5 Homological formulation of -regularity Each F i S acts on S by multiplication: F i : S S g F i g F i F j = F j F i. Let K.(S, F 0,, F n ) be the Koszul complex n 0 S e 0 e n S e i S 0 (a e i1 e ij ) = j ( 1) k F ik (a) e i1 ê ik e ij k=1 H 0 (K.) = S /(F 0, F 1, F n ) = R f, the Jacobian ring of f. Proposition ) f is -regular. TFAE (the following are equivalent): 2) {F 0, F 1,, F n } forms a regular sequence of S. 3) H i (K.) = 0, i 1. 4) dim K H 0 (K.) <. 5) dim K H 0 (K.) = d( ) = n! Vol( ) Z >0. Define 4

5 1.6 Hodge numbers Definition 1.5. Let R n be n-dimensional integral convex in R n. Let W (k) = #(Z n k ) = dim K (S ) k. and W (k)t k, the Poincare series of S. k=0 Definition 1.6. Define h (k) = dim K (R f ) k and h (k)t k, the Poincare series of R f, k 0 where f is -regular and / R f = S (F0, F 1,, F n ), dim R f = d( ) = n! Vol( ). (1 T ) n+1 W (k)t k = h (k)t k, of degree n. k 0 k 0 ( ) ( ) n + 1 n + 1 h (k) = W (k) W (k 1) + W (k 2) Theorem 1.7 (Ehrhart). There exists a polynomial Λ(t) of degree n such that 1) for k Z 0, W (k) = Λ(k); 2) for k Z >0, W (k) := #{interior lattice points in k } = ( 1) n Λ(k) ( (1 T ) n+1 k=0 W (k)t k = k 0 h (k)t k, a polynomial of degree n+1); 3) duality: h (k) = h (n + 1 k), k = 0, 1,, n + 1. Proposition 1.8. f, -regular over C h k (P H n 1 c (U f )) = h (k + 1). Definition 1.9. Let HP ( ) denote the Hodge polygon in R 2 with vertices (0, 0) and ( m k=0 h (k), m k=0 kh (k)), m = 0, 1,, n. 5

6 1.7 Reflexive and Calabi-Yau hypersurface Definition Let R n, convex, integral, n-dimensional. Assume O is in the interior of. Define { } = (x 1,, x n ) R n n x i y i 1, (y 1,, y n ). i=1 is also an n-dimensional convex polytope, not necessarily integral. Clearly, ( ) =. Definition is called reflexive if is also integral. Example a,b : b 0 a = a,b : 1 a a,b is reflexive iff a, b = 1. 0 Definition Let W be an irreducible normal n-dimensional projective variety with Gorenstein canonical singularities. Then W is called a Calabi- Yau variety if 1) the dualizing sheaf ˆΩ n W = O W is trivial; 2) H i (W, O W ) = 0, 0 < i < n. Elliptic curves and K3-surfaces are CY. Theorem 1.13 (Hibi, Batyrev). TFAE: 1) is reflexive. 2) For any hyperplane H = {(x 1,, x n ) R n n i=1 a ix i = 1} such that H is a codimension 1 face of, we have a i Z. 3) Hodge numbers are symmetric: h (k) = h (n k), 0 k n. 4) The closure U f of U f (f is -regular) in P is a CY variety with canonical singularities. Definition For reflexive; f -regular, U f is called an affine toric CY hypersurface. Definition Denote M p ( ) = {f/f p (f) =, f is -regular}. Let be reflexive. The family {f M p ( )} is called the mirror family of {g M p ( )} over F p. 1 b 6

7 Question: If g is the mirror of f, Z(f/F q, T ) Z(g/F q, T )? Definition A reflexive in R n is called Fano, if 1) is simplicial, i.e., each codimension 1 face of is a simplex. And 2) The vertices of each codimension 1 face of form a Z-basis of Z n in R n. Proposition Reflexive is Fano P is smooth. 1.8 A basic example Take = e 1, e 2,, e n, (e e n ), where e i s are the standard unit vectors in R n. Then = (n, 1,, 1), ( 1, n,, 1),, ( 1,, 1, n), ( 1, 1,, 1). is reflexive, Fano, but NOT Fano if n > 1. For n = 2, e 2 e 1 (e 1 + e 2 ) x 1 + x 2 = 1 = 1 1 Let f(λ, x) = x 1 + x x n + It s clear that (f) =. f(λ, x) is -regular λ (n + 1)α, α n+1 = 1. Mirror family: xn+1 1 x n+1 n x 1 x 2 x n + 1 x 1 x 2 x n λ. 1 x 1 x 2 x n λ g(λ, x) = + + x 1 x 2 x n 1 = (x n x n+1 n + 1 λx 1 x 2 x n ) x 1 x 2 x n x i x n x n+1 n λx 1 x 2 x n 7

8 Projective closure in P n : x n x n x n+1 n λx 0 x 1 x 2 x n = 0 (the well known family of CY hypersurfaces in P n.) Let { G = (Z/(n + 1)Z) n 1 = (ζ (1),, ζ (n) ) (ζ (i) ) n+1 = 1, Then G acts on U g(λ,x) : } n ζ (i) = 1. i=1 (ζ (1),, ζ (n) )(x 1,, x n ) = (ζ (1) x 1,, ζ (n) x n ). Proposition U f(λ,x) = U g(λ,x) /G. Proof. If g(λ, x) = 0 for some x, x i 0, let y 1 = x n+1 1 /x 1 x n. y n = x n+1 /x 1 x n n { x 1 x n = y 1 y n = y i y 1 y n x n+1 i y y n + 1 y 1 y n λ = 0. Exercise: = (x x n + 1 x 1 x n λ) h (0) = h (1) = = h (n) = 1. (Betti number d( ) = n + 1.) 2 Zeta Functions 2.1 L-functions of exponential sums For f F q [x ± 1,, x± n ], U f = {f = 0} G n m, we have Z(U f, T ) = exp( #U f (F q k) T k k ). k=1 Let Ψ : F p C x ψ(x) = exp( 2πix p ) 8

9 be a nontrivial character of F p. Then Ψ Tr Fq k /F p : F q k C induces a nontrivial character of F q k. The exponential sum S k (x 0 f) = Ψ Tr Fq k /F p (x 0 f). Then where It s easy to compute x i F q q k #U f (F q k) = x i F q k 1 i n x 0 F q Ψ Tr Fq k /F p (x 0 f) =(q k 1) n + S k (x 0 f) =#G n m(f q k) + S k (x 0 f). Z(U f, qt ) = Z(G n m, T )L(x 0 f, T ), L(x 0 f, T ) = exp( #S k (x 0 f) T k k ). It is enough to study L(x 0 f, T ). k=1 2.2 Dwork s p-adic analytic character Consider the Artin-Hasse series t + tp p + tp2 p 2 +. The Newton polygon of this tells us that there are exactly p 1 roots of 1 this series of slope p 1. Let π be one of these roots, and so ord p(π) = 1 p 1. Using this, we may define a splitting function ) θ(t) := exp ((πt) + (πt)p + Q p (π)[[t ]]. p Since exp (t + tpp ) + = (k,p)=1 (1 t k ) µ(k) k, it follows that θ(t) converges on t p < p 1 p 1. In particular, θ is defined at the Teichmüller points in C p. Splitting functions have the following remarkable properties: 9

10 Property 1. θ(1) is a primitive p-th root of unity. Property 2. We may define a nontrivial additive character ψ k : F p k C p by ψ k ( x) := θ(x)θ(x p ) θ(x pk 1 ) = ψ 1 (Tr Fp k /F p ( x)). where x is the Teichmüller representative of x. 2.3 Analytic representation of S k (x 0 f) Write x 0 f(x) = J j=1 ājx 0 x v j F q [x ±1 0,..., x±1 n ]. Then, with q = p a, we have S k (x 0 f) = ψ k (x 0 f( x)) x i F q k = = x i F q k j=1 x qk 1 i =1,x i Q p = x qk 1 i =1 J ψ k (ā j x 0 x v j ) J j=1 ak 1 θ((a j x 0 x v j ) pi ) (1) F a (f, x)f a (f, x q ) F a (f, x qk 1 ) (2) where we have lifted the coefficients of f to C p, that is, a j = T eich(ā j ), and, F a (f, x) := a 1 j=1 J θ(a j x 0 x v j ) pi. This is the p-adic analytic representation of S k (x 0 f) that we will use. 2.4 Frobenius endmorphism Recall S from section 1.3, with K replaced by Z p. Now, define S,p := { u L( ) A u π u 0 x u A u Z p, A u 0}. Note, S,p is isomorphic to the p-adic completion of S at p. Now, with a norm defined by A u π u 0 x u := sup u A u p, we see that S,p is a Banach Z p -module. By construction, we see that Γ := {π u 0 x u u L( )} is an orthonormal basis for S,p, that is, the coefficients tend to zero. Consider the field Q q (π) and its Galois group over Q p (π), which is cyclic of order a generated by τ. By definition, τ sends Teichmüller representatives to their p-th power. 10

11 and and Using notation from the last section, define F (f, x) := J θ(a j x 0 x v j ) j=1 G(x) := F (f, x)f τ (f, x p )F τ 2 (f, x p2 ). On the space S,p Z q [π], define the compact operators where, q = p a, and Note, we may formally write where φ 1 := ψ p F (f, x) φ a := ψ q F a (f, x) ψ p ( A u x u ) := A τ 1 pu x u. φ 1 = G(x) 1 ψ p G(x) and φ a = G(x) 1 ψ q G(x), ψ q ( A u x u ) := A qu x u. 2.5 Rationality of L(x 0 f, T ) and Z(U f /F q, T ) Now φ a has the following amazing property called the Dwork trace tormula: S k (x 0 f) = (q k 1) n+1 T r(φ k a) where T r denotes the trace of the operator. Recall the relation 1 det(i φ a T ) = exp k 1 T r(φ k a) T k. k Combining these with the binomial theorem, we see that L(x 0 f, T ) = exp k 1 = n+1 S k (x 0 f) T k k [ det(i q i φ a T ) ] ( 1) n i ( n+1 i ). This looks like rationality, however, remember that the operator φ a acts on S,p Z q [π], an infinite dimensional space and so the characteristic polynomials are actually power series. However, since this operator is compact, det(i q i φ a T ) is a p-adic entire function. Therefore, the L-function is p-adic meromorphic. To prove rationality, we need to use an extension of a theorem of Borel proven by Dwork. 11

12 Theorem 2.1 (Borel). Let g(t ) Z[[T ]]. Then g(t ) Q(T ) if and only if g(t ) satisfies both 1. g(t ) converges in some neighborhood of the origin in C. 2. g(t ) is p-adic meromorphic. We obtain Theorem 2.2 (Dwork). L(x 0 f, T ) Q(T ) and so Z(U f /F q, T ) Q(T ). To prove this, we need only show that L(x 0 f) converges in some neighbourhood of the origin in C. Now, and since S k (x 0 f) C (q k 1) n+1 q k(n+1) k 1 q k(n+1) T k converges for T C < 1/q n+1, we see that L(x 0 f, T ) converges for any T C < 1/q n+1. This proves the theorem. 2.6 p-adic Cohomological formula for L(x 0 f, T ) k As mentioned in section 2.5, we may define a compact operator φ a on a p-adic Banach module B := S,p Z q [π]. We may also define differential operators D i := G(x) 1 x i G(x) x i for each i = 0, 1,..., n acting on B. Since these commute, we may create a Koszul complex K (B, D 0,..., D n ), the top line of the commutative diagram below. Also, since φ a D i = qd i φ a, we may define a chain map between complexes: where 0 B (n+1 d B (n+1 n ) d d B (n+1 0 ) 0 q n+1 φ a q n φ a φ a n+1) 0 B (n+1 n+1) d B (n+1 n ) d B (n+1 i ) := B Λ i ( n j=0ze j ) and d : B (n+1 i ) B ( n+1 i 1) is defined by d(ae j1 e ji ) := d B (n+1 0 ) 0 i ( 1) k D jk (a)e j1 ê jk e ji. k=0 12

13 We may rewrite the L-function as follows. L(x 0 f, T ) ( 1)n = n+1 n+1 = n+1 det(i T q i φ a B) ( 1)i ( n+1 i ) det(i T q i φ a B (n+1 i ) ) ( 1) i = det(i T q i φ a H i (K (B, D 0,..., D n ))) ( 1)i. Now, if f is -regular, then all the homology spaces are trivial except for i = 0, in which case H 0 (K (B, D 0,..., D n )) = B/ n D i (B) is a free Z q [π]-module of rank d( ). That is the essence of the next two theorems. Theorem 2.3 (Adolphson-Sperber). If f is -regular, then L(x 0 f, T ) ( 1)n is a polynomial of degree d( ) = n! Vol( ). Theorem 2.4 (Denef-Loeser). If f is -regular, then L(x 0 f, T ) ( 1)n mixed of weight n + 1. That is, if is L(x 0 f, T ) ( 1)n = d( ) (1 α i T ), i=1 then α i = q w i, w i Z [0, n + 1]. Let e j = #{1 i d( ) w i = j}, 0 j n + 1. There exists a very complicated combinatorial formula for e j. Example: Let be a simplex and Then e 0 = 1, e j = c 0 = 1, c i = τ,face dim τ=i 1 Vol(τ), i 1. j ( ) n + 1 i ( 1) j i i! c i, j 1. n + 1 j Exercise: f(λ, x) = x x n + 1 x 1 x n λ, -regular. Compute e j. 13

14 2.7 Newton polygon for L(x 0 f, T ) ( 1)n Let f be -regular over F q. Write L(x 0 f, T ) ( 1)n = d( ) m=0 A m T m, A 0 = 1, A m Z. Define the q-adic Newton polygon of L(x 0 f, T ) ( 1)n to be the lower convex closure in R 2 of the points (m, ord q (A m )), m = 0, 1,, d( ). Denote this polygon by NP (f). NP (f) s }{{} h s =horizontal length of the slope s side Newton polygon of f Theorem 2.5. NP (f) has a side of slope s with horizontal length h s iff there are exactly h s reciprocal zeros α i s such that ord q (α i ) = s, i.e., α i = q s. Question. For s Q [0, n + 1], h s =? Theorem 2.6 (Adolphson-Sperber). f is -regular NP (f) HP ( ), with endpoints coincide, where HP ( ) is the Hodge polygon of. An outline of the proof is as follows. See section 2.6 for some relevant notions. We define an operator φ 1 on our Banach module B := S,p Z q [π]. This induces an operator on the finite dimensional homology space H 0 := H 0 (K (B, D 0,..., D n )) = B/ n D i (B) and so may be represented by a matrix if we provide a basis. Choosing a monomial basis Γ I := {π u 0 x u u I}, we may explicitly estimate the p- adic order of the entries of the matrix A 1 representing φ 1 to get a (Hodge) 14

15 filtration: φ 1 (Γ I ) = Γ I A 1, where M 00 M 01 M 02 pm 10 pm 11 pm 12 A 1 = p 2 M 20 p 2 M 21 p 2 M where M ij is a matrix with h (i) rows and h (j) columns. Further, the entries of M ij have ord p 0. Relating this to our operator φ a via the relation φ a 1 = φ a and using an argument of Dwork s, we may show the q-adic Newton polygon of det(i T φ a H 0 ) lies above the Hodge polygon, which is defined as the lower convex hull of the points ( m ) m h (i), i h (i). m=0,1,...,n Definition. If NP (f) = HP ( ), then f is called ordinary. In this case, L(x 0 f, T ) ( 1)n has exactly h (k) reciprocal zeros α i s such that ord q (α i ) = k for all 0 k n. 2.8 Variation of NP (f) with p Conjecture: Let f Z[x ± 1,, x± n ] be -regular. Then there exist infinitely many primes p such that NP (f mod p) = HP ( ) (p is then called ordinary). One further conjectures that the density δ(f) of ordinary primes exists and is positive. Example. If f = x 1 + x x 1 x 2 λ is -regular and hence defines an elliptic curve over Q, the density δ(f) is either 1/2 if f has CM (Deuring) or 1 if f has no CM (Serre). 2.9 Variation of NP (f) with f (p fixed) Let M p ( )(F p ) = {f F p [x ± 1,, x± n ] (f) =, f is -regular}. This set is non-empty if p > d( ). f M p ( )(F p ) f M p ( )(F q ) for some q. q-adic NP (f) is defined, independent of the choice of the defining field F q. The relatively cohomology is locally free and thus forms an overconvergent σ-module and in fact an overconvergent F -crystal on M p ( ). We obtain 15

16 Theorem 2.7 (Grothendieck-Katz). The global minimun GNP (, p) = inf NP (f) f M p( ) exists and is precisely attained for all f in a Zariski open dense subset U p ( ) M p ( ). This minimun polygon GNP (, p) is called the generic Newton polygon of the family M p ( ). Thus, Newton polygon goes up under specialization, that is, for f M p ( ), NP (f) GNP (, p) HP ( ). The first equality holds for all f U p ( ). Definition: If GNP (, p) = HP ( ), is called ordinary at p or generically ordinary at p. Question: Which primes p are ordinary for? 2.10 Generically ordinary primes Conjecture (Adolphson-Sperber): is ordinary for p 0. Proposition 2.8. Let be minimal (i.e., no lattice points on other than vertices). If p 1( modd( )), then is ordinary at p For minimal, x 0 f becomes a diagonal, the L-function can be computed directly using Gauss sums and the slopes can be found using the Stickelberger theorem. This is the local case. Note also for minimal, one has d( ) = 1 if n 2. Theorem 2.9 (Wan). 1) If n 3, is ordinary for p > d( ). 2) If n 4, there exists n-dimensional which is NOT ordinary for all primes p in a certain congruence class. 3) There exists D ( ) > 0 such that is ordinary for p 1( mod D ( )). Part 1) and part 3) follow from the collapsing decomposition (to be explained in the lectures) and a finer form of the above local proposition. Conjecture. There is a positive integer µ( ) such that the set of almost all (except for finitely many) ordinary primes for consists of the primes in certain congruence classes modulo µ( ) Generically ordinary Calabi-Yau hypersurfaces Theorem 2.10 (Wan). Let be reflexive. 1) If n = dim( ) 4, then is ordinary for p > d( ). 2) If is Fano, then is always ordinary for every p. 16

17 Part 2) follows from the star decomposition theorem. The case n = 4 of Part 1) follows from a combination of the star decomposition and the collapsing decomposition (to be explained in the lectures). Questions: 1) For reflexive with n = dim( ) 5, is ordinary for p > d( )? 2) If is reflexive and ordinary at p > d( ), is ordinary at p? (already yes if n 4 or is Fano) 2.12 Basic example Take Then f(λ, x) = x x n + 1 x 1 x n λ. = (f) = e 1, e 2,, e n, (e 1 + e e n ), = (n, 1,, 1),, ( 1, 1,, n), ( 1, 1,, 1). Theorem Both and are ordinary for all primes p. Theorem ) Let f(λ, x) be -regular over F q. Then L(x 0 f, T ) ( 1)n = 2) α 0 (λ) = 1, α i (λ) = q n+1. n (1 α i (λ)t ), d( ) = n ) is ordinary at p. That is, except for finitely many λ, we have ord q (α i (λ)) = i, 1 i n. Proof of 2). Since endpoints for NP (f) and HP ( ) coincide, we have ord(α 0 α 1 α n ) = ord(α 1 α n ) = n kh (k) = k=0 n(n + 1). 2 By Denef-Loeser (Theorem 2.2), α i q n+1 2. Now α 1 α n Z implies that (α 1 α n ) 2 = α 1 α n α 1 α n q n(n+1). But ord(α 1 α n ) 2 = n(n + 1), then (α 1 α n ) 2 = q n(n+1). Hence α i = q n+1 2 for all 1 i n. Let g(λ, x) = x n x n+1 n λx 0 x 1 x n. For almost all λ, it defines a smooth projective hypersurface in P n. Then Z(g(λ, x), T ) = P (T ) ( 1)n (1 T )(1 qt ) (1 q n 1 T ), 17

18 where P (T ) is a polynomial of degree n(nn ( 1) n ) n+1, pure of weight n 1. The Newton polygon of P (T ) HP (Dwork). Question: Is this family g(λ, x) of projective hypersurfaces in P n generically ordinary for all p > n + 1? (yes for n 3.) 2.13 Zeta functions of affine toric hypersurfaces Let f F q [x ± 1,, x± n ], = (f), f is -regular. Then trivially T = 1 is a root of L(x 0 f, T ) ( 1)n. And L(x 0 f, T ) ( 1)n 1 T = P (f, qt ) is a polynomial in 1 + T Z[T ] of degree d( ) 1 (with slope 1), where P (f, T ) is a polynomial in Z[T ]. We have where Z(U f, qt ) = n (1 q i T ) ( 1)n i 1 ( n i) L(x0 f, T ) n = (1 q i T ) ( 1)n i 1 ( n L(x i) 0 f, T ) ( 1)n ( 1 T i=1 n = (1 q i T ) ( 1)n i 1 ( n i) P (f, qt ) ( 1) n, i=1 n 1 Z(U f, T ) = (1 q i T ) ( 1)n i ( i+1) n P (f, T ) ( 1) n, P (f, T ) = and the β i s are algebraic numbers. d( ) 2 (1 β i T ) ) ( 1)n Definition The primitive Hodge polygon P HP ( ) is the polygon in R 2 with vertices (0, 0) and ( m k=0 h (k), m k=0 (k 1)h (k)), 1 m n. P HP ( ) 1 0 }{{}}{{} h (1) h (2) Primitive Hodge polygon of 18

19 P (f, T ) = det(i F rob q T PH n 1 c (U f F q, Q l )) = All results on L(x 0 f, T ) ( 1)n carry over to P (f, T ). Corollary If f is -regular over F q, then #U f (F q k) = (qk 1) n + ( 1) n+1 q k + ( 1) n+1 (β k 0 + β k β k d( ) 2 ), where β i q n p-adic Variation 3.1 p-adic Analytic formula for the Frobenius matrix Let f(λ, x) be the universal family of f M p ( ). For f(λ, x) M p ( )(F q ), P (f(λ, x), T ) = det(i F rob q T PH n 1 c ) = det(i F (λ)t ). Here F (λ is a matrix of size (d( ) 1) (d( ) 1). Is there any p-adic analytic formula for F (λ)? Since the relative cohomology forms a locally free overconvergent σ-module, one obtains Theorem 3.1. Zariski locally on M p ( ), there exists an overconvergent matrix A(λ) of size (d( ) 1) (d( ) 1) of the form A 00 (λ) A 00 (λ) pa 10 (λ) pa 11 (λ) A(λ) =.., p n 1 A n 1,0 (λ) p n 1 A n 1,1 (λ) where A ij (λ) has h (i + 1) rows and h (j + 1) columns, whose entries are overconvergent functions on the lifting of M p ( ) with norm 1, satisfies the following property: if f(λ, x) M p ( )(F p a) and λ = Teich(λ), then one can take F (λ) = A(λ pa 1 ) A(λ p )A(λ). That is, P (f(λ, x), T ) = det(i A(λ pa 1 ) A(λ p )A(λ)T ). 3.2 Deformation theory and Picard-Fuch equation Let p > 2. Since the relatively cohomology forms an overconvergent F - crystal whose underlying differential equation is the Picard-Fuch equation, we deduce that the matrix A(λ) as above can be expressed in terms of a fundamental solution matrix C(λ) of the Picard-Fuch equation: 19

20 A(λ) = C(λ p ) 1 A(λ 0 )C(λ), where λ 0 is a regular point. Remark: C(λ) is NOT analytic on the closed unit disk near λ 0. Example: Let π p 1 = p, λ pa = λ, θ(λ) = exp( πλ p ) exp(πλ) A(λ), Ψ Tr Fq k /F p (λ) = θ(λ pa 1 ) θ(λ p )θ(λ). If P (f(λ, x), T ) = d( ) 2 (1 α i (λ)t ) Z[T ] has h (k + 1) reciprocal roots with slope k, k = 0, 1,, n 1, then n 1 P (f(λ, x), T ) = P i (λ, T ), P i (λ, T ) Z p [T ], deg P i (λ, T ) =h (i + 1) i = slope of P i (λ, T ). Question: Any p-adic analytic formula for P i (λ, T )? 3.3 Hodge-Newton decomposition and unit root formula Let be ordinary at p > 2 and H p ( ) be the ordinary locus of M p ( ). One wishes to find a new basis such that the new matrix ( I00 E 01 (λ p ) ( ) ( ) ) I00 E A(λ) 01 (λ) B00 (λ) = 0 I 1 0 I 1 0 pa. (λ) This defines a p-adic contraction map and thus it has a unique solution matrix E 01 (λ) which is a convergent matrix of h (0) rows and n i=1 h (i) columns. By induction, one then shows that there exists a convergent matrix D(λ) on the lifting of H p ( ) such that B 00 (λ) D(λ p ) 1 A(λ)D(λ) = 0 pb 11 (λ) Then P (f(λ, x), T ) = n 1 ( ) det I p ai B ii (λ pa 1 ) B ii (λ p )B ii (λ)t, where f(λ, x) H p ( )(F p a), B ii (λ) is convergent (not overconvergent) on the closed unit disk, B ii (λ) = C ii (λ p ) 1 B ii (0)C ii (λ), 20

21 where C ii (λ) is a fundamental solution matrix of a piece of the Picard-Fuch equation. The p-adic analytic formula for P i (λ, T ) is then ( ) P i (λ, T ) = det I p ai B ii (λ pa 1 ) B ii (λ p )B ii (λ)t. 3.4 Unit root L-function and p-adic Galois representation Let be ordinary at p. As above, B 00 (λ) 0 pb 11 (λ) A(λ) p n 1 B n 1,n 1 (λ) Each B ii (λ) is invertible on the lifting of H p ( ) and hence it defined a unit root F-crystal on H p ( ). Alternatively, we have Theorem 3.2 (Katz). Each B ii defines a continuous p-adic representation ρ i : π arith 1 (H p ( )/F p ) GL h (i+1)(z p ), such that ρ i (Frob λ ) = B ii (λ pa 1 ) B ii (λ p )B ii (λ). It is clear that the L-function L(ρ i, T ) = λ Hp( ) closed point is analytic in T p < Dwork s unit root conjecture 1 det ( I T deg(λ) ρ i (Frob λ ) ) 1 + T (Q Z p)[[t ]] Theorem 3.3 (Wan). L(ρ i, T ) is p-adic meromorphic everywhere. Let ρ = ρ i. Write the p-adic Weierstrass factorization L(ρ i, T ) = j=1 (1 z(1) j T ) j=1 (1 z(2) j T ), z(1) j 0, z (2) j 0. Question: Let K p = Q p (z (1) j, z (2) j 1 j < ). Is [K p : Q p ] <? (p-adic RH for L(ρ i, T )). Definition 3.4. Let r + ρ = lim sup x log(1 + #{i ord q (z (1) i ) x} + #{j ord q (z (2) j ) x}) log x 21

22 This is called the order of the p-adic meromorphic function L(ρ, T ). It measures the size of L(ρ, T ). Clearly, we have 0 r ρ + +. Question: r + ρ < +? Theorem 3.5 (Wan). If rank(ρ) = 1 r + ρ < +. (true for the family x x n + 1 x 1 x n λ) Definition 3.6. Let r ρ = lim sup x log(1 + #{i ord q (z (1) i ) x} #{j ord q (z (2) j ) x} ) log x Clearly, 0 r ρ r + ρ +. Question: r ρ < +? (yes if rank(ρ) = 1) 3.6 p-adic Monodromy group Let be ordinary at p. Let ρ = ρ i : π arith 1 (H p ( )/F p ) GL h (i+1)(z p ). Then G p (, i) = ρ i (π arith 1 ) is a p-adic Lie-group. Question: G p (, i) =? Example (Igusa). For the elliptic family x 1 + x x 1 x 2 λ, one has G p (, 0) = G p (, 1) = Z p = GL 1 (Z p ). 22