A generalization of the Ross symbols in higher K-groups and hypergeometric functions II
aa r X i v : . [ m a t h . N T ] F e b A generalization of the Ross symbols in higher K -groups and hypergeometric functions II M. Asakura * Abstract
This is a sequel of the paper [As2] where we introduced higher Ross symbols inhigher K -groups of the hypergeometric schemes, and discussed the Beilinson regu-lators. In this paper we give its p -adic counterpart and an application to the p -adicBeilinson conjecture for K3 surfaces of Picard number . In [As2], we introduced a higher Ross symbol ξ Ross := (cid:26) − x − ν x , . . . , − x d − ν d x d (cid:27) in the Milnor K -group of the affine ring of a hypergeometric scheme U t : (1 − x n ) · · · (1 − x n d d ) = t (1.1)where ν k is a n k -th root of unity (cf. §3.1 and §5.1). This is a generalization of the Rosssymbol { − z, − w } in K of the Fermat curve z n + w n = 1 introduced in [R1], [R2](see [As2, 4.1] for the connection with our higher Ross symbols). In [As2] we discuss theBeilinson regulator map (cf. [S-Be]) reg B : K d +1 ( U t ) ( d +1) −→ H d +1 D ( U t , Q ( d + 1)) from Quillen’s higher K -group to the Deligne-Beilinson cohomology where K i ( − ) ( j ) ⊂ K i ( − ) ⊗ Q denotes the Adams weight piece. The main result [As2, Theorem 5.5] tells that reg( ξ Ross ) is a linear combination of complex analytic functions F a ( t ) := d X k =0 ( ψ ( a k ) + γ ) + log( t ) + a · · · a d t F a ( t ) (1.2) * Department of Mathematics, Faculty of Sciences, Hokkaido University, Sapporo 060-0810, JAPAN. [email protected] a = ( a , . . . , a d ) and F a ( t ) := d +3 F d +2 (cid:18) a + 1 , . . . , a d + 1 , , , . . . , t (cid:19) , is the hypergeometric function (see [As2, 5.1] for F a ( t ) , and [Sl] or [NIST, 15,16] for thegeneral theory of hypergeometric functions).The purpose of this paper is to provide the p -adic counterpart of [As2], namely we provesimilar theorems in p -adic situation by replacing the subjects as follows,Beilinson regulator syntomic regulator F a ( t ) F ( σ ) a ( t ) where F ( σ ) a ( t ) is a certain p -adic convergent function introduced in [As1]. Let us explainit more precisely. Let W be the Witt ring of a perfect field of characteristic p > , and K = Frac W the fractional field. For a smooth projective variety X over W , we denote by H • syn ( X, Z p ( j )) the syntomic cohomology of Fontaine-Messing (cf. [Ka, Chapter I]). Moregenerally, let U be a smooth W -scheme such that there is an embedding U ֒ → X into asmooth projective W -scheme X with Z = X \ U a simple relative normal crossing divisorover W . Then the log syntomic cohomology of ( X, Z ) is defined (cf. [Ts, §2.2]), which wedenote by H • syn ( U, Z p ( j )) . In their recent paper [N-N], Nekov´aˇr and Niziol established the syntomic regulator maps reg i,j syn : K i ( U ) ⊗ Q −→ H j − i syn ( U, Q p ( j )) := H j − i syn ( U, Z p ( j )) ⊗ Q from Quillen’s algebraic K -groups. These are the p -adic counterpart of the Beilinson regu-lator maps. Let us take U = U α the hypergeometric scheme (1.1) for α ∈ W and take thedegrees ( i, j ) = ( d + 1 , d + 1) , reg syn = reg d +1 ,d +1syn : K d +1 ( U α ) ( d +1) −→ H d +1syn ( U α , Q p ( d + 1)) ∼ = H d dR ( U α,K /K ) , (1.3)where U α,K := U α × W K . We then discuss the element reg syn ( ξ Ross ) in the de Rham co-homology. Although the author does not know whether the higher Ross symbol ξ Ross ∈ K d +1 ( U α ) lies in the image of K d +1 ( X α ) with X α ⊃ U α a smooth compactification (see[As2, 4.2] for more details), one can show that reg syn ( ξ Ross ) lies in W d H d dR ( U α,K /K ) (Lemma5.2), where W • H ∗ dR ( U α,K /K ) denotes the weight filtration by Deligne. The main theoremof this paper (=Theorem 5.5) describes reg syn ( ξ Ross ) by a linear combination of F ( σ ) a ( t ) introduced in [As1, §2], which is the p -adic counterpart of F a ( t ) (see §2.3 below for thereview on F ( σ ) a ( t ) ). The proof is based on the theory of F -isocrystals, especially the mainresult in [AM], and also uses the congruence relation for F ( σ ) a ( t ) proven in [As1, 3.2].Our main theorem has an application to the study of the p -adic Beilinson conjecture byPerrin-Riou. Conceptually saying, the conjecture asserts that the special values of the p -adic2 -functions are described by the syntomic regulators up to Q × . There are several previousworks by many people ([BK], [BD], [KLZ], [Ni], etc.). We also refer the recent paper [AC]where a number of numerical verifications for K of elliptic curves over Q are given apartfrom the method of the Beilinson-Kato elements. In this paper we shall discuss the p -adicBeilinson conjecture for singular K3 surfaces over Q (“singular” means the Picard number ). Let h ( X ) = h ( X ) / NS( X ) denote the transcendental motive (cf. [KMP, 7.2.2]).If X is a singular K3 surface over Q , it is 2-dimansional, and there is a Hecke eigenform f of weight with complex multiplication such that L ( h ( X ) , s ) = L ( f, s ) by a theoremof Livn´e [Li]. The p -adic Beilinson conjecture for K ( X ) (3) is formulated as the relationbetween the syntomic regulator and the special value of the p -adic L -function L p ( f, χ, s ) .See Conjecture 6.3 for the precise statement. Our hypergeometric schemes provide severalexplicit examples of singular K3 surfaces. The hypergeometric scheme (1 − x )(1 − x )(1 − x ) = α, α ∈ Q \ { , } is a K3 surface with the Picard number ≥ . This is isogenous to the K3 surface w = u u (1 + u )(1 + u )( u − αu ) , studied by Ahlgren, Ono and Penniston [AOP]. A complete list of α ’s such that the Picardnumber 20 (i.e. singular K3) is known. For such an example, one can employ the higherRoss symbol together with our main theorem (=Theorem 5.5), so that we have a formu-lation of the p -adic Beilinson conjecture in terms of our p -adic hypergeometric functions F ( σ ) a ( t ) ’s (Conjectures 6.8, 6.9). The advantage of our formulation is that it allows the nu-merical verifications, while we have not succeeded a theoretical proof. In §6.4, we constructother examples of singular K3 surfaces arising from the hypergeometric schemes, and give adescription of the p -adic regulators in terms of F ( σ ) a ( t ) (Theorem 6.15). p -adic Hypergeometric Functions Let K be a field of characteristic zero. Let ( α ) n = α ( α +1) · · · ( α + n − be the Pochhammersymbol. For a = ( a , . . . , a d ) ∈ K d +1 , the power series F a ( t ) = d +1 F d (cid:18) a , . . . , a d , . . . , t (cid:19) = ∞ X n =0 ( a ) n n ! · · · ( a d ) n n ! t n (2.1)is called the hypergeometric series . In case d = 1 , this is also referred to as the Gaussianhypergeometric series. The hypergeometric series is characterized as the unique solution(up to scalar) in K [[ t ]] of the hypergeometric differential operator (e.g. [Sl, 2.1.2], [NIST,16.8.3]), P HG ,a := D d +1 − t ( D + a ) · · · ( D + a d ) , D := t ddt . (2.2)3 .2 Dwork’s Hypergeometric Functions Let p be a prime. For a = ( a , . . . , a d ) ∈ Z d +1 p , the hypergeometric series F a ( t ) = ∞ X n =0 ( a ) n n ! · · · ( a d ) n n ! t n has coefficients in Z p . In his paper [Dw], Dwork discovered that a certain ratio of hypergeo-metric series is a uniform limit of a sequence of rational functions.For α ∈ Z p , let α ′ denote the Dwork prime, which is defined to be ( α + k ) /p where k ∈ { , , . . . , p − } such that α + k ≡ mod p . Define the i -th Dwork prime by α ( i ) =( α ( i − ) ′ and α (0) := α . Write a ′ = ( a ′ , . . . , a ′ s ) and a ( i ) = ( a ( i )1 , . . . , a ( i ) s ) . Dwork’s p -adichypergeometric function is defined to be a power series F Dw a ( t ) := F a ( t ) /F a ′ ( t p ) ∈ Z p [[ t ]] . Let W = W ( F p ) be the Witt ring of F p . Let c ∈ pW and σ a p -th Frobenius on W [[ t ]] given by σ c ( t ) = ct p , c ∈ pW . A slight modification of F Dw a ( t ) is ✞✝ ☎✆ Def. F Dw ,σ c a ( t ) := F a ( t ) /F a ′ ( t σ c ) ∈ W [[ t ]] . Dwork discovered certain congruence relations which we refer to as the
Dwork congru-ence . The precise statement is as follows. For a power series f ( t ) = P i ≥ a i t i , we denote [ f ( t )] For any n ≥ , we have F Dw ,σ c a ( t ) ≡ [ F a ( t )] ,n → z X ≤ k Let A be a commutative ring. Let d ≥ and n i ≥ i = 0 , , . . . , d ) be integers such that n · · · n d is invertible in A . We call an affine scheme U = Spec A [ x , . . . , x d ] / ((1 − x n ) · · · (1 − x n d d ) − t ) , t ∈ A (3.1)the hypergeometric scheme over A ([As2, §2.1]). It is easy to see that U is smooth over A if t (1 − t ) ∈ A × . Proposition 3.1 Assume that t (1 − t ) ∈ A × and A is an integral domain. Then there is asmooth compactification X ⊃ U such that Z = X \ U is a relative simple NCD over A .Proof. [As2, Proposition 2.1]. (cid:3) A is an integral domain and t (1 − t ) ∈ A × . Let µ n = µ n ( A ) denotethe group of n -th roots of unity in A . A finite abelian group G = µ n × · · · × µ n d acts on U in a way that ( x , . . . , x d ) ( ν x , . . . , ν d x d ) for ν = ( ν , . . . , ν d ) ∈ G . In this way the deRham cohomology groups H • dR ( U/A ) are endowed with the structure of A [ G ] -modules. Fora A [ G ] -module H and ( i , . . . , i d ) ∈ Z d +1 we write H ( i , . . . , i d ) = { x ∈ H | νx = ν i · · · ν i d d x, ∀ ν ∈ G } . (3.2)the simultaneous eigenspace.Let A = K [ t, ( t − t ) − ] with K a field of characteristic zero. Suppose that K containsprimitive n i -th roots of unity for all i . We denote by W • H i dR ( U/A ) the weight filtration. Inparticular W i H i dR ( U/A ) = Im[ H i dR ( X/A ) → H i dR ( U/A )] . Put I := { ( i , . . . , i d ) ∈ Z d +1 | ≤ i k < n k } and I + := { ( i , . . . , i d ) ∈ Z d +1 | < i k < n k } . There is the decomposition H • dR ( U/A ) = M ( i ,...,i d ) ∈ I H • dR ( U/A )( i , . . . , i d ) of the de Rham cohomology group. Theorem 3.2 ([As2, §3.2 Summary (1), (2)]) (1) The A -submodule M ( i ,...,i d ) ∈ I \ I + H • dR ( U/A )( i , . . . , i d ) is generated by exterior products of { dx k / ( x k − ν k ) | ν k ∈ µ n k } . (2) If i < d , then M ( i ,...,i d ) ∈ I + H i dR ( U/A )( i , . . . , i d ) = 0 . In particular W i H i dR ( U/A ) = 0 for < i < d by this and (1) . (3) M ( i ,...,i d ) ∈ I + H d dR ( U/A )( i , . . . , i d ) = W d H d dR ( U/A ) . For each ( i , . . . , i d ) ∈ I + , we put, cf. [As2, §2.4] ✞✝ ☎✆ Def. ω i ...i d := n − x i − n x i − · · · x i d − d dx · · · dx d (1 − x n ) · · · (1 − x n d d ) (3.3)a regular d -form in Γ ( X, Ω dX/A )( i , . . . , i d ) . Theorem 3.3 ([As2, Corollary 3.6]) Let D := K h t, ( t − t ) − , ddt i be the Wyle algebra of Spec A , which acts on H d dR ( U/A ) . Let ( i , . . . , i d ) ∈ I + and put a k := 1 − i k /n k . Let P HG ,a be the hypergeometric differential operator (2.2) . Then P HG ,a ( ω i ...i d ) = 0 and thehomomorphism D / D P HG ,a ∼ = −→ H d dR ( U/A )( i , . . . , i d ) , P P ( ω i ...i d ) of D -modules is bijective. In particular, this is an irreducible D -module. 6n view of Theorem 3.3, we think the piece H d dR ( U/A )( i , . . . , i d ) of being an hypergeomet-ric motive associated to the hypergeometric function (2.1). Convention ✓ ✏ For an integer n > , let Z ( n ) = S − Z denote the ring of fractions with respect to themultiplicative set S = { s ∈ Z | gcd( s, n ) = 1 } . For j ∈ Z ( n ) , let [ j ] n denote the uniqueinteger such that ≤ [ j ] n < n and [ j ] n ≡ j mod n Z ( n ) . We then extend the notation asfollows, H ( i , . . . , i d ) := H ([ i ] n , . . . , [ i d ] n d ) , ( i , . . . , i d ) ∈ d Y k =0 Z ( n k ) ,ω i ...i d := ω [ i ] n ... [ i d ] nd , ( i , . . . , i d ) ∈ d Y k =0 ( Z ( n k ) \ n k Z ( n k ) ) . ✒ ✑ Q on W d H d dR ( U/A ) Let K be a field of characteristic zero, and A = K [ t, ( t − t ) − ] . Let U be the hypergeometricscheme (3.1) over A (we do not assume that K contains primitive n i -th roots of unity). Itis a general fact that there is a ( − d -symmetric pairing on W d H d dR ( U/A ) induced fromthe polarization pairing on H d dR ( X/A ) (e.g. [PS, Corollary 2.12]). However we need moreexplicit description of it for the later use. Let H • dR ,c ( U/A ) denote the de Rham cohomologywith compact support. Lemma 3.4 The composition H d dR ,c ( U/A ) → H d dR ( X/A ) → W d H d dR ( U/A ) is surjective.Proof. We may replace K with K . Then G = µ n × · · · × µ n d acts on H d dR ,c ( U/A ) → W d H d dR ( U/A ) , and then the assertion is equivalent to claiming the surjectivity of the naturalmap u i ...i d : H d dR ,c ( U/A )( i , . . . , i d ) −→ W d H d dR ( U/A )( i , . . . , i d ) for each ( i , . . . , i d ) with < i k < n k . The map u i ...u d is a homomorphism of D -modules,and the right hand side is irreducible by Theorem 3.3. Therefore it is enough to show u i ...i d = 0 . Recall a smooth compactification X ⊃ U and the boundary Z = X \ U .There is an exact sequence H d − ( Z/A ) −→ H d dR ,c ( U/A ) −→ H d dR ( X/A ) −→ H d dR ( Z/A ) . (3.4)By [As2, Proposition 2.2], Z is a Tate motive and hence Gr pF Gr Wd H d dR ( Z/A ) = 0 unless p = d/ where F • denotes the Hodge filtration. In particular, F d H d dR ,c ( U/A ) → F d H d dR ( X/A ) is onto, and hence so is F d H d dR ,c ( U/A ) → F d W d H d dR ( U/A ) . Taking the eigen piece, onehas that F d H d dR ,c ( U/A )( i , . . . , i d ) → F d W d H d dR ( U/A )( i , . . . , i d ) ∼ = A is surjective, whichimplies u i ...i d = 0 as required. (cid:3) 7e now have the surjectivity of Gr Wd H d dR ,c ( U/A ) = H d dR ,c ( U/A ) /W d − −→ W d H d dR ( U/A ) . (3.5)There is the cup-product H d dR ,c ( U/A ) ⊗ A H d dR ( U/A ) −→ H d dR ,c ( U/A ) ∼ = → H d dR ( X/A ) ∼ = A, which gives rise to a pairing Q c : Gr Wd H d dR ,c ( U/A ) ⊗ A Gr Wd H d dR ,c ( U/A ) −→ A. This is compatible with the cup-product pairing on H d dR ( X/A ) ⊗ under the inclusion Gr Wd H d dR ,c ( U/A ) ֒ → H d dR ( X/A ) where the injectivity follows from the exact sequence (3.4). Lemma 3.5 The pairing Q c is non-degenerate. Moreover let T be the kernel of (3.5) . Then Q c restricted on T ⊗ A T is also non-degenerate.Proof. We may assume that the base field K is C . Let α ∈ C \ { , } be arbitrary. Let U α ⊂ X α be the complex manifolds of the fibers at t = α . Then it is enough to show that Q c : Gr Wd H dc ( U α , Q ) ⊗ Gr Wd H dc ( U α , Q ) −→ Q gives the polarization, namely h x, y i = (2 πi ) d Q c ( Cx, y ) is positive definite where C is theWeil operator (cf. [PS, 2.1.2]). Let [ ω ] ∈ H ( X, Q ) be a cohomology class such that [ ω ] | X α is ample. Let L : H • ( X α ) → H • +2 ( X α ) be the Lefschetz operator given by x x ∪ [ ω ] | X α .Let H d ( X α ) prim be the primitive part, namely the kernel of L : H d ( X α , Q ) → H d +2 ( X α , Q ) .Then it is enough to show Gr Wd H dc ( U α , Q ) ⊂ H d ( X α , Q ) prim . (3.6)The composition H dc ( U α ) → H d ( X α ) L → H d +2 ( X α ) agrees with the composition H dc ( U α ) L | Uα → H d +2 c ( U α ) → H d +2 ( X α ) where L | U α is the cup-product with [ ω ] | U α ∈ W H ( U α , Q ) . Itfollows from [As2, Theorems 3.7, 3.8] that H ( U α , Q ) ∼ = Q ( − ⊕ unless d = 2 . There-fore W H ( U α , Q ) = 0 if d = 2 . If d = 2 , each piece W H ( U α , C )( i , . . . , i d ) is a -dimensional irreducible π ( P \ { , , ∞} , α ) -module. Since [ ω ] | U α ∈ W H ( U α , C ) liesin the invariant part of π ( P \ { , , ∞} , α ) , we have [ ω ] | U α = 0 . In each case, we have L | U α = 0 , and hence the composition H dc ( U α ) → H d ( X α ) L → H d +2 ( X α ) is zero. Thismeans that (3.6) holds. (cid:3) Let T ⊥ ⊂ Gr Wd H d dR ,c ( U/A ) be the orthogonal complement with respect to Q c . It fol-lows from Lemma 3.5 and the surjectivity of (3.5) that one has the isomorphism T ⊥ ∼ = → W d H d dR ( U/A ) . Under this identification, Q c induces a pairing Q : W d H d dR ( U/A ) ⊗ A W d H d dR ( U/A ) −→ A. (3.7)By the construction, the following holds. 8 Q1) The pairing Q is ( − d -symmetric and non-degenerate, (Q2) Suppose that K contains primitive n i -th roots of unity for all i . Then Q ( σx, σy ) = Q ( x, y ) for σ ∈ G . (Q3) Q ( θx, y ) + Q ( x, θy ) = θ ( Q ( x, y )) for θ = ddt ∈ D . (Q4) Q ( F p , F q ) = 0 if p + q > d and Q ( F p , F q ) = A if p + q = d and p, q ≥ , where F • is the Hodge filtration.The property (Q2) implies that Q induces a perfect pairing Q : W d H d dR ( U/A )( i , . . . , i d ) ⊗ A W d H d dR ( U/A )( − i , . . . , − i d ) −→ A. (3.8) Let n , . . . , n d > be integers and p a prime such that p ∤ n · · · n d . Let W = W ( F p ) be the Witt ring, and F the p -th Frobenius. Put K := Frac( W ) the fractional field. Let A := W [ t, ( t − t ) − ] and U = Spec A [ x , . . . , x d ] / ((1 − x n ) · · · (1 − x n d d ) − t ) the hypergeometric scheme over A . We put A F := F [ t, ( t − t ) − ] , A K := K [ t, ( t − t ) − ] ,and U F := U × A A F , U K := U × A A K . Let D := K h t, ( t − t ) − , ddt i be the Wyle algebra. Put D := t ddt . Let ( i , . . . , i d ) ∈ Q dk =0 Z ( n k ) satisfy [ i k ] n k = 0 for all k (see Convention in §3.1 for the notation). Recall fromTheorems 3.2 and 3.3 the eigenspace H d dR ( U K /A K )( i , . . . , i d ) = W d H d dR ( U K /A K )( i , . . . , i d ) = d X k =0 A K D k ω i ...i d , which is stable under the action of D . We write H i ...i d ( U K /A K ) = H d dR ( U K /A K )( i , . . . , i d ) for simplicity of notation. Proposition 4.1 Let H i ...i d ( U K /A K ) K (( t )) := K (( t )) ⊗ A K H i ...i d ( U K /A K ) on which theaction of D extends in a natural way. Put a k := 1 − [ i k ] n k /n k . Let s k ∈ Q be definedby ( t + a ) · · · ( t + a d ) = t d +1 + s t d + · · · + s d +1 , and put q d − m := − s m +1 t/ (1 − t ) for m = 0 , , . . . , d . Put ˇ a := (1 − a , . . . , − a d ) and y d := (1 − t ) F ˇ a ( t ) = (1 − t ) d +1 F d (cid:18) − a , . . . , − a d , . . . , t (cid:19) . or ≤ i < d define y i by y i + Dy i +1 = q i +1 y d . Put b η i ...i d := y ω i ...i d + y Dω i ...i d + · · · + y d D d ω i ...i d ∈ H i ...i d ( U K /A K ) K (( t )) . (4.1) Then Ker[ D : H i ...i d ( U K /A K ) K (( t )) → H i ...i d ( U K /A K ) K (( t )) ] = K b η i ...i d . (4.2) Proof. Recall Theorem 3.3. H i ...i d ( U K /A K ) K (( t )) is a free K (( t )) -module with basis { D k ω i ...i d | k = 0 , , . . . , d } and the differential operator P HG ,a = D d +1 − t ( D + a ) · · · ( D + a d ) =(1 − t )( D d +1 + q d D d + · · · + q ) annihilates ω i ...i d . Therefore D d X k =0 z k D k ω i ...i d ! = d X k =0 D ( z k ) D k ω i ...i d + z k D k +1 ω i ...i d = d X k =1 ( z k − + D ( z k )) D k ω i ...i d + D ( z ) ω i ...i d + z d D d +1 ω i ...i d = d X k =1 ( z k − + D ( z k ) − q k z d ) D k ω i ...i d + ( D ( z ) − q d z d ) ω i ...i d vanishes if and only if z i + D ( z i +1 ) = q i +1 z d (0 ≤ i ≤ d − , D ( z ) = q d z d . (4.3)Put a differential operator P := D d +1 − D d ⋆ q d + · · · + ( − d D ⋆ q + ( − d +1 q where ⋆ denotes the composition of operators to make distinctions between D ⋆ f ∈ D and D ( f ) ∈ K (( t )) . Then (4.3) is equivalent to z i + D ( z i +1 ) = q i +1 z d (0 ≤ i ≤ d − , P ( z d ) = 0 , so that the assertion is reduced to show that y d = (1 − t ) F ˇ a ( t ) is the unique solution (up toscalar) in K (( t )) of the differential equation P ( y ) = 0 . One has P ⋆ (1 − t ) = D d +1 ⋆ (1 − t ) − d X m =0 ( − m X i < ···
10e define a unit root vector ✞✝ ☎✆ Def. η i ...i d := F ˇ a ( t ) − b η i ...i d = y F ˇ a ( t ) ω i ...i d + y F ˇ a ( t ) Dω i ...i d + · · · + (1 − t ) D d ω i ...i d . (4.4) Lemma 4.2 Let h a ( t ) be the polynomial as in (2.3) . Then we have y i F ˇ a ( t ) ∈ (cid:18) Z p [ t, h ˇ a ( t ) − ] ∧ (cid:19) [(1 − t ) − ] , i = 0 , , . . . , d where ( − ) ∧ denotes the p -adic completion, and hence η i ...i d ∈ A [ h ˇ a ( t ) − ] ∧ ⊗ A H i ...i d ( U K /A K ) . Proof. By the construction of y i , they are linear combinations of D j ( y d ) over a ring Z ( p ) [ t, (1 − t ) − ] and hence one can write y i = X j g j d j F ˇ a ( t ) dt j by some g j ∈ Z ( p ) [ t, ( t − t ) − ] . Now the assertion follows from (2.5). (cid:3) Let h ( t ) := Q a h a ( t ) where a = ( i /n , . . . , i d /n d ) runs over all ( d + 1) -tuple of integers ( i , . . . , i d ) such that < i k < n k . Put B := A [ h ( t ) − ] , B K := K ⊗ W B , and b B := A [ h ( t ) − ] ∧ = lim ←− n (cid:18) W/p n W [ t, ( t − t ) − , h ( t ) − ] (cid:19) (4.5)the p -adic completion, and b B K := K ⊗ W b B . Thanks to Lemma 4.2, the unit root vector η i ...i d belongs to H i ...i d ( U K /A K ) b B K := b B K ⊗ A K H i ...i d ( U K /A K ) . We call the direct summand ✞✝ ☎✆ Def. H unit i ...i d ( U K /A K ) b B K := B K η i ...i d of H i ...i d ( U K /A K ) b B K the unit root subspace . Recall from §3.2 the perfect pairing (3.8) Q : H i ...i d ( U K /A K ) ⊗ A K H − i ,..., − i d ( U K /A K ) −→ A K . (4.6)Tensoring with b B K , we have the pairing on the b B K -modules, which we also write by Q .Define a b B K -submodule ✞✝ ☎✆ Def. V H i ...i d ( U K /A K ) b B K ⊂ H i ...i d ( U K /A K ) b B K (4.7)to be the exact annihilator of the unit root part H unit − i ,..., − i d ( U K /A K ) b B K . By definition, thepairing H i ...i d ( U K /A K ) b B K /V H i ...i d ( U K /A K ) b B K ⊗ H unit − i ,..., − i d ( U K /A K ) b B K −→ b B K (4.8)is perfect. We shall later see η i ...i d ∈ V H i ...i d ( U K /A K ) b B K (Corollary 4.7). Let W (( t )) ∧ bethe p -adic completion and K (( t )) ∧ := K ⊗ W W (( t )) ∧ . We write V H i ...i d ( U K /A K ) K (( t )) ∧ := K (( t )) ∧ ⊗ A K V H i ...i d ( U K /A K ) . Note that b B K ⊂ K (( t )) ∧ .11 emma 4.3 (1) V H i ...i d ( U K /A K ) b B K is stable under the action of D . (2) H i ...i d ( U K /A K ) b B K /V H i ...i d ( U K /A K ) b B K ∼ = b B K is generated by ω i ...i d . (3) Q ( ω i ...i d , η − i ,..., − i d ) = C ∈ Q × .Proof. (1) is immediate from the fact that D b η i ...i d = 0 (Proposition 4.1). To see (2), it isenough to show Q ( ω i ...i d , η − i ,..., − i d ) = 0 . Write ω = ω i ...i d and ˇ ω = ω − i ,..., − i d . Recallfrom [As2, Corollary 3.10] the fact that Gr iF H i ...i d ( U K /A K ) is a free A K -module with basis D d − i ω . Therefore, thanks to the property (Q4) in §3.2, we have Q ( D i ω, D j ˇ ω ) = 0 (4.9)for any ( i, j ) such that i + j < d and i, j ≥ . In particular Q ( ω i ...i d , η − i ,..., − i d ) = (1 − t ) Q ( ω, D d ˇ ω ) . (4.10)If Q ( ω i ...i d , η − i ,..., − i d ) = 0 , then Q ( ω, D i ˇ ω ) vanishes for all i ∈ Z ≥ , which contradicts withthe fact that Q is a perfect pairing. Hence Q ( ω i ...i d , η − i ,..., − i d ) = 0 . We show (3). Since DQ ( D i ω, D j ˇ ω ) = Q ( D i +1 ω, D j ˇ ω ) + Q ( D i ω, D j +1 ˇ ω ) for i + j = d − and i, j ≥ by (4.9), one has ( − i Q ( ω, D d ˇ ω ) = Q ( D i ω, D d − i ˇ ω ) , i = 0 , , . . . , d. (4.11)Applying D on the both sides, one has ( − i DQ ( ω, D d ˇ ω ) = Q ( D i +1 ω, D d − i ˇ ω ) + Q ( D i ω, D d − i +1 ˇ ω ) . Taking the alternating sum of the both sides, one has ( d + 1) DQ ( ω, D d ˇ ω ) = ( − d Q ( D d +1 ω, ˇ ω ) + Q ( ω, D d +1 ˇ ω ) . (4.12)Using P HG ,a ω = 0 and P HG , ˇ a ˇ ω = 0 together with (4.9), one has Q ( D d +1 ω, ˇ ω ) = ( a + · · · + a d ) t − t Q ( D d ω, ˇ ω )= ( − d ( a + · · · + a d ) t − t Q ( ω, D d ˇ ω ) (by (4.11)) ,Q ( ω, D d +1 ˇ ω ) = ( d + 1 − ( a + · · · + a d )) t − t Q ( ω, D d ˇ ω ) . Apply the above to (4.12), then ( d +1) DQ ( ω, D d ˇ ω ) = ( d +1) t − t Q ( ω, D d ˇ ω ) ⇐⇒ ddt Q ( ω, D d ˇ ω ) = 11 − t Q ( ω, D d ˇ ω ) . This implies Q ( ω, D d ˇ ω ) = C (1 − t ) − with some C ∈ K . Hence Q ( ω i ...i d , η − i ,..., − i d ) = C by (4.10), and this is not zero by (2). Since the pairing Q and the elements ω, ˇ ω are definedover a ring Q ( t ) , it turns out that C ∈ K × ∩ Q ( t ) = Q × . This completes the proof of (3). (cid:3) .2 Rigid cohomology of Hypergeometric schemes Let F be the p -th Frobenius on W . Let A † be the weak completion of A , and put A † K := K ⊗ W A † . For c ∈ pW , let σ c be a F -linear p -th Frobenius on A † K given by σ c ( t ) = ct p .Let H • rig ( U F /A F ) denote the rigid cohomology, endowed with the p -th Frobenius Φ compatible with σ c . Werefer the book [LS] for general theory of rigid cohomology. There is the comparison isomor-phism H • rig ( U F /A F ) ∼ = H • dR ( U K /A K ) ⊗ A K A † K , and the action of G is compatible. Tensoring (4.6) with A † K , we have a pairing H i ...i d ( U K /A K ) A † K ⊗ A † K H − i ... − i d ( U K /A K ) A † K −→ A † K ⊗ A H dc, dR ( U K /A K ) ∼ = A † K which we also write by Q . Since the cup-product is compatible with the Frobenius, we have (Q5) Q (Φ( x ) , Φ( y )) = p d σ c ( Q ( x, y )) .The Frobenius σ c extends on b B K and K (( t )) ∧ in a natural way. According to this, we extend Φ to that on H i ...i d ( U K /A K ) b B K and H i ...i d ( U K /A K ) K (( t )) ∧ = K (( t )) ∧ ⊗ A H i ...i d ( U K /A K ) ,and denote by the same notation. Lemma 4.4 Let ( i , . . . , i d ) ∈ Q dk =0 Z ( n k ) satisfy i k mod n k for all k . Write H i ...i d = H i ...i d ( U K /A K ) R ⊃ V H i ...i d = V H i ...i d ( U K /A K ) R with R = b B K or K (( t )) ∧ (see Con-vention in §3.1 and (4.7) for the notation). Then (i) Φ( H p − i ...p − i d ) ⊂ H i ...i d . (ii) Φ( H unit p − i ...p − i d ) ⊂ H unit i ...i d . (iii) Φ( V H p − i ...p − i d ) ⊂ V H i ...i d .Proof. (i) follows from the fact that g Φ = Φ g for g ∈ G (see §3.1 for the group G ). Since Ker D in (4.2) is stable under the action of Φ , we have Φ( b η p − i ...p − i d ) = C b η i ...i d (4.13)for some C ∈ K × ( C is not zero by (Q5) on noticing that Q is a perfect pairing). Thisimplies Φ( η p − i ...p − i d ) = C F ˇ a ( t ) F ˇ a ′ ( t σ c ) η i ...i d = C F Dw ,σ c ˇ a ( t ) η i ...i d (4.14)where ˇ a = (1 − a , . . . , − a d ) and ˇ a ′ = (1 − a ′ , . . . , − a ′ d ) denotes the Dwork prime.Hence (ii) follows, and (iii) is immediate from (ii) and the definition of V H i ...i d . (cid:3) heorem 4.5 Let ( i , . . . , i d ) satisfy i k mod n k for all k . Then Φ( ω p − i ...p − i d ) ≡ p d F Dw ,σ c a ( t ) − ω i ...i d mod V H i ...i d ( U K /A K ) b B K Proof. Put b ω i ...i d := F a ( t ) − ω i ...i d . By Lemma 4.3 (3), Q ( b ω i ...i d , b η n − i ,...,n d − i d ) is a non-zero constant. By (4.13) together with (Q5) , we have Φ( b ω p − i ...p − i d ) ≡ p d C ′ b ω i ...i d ⇔ Φ( ω p − i ...p − i d ) ≡ p d C ′ F Dw ,σ c a ( t ) − ω i ...i d mod V H i ...i d with some C ′ ∈ K × . We show C ′ = 1 . For ν = ( ν , . . . , ν d ) ∈ G , let P ν denote thesubscheme of Y defined by { x − ν = · · · = x d − ν d = 0 } . R : H d log-crys (( Y , D ) / ( W [[ t ]] , −→ H ( Z/W ) = M ν ∈ G H ( P/W ) and R ◦ Φ ( Y , D ) = p d Φ Z ◦ R . Let j k be the unique integer such that j k ≡ p − i k mod n k and < j k < n k . R ( ω p − i ...p − i d ) = Res (cid:18) n − x j − n x j − · · · x j d − d dx · · · dx d (1 − x n ) · · · (1 − x n d d ) dtt (cid:19) = ( − d n · · · n d X ν ∈ G ν j · · · ν j d d P ν . Therefore R ◦ Φ ( Y , D ) ( ω p − i ...p − i d ) = p d Φ Z ( R ( ω p − i ...p − i d )) = p d ( − d n · · · n d X ν ∈ G ν pj · · · ν pj d d P ν = p d ( − d n · · · n d X ν ∈ G ν i · · · ν i d d P ν . On the other hand, the first term is R ( p d C ′ F Dw ,σ c a ( t ) − ω i ...i d ) = p d C ′ R ( ω i ...i d ) = p d C ′ ( − d n · · · n d X ν ∈ G ν i · · · ν i d d P ν , which concludes C ′ = 1 as required. (cid:3) Corollary 4.6 (Unit Root Formula) Φ( η p − i ...p − i d ) = F Dw ,σ c ˇ a ( t ) η i ...i d . roof. Recall (4.14), Φ( η p − i ...p − i d ) = C F Dw ,σ c ˇ a ( t ) η i ...i d . We want to show C = 1 . By (Q5) , we have Q (Φ( ω − p − i ,..., − p − i d ) , Φ( η p − i ...p − i d )) = p d σ c Q ( ω − i ,..., − i d , η i ...i d ) = p d Q ( ω − i ,..., − i d , η i ...i d ) , where the second equality follows from the fact Q ( ω − i ,..., − i d , η i ...i d ) ∈ Q × (Lemma 4.3 (3)).Applying Theorem 4.5 and (4.14) to this, we conclude C = 1 . (cid:3) Corollary 4.7 Q ( η − i ,..., − i d , η i ...i d ) = 0 . In other words, η i ...i d ∈ V H i ...i d ( U K /A K ) b B K .Proof. Applying Corollary 4.6 to the equality Q (Φ( η − p − i ,..., − p − i d ) , Φ( η p − i ...p − i d )) = p d σ c Q ( η − i ,..., − i d , η i ...i d ) in (Q5) , one has F Dw ,σ c a ( t ) F Dw ,σ c ˇ a ( t ) Q ( η − i ,..., − i d , η i ...i d ) = p d σ c Q ( η − i ,..., − i d , η i ...i d ) . Comparing the sup norm on the both side, it turns out that Q ( η − i ,..., − i d , η i ...i d ) = 0 . (cid:3) Remark 4.8 There is an alternative proof of Corollary 4.7 with use of the Hodge theory(monodromy weight filtration). In this section we shall discuss the syntomic regulator of the higher Ross symbols. The mainresults are Theorems 5.3 and 5.5, which are the p -adic counterparts of [As2, Theorem 5.5],and also a generalization of the results in [As1, §4.4] in higher dimension. Let n , . . . , n d > be integers and p a prime such that p ∤ n · · · n d . Let W = W ( F p ) be theWitt ring, and F the p -th Frobenius. Let A = W [ t, ( t − t ) − ] , S = Spec A and U = Spec A [ x , . . . , x d ] / ((1 − x n ) · · · (1 − x n d d ) − t ) the hypergeometric scheme over A (cf. §3.1). Then, for ν k ∈ µ n k ( A ) ( k = 0 , , . . . , d ), the higher Ross symbol is defined to be a Milnor symbol ξ Ross := (cid:26) − x − ν x , . . . , − x d − ν d x d (cid:27) ∈ K Md +1 ( O ( U )) , (5.1)in the Milnor K -group of O ( U ) . We also think it of being an element of Quillen’s higher K -group K d +1 ( U ) by the natural map K Mi ( O ( U )) → K i ( U ) , and denote by the same notation.Let X ⊃ U be a smooth compactification X ⊃ U such that Z = X \ U is a relative simpleNCD over A ([As2, Proposition 2.1]). Then we expect ξ Ross ∈ Im[ K d +1 ( X ) ( d +1) → K d +1 ( U ) ( d +1) ] . (5.2)See [As2, 4.2] for more details. This is true if d ≤ ([As2, Corollary 4.4]).15 .2 Category of filtered F -isocrystals [AM] In §5.3, we shall employ the category of filtered F -isocrystals introduced in [AM, §2.1] as afundamental material. We here recall the notation and some results which we shall need inbelow. For a moment, we work over an arbitrary smooth affine variety S = Spec( B ) over W = W ( F p ) . We denote by B † the weak completion of B . Namely if B = W [ T , · · · , T n ] /I ,then B † = W [ T , · · · , T n ] † /I where W [ T , · · · , T n ] † is the ring of power series P a α T α such that for some r > , | a α | r | α | → as | α | → ∞ . Let K := Frac( W ) be the fractionalfield, and write B † K = K ⊗ W B † .Let σ : B † → B † be a p -th Frobenius compatible with the Frobenius F on W . We definethe category Fil - F - MIC( S, σ ) (which we call the category of filtered F -isocrystals on S ) asfollows. The induced endomorphism σ ⊗ Z Q : B † K → B † K is also denoted by σ . An object of Fil - F - MIC( S, σ ) is a datum H = ( H dR , H rig , c, Φ , ∇ , Fil • ) , where• H dR is a coherent B K -module,• H rig is a coherent B † K -module,• c : H dR ⊗ B K B † K ∼ = −−→ H rig is a B † K -linear isomorphism,• Φ : σ ∗ H rig ∼ = −−→ H rig is an isomorphism of B † K -algebra,• ∇ : H dR → Ω B K ⊗ H dR is an integrable connection and• Fil • is a finite descending filtration on H dR of locally free B K -module (i.e. each gradedpiece is locally free),that satisfies ∇ (Fil i ) ⊂ Ω B K ⊗ Fil i − and the compatibility of Φ and ∇ , namely Φ ∇ rig = ∇ rig Φ where ∇ rig : H rig → Ω B † K ⊗ H rig is the connection induced from ∇ under the compar-ison map c . In what follows we write ∇ rig = ∇ for simplicity of notation.The category Fil - F - MIC( S, σ ) is an exact category ( not an abelian category) in whichthe tensor product ⊗ is defined in the customary way. There are the Tate objects O S ( n ) =( B K , B † K , c, p − n σ B , d, Fil • ) , which is the counterpart of the l -adic sheaf Q l ( n ) . We abbrevi-ate O S = O S (0) . We write H ( n ) := H ⊗ O S ( n ) for an object H ∈ Fil - F - MIC( S, σ ) .Let u : U → S = Spec( B ) be a smooth morphism of smooth W -schemes of pure relativedimension. Assume that there exists an open immersion U ֒ → X and a projective smoothmorphism u X : X → S that extends u . We also assume that D := X \ U is a relative normalcrossing divisor with smooth components over S . Let u : X → S be a projective completionof the composite morphism X u X −→ S ֒ → S : U u (cid:15) (cid:15) / / X u X (cid:15) (cid:15) / / X u (cid:15) (cid:15) S S / / S. Then one can construct an object H i ( U/S ) = ( H i dR ( U K /S K ) , H i rig ( U F p /S F p ) , c, ∇ , Φ , Fil • ) Fil - F - MIC( S, σ ) ([AM, 2.5]), where ∇ is the Gauss-Manin connection, Φ is the σ -linear p -th Frobenius and Fil • H i dR ( U K /S K ) is the Hodge filtration. Proposition 5.1 Let n ≥ be an integer. Suppose Fil n +1 H n +1dR ( U K /S K ) = 0 . Then there isthe symbol map K Mn +1 ( O ( U )) −→ Ext - F - MIC( S,σ ) ( O S , H n ( U/S )( n + 1)) (5.3) to the group of 1-extensions in Fil - F - MIC( S, σ ) .Proof. [AM, Proposition 2.12]. (cid:3) We turn to the setting in §5.1. Let A = W [ t, ( t − t ) − ] and U → S = Spec A be thehypergeometric scheme. Let σ c be the p -th Frobenius on W [[ t ]] given by σ c ( t ) = ct p with c ∈ pW . By virtue of the symbol map (5.3), the higher Ross symbol ξ Ross defines a1-extension −→ H d ( U/S )( d + 1) −→ M ξ Ross ( U/S ) −→ O S −→ (5.4)in Fil - F - MIC( S, σ c ) . Let Φ be the Frobenius on M ξ Ross ( U F p /S F p ) rig ∼ = M ξ Ross ( U K /S K ) dR ⊗ A K A † K . Let Φ U/S be the Frobenius on H d ( U F p /S F p ) rig (without Tate twist). By the definition ofthe Tate twist, Φ | H d ( U F p /S F p ) rig = p − d − Φ U/S . Let e ξ Ross ∈ Fil M ξ ( U/S ) dR be the unique lifting of ∈ O ( S ) . Then one has a class e ξ Ross − Φ( e ξ Ross ) ∈ H d dR ( U/S ) ⊗ A A † K . Lemma 5.2 e ξ Ross − Φ( e ξ Ross ) ∈ W d H d dR ( U/S ) ⊗ A A † K .Proof. Let σ i ( ε i ) = (1 , . . . , ε i , . . . , ∈ G = µ n × · · · × µ n d for a fixed primitive n i -th rootof unity ε i ∈ µ n i . Put Q := d Y i =0 (1 + σ i ( ε i ) + · · · + σ i ( ε i ) n i − ) ∈ Q [ G ] . Then it follows from Theorem 3.2 (3) that one has W d H d dR ( U/S ) = Ker( Q : H d dR ( U/S ) → H d dR ( U/S )) . Now the assertion follows from the fact that Qξ Ross = 0 in K Md +1 ( O ( U )) ⊗ Q . (cid:3) heorem 5.3 With the notation as above, we have e ξ Ross − Φ( e ξ Ross ) ≡ n − X i =1 · · · n d − X i d =1 (1 − ν i ) · · · (1 − ν i d d ) F ( σ c ) a ( t ) ω i ...i d mod M i ,...,i d V H i ...i d ( U/S ) b B K where we put a k := 1 − i k /n k and a := ( a , . . . , a d ) .Proof. One directly has (cf. [As2, Lemma 4.1]) d log( ξ Ross ) = ( − d n − X i =1 · · · n d − X i d =1 (1 − ν i ) · · · (1 − ν i d d ) ω i ...i d dtt . Let D := t ddt be the differential operator acting on M ξ ( U/S ) dR . It follows from [AM, (2.25)]that we have D ( e ξ Ross ) = n − X i =1 · · · n d − X i d =1 (1 − ν i ) · · · (1 − ν i d d ) ω i ...i d . Let us write e ξ Ross − Φ( e ξ Ross ) ≡ n − X i =1 · · · n d − X i d =1 E i ...i d ( t ) b ω i ...i d mod V H i ...i d ( U/S ) b B K where b ω i ...i d := F a ( t ) − ω i ...i d . Then D ( e ξ Ross − Φ( e ξ Ross )) = D ( e ξ Ross ) − p Φ D ( e ξ Ross )= X is an integer such that p m ≡ mod n k for all k . Note a ( m ) = a . It follows from the unit root formula (Corollary 4.6) thatwe have (Φ U/S,σ ) m ( η − i ,..., − i d ) = (Φ U/S,σ ) m ( η − p − m i ,..., − p − m i d )= m − Y i =0 F Dw ,σ a ( m − i − ( t p m − i − ) ! η − i ,..., − i d = m − Y i =0 F a ( m − i − ( t p m − i − ) F a ( m − i ) ( t p m − i ) ! η − i ,..., − i d = (cid:18) F a ( t ) F a ( t p m ) (cid:19) η − i ,..., − i d . Recall b B K = K ⊗ W ( W [ t, ( t − t ) − , h ( t ) − ] ∧ ) (see (4.5) for the notation). Choose a ℓ -th root ζ ∈ W × of unity with p ∤ ℓ such that (1 − ζ ) h ( ζ ) mod p . Let U F ,ζ := U F × S F Spec F [ t ] / ( t − ζ ) be the fiber at t = ζ . Replacing m with mϕ ( ℓ ) , we may assume ℓ | p m − and hence ζ p m = ζ . Then the above formula implies that the evaluation F a ( t ) F a ( t p m ) (cid:12)(cid:12)(cid:12)(cid:12) t = ζ is the eigenvalue of the p m -th Frobenius on the rigid cohomology W d H d rig ( U F ,ζ / F ) with re-spect to the unit root vector η − i ,..., − i d . Now suppose F a ( t ) ∈ b B K . Then F a ( t ) F a ( t p m ) (cid:12)(cid:12)(cid:12)(cid:12) t = ζ = F a ( t ) | t = ζ F a ( t p m ) | t = ζ = 1 which contradicts with the Riemann-Weil hypothesis. This shows F a ( t ) b B K as required. (cid:3) Remark 5.4 The main result of [As2] is an explicit formula of the pairing h reg B ( ξ Ross ) | ∆ t i of the Beilinson regulator reg B ( ξ Ross ) with a certain homology cycle ∆ t . See loc. cit. Theo-rem 5.5 for the details. One can think Q (reg syn ( ξ Ross ) , b η i ...i d ) f being a p -adic counterpart of h reg B ( ξ Ross ) | ∆ t i in the following way. Let F an a ( t ) denotethe complex analytic function defined by the hypergeometric series F a ( t ) . Let y an i be theanalytic functions defined in the same way as in Proposition 4.1, and put b η an i ...i d := y an ω i ...i d + y an Dω i ...i d + · · · + y an d D d ω i ...i d , and η an i ...i d := F an ˇ a ( t ) − b η an i ...i d . Proposition 4.1 asserts D b η an i ...i d = 0 . It follows from Lemma4.3 (3) that Q ( ω i ...i d , η an − i ,..., − i d ) is a constant and hence Q ( D j ω i ...i d , b η an − i ,..., − i d ) = D j Q ( ω i ...i d , b η an − i ,..., − i d ) = D j F an a ( t ) × (const.)for all j ≥ . On the other hand, the homology cycle ∆ α satisfies ([As2, Thm.3.1]) h ω i ...i d | ∆ t i = F an a ( t ) × (const.)which implies h D j ω i ...i d | ∆ t i = D j F an a ( t ) × (const.)for all j ≥ . We thus have Q ( − , b η an − i ,..., − i d ) = h− | ∆ t i × (const.) Theorem 5.5 Suppose p > d + 1 . Let α ∈ W satisfy α , and h ( α ) modulo p ,so that ( t − α ) is a maximal ideal of b B K . Let σ be the p -th Frobenius on W [[ t ]] given by σ ( t ) = α − p t p . Let reg syn : K d +1 ( U α ) −→ H d +1syn ( U α , Q p ( d + 1)) ∼ = H d dR ( U α,K /K ) be the syntomic regulator map, cf. (1.3) . Then reg syn ( ξ Ross | U α ) ∈ W d H d dR ( U α,K /K ) and Q (reg syn ( ξ Ross | U α ) , η − i ,..., − i d ) Q ( ω i ...i d , η − i ,..., − i d ) = (1 − ν i ) · · · (1 − ν i d d ) F ( σ ) a ( t ) | t = α . Proof. With use of [AM, Theorem 3.8], this is immediate from Theorem 5.3 together withLemma 5.2. (cid:3) p -adic Beilinson conjecture for K of K3 surfaces We discuss the syntomic regulators for singular K3 surfaces over Q . For such K3 surfaces,there are the Hecke eigenforms which provide the L -functions and the p -adic L -functions ofK3, and then one can formulate the p -adic Beilinson conjecture.For a smooth projective variety S over a field k , we denote by NS( S ) the Neron-Severigroup. Note NS( S ) ⊗ Q = (NS( S × k k ) ⊗ Q ) Gal( k/k ) the fixed part by the Galois group.Let h ( S, Q ( m )) = h ( S, Q ( m )) / NS( S ) ⊗ Q ( m − denote the transcendental part of themotive h ( S, Q ( m )) (cf. [KMP, 7.2.2]). When k = Q , one can define the L -function of h ( S, Q ) , which we denote by L ( h ( S ) , s ) . 20 .1 Singular K3 surfaces over Q A K3 surface X over a field k is called singular if the rank of NS( X × k k ) is the largest. Wework over the base field k = Q , then X is singular if and only if the rank is , and hencethe transcendental motive h ( X ) = h ( X, Q ) is 2-dimensional. Every such K3 admits the Shioda-Inose structure by elliptic curves E, E ′ with complex multiplications ([I-S], [Mo]), X ρ Z (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ E × E ′ ρ E × E ′ { { ✇✇✇✇✇✇✇✇✇ Z (6.1)in which the arrows are rational dominant maps of degree 2, and Z is a K3 surface andthe base field is Q . Moreover E and E ′ are isogenous. Thus the L -function L ( h ( X ) , s ) is essentially the L -function of the symmetric square of the elliptic curves. Although theShioda-Inose structure usually requires the base field extension, the L -function has the de-scent to Q . Theorem 6.1 (Livn´e, [Li]) Let X be a singular K3 surface over Q . Then there is a Heckeeigenform f of weight with complex multiplication such that L ( h ( X ) , s ) = L ( f, s ) . The p -adic Beilinson conjecture asserts that the special values of p -adic L -functions aredescribed by syntomic regulators. See [P, 4.2.2] and also [Co, Conj.2.7]. Concerning asingular K3 surface, one can take the p -adic L -function to be the p -adic L -function L p ( f, χ, s ) of the associated Hecke eigenform f and some Dirichlet character χ : Z × p → Q × p ([MTT]).See [AC, Conjecture 3.1] for the general statement of the p -adic Beilinson conjecture for amodular form. In case of singular K3 surfaces, one can write down (a part of) the conjectureas follows.Let X be a K3 surface over Q . Let p be a prime at which X has a good reduction. Let X Z p be the smooth model over Z p and put X Q p := X × Q Q p and X F p := X Z p × Z p F p . Let reg syn : K ( X Z p ) −→ H ( X Z p , Q p (3)) ∼ = H ( X Q p / Q p ) be the syntomic regulator map ([N-N]). Let X Q p := X Q p × Q p Q p and put NS dR ( X Q p ) := H ( X Q p / Q p ) ∩ NS( X Q p ) ⊗ Q p . Let NS dR ( X Q p ) ⊥ denote the orthogonal complement of NS dR ( X Q p ) ⊂ H ( X Q p / Q p ) withrespect to the cup-product. It follows NS dR ( X Q p ) ⊥ ⊕ NS dR ( X Q p ) ∼ −→ H ( X Q p / Q p ) . The cup-product induces a non-degenerate pairing on (cid:0) H ( X Q p / Q p ) / NS dR ( X Q p ) (cid:1) ⊗ NS dR ( X Q p ) ⊥ ∼ = NS dR ( X Q p ) ⊥ ⊗ NS dR ( X Q p ) ⊥ , which we write by h− , −i . 21 emma 6.2 Suppose that X has a good ordinary reduction. Let α p be the eigenvalue ofthe p -th Frobenius on H ( X Q p / Q p ) which is a p -adic unit (such α p is unique), and η ∈ H ( X Q p / Q p ) the eigenvector. Then η ∈ NS dR ( X Q p ) ⊥ . Let ω ∈ Γ ( X Q p , Ω X Q p / Q p ) be anon-zero regular -form. If X is singular, then dim NS dR ( X Q p ) ⊥ = 2 and hence NS dR ( X Q p ) ⊥ = Q p ω + Q p η. Proof. We want to show that the cup-product η ∪ z vanishes for any z ∈ NS dR ( X Q p ) . Theeigenvalues of Φ on NS dR ( X Q p ) are p × (roots of unity). Take m > such that Φ m = p m on NS( X Q p ) . Since Φ( x ) ∪ Φ( y ) = Φ( x ∪ y ) = p x ∪ y , we have Φ m ( η ) ∪ Φ m ( z ) = p m ( η ∪ z ) ⇐⇒ α mp p m ( η ∪ z ) = p m ( η ∪ z ) . Since α p ∈ Z × p , it follows that η ∪ z = 0 . (cid:3) Conjecture 6.3 (weak p -adic Beilinson conjecture for ordinary singular K3 surfaces) Let X be a singular K3 surface over Q and f the corresponding Hecke eigenform of weight .Fix ω ∈ Γ ( X, Ω X/ Q ) a regular 2-form. Suppose that X has a good ordinary reduction at p , which is equivalent to that p ∤ a p where f = P ∞ n =1 a n q n . Let α p be the unit root of T − a p T + p , and η ∈ NS( X Q p ) its eigenvector (note that ω is unique up to Q × and η is unique up to Q × p ). Then there is an integral element ξ ∈ K ( X ) (3) Z (cf. [S-Int]) and aconstant C ∈ Q × not depending on p such that L p ( f, ω − , 0) = C (1 − p α − p ) h reg syn ( ξ ) , η ih ω, η i (6.2) where ω Tei is the Teichm¨uller character. Remark 6.4 For a singular K3 surface X , one can expect K i ( X ) ( j ) = K i ( X ) ( j ) Z with i = 1 thanks to the Shioda-Inose structure (6.1) . See [As2, Remark 6.8] for details. In [AOP], Ahlgren, Ono and Penniston study the L -function of a K3 surface Y a over Q defined by an affine equation w = u u (1 + u )(1 + u )( u − au ) , a ∈ Q \ { , } . Their first main result is the following. Theorem 6.5 ([AOP, Theorem 1.1]) Let E a : y = x (cid:18) x + 2 x − a − a (cid:19) E ′ a : (1 − a ) y = x (cid:18) x + 2 x − a − a (cid:19) be ellptic curves over Q . Then L ( h ( Y a ) , s ) = L ( h ( E a × E ′ a ) , s ) . Y a and E a × E ′ a which gives an isomorphism h ( Y a ) ∼ = h ( E a × E ′ a ) of motives over Q is constructed by van Geemen and Top [vGT, Theorem 1.2].Following [AOP], we call Y a modular if L ( h ( Y a ) , s ) is the L -function of a Hecke eigen-form of weight with complex multiplication, or equivalently E a has a complex multiplica-tion ([vGT, Theorem 1.2]). The second main result of [AOP] gives the complete list of a ’sfor Y a to be modular. Theorem 6.6 ([AOP, Theorem 1.2]) The K3 surface Y a is modular if and only if a = − , ± , − ± , ± . In each case, E a has complex multiplication. The corresponding Hecke eigenforms are as follows ([AOP, p.366–367]). Let η ( z ) be theDedekind eta function. Let A = η (4 z ) , B = η ( z ) η (2 z ) η (4 z ) η (8 z ) , C = η (2 z ) η (6 z ) , D = η ( z ) η (7 z ) be weight newforms of level , , , respectively. Let χ D denote the quadratic characterassociated to the quadratic field Q ( √ D ) . Then the corresponding Hecke eigenforms aregiven as follows. a − / − − / / Hecke eigenform B ⊗ χ − C C ⊗ χ − A A ⊗ χ D D ⊗ χ − (6.3)where f ⊗ χ denotes the χ -twist of the modular form. p -adic Beilinson conjecture for K3 surfaces in [AOP] Let U a : (1 − x )(1 − x )(1 − x ) = a the hypergeometric scheme over Q , and let X a be a smooth compactification of U a whichis a K3 surface. Put Z a := X a \ U a . We discuss the p -adic Beilinson conjecture for X a .Let p > be a prime at which X a has a good ordinary reduction. We take integral models X a, Z ( p ) ⊃ U a, Z ( p ) which are smooth over Z ( p ) (Proposition 3.1). For a Z ( p ) -ring R , we write X a,R := X a, Z ( p ) × Z ( p ) R etc. Recall the 2-forms ω , , , η , , ∈ W H ( U a / Q p ) ∼ = H ( X a / Q p ) /H ,Z a ( X a / Q p ) from (3.3) and (4.4). Let ξ Ross = (cid:26) − x x , − x x , − x x (cid:27) ∈ K M ( O ( U a, Z ( p ) )) be the higher Ross symbol. We think ξ Ross to be an element of K ( U a, Z ( p ) ) (3) under thenatural map K M ( O ( U a, Z ( p ) )) → K ( U a, Z ( p ) ) (3) . There is the exact sequence K Z a, Z ( p ) ( X a, Z ( p ) ) (3) −→ K ( X a, Z ( p ) ) (3) −→ K ( U a, Z ( p ) ) (3) . ξ Ross lies in the image of K ( X a, Z ( p ) ) (3) ([As2, Corollary 4.4]), so that there is a lifting g ξ Ross ∈ K ( X a, Z ( p ) ) (3) . It is a standard argument on the syntomic regulator maps to see thatthere is a commutative diagram K ( X a, Z ( p ) ) (3) reg syn / / (cid:15) (cid:15) H ( X a, Z p , Q p (3)) ∼ = H ( X a, Q p / Q p ) (cid:15) (cid:15) K ( X a, Z ( p ) ) (3) /K Z a, Z ( p ) ( X a, Z ( p ) ) (3) / / H ( X a, Q p / Q p ) / NS dR ( X a, Q p ) h− ,η , , i / / Q p (6.4)where h− , η , , i is the cup-product pairing which is well-defined by Lemma 6.2. The dia-gram implies that h reg syn ( g ξ Ross ) , η , , i does not depend on the choice of the lifting. Apply-ing Theorem 5.5, we have the description of the right hand side of (6.2) in Conjecture 6.3 interms of our p -adic function F ( σ ) a ( t ) . Theorem 6.7 Let σ ( t ) = a − p t p . Then h reg syn ( g ξ Ross ) , η , , ih ω , , , η , , i = 8 F ( σ ) , , ( t ) | t = a . Next, we see the left hand side of (6.2) in Conjecture 6.3, namely the p -adic L -function.Recall the K3 surface Y a from §6.2. There is a dominant rational map ρ : X a −→ Y a (6.5)over Q ([As2, (6.7)]), which induces an isomorphism h ( Y a , Q ) ∼ = h ( X a , Q ) (6.6)of motives over Q . Therefore the p -adic Beilinson conjecture for Y a is equivalent to that for X a . The L -functions of X a and Y a agree, and if a = − , ± , − ± , ± , then they arethe L -functions of the Hecke eigenforms as in the table (6.3). Together with Theorem 6.7,Conjecture 6.3 for K ( X a ) or K ( Y a ) can be formulated as follows. Conjecture 6.8 Let a = − , ± , − ± , ± . Let f a be the Hecke eigenform correspondingto Y a , cf. table (6.3) . Let p > be a prime such that p ∤ a p and let α p be the unit root of T − a p T + p . Let σ be the p -th Frobenius given by σ ( t ) = a − p t p . Then there is a constant C a ∈ Q × not depending on p such that L p ( f a , ω − , 0) = C a (1 − p α − p ) F ( σ ) , , ( t ) | t = a . In view of [As2, Theorem 6.9], it is also plausible to expect that Conjecture 6.8 remainstrue for a = 1 . Conjecture 6.9 Let p ≡ mod be a prime. Let α p ∈ Z p be the root of T − a p T + p suchthat α p ≡ a p mod p where A = η (4 z ) = P a n q n . Then there is a constant C ∈ Q × notdepending on p such that L p ( A, ω − , 0) = C (1 − p α − p ) F ( σ ) , , ( t ) | t =1 where σ ( t ) = t p . .4 Some other elliptic K3 surfaces and p -adic regulators Let A = Q [ s, ( s − s ) − ] and V n = Spec A [ x , x ] / ((1 − x n )(1 − x n ) − s ) a hypergeometric scheme for n ≥ an integer. Let C n ⊃ V n be a smooth compactificationover A (Proposition 3.1). The relative dimension of C n /A is . Let J ( C n ) → Spec A be thejacobian scheme. Put H := H ( C n /A ) . For < i , i < n , we denote by H Q ( i , i ) ⊂ H Q := H ⊗ Q Q the eigenspace defined in (3.2). Then the subspace X gcd( r,n )=1 H Q ( ri , ri ) ⊂ H Q is endowed with A -module structure, which we denote by H A ( i , i ) , H A ( i , i ) ⊗ Q Q = X gcd( r,n )=1 H Q ( ri , ri ) . Note H A ( i , i ) = H A ( i ′ , i ′ ) if and only if i ≡ ri ′ and i ≡ ri ′ mod n for some r prime to n . Let H = M i ,i H A ( i , i ) be the decomposition where ( i , i ) runs over representatives of the set { (¯ i , ¯ i ) ∈ ( Z /n Z ) | ¯ i = 0 , ¯ i = 0 } / ∼ with (¯ i , ¯ i ) ∼ (¯ i ′ , ¯ i ′ ) ⇔ (¯ i , ¯ i ) = ( r ¯ i ′ , r ¯ i ′ ) for some r ∈ ( Z /n Z ) × .Thanks to the Poincare reducibility theorem ([Mu, §19 Thm. 1]), the above decompositioninduces an isogeny J ( C n ) −→ Y i ,i J i ,i of abelian A -schemes. We note that X 1) = ⇒ ( F ± n ) ∗ ( H Q ( i , i )) = 0 . (6.10)Let B = Q [ t, ( t − t ) − ] , and U n = Spec B [ x , x , x ] / ((1 − x n )(1 − x n )(1 − x ) − t ) . Let X n ⊃ U n be a smooth compactification over B . Let S := Spec B [ x , (1 − x ) − , (1 − t − x ) − ] . Let φ : S → Spec A be the morphism given by φ ∗ ( s ) = t (1 − x ) − , and V n,S ⊂ C n,S % % ❏❏❏❏❏❏❏❏❏❏❏ F ± n,S / / E ± n,Sh ± n,S ~ ~ ⑤⑤⑤⑤⑤⑤⑤⑤ S the base change of (6.8) by φ , where V n,S := V n × A S (6.11) = Spec B [ x , x , x , (1 − x ) − , (1 − t − x ) − ] / ((1 − x n )(1 − x n ) − t (1 − x ) − )= Spec B [ x , x , x , (1 − t − x ) − ] / ((1 − x n )(1 − x n )(1 − x ) − t ) ֒ → U n . The morphisms h ± n,S give rise to relatively minimal elliptic fibrations E ± n → P B ( x ) over B ,so that we have a commutative diagram V n,S g n ●●●●●●●●● f ± n / / E ± nh ± n { { ✇✇✇✇✇✇✇✇✇ P B ( x ) (6.12)of smooth B -schemes. Proposition 6.10 E ± n ( n = 3 , , are elliptic K3 surfaces over B .Proof. Let x ∈ Spec B be a closed point, and k ( x ) the residue field. We denote by E ± n,x thefiber at x . We first show that the proposition is reduced to show that E ± n,x are K3 surfaces over k ( x ) for any x . Indeed, suppose that they are true, namely H ( E ± n,x , O ) = 0 and Ω E ± n,x /k ( x ) ∼ = O . Then it follows from [Ha, III, 12.9] that one has the vanishing H ( E ± n , O ) = 0 . Moreover26 (Ω E ± n /B ) is locally free B -module of rank one (and hence H (Ω E ± n /B ) ∼ = B ), and thenatural map H (Ω E ± n /B ) ⊗ B k ( x ) −→ H (Ω E ± n,x /k ( x ) ) is bijective. We have a commutative diagram ( H (Ω E ± n /B ) ⊗ B O E ± n ) ⊗ B k ( x ) ∼ = (cid:15) (cid:15) / / Ω E ± n /B ⊗ B k ( x ) ∼ = (cid:15) (cid:15) H (Ω E ± n,x /k ( x ) ) ⊗ k ( x ) O E ± n,x / / Ω E ± n,x /k ( x ) and the bottom arrow is bijective as Ω E ± n,x /k ( x ) ∼ = O . Hence this implies an isomorphism O E ± n ∼ = H (Ω E ± n /B ) ⊗ B O E ± n ∼ = / / Ω E ± n /B as required.We may now replace the base ring B with a field k of characteristic zero. Then wewant to show the vanishing H ( E ± n , O ) = 0 and an isomorphism Ω E ± n /k ∼ = O . To dothis, we may further replace k with C . There is the injective map ( f − n ) ∗ : H ( E ± n ) ֒ → H ( X n ) = W H ( U n ) . However since W H ( U n ) = 0 (Theorem 3.2 (2)), it turns outthat H ( E ± n ) = 0 and hence the vanishing H ( E ± n , O ) = 0 follows. The rest is to show anisomorphism Ω E ± n / C ∼ = O . To do this, we employ the canonical bundle formula. There is a A -rational point of C n /A , and hence there is a section of the elliptic fibration h ± n : E ± n → P k .Hence it follows from the canonical bundle formula [BPV, V, (12.3)] that one has K E ± n := Ω E ± n /k ∼ = ( h ± n ) ∗ O P ( e ) (6.13)with e = χ ( O E ± n ) − χ ( O P ) = χ ( O E ± n ) − . We show e = 0 . We denote the topologicalEuler number of X by e ( X ) . χ ( O E ± n ) = 112 ( K E ± n + e ( E ± n )) (Noether’s formula, [BPV, I, (5.5)]) = 112 e ( E ± n ) (by (6.13)) = 112 X s e ( Z ± n,s ) ([BPV, III, (11.4)])where Z ± n,s runs over all singular fibers of h ± n : E ± n → P ( x ) . To compute the last term,we write up the singular fibers for each n = 3 , , . The singlar fibers of h ± n appear at x = ± , ±√ − t and x = ∞ (we think t ∈ C \ { , } of being a constant). The Kodairatype of each singular fiber is determined by the local monodromy on R ( h ± n ) ∗ Q , and it can becomputed from the local monodromy for the fibration E ± n → Spec A = A ( s ) \ { , } (here27e think s of being a parameter). By virtue of (6.9), this is isomorphic to the monodromyof the Gaussian hypergeometric function F (cid:16) n , − n ; s (cid:17) , which is well-understood. In thisway, we obtain the complete list of singular fibers of h ± n , x = ± x = ±√ − t x = ∞ h ± IV ∗ I I h ± III ∗ I I h ± II ∗ I I It is now immediate to have P s e ( Z ± n,s ) = 24 in all cases. Hence we have e = 0 as required. (cid:3) Lemma 6.11 Let V n,S = V n × A S ֒ → U n be as in (6.11) . One has W H ( U n /B ) = W H ( V n,S /B ) . Hence the pull-back ( f ± n ) ∗ : H ( E ± n /B ) −→ W H ( U n /B ) is defined.Proof. (This is implicitly proven in the proof of [As2, Theorem 3.4]). It is enough to showthe lemma at each fiber, so that we may replace B with C and may assume that U n and V n,S are smooth complex affine varities. Let F , F ⊂ U n be the complement of V n,S whichare irreducible affine curves with unique singular points P ∈ F and P ∈ F . Put U ◦ n := U n \ { P , P } and F ◦ i := F i \ { P i } . Then there is an exact sequence H ( U n ) ∼ = (cid:15) (cid:15) H ( V n,S ) Res / / H ( F ◦ i ) ⊕ ⊗ Q ( − / / H ( U ◦ n ) / / H ( V n,S ) Res / / H ( F ◦ i ) ⊕ ⊗ Q ( − of mixed Hodge structures, where H i ( − ) = H i ( − , Q ) denotes the Betti cohomology. Since Res is surjective and the weight of H ( F ◦ i ) ⊗ Q ( − is ≥ , we have W H ( U n ) = W H ( V n,S ) as required. (cid:3) Put H ( U n ) := W H ( U n /B ) . The sum X r H ( U n ) Q ( ri , ri , ⊂ H ( U n ) Q := H ( U n ) ⊗ Q Q of the subspaces is endowed with B -module structure, which we denote by H ( U n ) B ( i , i , , H ( U n ) B ( i , i , ⊗ Q = X r H ( U n ) Q ( ri , ri , ,H ( U n ) = M i ,i H ( U n ) B ( i , i , . Moreover we denote by H ( U n ) B ( i , i , + (resp. H ( U n ) B ( i , i , − ) the fixed part (resp.anti-fixed part) by the involution τ ( x , x , x ) = ( x , x , x ) .28 roposition 6.12 The image of ( f ± n ) ∗ : H ( E ± n /B ) −→ H ( U n ) = W H ( U n /B ) agrees with the component H ( U n ) B (1 , n − , ± .Proof. There is a commutative diagram H ( E ± n /B ) ( f ± n ) ∗ / / (cid:15) (cid:15) W H ( V n,S /B ) ( ⋆ ) = H ( U n ) ∩ (cid:15) (cid:15) H ( E ± n,S /B ) ( F ± n,S ) ∗ / / H ( V n,S /B ) H ( S, R ( h ± n ) ∗ Ω • E ± n,S /S ) i O O / / H ( S, R g n, ∗ Ω • V n,S /S ) ∼ = O O (6.14)where the bottom arrow is induced from F ± n in (6.8) and ( ⋆ ) follows from Lemma 6.11.The map i is injective and the cokernel of i is generated by the cycle class e of a section of E ± n,S → S . One easily sees ( F ± n,S ) ∗ ( e ) = 0 . Therefore the image of F ∗ n,S agrees with theimage of H ( S, R ( h ± n ) ∗ Ω • E ± n,S /S ) , and hence we have ( f ± n ) ∗ H ( E ± n /B ) ⊂ H ( U n ) B (1 , n − , ± . Since H ( U n ) B (1 , n − , ± is an irreducible connection (Theorem 3.3), the equality holdsin the above, as required. (cid:3) Corollary 6.13 For a geometric point ¯ x → Spec B , let E ± n, ¯ x be the fiber at ¯ x . Then the rankof the Neron-Severi group NS( E ± n, ¯ x ) is ≥ .Proof. This follows from Proposition 6.12 and the fact rank H ( U n ) B (1 , n − , ± = 3 . (cid:3) Taking the Hodge filtration F of the diagram (6.14), we have a commutative diagram H ( E ± n , Ω E ± n /B ) ( f ± n ) ∗ / / H ( X n , Ω X n /B ) ( f ± n ) ∗ o o F H ( S, R g n, ∗ Ω • V n,S /S ) ∼ = O O F H ( S, R ( h ± n ) ∗ Ω • E ± n,S /S ) ∼ = O O / / F H ( S, R g n, ∗ Ω • C n,S /S ) ∼ = O O o o (6.15)together with the push-forward maps. 29 emma 6.14 Let ω i ,i , ∈ H ( X n , Ω X n /B ) br the regular 2-forms in (3.3) . Then ( f ± n ) ∗ H ( E ± n , Ω E ± n /B ) = B ( ω ,n − , ∓ ω n − , , ) (6.16) ( i , i ) = (1 , n − , ( n − , 1) = ⇒ ( f ± n ) ∗ ( ω i ,i , ) = 0 , (6.17) ( f ± n ) ∗ ( ω ,n − , ) = ∓ ( f ± n ) ∗ ( ω n − , , ) = 0 . (6.18) Proof. Since the involution τ ( x , x , x ) = ( x , x , x ) on U n satisfies τ ω ,n − , = − ω n − , , ,one has F H ( U n ) B (1 , n − , ± = B ( ω ,n − , ∓ ω n − , , ) . Now (6.16) is immediate from Proposition 6.12. Moreover (6.17) follows from (6.10) byvirtue of the diagram (6.15). By the construction of F ± n , they satisfy ( F + n ) ∗ ( F − n ) ∗ = ( F − n ) ∗ ( F + n ) ∗ =0 , which implies ( f + n ) ∗ ( f − n ) ∗ = ( f − n ) ∗ ( f + n ) ∗ = 0 . Therefore, ( f + n ) ∗ ( ω ,n − , + ω n − , , ) = 0 , ( f − n ) ∗ ( ω ,n − , − ω n − , , ) = 0 . There remains to show the non-vanishing ( f ± n ) ∗ ( ω ,n − , ) = 0 . Suppose ( f ± n ) ∗ ( ω ,n − , ) = 0 and hence ( f ± n ) ∗ ( ω n − , , ) = 0 as well. Then ( f ± n ) ∗ H (Ω X n /B ) = ( f ± n ) ∗ F W H ( U n /B ) = 0 by (6.17). Since ( f ± n ) ∗ is surjective onto H (Ω E ± n /B ) , this contradicts with that E ± n are K3surfaces (Proposition 6.10). This completes the proof. (cid:3) We discuss the p -adic Beilinson conjecture for K ( E − n ) with n = 3 , , . Let f − n : U n → E − n be the morphism (6.12) (recall that we only consider the cases n = 3 , , ). For a ∈ Q \ { , } , we denote by f n,a : U n,a → E − n,a and X n,a the fibers at the closed point t = a of Spec B . Let ζ n be a primitive n -th root of unity. Let p > be a prime at which X n,a hasa good ordinary reduction. Let U a,n, Q ( ζ n ) = U a,n × Q Spec Q ( ζ n ) and U a,n, Z ( p ) [ ζ n ] a smoothmodel over Z ( p ) [ ζ n ] . Let ξ Ross ( ζ n ) = (cid:26) − x − ζ n x , − x − ζ n x , − x x (cid:27) ∈ K M ( O ( U a,n, Z ( p ) [ ζ n ] )) (6.19)be the higher Ross symbol, which we think of being an element of Quillen’s K . Put ξ n := N Q ( ζ n ) / Q ( ξ Ross ( ζ n )) ∈ K ( U n,a, Z ( p ) ) where N Q ( ζ n ) / Q is the norm map in Quillen’s K -theory. Since ξ Ross ( ζ n ) lies in the imageof K ( X n,a, Z ( p ) [ ζ n ] ) ([As2, Corollary 4.4]), so does ξ n , and hence there is a lifting e ξ n ∈ K ( X n,a, Z ( p ) ) . Put ω n := ( f − n,a ) ∗ ω ,n − , , η n := ( f − n,a ) ∗ η ,n − , ∈ H ( E − n,a ) . reg syn : K ( E − n,a, Z ( p ) ) / / H ( E − n,a, Z ( p ) , Q p (3)) ∼ = H ( E − n,a / Q ) be the syntomic regulator map where E − n,a, Z ( p ) is a smooth integral model of E − n,a over Z ( p ) .Since η n is a unit root vector, one can show that h reg syn ( f − n ∗ e ξ n ) , η n i does not depend on thechoice of lifting in the same way as in §6.2 (see the diagram (6.4)). We have h reg syn (( f − n,a ) ∗ e ξ n ) , η n i = h reg syn ( e ξ n ) , ( f − n,a ) ∗ η n i = h reg syn ( ξ Ross ( ζ n )) , ( f − n,a ) ∗ η n i + h reg syn ( ξ Ross ( ζ − n )) , ( f − n,a ) ∗ η n i . Since reg syn ( ξ Ross ( ζ ± n )) = (1 − ζ n )(1 − ζ − n ) F ( σ ) n , n − n , ( t ) | t = a ( ω ,n − , + ω n − , , ) + (other terms)by Theorem 5.3, we have h reg syn (( f − n,a ) ∗ e ξ n ) , η n i = 2(1 − ζ n )(1 − ζ − n ) F ( σ ) n , n − n , ( t ) | t = a h ω ,n − , + ω n − , , , ( f − n,a ) ∗ η n i = 2(1 − ζ n )(1 − ζ − n ) F ( σ ) n , n − n , ( t ) | t = a × h ω n , η n i (by (6.18)) = 4(1 − ζ n )(1 − ζ − n ) F ( σ ) n , n − n , ( t ) | t = a h ω n , η n i . Finally we note h ω n , η n i = h ( f − n,a ) ∗ ( f − n,a ) ∗ ω ,n − , , η ,n − , i = c h ω ,n − , + ω n − , , , η ,n − , i (by (6.16)) = c h ω n − , , , η ,n − , i with c = 0 a constant, and this does not vanish by Lemma 4.3 (3). Summing up the above,we have the description of the p -adic regulator for K ( E − n,a ) . Theorem 6.15 Let σ ( t ) = a − p t p with p > . Suppose that E − n,a has a good ordinaryreduction at p . Then h reg syn (( f − n,a ) ∗ e ξ n ) , η n ih ω n , η n i = 4(1 − ζ n )(1 − ζ − n ) F ( σ ) n , n − n , ( t ) | t = a with n = 3 , , . Conjecture 6.16 Suppose that E − n,a is a singular K3 surface over Q . Let A n,a = P a n q n bethe corresponding Hecke eigenform of weight , and α p the unit root of T − a p T + p . Thenthere is a constant C n,a ∈ Q × not depending on p such that L p ( A n,a , ω − , 0) = C n,a (1 − p α − p ) F ( σ ) n , n − n , ( t ) | t = a . Unlike the K3 surfaces in [AOP], the author has not worked out on the ( p -adic) L -functionsof E − n,a . 31 eferences [AOP] Ahlgren, S.; Ono, K.; Penniston, D., Zeta Functions of an infinite family of K3surfaces . American Journal of Mathematics, Vol. 124, No. 2 (Apr., 2002), pp.353–368.[As1] , New p -adic hypergeometric functions and syntomic regulators .arXiv.1811.03770.[As2] , A generalization of the Ross symbols in higher K -groups and hypergeo-metric functions I , arXiv.2003.10652[AC] Asakura, M., Chida, M., A numerical approach toward the p -adic Beilinson con-jecture for elliptic curves over Q . arXiv:2003.08888[AM] Asakura, M., Miyatani, K., Milnor K -theory, F -isocrystals and syntomic regula-tors . arXiv:2007.14255.[BK] Bannai, K., Kings, G., p -adic Beilinson conjecture for ordinary Hecke motives as-sociated to imaginary quadratic fields . Algebraic number theory and related topics2009, 9–30, RIMS Kˆokyˆuroku Bessatsu, B25, Res. Inst. Math. Sci. (RIMS), Ky-oto, 2011.[BPV] Barth, W., Peters, C., Van de Ven, A. Compact complex surfaces. Second edition. Springer-Verlag, Berlin. 2004. 436 pp.[BD] Bertolini, M., Darmon, H., Kato’s Euler system and rational points on ellipticcurves I: A p-adic Beilinson formula . Israel J. Math. (2014), no. 1, 163–188.[KMP] Kahn, B., Murre, J., Pedrini, C., On the transcendental part of the motive of a sur-face . Algebraic cycles and motives. Vol. 2, 143–202, London Math. Soc. LectureNote Ser., , Cambridge Univ. Press, Cambridge, 2007.[Co] Colmez, P., Fonctions L p -adiques . S´eminaire Bourbaki, Vol. 1998/99, Ast´erisqueNo. 266 (2000), Exp. No. 851, 3, 21–58.[Dw] Dwork, B., p -adic cycles . Publ. Math. IHES, tome 37 (1969), 27–115.[vGT] van Geemen, B., Top, J., An isogeny of K3 surfaces. Bull. London Math. Soc. (2006), no. 2, 209–223.[Ha] Hartshorne, R.: Algebraic Geometry . (Grad. Texts in Math. 52), New Tork,Springer, 1977,[I-S] Inose, H., Shioda, T., On singular K3 surfaces , In: Complex Analysis and Alge-braic Geometry (W. Baily and T. Shioda, eds.), Iwanami Shoten, Tokyo, 1977, pp.119–136. 32Ka] Kato, K., On p -adic vanishing cycles (application of ideas of Fontaine-Messing) .In: Algebraic geometry, Sendai, 1985 (Adv. Stud. Pure Math. 10), pp. 207–251,Amsterdam, North-Holland, 1987.[KLZ] Kings, G., Loeffler, D. and Zerbes, S.-L., Rankin-Eisenstein classes for modularforms . Amer. J. Math. (2020), no. 1, 79–138.[Li] Livn´e, R., Motivic Orthogonal Two-dimensional Representations of Gal( Q / Q ) .Israel J. of Math. 92 (1995), 149–156.[MTT] Mazur, B., Tate, J., Teitelbaum, J., On p -adic analogues of the conjectures of Birchand Swinnerton-Dyer . Invent. Math. (1986), 1–48.[Mo] Morrison, D. R., On K3 surfaces with large Picard number , Invent. Math. ,(1984), 105–121.[Mu] Mumford, D., Abelian varieties. (With appendices by C. P. Ramanujam and YuriManin). Corrected reprint of the second (1974) edition. Tata Institute of Funda-mental Research Studies in Mathematics, 5. 2008. 263 pp.[N-N] Nekov´aˇr, J., Niziol, W., Syntomic cohomology and p -adic regulators for varietiesover p -adic fields . With appendices by Laurent Berger and Fr´ed´eric D´eglise. Al-gebra Number Theory (2016), no. 8, 1695–1790.[Ni] Niklas M., Rigid syntomic regulators and the p -adic L -function of a modular form .Regensburg PhD Thesis, 2010, available at http://epub.uni-regensburg.de/19847/[P] Perrin-Riou, B., Fonctions L p -adiques des repr´esentations p -adiques. Ast´erisque (1995).[PS] Peters, C., J. Steenbrink, J., Mixed Hodge structures. Ergebnisse der Mathematikund ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, .Springer-Verlag, Berlin, 2008.[R1] Ross, R., K of Fermat curves and values of L -functions , C.R. Acad. Sci. Paris,Serie I. (1991), 1–5.[R2] , K of Fermat curves with divisorial support at infinity , Compositio Math. (1994), no. 3, 223–240.[Sa] Samart, D., Three-variable Mahler measures and special values of modular andDirichlet L -series . Ramanujan J. (2013), no. 2, 245–268.[S-Be] Schneider, P., Introduction to the Beilinson conjectures. In Beilinson’s Conjectureson Special Values of L -Functions (M. Rapoport, N. Schappacher and P. Schneider,ed), Perspectives in Math. Vol.4, 1–35, 1988.33S-Int] Scholl, A. J., Integral elements in K -theory and products of modular curves . In:Gordon, B. B., Lewis, J. D., M ¨uller-Stach, S., Saito, S., Yui, N. (eds.) The arith-metic and geometry of algebraic cycles, Banff, 1998, (NATO Sci. Ser. C Math.Phys. Sci., 548), pp. 467–489, Dordrecht, Kluwer, 2000.[Sl] Slater, L., Generalized hypergeometric functions , Cambridge Univ. Press, Cam-bridge 1966.[LS] Le Stum, B., Rigid cohomology.