zeta functions of certain k3 families : application of the
TRANSCRIPT
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Zeta functions of certain K3 families : application of
the formula of Clausen
M. Asakura*
Abstract
Based on the theory of rigid cohomology, we provide an explicit formula of zeta
functions of certain K3 families, which we call the hypergeometric type. The central
point of our argument is the comparison between the 2nd rigid cohomology of a K3 and
the symmetric product of an elliptic curve, that is brought from the classical formula of
Clausen.
1 Introduction
A projective smooth surface X is called a K3 surface if
H1(X,OX) = 0, KX∼= OX .
The subject of this paper is the l-adic Galois representation
GQ = Gal(Q/Q) −→ Aut(H2et(X,Ql)), X := X ×Q Q
for a K3 surface X overQ such that the Picard number ρ(X) := rankNS(X) is≥ 19. Thanks
to the theorem of Morrison [Mo], X is isogenous to a Kummer K3 surface Km(E × E) of
an elliptic curve E (not necessarily defined over Q), which is often referred to as the Shioda-
Inose structure. Then one finds that the Galois representation H2et(X,Ql) is potentially iso-
morphic to the symmetric product Sym2H1et(E,Ql) up to a simple factor. Morrison’s theo-
rem asserts only the existence of the Kummer K3, the problem on finding an explicit E is
nontrivial. Besides, the isogeny is not necessarily defined over Q (even when so is E), and
then it is another new task to explore the GQ-representation.
In this paper, we study the characteristic polynomial
det(1− φ−1p T | H2
et(X,Ql)) (1.1)
*Hokkaido University, Sapporo 060-0810, JAPAN. [email protected]
1
of the p-th Frobenius φp ∈ GQ for a K3 surface X whose “period” is the hypergeometric
series
Fα(t) = 3F2
(α0, α1, α2
1, 1; t
)=
∞∑
n=0
(α0)nn!
(α1)nn!
(α2)nn!
tn, (α)n := α(α+1) · · · (α+n−1)
where α = (α0, α1, α2) is either of the following,
(1
2,1
2,1
2
),
(1
3,2
3,1
2
),
(1
4,3
4,1
2
),
(1
6,5
6,1
2
). (1.2)
In precise, we consider a projective smooth family
f : X −→ T = SpecQ[t, (t− t2)−1]
of K3 surfaces such that the generic fiber X t = X ×T Q(t) satisfies ρ(X t) = 19. Put
VdR(X /T ) = Coim[H2dR(X /T ) → H2
dR(X t)/NS(X t)⊗Q(t)]
a free O(T )-module of rank 3. Let D = Q[t, (t− t2)−1, ddt] be the Wyle algebra of T , and let
Pα = D3 − t(D + α0)(D + α1)(D + α2), D := td
dt
be the hypergeometric differential operator, which annihilates Fα(t). Then we call f of
hypergeometric type Fα(t) if there is an isomorphism
VdR(X /T ) ∼= D/DPα
of D-modules (Definition 3.3). Here are examples of K3 families of hypergeometric type.
(i) The Dwork family (cf. [Ka])
tx40 + x4
1 + x42 + x4
3 − 4x0x1x2x3 = 0 (1.3)
of quartic surfaces is of hypergeometric type F 14, 34, 12(t).
(ii) The K3 family (cf. [AOP])
z2 = xy(1 + x)(1 + y)(x− ty) (1.4)
is of hypergeometric type F 12, 12, 12(t).
(iii) The K3 family (cf. [As1, §6.3])
(1− x2)(1− y2)(1− z2) = t (1.5)
is of hypergeometric type F 12, 12, 12(t). This is isogenous to the family (1.4) overQ ([As1,
Lemma 6.3]).
2
(iv) Let n = 3, 4, 6, and E ±n → T the elliptic K3 surface constructed in [As2, 6.4]. Then
this is of hypergeometric type F 1n,n−1
n, 12(t).
The purpose of this paper is that for each α in (1.2), we describe the characteristic poly-
nomial (1.1) by a specific elliptic curve
Eα,s =
y2 = x(x− 1)(x− s) α = (12, 12, 12)
y2 = x3 + (3x+ 4− 4s)2 α = (13, 23, 12)
y2 = x(x2 − 2x+ 1− s) α = (14, 34, 12)
y2 = 4x3 − 3x+ 1− 2s α = (16, 56, 12).
(1.6)
Theorem 1.1 (Theorem 3.4) Let f : X → T be a K3 family of hypergeometric type Fα(t).Let p > 3 be a prime at which there is an integral regular flat model
fZ(p): XZ(p)
−→ TZ(p)
over the ring Z(p) ⊂ Q such that fZ(p)is smooth projective. Let a ∈ Zp such that a(1−a) 6≡ 0
mod p, and Xa the fiber at t = a. Let
Vet(Xa)Ql:= Coim[H2
et(Xa,Ql) → H2et(Xt,Ql)/NS(X t)⊗Ql] ∼= Q3
l .
Put b = 12(1 −
√1− a), and let Eα,b be the elliptic curve (1.6). Let 1 − ap2(Eα,b)T + p2T 2
be the characteristic polynomial of the p2-th Frobenius φp2 ∈ GQ, namely ap2(Eα,b) ∈ Z
satisfies
1− ap2(Eα,b) + p2 = ♯Eα,b(Fp2).
Let
dα =
−1 α = (12, 12, 12), (1
6, 56, 12)
−2 α = (14, 34, 12)
−3 α = (13, 23, 12)
and put
Aa,p =
ap2(Eα,b)√1− a ∈ Zp
(dαp)ap2(Eα,b)
√1− a 6∈ Zp, Eα,b: ordinary at p
2p√1− a 6∈ Zp, Eα,b: supersingular at p.
where (∗p) denotes the Legendre symbol. Then
det(1−φ−1p T | Vet(Xa)Ql
) =
(1−
(1− a
p
)χX /T (φp)pT
)(1−χX /T (φp)Aa,pT + p2T 2)
where χX /T : GQ → {±1} is the character (3.2) defined in §3.2.
Notice that Aa,p does not depend on the choice of b as Eα,b and Eα,1−b are isogenous over
Fp(b,√
dα) (see (2.37), . . . ,(2.40) below). For the families (i), . . . , (iv), the character χX /T
is trivial (Remark 3.9).
As a byproduct of the proof of Theorem 1.1, we have the following description of the
Galois representation over Q(√1− a).
3
Corollary 1.2 (Theorem 3.5) Let F be a number field and let a ∈ F \ {0, 1} be arbitrary.
Then there is an isomorphism
Vet(Xa)Ql∼= Sym2H1
et(Eα,b,Ql)⊗ χX /T (1.7)
of GF (√1−a)-representations.
If α = (12, 12, 12) or (1
6, 56, 12), we have an alternative description of the characteristic polyno-
mial by another elliptic curve,
Cα,t =
{y2 = x3 − 2x2 + t
t−1x α = (1
2, 12, 12)
y2 = 4x3 − 3(1− t)x+ (1− t)2 α = (16, 56, 12).
(1.8)
Theorem 1.3 (Theorem 3.6) Let α be either of (12, 12, 12) or (1
6, 56, 12). Then
det(1− φ−1p T | Vet(Xa)Ql
)
=
(1−
(1− a
p
)χX /T (φp)pT
)(1−
(1− a
p
)χX /T (φp)ap2(Cα,a)T + p2T 2
).
Hence, for arbitrary a ∈ Q \ {0, 1}, there is an isomorphism
Vet(Xa)Ql∼= Sym2H1
et(Cα,b,Ql)⊗ χX /T ⊗ χ1−a (1.9)
of GQ-representations where χ1−a denotes the Kronecker character for Q(√1− a).
There are lots of works concerning zeta functions of K3 surfaces or Calabi-Yau mani-
folds with hypergeometric functions, the author does not catch up all of them though. For
instance, there are a number of papers describing the zeta functions in terms of finite hyper-
geometric series, [Go1], [Go2], [Ko], [Mc], [Mi], [O] etc. On the other hand, the author finds
only a few papers which exhibit a characteristic polynomial of φp (not φpm for a particular
m) for all but finitely many p. Concerning the Dwork family (i), the Shioda-Inose structure
is provided by Elkies-Schutt [E-S], which imposes the Galois representation Vet(Xa)Qlpo-
tentially. Corollary 1.2 for X the Dwork family can be derived from Naskrecki [Na, Cor.6.7]
or Otsubo [O, Thm.7.4] where they discuss the family in a context of finite hypergeometric
series. However these results are not enough to determine the GQ-representation. As long
as the author sees, Theorem 1.1 is a new formula for the Dwork family. Concerning the K3
family (ii), the isomorphism (1.9) is proved in [AOP], and the Shioda-Inose structure (de-
fined over Q) is exhibited in [vG-T]. For this family, Theorem 1.3 is nothing new, while our
proof is entirely different from theirs.
For the proof of Theorems 1.1 and 1.3, we follow the argument of Dwork [Dw2]. The
key tool is the rigid cohomology (the Monsky-Washnitzer cohomology) (cf. [LS]). The
characteristic polynomial of Frobenius can be obtained from the Frobenius structure on the
rigid cohomology
H2rig(XFp
/TFp)
4
where XFp:= XZ(p)
×Z(p)Fp etc. We then compare it with the symmetric product
Sym2H1rig(Eα,Fp
/SFp)
in a direct way, where S = SpecQ[t,√1− t, (t− t2)−1] and Eα → S is the family of elliptic
curves Eα,s with s = 12(1 −
√1− t). See Theorem 2.8 for the detail. The comparison is
brought from the classical formula of Clausen ([NIST, 16.12.2])
3F2
(2a, 2b, a+ b
a+ b+ 12, 2a+ 2b
; t
)= 2F1
(a, b
a+ b+ 12
; t
)2
. (1.10)
The idea is sketched in [Dw2, p.92–93] for the Dwork family, while we take more thorough
discussion in this paper. Besides, to work out on the “sign” such as (dαp), we need addi-
tional argument that is not suggested in loc.cit. As a final comment, Otsubo’s approach is
comparable with ours. He obtains an analogue of Clausen’s formula in a context of finite
hypergeometric series ([O, Thm. 6.5]), and proves a similar (but weaker) result to Theorem
1.1 for the Dwork family ([O, Thm. 7.4]).
Acknowledgement. The author is grateful to Noriyuki Otsubo for the stimulating discussion
on the Dwork family and for encouraging him to write this paper.
2 Characteristic polynomial for Hypergeometric differen-
tial equations
Let p be a prime number. Let W = W (Fp) be the Witt ring of the algebraic closure Fp of Fp.
Let K = Frac(W ) be the fractional field.
2.1 F -isocrystals of hypergeometric differential equations
For α ∈ Zp, we denote by α′ the Dwork prime which is defined to be (α + l)/p where
l ∈ {0, 1, . . . , p− 1} such that a + l ≡ 0 mod p. We define α(i) := (α(i−1))′ with α(0) = α.
For α = (α0, α1, . . . , αn) ∈ Zn+1p , we denote α(i) = (α
(i)0 , α
(i)1 , . . . , α
(i)2 ). We write
Fα(t) = n+1Fn
(α0, . . . , αn
1, . . . , 1; t
)=
∞∑
i=0
(α0)ii!
· · · (αn)ii!
ti ∈ Zp[[t]]
the hypergeometric series where (α)i = α(α+1) · · · (α+ i−1) is the Pochhammer symbol,
and
Pα = Dn+1 − t(D + α0)(D + α1) · · · (D + αn), D := td
dt(2.1)
the hypergeometric differential operator whose solution is Fα(t).
Let n = 2 and α = (α0, α1, α2) ∈ Z3p. Let D = K[t, (t− t2)−1, d
dt] be the Wyle algebra,
and let
Vα := D/DPα = D ωα (2.2)
5
be a left D-module where the symbol ωα denotes 1D + DPα a generator of Vα. The module
Vα is a free K[t, (t− t2)−1]-module of rank 3,
Vα =
2⊕
i=0
K[t, (t− t2)−1]Diωα.
We denote by ∇ the connection on Vα induced by ddt
. Put
y2 = (1−t)Fα(t), y1 = −(α0+α1+α2)t
1− ty2−Dy2, y0 = −(α0α1+α1α2+α0α2)
t
1− ty2−Dy1
and letting α = (1− α0, 1− α1, 1− α2),
ωα =1
Fα(t)ωα (2.3)
ξα = −(1− t)α0+α1+α2−1 1
Fα(t)(DFα(t) · ωα − Fα(t) ·Dωα) (2.4)
ηα = y0ωα + y1Dωα + y2D2ωα. (2.5)
Since
ωα = Fαωα
Dωα = DFα · ωα + (1− t)1−α0−α1−α2Fα
Fα
ξα
D2ωα = −y−12 (y0Fα + y1DFα)ωα − (1− t)−α0−α1−α2
y1Fα
ξα + y−12 ηα
it turns out that {ωα, ξα, ηα} forms a free basis of W [[t]]⊗W [t] Vα. Let
Gα(t) := (1− t)1−α0−α1−α2Fα(t)−2Fα(t), Gα(t) := (1− t)α0+α1+α2−2Fα(t)
−2Fα(t),
then it is straightforward to see
Dωα = Gα(t)ξα, Dξα = Gα(t)ηα, Dηα = 0. (2.6)
Let σ be the p-th Frobenius on W [[t]] given by σ(t) = ctp with c ∈ 1 + pW . Letting A†
denote the weak completion of a ring A (e.g. [LS, p.135]), we put
V †α := K[t, (t− t2)−1]† ⊗K[t,(t−t2)−1] Vα. (2.7)
Lemma 2.1 Suppose αi ∈ Zp∩Q and 0 < αi < 1 for each i. Let m ≥ 1 be the least integer
such that α(m) and α agree up to permutation. Then there is unique σ-linear p-th Frobenius
endomorphism Φ on⊕m−1
i=0 V †α(i) satisfying pΦD = DΦ, Φ(V †
α(i)) ⊂ V †α(i−1) and
Φ(ωα(i)) ≡ p2ωα(i−1) mod 〈ξα(i−1) , ηα(i−1)〉. (2.8)
6
Proof. The uniqueness follows from [Dw2]. The existence follows from the fact that there is
a motive corresponding to ⊕V †α(i) . Such a motive is given in [As2]. Let
U = SpecW [t, (t− t2)−1][x0, x1, x2]/((1− xn00 )(1− xn1
1 )(1− xn22 )− t)
be an affine scheme over T = SpecW [t, (t− t2)−1]. Then for 0 < ik < nk,
V i0n0
,i1n1
,i2n2
∼= W2H2dR(UK/TK)(i0, i1, i2)
as D-module, and the Frobenius on
⊕W2H2rig(UFp
/TFp)(i0, i1, i2)
satisfies the conditions in the lemma ([As2, Theorems 3.3, 4.5]). �
Definition 2.2 We define an F -isocrystal
V crys(α, σ) =
(m−1⊕
i=0
Vα(i) ,
m−1⊕
i=0
V †α(i),∇,Φ, σ
).
Note V crys(α, σ) = V crys(α(i), σ). Define Frob(σ)α to be the 3× 3-matrix defined by
(Φ(ωα(1)) Φ(Dωα(1)) Φ(D2ωα(1))
)=(ωα Dωα D2ωα
)Frob(σ)α .
By a simple computation, it is possible to describe the matrix Frob(σ)α explicitly except the
“constant terms”. Let
Φ(ωα(1)) = p2ωα − p2E1(t)ξα + p2E2(t)ηα, (2.9)
Φ(ξα(1)) = pǫξα − pE3(t)ηα (2.10)
Φ(ηα(1)) = ǫ′ηα (2.11)
with ǫ, ǫ′ ∈ K and Ei(t) ∈ K[[t, t−1]]. Apply D on (2.9). It follows from (2.6) that we have
pGα(1)(tσ)Φ(ξα) = p2(Gα(t)− tE ′1(t))ξα mod 〈ηα〉.
This implies ǫ = 1 by (2.10), and
tE ′1(t) = Gα(t)−Gα(1)(tσ)
which characterizes the series expansion of E1(t) except the constant term. Apply D on
(2.10). We have
pGα(1)(tσ)Φ(ηα) = p(Gα(t)− tE ′3(t))ηX ,
which implies ǫ′ = 1 by (2.11) and
tE ′3(t) = Gα(t)−Gα(1)(tσ).
We again apply D on (2.9). We have
tE ′2(t) = Gα(t)E1(t)−Gα(1)(tσ)E3(t).
This implies E1(0) = E3(0). Summing up the above, we have
7
Proposition 2.3 We have ǫ = ǫ′ = 1 and Ei(t) ∈ K[[t]] satisfy
tE ′1(t) = Gα(t)−Gα(1)(tσ)
tE ′3(t) = Gα(t)−Gα(1)(tσ)
tE ′2(t) = Gα(t)E1(t)−Gα(1)(tσ)E3(t)
and E1(0) = E3(0).
2.2 Formula on Characteristic polynomial of Frob(σ)α
In what follows, let α be either of the following,(1
2,1
2,1
2
),
(1
3,2
3,1
2
),
(1
4,3
4,1
2
),
(1
6,5
6,1
2
). (2.12)
In each case, since α(1) and α agree up to permutation, the F -isocrystal V crys(α, σ) is of rank
3,
Vα =2⊕
i=0
K[t, (t− t2)−1]Diωα
The following is the main result in this section.
Theorem 2.4 Suppose p > 3. Let a ∈ Zp satisfy a(1−a) 6≡ 0 mod p. Put b = 12(1−
√1− a).
Let Eα,s be the elliptic curves as in (1.6). Let 1 − ap2(Eα,b)T + p2T 2 be the characteristic
polynomial of the p2-th Frobenius φ−1p2 ∈ GF
p2, namely ap2(Eα,b) ∈ Z satisfies
1− ap2(Eα,b) + p2 = ♯Eα,b(Fp2).
Let
dα =
−1 α = (12, 12, 12), (1
6, 56, 12)
−2 α = (14, 34, 12)
−3 α = (13, 23, 12)
and put
Aa,p =
ap2(Eα,b)√1− a ∈ Zp(
dαp
)ap2(Eα,b)
√1− a 6∈ Zp, Eα,b: ordinary at p
2p√1− a 6∈ Zp, Eα,b: supersingular at p.
Let σa(t) = a1−ptp. Then
det(I − Frob(σa)α |t=aT ) =
(1−
(1− a
p
)pT
)(1−Aa,pT + p2T 2).
By isogeny (2.37), (2.38), (2.39) and (2.40) in below, one finds ap2(Eα,b) = ap2(Eα,1−b), and
hence Aa,p does not depend on the choice of b.
The proof of Theorem 2.4 shall be given in §2.5.
8
2.3 Frobenius on the elliptic curve Eα,s
Let s = 12(1 −
√1− t), and S := SpecW [s, t, (t − t2)−1] → T a finite etale morphism of
degree 2. Let σ be the p-th Frobenius given by tσ = ctp as in §2.1. This extends on O(S)†
as follows,
σ(√1− t) =
√1− tσ (2.13)
=√1− t · (1− t)
p−12
√1− tσ
(1− t)p(2.14)
=√1− t · (1− t)
p−12
(1 +
1
2pw(t)− 1
8(pw(t))2 + · · ·
)(2.15)
where (1 − tσ)/(1 − t)p = 1 + pw(t). For α ∈ W such that (1 − 2α)α(1 − α) 6≡ 0mod p, one can define the evaluation at s = α of an element of K[t, s, (t − t2)−1]† or
Q ⊗ (W [t, s, (t − t2)−1]∧) the p-adic completion. For example, if t = a ∈ Zp satisfies
tσ|t=a = a, then for b = 12(1−
√1− a)
sσ|s=b =
{b
√1− a ∈ Zp
1− b√1− a 6∈ Zp.
(2.16)
Let
g : Eα −→ S
be the elliptic fibration whose general fiber is Eα,s. We denote the fiber at s = b by Eα,b, and
write Eα,K = Eα ×W K, Eα,Fp= Eα ×W Fp etc. Let
H•rig(Eα,Fp
/TFp)
be the rigid cohomology (cf. [LS]). The σ-linear p-th Frobenius endomorphismΦE is defined
on the rigid cohomology, and hence on
K[t, s, (t− t2)−1]† ⊗O(SK) H•dR(Eα,K/SK),
thanks to the comparison theorem. Since σ(s) = 4p−1csp + · · · ∈ spW [[s]], the Frobenius
σ extends on W [[s]] = W [[t]] in a natural way, which we write by the same notation. The
p-th Frobenius ΦE extends on W [[s]] ⊗O(S) H1dR(Eα,K/SK) as well. For each α, we choose
a Weierstrass equation
Eα,s : Y2 = 4X3 − g2(s)X − g3(s) (2.17)
such that g2(s), g3(s) ∈ W [s] satisfy ∆ := g32 − 27g23 ∈ sW [s] and g2(0)g3(0) 6= 0. We then
put
ωE ,α =dX
Y, ηE ,α =
XdX
Y.
9
Lemma 2.5 Let Ds = s dds
and put E := 2g2Dsg3 − 3Dsg2 · g3. Then
DsωE ,α = −Ds∆
12∆ωE ,α +
3E
2∆ηE ,α,
DsηE ,α = −g2E
8∆ωE ,α +
Ds∆
12∆ηE ,α.
Proof. This is well-known, e.g. [A-C, Theorem 7.1]. �
Lemma 2.6 Let DS = K[s, t, ddt] be the Wyle algebra of S. Let Ds := s d
ds. Then for
α = (α0, α1,12), we have isomorphisms
DS/DSPα0α1
∼=−→ H1dR(Eα,K/SK), 1DS
7−→ ωE ,α, (2.18)
O(SK)⊗O(T ) Vα = DS · ωα
∼=−→ Sym2H1dR(Eα,K/SK), ωα 7−→ ω2
E ,α (2.19)
of left DS-modules, see (2.1) for the definition of Pα0α1 and (2.2) for Vα.
Proof. We may replace K with C. It is a standard exercise to show that there is a homology
cycle γ ∈ H1(Eα,s,Q) that is a vanishing cycle at s = 0, such that
∫
γ
ωE ,α = 2πi 2F1
(α0, α1
1; s). (2.20)
This implies DSωE ,α ( H1dR(Eα,K/SK), and hence DSωE ,α = 0 as H1
dR(Eα,K/SK) is ir-
reducible. This shows that the arrow (2.18) is well-defined. Since both side of (2.18) are
irreducible, it turns out to be bijective.
Next we show (2.19). Employing Clausen’s formula (1.10) together with the transforma-
tion formula [NIST, 15.8.18], we have
3F2
(α0, α1,
12
1, 1; t
)= 2F1
(α0, α1
1; s)2
, s =1
2(1−
√1− t). (2.21)
Therefore we have ∫
γ⊗γ
Pα(ω2E ,α) = 0
and hence that the map
O(S)⊗O(T ) Vα∼= DS/DSPα −→ Sym2H1
dR(Eα,K/SK), A 7−→ A(ω2E ,α)
is well-defined. Since both sides are irreducible, this is bijective. �
10
Let us describe the Frobenius ΦE . Let W ((t))∧ be the p-adic completion of W ((t)) =W [[t]][t−1]. Let α = (α0, α1,
12). According to [A-M, §5.1], there is a basis (de Rham
symplectic basis)
ωα,E , ηα,E ∈ W ((t))∧ ⊗O(S) H1dR(Eα,K/SK)
which satisfies (cf. [A-C, Propositions 4.1, 4.2])
(Ds(ωE ,α) Ds(ηE ,α)
)=(ωE ,α ηE ,α
)( 0 0Dsqq
0
)(2.22)
where q ∈ W [[s]] is defined by j(Eα,s) = 1/q + 744 + 196884q + · · · , and
(ΦE (ωE ,α) ΦE (ηE ,α)
)=(ωE ,α ηE ,α
)( p 0−pτ (σ)(s) 1
)(2.23)
where τ (σ)(s) ∈ W [[s]] is the power series given in [A-C, Proposition 4.2]. The explicit
relation with the basis {ωE ,α, ηE ,α} is
ωE ,α =κ
Fα0,α1(s)ωE ,α, ηE ,α = −H(s)ωE ,α + κ−1Fα0,α1(s)ηE ,α, κ :=
√− g2(0)
18g3(0)(2.24)
where H(s) is the power series given in [A-C, Propositions 7.3].
We pick up the results which shall be necessary in our proof of Theorem 2.4.
Proposition 2.7 Let
(ΦE (ωE ,α) ΦE (ηE ,α)
)=(ωE ,α ηE ,α
)(pA0 B0
pA1 B1
). (2.25)
Then all of A0, . . . , B1 belong to W [s, t,√1− t]† and they satisfy
A0B1 −A1B0 = 1 (2.26)
and
B1 ≡ ǫFα0α1(s)
Fα0α1(sσ)
mod p, ǫ := F (κ)/κ = ±1. (2.27)
Proof. See [A-C, Propositions 4.3]. Note that (2.26) and (2.27) directly follow from the
explicit descriptions of A0, . . . , B1 in loc.cit. �
2.4 Comparison Theorem
Theorem 2.8 Under the isomorphism
O(SK)† ⊗O(TK ) Vα
∼= O(SK)† ⊗O(SK ) Sym
2H1dR(Eα,K/SK), (2.28)
induced from (2.19), the Frobenius Φ on the left agrees with Sym2ΦE on the right.
11
Proof. It follows from the main theorem of [Dw2] that they agree up to a constant c, namely
Φ = c Sym2ΦE . We show c = 1. To do this we employ the basis {ωE ,α, ηE ,α}. Since
〈ξα, ηα〉 = 〈Dωα, D2ωα〉 ⊂ W ((t))∧ ⊗O(T ) Vα
by (2.6), one finds that under the isomorphism (2.28), this corresponds to the subspace
〈D(ω2E ,α), D
2(ω2E ,α)〉 = 〈ωE ,αηE ,α, η
2E ,α〉 ⊂ W ((t))∧ ⊗O(S) Sym
2H1dR(Eα,K/SK)
where the above equality follows from (2.22). Hence the isomorphism (2.28) induces an
isomorphism
(W ((t))∧ ⊗O(T ) Vα)/〈ξα, ηα〉∼=
// (W ((t))∧ ⊗O(S) Sym2H1
dR(Eα,K/SK))/〈ωE ,αηE ,α, η2E ,α〉
W ((t))∧ ωα W ((t))∧ ω2E ,α
compatible with respect to Φ and c Sym2ΦE . It sends ωα to κ2ω2E ,α by (2.3) and (2.24)
together with (2.21). Since
Φ(ωα) ≡ p2ωα mod 〈ξα, ηα〉 by (2.8)
Sym2ΦE (ω2E ,α) ≡ p2ω2
E ,α mod 〈ωE ,αηE ,α, η2E ,α〉 by (2.23)
one has
c =F (κ2)
κ2= 1
as required. �
Corollary 2.9 Let Frob(σ)α be the matrix defined in Definition 2.2, and let A0, . . . , B1 be as
in (2.25). Let X be the 3× 3 matrix defined by
(ωα Dωα D2ωα
)=(ω2
E ,α ωE ,αηE ,α η2E ,α
)X
under the identification (2.28). Then
Frob(σ)α = X−1
p2A20 pA0B0 B2
0
2p2A0A1 p(A0B1 + A1B0) 2B0B1
p2A21 pA1B1 B2
1
Xσ
Proof. Immediate from Theorem 2.8. �
12
2.5 Proof of Theorem 2.4
Lemma 2.10 Let q = pm, and let φq = (φp)m ∈ GFp
be the q-th Frobenius. Let a ∈ W (Fq)satisfy that a(1−a) 6≡ 0 mod p. Put b = 1
2(1−
√1− a). Let σa be given by tσa = F (a)a−ptp
where F is the p-th Frobenius on W (Fq). Then
det(I − (Frob(σa)α )2m|t=aT ) = det(1− (φ2
q)−1T | Sym2H1
et(Eα,b,Fp,Ql)) (2.29)
where GFq2
acts on H•et(Eα,b,Fp
,Ql) as the reduction of Eα,b is defined over Fq2 . If√1− a ∈
W (Fq), then
det(I − (Frob(σa)α )m|t=aT ) = det(1− (φq)
−1T | Sym2H1et(Eα,b,Fp
,Ql)) (2.30)
where GFqacts on H•
et(Eα,b,Fp,Ql) as the reduction of Eα,b is defined over Fq.
Proof. Let K0 be the fractional field of W (Fq), and we denote by V †α,K0
the module (2.7)
replacing K with K0. Then Φ|t=a acts on V †α,K0
|t=a as a F -linear map, and the m-fold com-
position Φm|t=a is a K0-linear map providing the characteristic polynomial of (Frob(σa)α )m.
On the other hand, Φ2E|s=b acts on H1
dR(Eα,b/K0) as a F -linear map, and the K0-linear map
Φ2mE
|s=b provides the right hand side of (2.29). If√1− a ∈ W (Fq), then ΦE |s=b acts on
H1dR(Eα,b/K0), and Φm
E|s=b provides the right hand side of (2.30). Therefore the lemma
follows by Theorem 2.8. �
Lemma 2.11 Let a ∈ Zp satisfy that a(1− a) 6≡ 0 mod p, and b := 12(1−
√1− a). Then
det Frob(σ)α |s=b =
(1− a
p
)p3. (2.31)
Let A0, . . . , B1 be as in (2.25). If√1− a 6∈ Zp, then
TrFrob(σa)α |t=a =
(pA0 − B1)2|s=b + p α = (1
2, 12, 12), (1
6, 56, 12)
(p(A0 + 2A1)−B1)2|s=b + p α = (1
3, 23, 12)
(p(A0 − 13A1)−B1)
2|s=b + p α = (14, 34, 12).
(2.32)
Proof. One has det Frob(σ)α = p3(1− 2s)3/(1− 2sσ)3 by (2.26) and Corollary 2.9. So (2.31)
is immediate by (2.16). To show (2.32), we need an explicit computation of the matrix X .
For example, in case α = (12, 12, 12), one has
X =
1 −1
37s2−7s+118(2s−1)2
0 12s−1
−10s2+10s−13(2s−1)3
0 0 12(2s−1)2
by Lemma 2.5, and then (2.32) follows. For other α’s, the proof is similar (left to the reader).
�
13
Proof of Theorem 2.4. We want to compute the characteristic polynomial
det(I − Frob(σa)α |t=aT ) = (1− γ1T )(1− γ2T )(1− γ3T ) ∈ Z[T ].
If√1− a ∈ Zp, then Theorem 2.4 is immediate from (2.30). We suppose
√1− a 6∈ Zp until
the end of the proof. We have
γ1γ2γ3 = −p3 (2.33)
by (2.31). Let
det(1− φ−1p2 T | H1
et(Eα,b,Fp,Ql)) = (1− αp2(Eα,b)T )(1− βp2(Eα,b)T ) ∈ Z[T ]
be the characteristic polynomial for the p2-th Frobenius φp2 ∈ GFp2
. We know from (2.29)
that the triplet (γ21 , γ
22 , γ
23) agrees with (p2, αp2(Eα,b)
2, βp2(Eα,b)2) up to permutation. We
may let
γ1 = ±p, γ2 = εαp2(Eb), γ3 = ε′βp2(Eα,b) (2.34)
where ε, ε′ = ±1.
If Eα,b has a supersingular reduction at p, then ordp(αp2(Eα,b)) > 0 and ordp(βp2(Eb)) >0, and hence ordp(γi) > 0 for every i. By (2.32), there is e ∈ W such that
γ1 + γ2 + γ3 = e2 + p
which is zero modulo p, whence
γ1 + γ2 + γ3 ≡ p mod p2. (2.35)
Suppose γ1 = p. Then γ2 + γ3 ≡ 0 mod p2 by (2.35) and γ2γ3 = −p2 by (2.33). Since
γ2 + γ3 is a rational integer satisfying |γ2 + γ3| ≤ 2p, it turns out that γ2 + γ3 = 0 and
hence (γ2, γ3) = (p,−p) or (−p, p). Suppose γ1 = −p. Then γ2 + γ3 ≡ 2p mod p2 and
γ2γ3 = p2. Since |2p+np2| ≤ 2p⇔ n = 0 as p ≥ 5, one concludes γ2+ γ3 = 2p and hence
γ2 = γ3 = p. In both cases the triplet (γ1, γ2, γ3) agrees with (p, p,−p) up to permutation.
This completes the proof of Theorem 2.4 when Eα,b has a supersingular reduction.
The rest is the case that Eb has an ordinary reduction. We may let ordp(αp2(Eα,b)) = 0and ordp(βp2(Eα,b)) > 0. We first see that ε = ε′ in (2.34). Indeed, if ε 6= ε′, then
αp2(Eα,b)−βp2(Eα,b) is a rational integer as γ1+γ2+γ3 ∈ Z. Since αp2(Eα,b)+βp2(Eα,b) ∈Z, it turns out that αp2(Eα,b) is a rational integer satisfying |αp2(Eb)| = p, and hence
αp2(Eα,b) = ±p. This contradicts with the assumption ordp(αp2(Eα,b)) = 0. We thus
have ε = ε′. We want to show
ε =
(dαp
), (2.36)
which finishes the proof of Theorem 2.4 by (2.33). We denote f(s)<n =∑
i<n aisi the
truncated polynomial for a series f(s) =∑∞
i=0 aisi. It follows from (2.27) and (2.32) that
we have
εαp2(Eα,b) ≡(
Fα0α1(s)
Fα0α1(sσ)
)2 ∣∣∣∣s=b
≡ (Fα0α1(s)<p)2|s=b mod p
14
where the second congruence follows from the Dwork congruence ([Dw1, Theorem 3]). On
the other hand, Dwork’s unit root formula ([VdP, (7.14)]) yields
αp2(Eα,b) =Fα0α1(s)
Fα0α1(sσ2)
∣∣∣∣s=b
=Fα0α1(s)
Fα0α1(sσ)
Fα0α1(sσ)
Fα0α1(sσ2)
∣∣∣∣s=b
(2.16)=
Fα0α1(s)
Fα0α1(sσ)
∣∣∣∣s=b
· Fα0α1(s)
Fα0α1(sσ)
∣∣∣∣s=1−b
≡ Fα0α1(s)<p|s=b · Fα0α1(s)<p|s=1−b mod p
Therefore we have
ε ≡ Fα0α1(s)<p|s=b · (Fα0α1(s)<p|s=1−b)−1 mod p.
We compute the right hand side. To do this, we see the Cartier operator C for the reductions
of Eα,b and Eα,1−b respectively. Let α = (12, 12, 12). Recall that the elliptic curve Eα,s is
defined by the Weierstrass equation y2 = x(x− 1)(x− s). Then one has
C
(dx
y
)= (−1)
p−12 (F 1
2, 12(s)<p|s=b)
dx
y, C
(dx
y
)= (−1)
p−12 (F 1
2, 12(s)<p|s=1−b)
dx
y
for Eα,b and Eα,1−b respectively. Using an isomorphism
E 12, 12, 12,b : y
2 = x(x− 1)(x− b) −→ E ′12, 12, 12,1−b
: −y2 = x(x− 1)(x− 1 + b) (2.37)
(x, y) 7−→ (1− x, y),
it follows
ε ≡F 1
2, 12(s)<p|s=b
F 12, 12(s)<p|s=1−b
≡ (√−1)p/
√−1 =
(−1
p
)mod p.
This completes the proof of (2.36) in case α = (12, 12, 12). The proof in the other cases goes
in the same way with use of the isogeny
E 13, 23, 12,b : y
2 = y2 = x3 + (3x+ 4b)2 −→ E ′13, 23, 12,1−b
: −3y2 = y2 = x3 + (3x+ 4− 4b)2
(2.38)
(x, y) 7−→(x3 + 12x2 + 48bx+ 64b2
−3x2,x3 − 48bx− 128b2
9x3y
),
E 14, 34, 12,b : y
2 = x(x2 − 2x+ 1− b) −→ E ′14, 34, 12,1−b
: −2y2 = x(x2 − 2x+ b) (2.39)
(x, y) 7−→(−1
2x+ 1 +
b− 1
2x,y(x2 + b− 1)
4x2
)
E 16, 56, 12,b : y
2 = 4x3 − 3x+ 1− 2b −→ E ′16, 56, 12,1−b
: −y2 = 4x3 − 3x− 1 + 2b (2.40)
(x, y) 7−→ (−x, y).
This completes the proof of Theorem 2.4.
15
2.6 Cases α = (12,12,
12), (
16,
56,
12)
Theorem 2.12 Let α be either of (12, 12, 12), (1
6, 56, 12). Let Cα,t be the elliptic curves in (1.8).
Then
det(I − Frob(σa)α |t=a) =
(1−
(1− a
p
)pT
)(1−
(1− a
p
)ap2(Cα,a)T + p2T 2
)
for any a ∈ Zp such that a(1− a) 6≡ 0 mod p.
Proof. The proof goes in a similar way to that of Theorem 2.4 and it is simpler. We sketch
the outline. The details are left to the reader.
The key formula on the hypergeometric function is
3F2
(α0, α1,
12
1, 1; t
)= 2F1
( 12α0,
12α1
1; t
)2
(Clausen)
= (1− t)−α02F1
(12α0, 1− 1
2α1
1;
t
t− 1
)2
([NIST, 15.8.1]).
Let α = (12, 12, 12). Then
3F2
(12, 12, 12
1, 1; t
)= (1− t)−
12 2F1
(14, 34
1;
t
t− 1
)2
.
Since the “2F1” in the right is the period of the elliptic curve Cα,t, one can show an isomor-
phism
Vα∼= Sym2H1
dR(Cα,t/T )⊗ O(T )√1− t
of F -isocrystals in a similar way to the proof of Lemma 2.6, where O(T )√1− t denotes
the minus part H0dR(S/T )
− with respect to the involution in Aut(S/T ) ∼= Z/2Z. The rest is
automatic.
Let α = (16, 56, 12). Then
3F2
(16, 56, 12
1, 1; t
)= 2F1
(112, 512
1; t
)2
and the period of Cα,t is
(1− t)−14 2F1
(112, 512
1; t
).
Therefore one can show
Vα∼= Sym2H1
dR(Cα,t/T )⊗ O(T )√1− t
as well.
�
16
3 Galois representations of K3 surfaces of Hypergeometric
type
3.1 Lemmas on Galois representations of Kummer surfaces
For an abelian surface A over a field F of characteristic 0, the minimal resolution of the quo-
tient A/〈−1〉 is a K3 surface, which we write by Km(A) and call the Kummer K3 surface.
Then one finds that there is a natural bijection
H2et,tr(Km(A),Ql)
∼=−→ H2et,tr(A,Ql)
on the transcendental part H2et,tr(X,Ql) := H2
et(X,Ql)/NS(X)⊗Ql of the l-adic cohomol-
ogy.
Lemma 3.1 Suppose that F is a number field.
(1) The GF -representation H2et(Km(A),Ql) is semisimple.
(2) Let VQl⊂ H2
et(Km(A),Ql) and V ′Ql
⊂ H2et(Km(A′),Ql) be sub GF -representations.
Suppose that there is a set S of primes of F with density 1 such that
Tr(φ℘ | VQl) = Tr(φ℘ | V ′
Ql) (3.1)
for every ℘ ∈ S where φ℘ is the Frobenius at ℘. Then there is an isomorphism VQl∼=
V ′Ql
of GF -representations.
Proof. (1) follows from a theorem of Faltings which asserts that H1et(A,Ql) is semisimple.
(2) follows from the fact that VQland V ′
Qlare semisimple (cf. [Se, I, 2.3]). �
Lemma 3.2 Let X be a K3 surface over a number field F such that ρ(X) ≥ 19. Then Xis singular (i.e. ρ(X) = 20) if and only if dimQl
H2et(X,Ql(1))
GF ′ = 20 for some finite
extension F ′/F .
If we admit the Tate conjecture for cycles on X ×X of codimension 2, then we can drop the
condition “ρ(X) ≥ 19”.
Proof. By the assumption, X has the Shioda-Inose structure, namely there is an elliptic curve
E over a number field such that the Kummer surface Km(E × E) is isogenous to X. Then
X is singular if and only if E has a CM. The isogeny induces a surjective map
Sym2H1et(E,Ql) −→ H2
et(X,Ql)/NS(X)⊗Ql
of GL-representations for some finite extensionL/F . Suppose that dimQlH2
et(X,Ql(1))GF ′ =
20 for some F ′/F . We may assume L ⊂ F ′. We want to show ρ(X) = 20. Suppose
that ρ(X) = 19. Then since the above map is bijective, the fixed part (Sym2H1et(E,Ql) ⊗
Ql(1))GF ′ is 1-dimensional. This is equivalent to that dimQl
End(H1et(E,Ql))
GF ′ = 2. Since
End(E)⊗Ql∼= End(H1
et(E,Ql))GF ′ by a theorem of Faltings, this implies that E has a CM
and hence that X is singular. This is a contradiction. This completes the proof of the “if”
part. The “only if” part is obvious. �
17
3.2 K3 surfaces of Hypergeometric type
Let
f : X −→ T = SpecQ[t, (t− t2)−1]
be a projective smooth family of K3 surfaces. Let D = Q[t, (t − t2)−1, ddt] be the Wyle
algebra of T . Put X t := X ×T Q(t) and set
VdR(X /T ) = Coim[H2dR(X /T ) → H2
dR(X t)/NS(X t)⊗Q(t)]
a D-module which is free of finite rank over O(T ).
Definition 3.3 We call f of hypergeometric type Fα(t) if the following conditions hold.
(i) ρ(X t) = 19 or equivalently VdR(X /T ) is of rank 3 over O(T ).
(ii) There is an isomorphism VdR(X /T ) ∼= D/DPα of left D-modules, where Pα is the
hypergeometric differential operator (2.1).
Let α be either of the following as before,
(1
2,1
2,1
2
),
(1
3,2
3,1
2
),
(1
4,3
4,1
2
),
(1
6,5
6,1
2
).
Let p > 3 be a prime and put W = W (Fp) and K = FracW . Suppose that there is an
integral regular flat model
fZ(p): XZ(p)
−→ TZ(p)
over the ring Z(p) ⊂ Q such that fZ(p)is smooth projective. For a ∈ Z(p) such that a(1 −
a) 6≡ 0 mod p, we denote by Xa the fiber at t = a. We denote XW = XZ(p)×Z(p)
W ,
XK = XW ×W K, . . . as before. Recall from Definition 2.2 the F -isocrystal
V crys(α, σ) =(Vα, V
†α ,∇,Φ, σ
).
Thanks to the comparison
K[t, (t− t2)−1]† ⊗O(TK ) H2dR(X /T ) ∼= H2
rig(XFp/TFp
)
of the de Rham and rigid cohomology, the p-th Frobenius on
K[t, (t− t2)−1]† ⊗O(TK ) VdR(XK/TK)
is defined, which we denote by ΦX . We fix an isomorphism VdR(X /T ) ∼= D/DPα (this
is unique up to scalar as both sides are irreducible). It follows from [Dw2] that there is a
constant εp such that
Φ = εpΦX .
18
Let a ∈ Z(p) such that a(1 − a) 6≡ 0 mod p and let σ = σa given by σa(t) = a1−ptp. By
Theorem 2.4, the characteristic polynomial of ΦX |t=a on VdR(Xa/Qp) is
(1−
(1− a
p
)pεpT
)(1− εpAa,pT + (εppT )
2).
Since this is a polynomial with coefficients in Z and p± Aa,p 6= 0, it turns out that εp ∈ Q×
(not depend on a). Moreover since (det VdR(Xa/Qp))⊗2 is isomorphic to the Tate object
Qp(−6), one has ε6p = 1 and hence εp = ±1. Using the l-adic cohomology,
Vet(Xa)Ql:= Coim[H2
et(Xa,Ql) → H2et(X t,Ql)/NS(X t)⊗Ql], Xa := Xa ×Z(p)
Q,
one has
det(φ−1p | (det Vet(Xa)Ql
)⊗Ql(3)) = ε3p
(1− a
p
)= εp
(1− a
p
).
Define a quadratic character χX /T by
χX /T : GQ −→ Aut((detVet(Xa)Ql)⊗Ql(3)⊗ χ1−a) ∼= Q×
l (3.2)
where χ1−a denotes the quadratic character for Q(√1− a). This does not depend on either
a or l, takes value in {±1} and satisfies χX /T (φp) = εp for almost all p.
Summing up the above, we have
Theorem 3.4 Let the notation and assumption be as above. Then
det(1−φ−1p T | Vet(Xa)Ql
) =
(1−
(1− a
p
)χX /T (φp)pT
)(1−χX /T (φp)Aa,pT +p2T 2).
Theorem 3.5 Let F be a number field. Let a ∈ F \ {0, 1} be arbitrary. Then there is an
isomorphism
Vet(Xa)Ql∼= Sym2H1
et(Eα,b,Ql)⊗ χX /T
of GF (√1−a)-representations where Eα,b is the elliptic curve in Theorem 2.4.
Proof. By virtue of Lemma 3.1 (2), it is enough to show that
Tr(φ℘ | Vet(Xa)Ql) = Tr(φ℘ | Sym2H1
et(Eα,b,Ql)⊗ χX /T ).
for almost all primes ℘ of F (√1− a). However this follows from Lemma 2.10 (2.30). �
In the case α = (12, 12, 12) or (1
6, 56, 12), the same discussion replacing Theorem 2.4 with
Theorem 2.12 yields the following.
Theorem 3.6 Let α be either of (12, 12, 12) or (1
6, 56, 12). Then, for a ∈ F \ {0, 1}, there is an
isomorphism
Vet(Xa)Ql∼= Sym2H1
et(Cα,a,Ql)⊗ χX /T ⊗ χ1−a
of GF -representations where Cα,a is the elliptic curve in Theorem 2.12.
19
Theorem 3.7 Let a ∈ Q \ {0, 1}. Then Xa is singular (i.e. the Picard number 20) if and
only if Eα,b has a CM.
Proof. Immediate from Theorem 3.5 and Lemma 3.2. �
It is known that there are 13 rational CM j-invariants, and 58 CM j-invariants that are
quadratic numbers but not rational ([D-LR]). The complete list of them is available by
SAGE. We thus have the complete list of rational a’s such that Xa is singular.
a ∈ Q \ {0, 1} such that Xa is singular
α =
(1
2,1
2,1
2
)−1, 4,
1
4,−8,−1
8, 64,
1
64
α =
(1
3,2
3,1
2
)−4,
1
2,−1
24,−1
210,−32
24,33
24,2
33,33
2,−1
245,4
53,−1
2456,
−1
23337
α =
(1
4,3
4,1
2
) −1
22,1
32,1
34,−1
243,−24
32,−1
2234,25
34,28
34,
−1
26345,1
74,
−1
223474,−28
3472,
1
34112,
1
38114
α =
(1
6,5
6,1
2
) −1
29,−33
29,22
53,−26
53,
−1
21253,−32
2953,33
53,23
113,
−1
2953113,
26
53173,
−1
21253233293
Remark 3.8 It is natural to expect that the Shioda-Inose structure ofXa is given by Km(Eα,b×Eα,b), the author does not have a proof though.
Remark 3.9 For the K3 families (i), . . . , (iv) in §1, the character χX /T is trivial. For exam-
ple, let X → T be the family (iii). There is an isomorphism ([As1, Corollary 3.6])
D/DPα
∼=−→ VdR(X /T ), A 7−→ Aω1,1,1
where ωi,j,k is the regular 2-form defined in [As1, (2.7)]. Letting ω := ω1,1,1, we construct a
basis {ω, ξ, η} in the same as in §2.1. Then it is shown that
ΦX (ω) ≡ p2ω mod 〈ξ, η〉
in the proof of [As2, Theorem 4.5]. This shows εp = 1, namely χX /T is trivial. For the other
K3 families, one can show that χX /T is trivial in the same way (the argument in the proof of
[As2, Theorem 4.5] works).
20
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