ZETA FUNCTIONS OF AN INFINITE FAMILY OF K3 ... - CiteSeerX
ZETA FUNCTIONS OF AN INFINITE FAMILY OF K3 SURFACES SCOTT AHLGREN, KEN ONO AND DAVID PENNISTON
1. Introduction Given a projective variety defined over a number field, a central problem in number theory is to write down all of its local zeta functions. The case of elliptic curves over Q has recently been settled, and here we study the problem for K3 surfaces, a natural two-dimensional analog of elliptic curves. This is in general a difficult problem, since at any prime, the nontrivial factor of the local zeta function of a K3 surface is a polynomial of degree as high as 22. An important work in this area is due to Shioda and Inose. If a K3 surface has Picard number 20 (the largest it can be), they show in [I-S] that its L-function is essentially the symmetric square of the L-function of an elliptic curve with complex multiplication. Moreover, if the surface is defined over Q, its L-function is the L-function of a weight 3 Hecke eigenform with complex multiplication. We shall refer to such surfaces as modular. Although this result is general, the problem of computing explicit examples is nontrivial. In fact, only a few examples exist in the literature (see, for example, [B-S], [K-T], [L]), and they were obtained by a wide variety of methods, both geometric and transcendental. There are several natural difficulties. For instance, there is the problem of identifying suitable candidates for such surfaces. Then there is the issue of demonstrating that the L-function of an alleged modular surface agrees with the L-function of an explicit weight 3 modular form. Work of Morrison [M] suggests that one can obtain similar arithmetic results for K3 surfaces with Picard number ≥ 19. In this vein we show, for a certain family of K3 surfaces, that the general problem of computing zeta functions can be quite simple, and involve little more than elementary character sums. To clarify the merit of such an approach, we note that our family includes, up to isogeny and quadratic twist, all of the modular examples alluded to above. We consider the one parameter family {Xλ }, for λ ∈ Q \ {0, −1}, of K3 surfaces whose function fields are given by Xλ
s2 = xy(x + 1)(y + 1)(x + λy)
:
(1)
in relation to the family of elliptic curves Eλ
:
y 2 = (x − 1) x2 −
1 λ+1
.
For convenience, if p is an odd prime, then let φp (x) denote the Legendre symbol
(2) x p
.
Date: May 9, 2000. The first two authors thank the National Science Foundation for its generous support. The second author also thanks the Alfred P. Sloan Foundation and the David and Lucile Packard Foundation for their support. 1
2
SCOTT AHLGREN, KEN ONO AND DAVID PENNISTON
Theorem 1.1. If λ ∈ Q \ {0, −1} and p is an odd prime at which Eλ has good reduction, then the local zeta function of Xλ at p is Z(Xλ /Fp , T ) =
(1 − T )(1 −
p2 T )(1
−
pT )19 (1
1 , 2 − pT )(1 − πλ,p T )(1 − π 2λ,p T )
where πλ,p and π λ,p are the eigenvalues of the Frobenius at p on Eλ , and = φp (λ + 1). In view of Theorem 1.1, it is easy to characterize those λ for which Xλ is modular. Theorem 1.2. If λ ∈ Q \ {0, −1}, then the surface Xλ is modular if and only if λ ∈ {1, 8, 1/8, −4, −1/4, −64, −1/64}. For completeness, we note that Beukers and Stienstra [B-S] show that X−1 is also a modular K3 surface. This follows easily from Theorem 1.2 since X−1 is a quadratic twist of both X8 and X1/8 . Even though E−1 is singular, the modularity of X−1 can be proved using a minor modification of the arguments contained here. We prefer to omit them for brevity. In §5 we shall show that all of the surfaces in Theorem 3 are related to weight 3 newforms that are products of Dedekind’s eta-function. It is interesting to note that these forms constitute the complete list of weight 3 newforms which are eta-products. Proving Theorem 3 is tantamount to computing the Picard number of all of the Xλ . Finding the Picard number of a K3-surface is in general a difficult problem, and is often equivalent to calculating the rank of an elliptic curve over a function field. In our situation, the computation follows easily from a well known theorem of Ribet [R]. 2. A theorem on character sums In this section we prove a theorem about character sums which is essential for all of the results in this paper. Suppose that λ ∈ Q \ {0, −1} and that p is an odd prime such that λ 6≡ 0, −1 (mod p). If q = pr , then let φq be the extended Legendre symbol on Fq (we will often write φ = φq for simplicity). Further, define quantities a(λ, q) and A(λ, q) by X 1 a(λ, q) := − , φq (x − 1) x2 − λ+1 x∈Fq
A(λ, q) :=
X
(3)
φq (xy(x + 1)(y + 1)(x + λy)).
x,y∈Fq
Theorem 2.1. If λ ∈ Q, and λ 6≡ 0, −1 (mod p), then A(λ, q) = φq (λ + 1)(a(λ, q)2 − q). Before starting the proof, we mention that results of this type have been obtained by Greene [G] in the context of finite field analogues of hypergeometric functions. We will denote by χ, ψ, and A multiplicative characters on Fq , and by the trivial character on Fq (we agree P that (0) = 0). Given characters χ and ψ, we define the usual Jacobi sum J(χ, ψ) := x∈Fq χ(x)ψ(1 − x). Throughout this section, all sums (unless otherwise noted) will run over all elements or over all characters of Fq . Throughout, we will repeatedly use the following facts, which are valid for any characters. For the first, see [B-E-W] (recall our convention that (0) = 0). The second and third are easy consequences of orthogonality.
ZETA FUNCTIONS OF AN INFINITE FAMILY OF K3 SURFACES
3
J(χ, ψ) = χ(−1)J(χ, χψ). ( X q − 1 if x = 1 χ(x) = 0 else. χ ( X 1 if x = 0 1 ψ(1 − x) = q−1 J(ψ, χ)χ(x) + 0 else. χ
(4) (5)
(6)
We begin with an easy lemma. If λ ∈ Q and λ 6≡ 0, −1 (mod p), then define X 1 λ g(λ) := q−1 J(χ, φ)J(χ, φχ)χ( 4(λ+1) ). χ −λ Lemma 2.2. If λ ∈ Q, λ 6≡ 0, −1 (mod p), then −φ(2)a(q, λ) = g(λ) + φ( λ+1 ).
Proof. Expanding the Jacobi sums and using (5) gives X X 4(λ+1)xy(1−y) X 1 g(λ) = q−1 φ(1 − x)φ(1 − y) χ( ) = φ(1 − y)φ(1 − λ x,y
χ
λ ). 4(λ+1)y(1−y)
y6=0,1
Simplifying the last sum (and accounting for the term y = 1), we find that X λ −λ − φ( λ+1 g(λ) = φ(y)φ 4y(1 − y) − λ+1 ). y
The lemma follows after the change of variables y →
1−y . 2
To prove Theorem 2.1, we must expand the quantity g(λ)2 . In the course of the computation, several error terms will arise. Since it is straightforward, we will delay evaluation of these terms (which will be denoted by E1 , E2 , etc.) until the end. Expanding g(λ)2 as a sum over χ and ψ and making the change ψ → ψχ, we find that X 1 λ ψ J(χ, φ)J(χ, φχ)J(ψχ, φ)J(ψχ, φψχ). g(λ)2 = (q−1) 2 4(λ+1) χ,ψ
Expanding the final Jacobi sum gives X X 1 λ ψ( 4(λ+1) )ψ(x)φψ(1 − x) J(χ, φ)J(ψχ, φ)J(χ, φχ)χ(x)χ(1 − x). g(λ)2 = (q−1) 2 χ
ψ,x
(7)
Notice that (replacing w with y/w in the second step) we have X X J(χ, φ)J(ψχ, φ) = χ(w)φ(1 − w)ψχ(y)φ(1 − y) = φχ(w)φ(w − y)ψ(y)φ(1 − y). w,y
w,y
Rearranging and using (6), this becomes X
φχ(w)φ(1 −
1−w )ψ(y) 1−y
=
1 q−1
w,y y6=1
X w,y y6=1
=
1 q−1
X A
φχ(w)ψ(y)
X
J(A, φ)A( 1−w )+ 1−y
A
J(A, φ)J(φχ, A)J(ψ, A) +
X y6=1
X y6=1
ψ(y).
ψ(y)
4
SCOTT AHLGREN, KEN ONO AND DAVID PENNISTON
This together with (7) shows that g(λ)2 is given by the expression X X X 1 λ ψ( )ψ(x)φψ(1 − x) J(A, φ)J(ψ, A) J(φχ, A)J(χ, φχ)χ(x)χ(1 − x) + E1 , 3 (q−1) 4(λ+1) χ (8) A ψ,x where E1 :=
1 (q−1)2
X
λ ψ( 4(λ+1) )ψ(x)φψ(1 − x)
X
J(χ, φχ)χ(x)χ(1 − x)
χ
ψ,x
X
ψ(y).
(9)
y6=1
By (7), we may assume that x 6= 0, 1. Therefore the sum on χ in (8) can be written X X X w(1−w) x x φ( 1−x = (q − 1)φ( φ(w). ) A(1 − y)φ(w) φχ x(1−x)y ) A 1 − w(1−w) 1−x x(1−x) w,y w6=0,1
χ
w6=1
x When w = 1, the summand above equals (q − 1)φ( 1−x ). Accounting for this term, making the change of variables w → wx and factoring, the last displayed equation becomes X wx x (q − 1)φ(1 − x) φ(w)A(1 − w)A(1 − 1−x ) − (q − 1)φ( 1−x ). w
Together with (8) this yields g(λ)2 = E1 + E2 + X X X 1 λ ψ( )ψ(x)ψ(1 − x) J(A, φ)J(ψ, A) φ(w)A(1 − w)A(1 − 2 (q−1) 4(λ+1) ψ,x
w
A
where E2 :=
−1 (q−1)2
X
λ ψ( 4(λ+1) )ψφ(x)ψ(1 − x)
ψ,x
wx ), 1−x
X
J(A, φ)J(ψ, A).
(10)
(11)
A
Using (6), the sum on A in (10) can be written X X wx φ(1 − z)φ(w) J(ψ, A)A( 1z (1 − w)(1 − 1−x ) w,z w6=1,z6=0
A
X
= (q − 1)
φ(1 − z)φ(w)ψ 1 − z1 (1 − w)(1 −
wx ) 1−x
− (q − 1)
w,z w6=1,z6=0
X
φ(1 − z)φ( 1−x ). x
z6=0
Replacing z by (1 − w)/z and simplifying, this becomes X wx (q − 1) φ(z)φ(z − 1 + w)φ(w)ψ 1 − z(1 − 1−x ) + (q − 1)φ( 1−x ) x w,z w6=1
= (q − 1)
X
φ(z)φ(z − 1 + w)φ(w)ψ 1 − z(1 −
wx ) 1−x
+ F (ψ, x),
(12)
w,z z6=1
where F (ψ, x) contains the error arising from the terms w = 1 and z = 1; i.e. X X x F (ψ, x) = (q − 1)φ( 1−x ) − (q − 1) ψ 1 − z(1 − ) + (q − 1) ψ x 1−x z6=0
w
wx 1−x
. (13)
ZETA FUNCTIONS OF AN INFINITE FAMILY OF K3 SURFACES
5
Making the change of variables w → (1 − z)w and applying (6), the sum in (12) becomes X wzx (q − 1) φ(z)φ(−1 + w)φ(w)ψ(1 − z)ψ 1 + 1−x ) + F (ψ, x) w,z
=
X
φ(z)φ(−1 + w)φ(w)ψ(1 − z)
X
w,z
=
X
J(χ, ψ)χ( −wzx ) + F (ψ, x) 1−x
χ x J(χ, φ)J(ψ, χ)J(ψ, φχ)χ( x−1 ) + F (ψ, x).
χ
With (10) this yields X 1 λ g(λ)2 = (q−1) ψ( 4(λ+1) )χ(−1)J(ψχ, ψχ)J(ψ, χ)J(χ, φ)J(ψ, φχ) + E1 + E2 + E3 , 2 ψ,χ
where E3 :=
1 (q−1)2
X
λ ψ( 4(λ+1) )ψ(x)ψ(1 − x)F (ψ, x).
(14) (15)
ψ,x
To simplify the main term in (14), we apply (4) to find that X χ(−1)(ψχ, ψχ)J(ψ, χ) = J(ψχ, ψ 2 )J(ψ, χψ) = ψχ(w)ψ 2 (1 − w)ψ(1 − y)ψχ(y). w,y
Making the change w → X
w y
and simplifying, this becomes
ψχ(w)ψ 2 (y − w)ψ(1 − y) =
w,y y6=0
X
ψχ(w)ψ 2(y − w)ψ(1 − y) −
X
w,y w6=1
ψχ(w) +
w
X y
ψ(1 − y). (16)
For ease of notation, define a function δ by ( 0 if χ 6= δ(χ) := 1 if χ = . Making the change y → 1 − (1 − w)y and simplifying, the expression in (16) becomes ψ(−1)J(ψχ, ψ)J(ψ, ψ) − (q − 1)δ(ψχ) + (q − 1)δ(ψ). It is well-known (see[B-E-W]) that ψ(4)J(ψ, ψ) = J(ψ, φ) + (q − 1)δ(ψ). This, together with (14) and the last displayed equation, shows that X X 1 −λ g(λ)2 = (q−1) J(χ, φ) ψ( λ+1 )J(ψ, φ)J(ψχ, ψ)J(ψ, φχ) + E1 + E2 + E3 + E4 , 2 χ (17) ψ where E4 :=
−1 q−1
X λ ψ( 4(λ+1) )J(ψ, φ)J(ψ, φψ) +
1 q−1
X J(χ, φ)J(, φχ) + χ
ψ
1 q−1
X χ
J(χ, φ)J(χ, )J(, φχ). (18)
The main term in (17) is X X X (λ+1)wyz 1 J(χ, φ) φ(1 − w)φχ(1 − y)χ(z) ψ −λ(1−z) . 2 (q−1) χ
w,y,z wyz6=0
ψ
(19)
6
SCOTT AHLGREN, KEN ONO AND DAVID PENNISTON
Notice that w→
z 1−z
=
−λ (λ+1)wy
if and only if z =
−λ λ+1 λ wy− λ+1
. Using this fact, making the changes
y y → y−1 , and then applying (6), the main term becomes X X 1 wλ J(χ, φ) φ(1 − λ+1 )φχ(1 − y)χ(1 − wy) (q−1)
wλ , λ+1
w,y wy6=0
χ
1 (q−1)
=
X
J(χ, φ)
1 (q−1)
X
=
φ(1 −
wλ )φχ(1 λ+1
w,y wy6=0
χ
=
X
X
φ(1 −
wλ )φ(1 λ+1
− y)
X
w,y wy6=0
− y)χ(1 −
wy ) y−1
1 J(χ, φ)χ( 1−y+wy )
χ
φ(1 −
wλ )φ(1 λ+1
− y)φ(1 − y(1 − w))φ(−y)φ(1 − w)
φ(1 −
wλ )φ(1 λ+1
− y)φ(1 − y(1 − w))φ(−y)φ(1 − w) + φ(−1).
w,y w6=0
X
=
w,y
Taking w to 1 − and then replacing w, y by −w, −y, this becomes X φ(1 + λ) φ(1 + wλ )φ(1 − y)φ(1 − w)φ(−w) + φ(−1) = φ(1 + λ)A(λ) + φ(−1). y w y
w,y
Together with (17), this yields g(λ)2 = φ(1 + λ)A(λ) + E1 + E2 + E3 + E4 + φ(−1). A calculation (see below) shows that −λ E1 + E2 + E3 + E4 = q − 1 − φ(−1) − 2φ( λ+1 )g(λ).
(20)
Together with Lemma 2.2, the last two formulae give the statement in Theorem 2.1; i.e. a(λ, q)2 = q + φ(1 + λ)A(λ). To finish the proof, it is necessary to verify (20); we will only provide details for the computation of E1 (which is given by (9)). We have X X X w(1−w) J(χ, φχ)χ(x)χ(1 − x) = φ(w) χ x(1−x) . χ
χ
w6=1
The argument of χ equals one when w = x or w = 1 − x. Taking into account the case when x = 12 , we find that X J(χ, φχ)χ(x)χ(1 − x) = (q − 1) φ(x) + φ(1 − x)(x − 12 ) . χ
P Returning to (9), we note that y6=1 ψ(y) = −1 + (q − 1)δ(ψ). Therefore, X X X X −1 λ 1 λ ψ( 4(λ+1) )J(φψ, φψ) − q−1 ψ( 4(λ+1) )ψ(x)ψ(1 − x) + 1+ φ(x)φ(1 − x) E1 := q−1 ψ
=
−1 q−1
X ψ
ψ
λ 4(λ+1)
x6= 12 ,0,1
ψ,x x6= 21
J(φψ, φψ) −
1 q−1
X ψ
ψ
λ 4(λ+1)
x
J(ψ, ψ) + q − 3 − φ(−1).
ZETA FUNCTIONS OF AN INFINITE FAMILY OF K3 SURFACES
7
In the second sum we have used the fact that the summand is zero when x = 12 . For the remaining error terms (which are given by (11), (15), (18)), we use similar arguments to find that −λ E2 = −φ( λ+1 )g(λ), X X 1 λ 1 λ E3 = 1 + q−1 J(φψ, φψ) + q−1 J(ψ, ψ), ψ 4(λ+1) ψ 4(λ+1)
ψ
ψ
E4 = 1 −
−λ φ( λ+1 )g(λ).
Equation (20) follows, and the proof of Theorem 2.1 is complete. 3. The Geometry of these K3 surfaces In this section and the next we follow closely the exposition in [B-S]. Let K denote either Q or Fp (p an odd prime). For each λ ∈ K \ {0, −1}, we define the surface Xλ /K to be the smooth complete model of the double cover of P2 branched over the union Uλ of six lines given by Uλ : XY Z(X + λY )(Y + Z)(Z + X) = 0. To see that Xλ is a K3 surface, one can refer to [B-P-V], or check that the elliptic fibration of Xλ we give below satisfies the conditions given in [B-S], §4. We now give a description of the natural map ψ : Xλ → P2 . Let Bλ be the surface obtained in the following way: first blow up P2 at the triple points of Uλ , then blow up the resulting surface at each of the double points of the total transform of Uλ . The total transform Tλ ⊂ Bλ of Uλ consists of a connected union of rational curves whose only singularities are ordinary double points, and at each of these points a component of odd multiplicity and one of even multiplicity meet. The branch locus Uλ and the dual graph of Tλ are depicted here (• denotes a component of odd multiplicity, and denotes a component of even multiplicity): 22 *? •**?? 2 ? *** ??? ** ??? ?? • • ? •P***PPPP • P P ** 2 • •22 ** 22 ** 22 * • •
2 222 FF 22 F F FF 22 FF mm22m Fm mmmmm FFFF222 m FF22 mmm F2F m m m 2
Figure 1 Then Xλ is the double cover of Bλ branched above the components of Tλ having odd multiplicity. Moreover, ψ −1 (Uλ ) is a union of rational curves whose dual graph is exactly that of Tλ , and Xλ \ ψ −1 (Uλ ) is a smooth affine surface given by the equation s2 = xy(x + 1)(y + 1)(x + λy).
(21)
Now let us show that the Xλ are elliptic surfaces over P1 . For a fixed λ, consider the family of cubics (with parameter τ ) in P2 given by X(Y + Z)(Z + X) − τ (X + λY )Y Z = 0.
(22)
8
SCOTT AHLGREN, KEN ONO AND DAVID PENNISTON
One can check that the generic cubic in this family intersects Uλ at all of its triple points, and at the three double points (1, 0, −1), (λ, −1, 1) and (λ, −1, −λ). At the double points, the cubic is transversal to both of the components of Uλ meeting there, and at the triple points is transversal to two of the components meeting there and tangential to the third. It follows that the proper transform of the generic cubic in Bλ intersects only components of Tλ of even multiplicity, and therefore the family (22) lifts through ψ to Xλ . To see the elliptic fibration explicitly, the change of variables X = λt2 V W,
Y = U 2 + λt2 UW,
Z = UV
(where τ = t2 , reflecting the double cover) transforms (22) into Cλ : V 2 W + (1 − t2 )UV W + λt2 V W 2 = U 3 + λt2 U 2 W.
(23)
The discriminant of Cλ is −λ3 t8 [t6 + (8λ − 3)t4 + (16λ2 + 20λ + 3)t2 − (λ + 1)]. Using Tate’s algorithm, we find that the singular fibers are generically of the following types: I10 I8 I1 I1 I1 I1 I1 I1 .
(24)
The singular fibers above t = ∞ and t = 0 are visible in the dual graph of Tλ (see Figure 1) as the outer and inner cycles, respectively. The N´eron-Severi group of Xλ /K is the group of algebraic equivalence classes of divisors on Xλ . Shioda [S] showed that the rank rλ of the N´eron-Severi group of Xλ /K is given by X (mj − 1) + 2 + rank(Cλ (K(t)), rλ = where the sum is over the singular fibers, and mj is the number of components of the singular fiber j not meeting the zero section. In particular, he showed that the equivalence classes of the generic fiber, the zero section and the components of the singular fibers not meeting the zero section are linearly independent elements of the N´eron-Severi group modulo torsion. From (24), then, these give 18 linearly independent elements. Also note that (0, 0) is an element of Cλ (K(t)), and one can check that it has infinite order. Therefore rλ ≥ 19. 4. Proof of Theorem 1.1 Given a prime p, the zeta function of a projective variety Y /Fp is ! ∞ X Z(Y /Fp, T ) = exp #Y (Fpn ) · T n /n . n=1
The following proposition is our first step in proving Theorem 1.1. Proposition 4.1. If p is an odd prime, λ ∈ Fp \ {0, −1}, and q = pr , then X #Xλ (Fq ) = 1 + q 2 + 19q + φq (xy(x + 1)(y + 1)(x + λy)). x,y∈Fq
Proof. Recall (see §3) that we have a map ψ : Xλ → P2 ramified above Uλ , that Xλ \ ψ −1 (Uλ ) is a smooth affine surface given by (21) and that ψ −1 (Uλ ) is a union of rational curves whose dual graph is given in Figure 1. Since all the components of Uλ are lines defined over Fp , we can read from Figure 1 that the number of Fq -rational points on ψ −1 (Uλ ) is 9(q + 1) + 15(q − 1) = 24q − 6.
ZETA FUNCTIONS OF AN INFINITE FAMILY OF K3 SURFACES
9
Moreover, from (21) we see that the number of Fq -rational points on Xλ \ψ −1 (Uλ ) with s = 0 is 5q − 7. Therefore, we have that X #Xλ (Fq ) = (φq (xy(x + 1)(y + 1)(x + λy)) + 1) + (24q − 6) − (5q − 7). x,y∈Fq
Recall that if Eλ (λ ∈ Q) has good reduction at p, then the zeta function of Eλ at p is Z(Eλ /Fp , T ) =
(1 − πλ,p T )(1 − π λ,p T ) 1 − a(λ, p)T + pT 2 = . (1 − T )(1 − pT ) (1 − T )(1 − pT )
Then Proposition 4.1 and Theorem 2.1 imply Theorem 4.2. If λ ∈ Q \ {0, −1} and Eλ has good reduction at a prime p, then #Xλ (Fp ) = 1 + p2 + 19p + φp (λ + 1) a(λ, p)2 − p and #Xλ (Fp2 ) = 1 + p4 + 18p2 + a(λ, p)2 − 2p
2
.
(25) (26)
Now let us prove Theorem 1.1. The Weil conjectures tell us that Z(Xλ /Fp , T ) ∈ Q(T ). Indeed, since Xλ is a K3 surface, its zeta function has the form 1 , Z(Xλ /Fp , T ) = (1 − T )(1 − p2 T )Pλ,p (T ) with Pλ,p (T ) =
22 Y (1 − αi T ),
αi ∈ C,
|αi | = p.
i=1
As in [B-S], §12, we will use the N´eron-Severi group of Xλ /Fp to find a large factor of Pλ,p (T ). Recall from §3 that on the elliptic surface Xλ , the generic fiber, zero section and the components of the singular fibers not meeting the zero section are linearly independent elements of the N´eron-Severi group modulo torsion. By (24) these comprise 18 linearly independent elements. As all of these are fixed by the action of Frobenius, we have that 4 Y Pλ,p (T ) = (1 − pT ) · (1 − αi T ). 18
i=1
Hence, for any positive integer r, #Xλ (Fpr ) = 1 + p2r + 18pr + α1r + α2r + α3r + α4r .
(27)
Since rλ ≥ 19, it follows that one of the αi (1 ≤ i ≤ 4) is equal to p times a root of unity. If we denote by ωr a primitive r-th root of unity, then α1 = pωr for some r. √ In what follows, for brevity write πλ,p := π and π λ,p := π, and recall that |π| = |π| = p. Note that (26) and (27) give 4 X i=1
αi2 = (a(λ, p)2 − 2p)2 = π 4 + π 4 + 2p2 .
(28)
10
SCOTT AHLGREN, KEN ONO AND DAVID PENNISTON
We claim that one of the αi is real. To see this, suppose that none of the αi is real. Then α1 is either quadratic or quartic over Z. If it is quartic, then r ∈ {5, 8, 10, 12}. If r = 5 or P P 2 r = 10, then 4i=1 αi2 = −p2 , a contradiction since by (28), αi is the square of an integer. P4 2 2 If r = 8, then i=1 αi = 0, and (28) gives a(λ, p) = 2p, a contradiction since p is odd. P P 2 Finally, if r = 12, then 4i=1 αi2 = 2p2 , again contradicting the fact that αi is the square of an integer. We conclude that α1 is not quartic over Z. Similar arguments show that α1 is not quadratic over Z, and our claim follows. Since one of the αi is real, and the αi satisfy the same quartic polynomial over Z, it follows that two of the αi , say α1 and α2 , are real. Equations (25), (26) and (27) now yield α1 + α2 + α3 + α4 = (1 + φp (λ + 1)) p + φp (λ + 1)(π 2 + π 2 )
(29)
α32 + α42 = π 4 + π 4 .
(30)
(α3 + α4 )2 − (π 4 + π 4 ) = 2α3 α4 ∈ {2p2 , −2p2 }.
(31)
and These imply that We now claim that there exist 1 ≤ i, j ≤ 4, i 6= j, with αi = p and αj = φp (λ + 1)p. We argue by contradiction. If this does not hold, then we have four possible outcomes. Case 1: φp (λ + 1) = 1, α1 = α2 = −p. Then (29) gives α3 + α4 = 4p + π 2 + π 2 . Since |αi | = p = |π 2 |, this implies that α3 = α4 = p, contradicting our assumption. Case 2: φp (λ + 1) = 1, α1 = p, α2 = −p. Then (29) gives α3 + α4 = 2p + π 2 + π 2 , and the left hand side of (31) is 6p2 +4p(π 2 +π 2 ). Hence π 2 +π 2 ∈ {−p, −2p}. If π 2 +π 2 = −p, then a(λ, p)2 = p, a contradiction since a(λ, p) ∈ Z. If π 2 + π 2 = −2p, then π 2 = π 2 = −p, which by (29) gives α3 + α4 = 0, implying that either α3 = ip, α4 = −ip (contradicting (30), since π 4 + π 4 = 2p2 ), or α3 = p, α4 = −p, which contradicts our assumption. Case 3: φp (λ + 1) = −1, α1 = α2 = p. Then (29) gives α3 + α4 = −2p − π 2 − π 2 , which gives a contradiction by the same argument as in Case 2. Case 4: φp (λ + 1) = −1, α1 = α2 = −p. Then (29) gives α3 + α4 = 2p − π 2 − π 2 , and the left hand side of (31) becomes 6p2 − 4p(π 2 + π 2 ), so π 2 + π2 ∈ {p, 2p}. If π 2 + π2 = p, then π 2 = pω6 , which contradicts the fact that π is at most quadratic over Z. So π 2 +π 2 = 2p, which implies that a(λ, p)2 = 4p, also a contradiction. Upon renumbering, then, we have α1 = p, α2 = φp (λ + 1)p. Then (29) becomes α3 + α4 = φp (λ + 1)(π 2 + π 2 ).
(32)
To finish, we just need to show that {α3 , α4 } = {φp (λ + 1)π 2 , φp (λ + 1)π2 }. First suppose that α3 ∈ R. Then α4 ∈ R, and either α3 = α4 (which by (32) immediately gives our result since |π 2 | = |π 2 | = p), or α3 = −α4 , which by (32) gives π 2 + π 2 = 0. This latter implies that a(λ, p)2 = 2p, a contradiction since p is odd.
ZETA FUNCTIONS OF AN INFINITE FAMILY OF K3 SURFACES
11
Finally, suppose that α3 ∈ / R. We have seen that π 2 + π 2 6= 0, so by (32), π 2 ∈ / R and 2 Re(α3 ) = φp (λ + 1)Re(π ), and this gives our result since |α3 | = |π 2 | = p. 5. Proof of Theorem 1.2
P n 2πiz Recall that Xλ is modular if there is a weight 3 cusp form fλ (z) = ∞ n=1 aλ (n)q (q := e throughout) which is an eigenform of the Hecke operators and has the property that L(Xλ , s) :=
Y
∗
−s
Z (Xλ /Fp , p ) =
p
∞ X aλ (n) n=1
ns
.
(33)
Here the product is over the primes p at which Eλ has good reduction, and Z ∗ (Xλ /Fp , T ) denotes the product of the two nontrivial factors of Z(Xλ /Fp , T ). In view of Theorem 1.1 and our formulation above, care must be taken in defining Z ∗ (Xλ /Fp , T ) for the primes p at which Eλ has supersingular reduction. If η(z) is Dedekind’s eta-function η(z) := q
1/24
∞ Y
(1 − q n ),
(34)
n=1
then the four weight 3 newforms we require are: A(z) := η 6 (4z) = q − 6q 5 + 9q 9 + · · · ∈ S3 (Γ0 (16), χ−1 ),
(35)
B(z) := η 2 (z)η(2z)η(4z)η 2 (8z) = q − 2q 2 − 2q 3 + · · · ∈ S3 (Γ0 (8), χ−2 ),
(36)
C(z) := η (2z)η (6z) = q − 3q + 2q + 9q − · · · ∈ S3 (Γ0 (12), χ−3 ),
(37)
D(z) := η (z)η (7z) = q − 3q + 5q − 7q − · · · ∈ S3 (Γ0 (7), χ−7 ).
(38)
3
3
3
3
3
2
7
4
9
7
We begin by considering the “if” direction of Theorem 1.2 (i.e. that Xλ is modular for the seven exceptional values of λ). If λ ∈ Q \ {0, −1}, then the j-invariant of Eλ is 64(λ + 4)3 . λ2 Using the fact that an elliptic curve E/Q has complex multiplication if and only if j(Eλ ) =
(39)
j(E) ∈ {123, 663 , 203 , 0, 2 · 303 , −3 · 1603, −153 , 2553 , −323 , −963 , −9603 , −52803 , −6403203}, (39) implies that Eλ has complex multiplication if and only if λ is one of the seven exceptional values. Arguing as in [B-S], one can show that f1 = B ⊗χ−4 , f8 = A, f1/8 = A⊗χ8 , f−4 = C, f−1/4 = C ⊗ χ−4 , f−64 = D, and f−1/64 = D ⊗ χ−4 . Here f ⊗ χ denotes the χ-twist√of the modular form f , and χD denotes the Kronecker character for the quadratic field Q( D). Now we prove the “only if” direction in Theorem 1.2. If λ ∈ Q \ {0, −1} is not one of these exceptional values, then it suffices to show that L(Xλ , s) cannot be a modular L-function. For these λ, it is well known that the set of primes p at which Eλ has supersingular reduction has density zero. Moreover, if p is a prime of good reduction which is not supersingular for Eλ , then the local Euler factor at p for L(Xλ , s) is 1 . 1 − φp (λ + 1) (a(λ, p)2 − 2p) p−s + p2−2s
(40)
12
SCOTT AHLGREN, KEN ONO AND DAVID PENNISTON
Therefore, for such p the coefficient aλ (p) in L(Xλ , s) is φp (λ + 1)(a(λ, p)2 − 2p). If Xλ is modular, then for every prime ` there is a Galois representation ρ` : Gal(Q/Q) → GL2 (F` ) with the property that Tr(ρ` (Frob(p)) ≡ φp (λ + 1)(a(λ, p)2 − 2p) (mod `)
(41)
for a set of primes with density one. This contradicts a well known theorem due to Ribet which implies that all but finitely many such ρ` are large for eigenforms without complex multiplication (for example see [O-S], Lemma, p. 459). acknowledgements The authors thank Ling Long for an early version of her manuscript [L]. References [B-E-W] [B-S] [B-P-V] [G] [I-S] [K-T] [L] [M] [O-S] [R] [S] [Y]
B. Berndt, R. Evans, and K. Williams, Gauss and Jacobi sums, Canadian Math. Soc. Monographs 21, Wiley Interscience, 1998. F. Beukers and J. Stienstra, On the Picard-Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces, Math. Ann. 271 (1985), 269-304. W. Barth, C. Peters and A. van de Ven, Compact complex surfaces, Springer, Berlin-HeidelbergNew York, 1984. J. Greene, Character sum analogues for hypergeometric and generalized hypergeometric functions over finite fields, Ph.D. thesis, University of Minnesota, 1984. H. Inose and T. Shioda, On singular K3 surfaces, Complex analysis and algebraic geometry (eds. W. Baily and T. Shioda), Cambridge (1977), 119-136. M. Kuwata and J. Top, A singular K3 surface related to sums of consecutive cubes, preprint. L. Long, The modularity conjecture for a class of elliptic modular surfaces, preprint. D. Morrison, On K3 surfaces with large Picard number, Invent. Math. 75 (1984), 105-121. K. Ono and C. Skinner, Fourier coefficients of half-integral weight modular forms modulo `, Annals of Math. 147 (1998), 453-470. K. Ribet, Galois representations attached to eigenforms with Nebentypus, Springer Lecture Notes 601 (1976), 17-51. T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24, no. 1 (1972), 20-59. N. Yui, The arithmetic of certain Calabi-Yau varieties over number fields, Arithmetic and geometry of algebraic cycles, Banff (1998).
Department of Mathematics, Colgate University, Hamilton, NY 13346 E-mail address: [email protected] Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706 E-mail address: [email protected] Department of Mathematics, Pennsylvania State University, University Park, PA 16802 E-mail address: [email protected]
1. Introduction Given a projective variety defined over a number field, a central problem in number theory is to write down all of its local zeta functions. The case of elliptic curves over Q has recently been settled, and here we study the problem for K3 surfaces, a natural two-dimensional analog of elliptic curves. This is in general a difficult problem, since at any prime, the nontrivial factor of the local zeta function of a K3 surface is a polynomial of degree as high as 22. An important work in this area is due to Shioda and Inose. If a K3 surface has Picard number 20 (the largest it can be), they show in [I-S] that its L-function is essentially the symmetric square of the L-function of an elliptic curve with complex multiplication. Moreover, if the surface is defined over Q, its L-function is the L-function of a weight 3 Hecke eigenform with complex multiplication. We shall refer to such surfaces as modular. Although this result is general, the problem of computing explicit examples is nontrivial. In fact, only a few examples exist in the literature (see, for example, [B-S], [K-T], [L]), and they were obtained by a wide variety of methods, both geometric and transcendental. There are several natural difficulties. For instance, there is the problem of identifying suitable candidates for such surfaces. Then there is the issue of demonstrating that the L-function of an alleged modular surface agrees with the L-function of an explicit weight 3 modular form. Work of Morrison [M] suggests that one can obtain similar arithmetic results for K3 surfaces with Picard number ≥ 19. In this vein we show, for a certain family of K3 surfaces, that the general problem of computing zeta functions can be quite simple, and involve little more than elementary character sums. To clarify the merit of such an approach, we note that our family includes, up to isogeny and quadratic twist, all of the modular examples alluded to above. We consider the one parameter family {Xλ }, for λ ∈ Q \ {0, −1}, of K3 surfaces whose function fields are given by Xλ
s2 = xy(x + 1)(y + 1)(x + λy)
:
(1)
in relation to the family of elliptic curves Eλ
:
y 2 = (x − 1) x2 −
1 λ+1
.
For convenience, if p is an odd prime, then let φp (x) denote the Legendre symbol
(2) x p
.
Date: May 9, 2000. The first two authors thank the National Science Foundation for its generous support. The second author also thanks the Alfred P. Sloan Foundation and the David and Lucile Packard Foundation for their support. 1
2
SCOTT AHLGREN, KEN ONO AND DAVID PENNISTON
Theorem 1.1. If λ ∈ Q \ {0, −1} and p is an odd prime at which Eλ has good reduction, then the local zeta function of Xλ at p is Z(Xλ /Fp , T ) =
(1 − T )(1 −
p2 T )(1
−
pT )19 (1
1 , 2 − pT )(1 − πλ,p T )(1 − π 2λ,p T )
where πλ,p and π λ,p are the eigenvalues of the Frobenius at p on Eλ , and = φp (λ + 1). In view of Theorem 1.1, it is easy to characterize those λ for which Xλ is modular. Theorem 1.2. If λ ∈ Q \ {0, −1}, then the surface Xλ is modular if and only if λ ∈ {1, 8, 1/8, −4, −1/4, −64, −1/64}. For completeness, we note that Beukers and Stienstra [B-S] show that X−1 is also a modular K3 surface. This follows easily from Theorem 1.2 since X−1 is a quadratic twist of both X8 and X1/8 . Even though E−1 is singular, the modularity of X−1 can be proved using a minor modification of the arguments contained here. We prefer to omit them for brevity. In §5 we shall show that all of the surfaces in Theorem 3 are related to weight 3 newforms that are products of Dedekind’s eta-function. It is interesting to note that these forms constitute the complete list of weight 3 newforms which are eta-products. Proving Theorem 3 is tantamount to computing the Picard number of all of the Xλ . Finding the Picard number of a K3-surface is in general a difficult problem, and is often equivalent to calculating the rank of an elliptic curve over a function field. In our situation, the computation follows easily from a well known theorem of Ribet [R]. 2. A theorem on character sums In this section we prove a theorem about character sums which is essential for all of the results in this paper. Suppose that λ ∈ Q \ {0, −1} and that p is an odd prime such that λ 6≡ 0, −1 (mod p). If q = pr , then let φq be the extended Legendre symbol on Fq (we will often write φ = φq for simplicity). Further, define quantities a(λ, q) and A(λ, q) by X 1 a(λ, q) := − , φq (x − 1) x2 − λ+1 x∈Fq
A(λ, q) :=
X
(3)
φq (xy(x + 1)(y + 1)(x + λy)).
x,y∈Fq
Theorem 2.1. If λ ∈ Q, and λ 6≡ 0, −1 (mod p), then A(λ, q) = φq (λ + 1)(a(λ, q)2 − q). Before starting the proof, we mention that results of this type have been obtained by Greene [G] in the context of finite field analogues of hypergeometric functions. We will denote by χ, ψ, and A multiplicative characters on Fq , and by the trivial character on Fq (we agree P that (0) = 0). Given characters χ and ψ, we define the usual Jacobi sum J(χ, ψ) := x∈Fq χ(x)ψ(1 − x). Throughout this section, all sums (unless otherwise noted) will run over all elements or over all characters of Fq . Throughout, we will repeatedly use the following facts, which are valid for any characters. For the first, see [B-E-W] (recall our convention that (0) = 0). The second and third are easy consequences of orthogonality.
ZETA FUNCTIONS OF AN INFINITE FAMILY OF K3 SURFACES
3
J(χ, ψ) = χ(−1)J(χ, χψ). ( X q − 1 if x = 1 χ(x) = 0 else. χ ( X 1 if x = 0 1 ψ(1 − x) = q−1 J(ψ, χ)χ(x) + 0 else. χ
(4) (5)
(6)
We begin with an easy lemma. If λ ∈ Q and λ 6≡ 0, −1 (mod p), then define X 1 λ g(λ) := q−1 J(χ, φ)J(χ, φχ)χ( 4(λ+1) ). χ −λ Lemma 2.2. If λ ∈ Q, λ 6≡ 0, −1 (mod p), then −φ(2)a(q, λ) = g(λ) + φ( λ+1 ).
Proof. Expanding the Jacobi sums and using (5) gives X X 4(λ+1)xy(1−y) X 1 g(λ) = q−1 φ(1 − x)φ(1 − y) χ( ) = φ(1 − y)φ(1 − λ x,y
χ
λ ). 4(λ+1)y(1−y)
y6=0,1
Simplifying the last sum (and accounting for the term y = 1), we find that X λ −λ − φ( λ+1 g(λ) = φ(y)φ 4y(1 − y) − λ+1 ). y
The lemma follows after the change of variables y →
1−y . 2
To prove Theorem 2.1, we must expand the quantity g(λ)2 . In the course of the computation, several error terms will arise. Since it is straightforward, we will delay evaluation of these terms (which will be denoted by E1 , E2 , etc.) until the end. Expanding g(λ)2 as a sum over χ and ψ and making the change ψ → ψχ, we find that X 1 λ ψ J(χ, φ)J(χ, φχ)J(ψχ, φ)J(ψχ, φψχ). g(λ)2 = (q−1) 2 4(λ+1) χ,ψ
Expanding the final Jacobi sum gives X X 1 λ ψ( 4(λ+1) )ψ(x)φψ(1 − x) J(χ, φ)J(ψχ, φ)J(χ, φχ)χ(x)χ(1 − x). g(λ)2 = (q−1) 2 χ
ψ,x
(7)
Notice that (replacing w with y/w in the second step) we have X X J(χ, φ)J(ψχ, φ) = χ(w)φ(1 − w)ψχ(y)φ(1 − y) = φχ(w)φ(w − y)ψ(y)φ(1 − y). w,y
w,y
Rearranging and using (6), this becomes X
φχ(w)φ(1 −
1−w )ψ(y) 1−y
=
1 q−1
w,y y6=1
X w,y y6=1
=
1 q−1
X A
φχ(w)ψ(y)
X
J(A, φ)A( 1−w )+ 1−y
A
J(A, φ)J(φχ, A)J(ψ, A) +
X y6=1
X y6=1
ψ(y).
ψ(y)
4
SCOTT AHLGREN, KEN ONO AND DAVID PENNISTON
This together with (7) shows that g(λ)2 is given by the expression X X X 1 λ ψ( )ψ(x)φψ(1 − x) J(A, φ)J(ψ, A) J(φχ, A)J(χ, φχ)χ(x)χ(1 − x) + E1 , 3 (q−1) 4(λ+1) χ (8) A ψ,x where E1 :=
1 (q−1)2
X
λ ψ( 4(λ+1) )ψ(x)φψ(1 − x)
X
J(χ, φχ)χ(x)χ(1 − x)
χ
ψ,x
X
ψ(y).
(9)
y6=1
By (7), we may assume that x 6= 0, 1. Therefore the sum on χ in (8) can be written X X X w(1−w) x x φ( 1−x = (q − 1)φ( φ(w). ) A(1 − y)φ(w) φχ x(1−x)y ) A 1 − w(1−w) 1−x x(1−x) w,y w6=0,1
χ
w6=1
x When w = 1, the summand above equals (q − 1)φ( 1−x ). Accounting for this term, making the change of variables w → wx and factoring, the last displayed equation becomes X wx x (q − 1)φ(1 − x) φ(w)A(1 − w)A(1 − 1−x ) − (q − 1)φ( 1−x ). w
Together with (8) this yields g(λ)2 = E1 + E2 + X X X 1 λ ψ( )ψ(x)ψ(1 − x) J(A, φ)J(ψ, A) φ(w)A(1 − w)A(1 − 2 (q−1) 4(λ+1) ψ,x
w
A
where E2 :=
−1 (q−1)2
X
λ ψ( 4(λ+1) )ψφ(x)ψ(1 − x)
ψ,x
wx ), 1−x
X
J(A, φ)J(ψ, A).
(10)
(11)
A
Using (6), the sum on A in (10) can be written X X wx φ(1 − z)φ(w) J(ψ, A)A( 1z (1 − w)(1 − 1−x ) w,z w6=1,z6=0
A
X
= (q − 1)
φ(1 − z)φ(w)ψ 1 − z1 (1 − w)(1 −
wx ) 1−x
− (q − 1)
w,z w6=1,z6=0
X
φ(1 − z)φ( 1−x ). x
z6=0
Replacing z by (1 − w)/z and simplifying, this becomes X wx (q − 1) φ(z)φ(z − 1 + w)φ(w)ψ 1 − z(1 − 1−x ) + (q − 1)φ( 1−x ) x w,z w6=1
= (q − 1)
X
φ(z)φ(z − 1 + w)φ(w)ψ 1 − z(1 −
wx ) 1−x
+ F (ψ, x),
(12)
w,z z6=1
where F (ψ, x) contains the error arising from the terms w = 1 and z = 1; i.e. X X x F (ψ, x) = (q − 1)φ( 1−x ) − (q − 1) ψ 1 − z(1 − ) + (q − 1) ψ x 1−x z6=0
w
wx 1−x
. (13)
ZETA FUNCTIONS OF AN INFINITE FAMILY OF K3 SURFACES
5
Making the change of variables w → (1 − z)w and applying (6), the sum in (12) becomes X wzx (q − 1) φ(z)φ(−1 + w)φ(w)ψ(1 − z)ψ 1 + 1−x ) + F (ψ, x) w,z
=
X
φ(z)φ(−1 + w)φ(w)ψ(1 − z)
X
w,z
=
X
J(χ, ψ)χ( −wzx ) + F (ψ, x) 1−x
χ x J(χ, φ)J(ψ, χ)J(ψ, φχ)χ( x−1 ) + F (ψ, x).
χ
With (10) this yields X 1 λ g(λ)2 = (q−1) ψ( 4(λ+1) )χ(−1)J(ψχ, ψχ)J(ψ, χ)J(χ, φ)J(ψ, φχ) + E1 + E2 + E3 , 2 ψ,χ
where E3 :=
1 (q−1)2
X
λ ψ( 4(λ+1) )ψ(x)ψ(1 − x)F (ψ, x).
(14) (15)
ψ,x
To simplify the main term in (14), we apply (4) to find that X χ(−1)(ψχ, ψχ)J(ψ, χ) = J(ψχ, ψ 2 )J(ψ, χψ) = ψχ(w)ψ 2 (1 − w)ψ(1 − y)ψχ(y). w,y
Making the change w → X
w y
and simplifying, this becomes
ψχ(w)ψ 2 (y − w)ψ(1 − y) =
w,y y6=0
X
ψχ(w)ψ 2(y − w)ψ(1 − y) −
X
w,y w6=1
ψχ(w) +
w
X y
ψ(1 − y). (16)
For ease of notation, define a function δ by ( 0 if χ 6= δ(χ) := 1 if χ = . Making the change y → 1 − (1 − w)y and simplifying, the expression in (16) becomes ψ(−1)J(ψχ, ψ)J(ψ, ψ) − (q − 1)δ(ψχ) + (q − 1)δ(ψ). It is well-known (see[B-E-W]) that ψ(4)J(ψ, ψ) = J(ψ, φ) + (q − 1)δ(ψ). This, together with (14) and the last displayed equation, shows that X X 1 −λ g(λ)2 = (q−1) J(χ, φ) ψ( λ+1 )J(ψ, φ)J(ψχ, ψ)J(ψ, φχ) + E1 + E2 + E3 + E4 , 2 χ (17) ψ where E4 :=
−1 q−1
X λ ψ( 4(λ+1) )J(ψ, φ)J(ψ, φψ) +
1 q−1
X J(χ, φ)J(, φχ) + χ
ψ
1 q−1
X χ
J(χ, φ)J(χ, )J(, φχ). (18)
The main term in (17) is X X X (λ+1)wyz 1 J(χ, φ) φ(1 − w)φχ(1 − y)χ(z) ψ −λ(1−z) . 2 (q−1) χ
w,y,z wyz6=0
ψ
(19)
6
SCOTT AHLGREN, KEN ONO AND DAVID PENNISTON
Notice that w→
z 1−z
=
−λ (λ+1)wy
if and only if z =
−λ λ+1 λ wy− λ+1
. Using this fact, making the changes
y y → y−1 , and then applying (6), the main term becomes X X 1 wλ J(χ, φ) φ(1 − λ+1 )φχ(1 − y)χ(1 − wy) (q−1)
wλ , λ+1
w,y wy6=0
χ
1 (q−1)
=
X
J(χ, φ)
1 (q−1)
X
=
φ(1 −
wλ )φχ(1 λ+1
w,y wy6=0
χ
=
X
X
φ(1 −
wλ )φ(1 λ+1
− y)
X
w,y wy6=0
− y)χ(1 −
wy ) y−1
1 J(χ, φ)χ( 1−y+wy )
χ
φ(1 −
wλ )φ(1 λ+1
− y)φ(1 − y(1 − w))φ(−y)φ(1 − w)
φ(1 −
wλ )φ(1 λ+1
− y)φ(1 − y(1 − w))φ(−y)φ(1 − w) + φ(−1).
w,y w6=0
X
=
w,y
Taking w to 1 − and then replacing w, y by −w, −y, this becomes X φ(1 + λ) φ(1 + wλ )φ(1 − y)φ(1 − w)φ(−w) + φ(−1) = φ(1 + λ)A(λ) + φ(−1). y w y
w,y
Together with (17), this yields g(λ)2 = φ(1 + λ)A(λ) + E1 + E2 + E3 + E4 + φ(−1). A calculation (see below) shows that −λ E1 + E2 + E3 + E4 = q − 1 − φ(−1) − 2φ( λ+1 )g(λ).
(20)
Together with Lemma 2.2, the last two formulae give the statement in Theorem 2.1; i.e. a(λ, q)2 = q + φ(1 + λ)A(λ). To finish the proof, it is necessary to verify (20); we will only provide details for the computation of E1 (which is given by (9)). We have X X X w(1−w) J(χ, φχ)χ(x)χ(1 − x) = φ(w) χ x(1−x) . χ
χ
w6=1
The argument of χ equals one when w = x or w = 1 − x. Taking into account the case when x = 12 , we find that X J(χ, φχ)χ(x)χ(1 − x) = (q − 1) φ(x) + φ(1 − x)(x − 12 ) . χ
P Returning to (9), we note that y6=1 ψ(y) = −1 + (q − 1)δ(ψ). Therefore, X X X X −1 λ 1 λ ψ( 4(λ+1) )J(φψ, φψ) − q−1 ψ( 4(λ+1) )ψ(x)ψ(1 − x) + 1+ φ(x)φ(1 − x) E1 := q−1 ψ
=
−1 q−1
X ψ
ψ
λ 4(λ+1)
x6= 12 ,0,1
ψ,x x6= 21
J(φψ, φψ) −
1 q−1
X ψ
ψ
λ 4(λ+1)
x
J(ψ, ψ) + q − 3 − φ(−1).
ZETA FUNCTIONS OF AN INFINITE FAMILY OF K3 SURFACES
7
In the second sum we have used the fact that the summand is zero when x = 12 . For the remaining error terms (which are given by (11), (15), (18)), we use similar arguments to find that −λ E2 = −φ( λ+1 )g(λ), X X 1 λ 1 λ E3 = 1 + q−1 J(φψ, φψ) + q−1 J(ψ, ψ), ψ 4(λ+1) ψ 4(λ+1)
ψ
ψ
E4 = 1 −
−λ φ( λ+1 )g(λ).
Equation (20) follows, and the proof of Theorem 2.1 is complete. 3. The Geometry of these K3 surfaces In this section and the next we follow closely the exposition in [B-S]. Let K denote either Q or Fp (p an odd prime). For each λ ∈ K \ {0, −1}, we define the surface Xλ /K to be the smooth complete model of the double cover of P2 branched over the union Uλ of six lines given by Uλ : XY Z(X + λY )(Y + Z)(Z + X) = 0. To see that Xλ is a K3 surface, one can refer to [B-P-V], or check that the elliptic fibration of Xλ we give below satisfies the conditions given in [B-S], §4. We now give a description of the natural map ψ : Xλ → P2 . Let Bλ be the surface obtained in the following way: first blow up P2 at the triple points of Uλ , then blow up the resulting surface at each of the double points of the total transform of Uλ . The total transform Tλ ⊂ Bλ of Uλ consists of a connected union of rational curves whose only singularities are ordinary double points, and at each of these points a component of odd multiplicity and one of even multiplicity meet. The branch locus Uλ and the dual graph of Tλ are depicted here (• denotes a component of odd multiplicity, and denotes a component of even multiplicity): 22 *? •**?? 2 ? *** ??? ** ??? ?? • • ? •P***PPPP • P P ** 2 • •22 ** 22 ** 22 * • •
2 222 FF 22 F F FF 22 FF mm22m Fm mmmmm FFFF222 m FF22 mmm F2F m m m 2
Figure 1 Then Xλ is the double cover of Bλ branched above the components of Tλ having odd multiplicity. Moreover, ψ −1 (Uλ ) is a union of rational curves whose dual graph is exactly that of Tλ , and Xλ \ ψ −1 (Uλ ) is a smooth affine surface given by the equation s2 = xy(x + 1)(y + 1)(x + λy).
(21)
Now let us show that the Xλ are elliptic surfaces over P1 . For a fixed λ, consider the family of cubics (with parameter τ ) in P2 given by X(Y + Z)(Z + X) − τ (X + λY )Y Z = 0.
(22)
8
SCOTT AHLGREN, KEN ONO AND DAVID PENNISTON
One can check that the generic cubic in this family intersects Uλ at all of its triple points, and at the three double points (1, 0, −1), (λ, −1, 1) and (λ, −1, −λ). At the double points, the cubic is transversal to both of the components of Uλ meeting there, and at the triple points is transversal to two of the components meeting there and tangential to the third. It follows that the proper transform of the generic cubic in Bλ intersects only components of Tλ of even multiplicity, and therefore the family (22) lifts through ψ to Xλ . To see the elliptic fibration explicitly, the change of variables X = λt2 V W,
Y = U 2 + λt2 UW,
Z = UV
(where τ = t2 , reflecting the double cover) transforms (22) into Cλ : V 2 W + (1 − t2 )UV W + λt2 V W 2 = U 3 + λt2 U 2 W.
(23)
The discriminant of Cλ is −λ3 t8 [t6 + (8λ − 3)t4 + (16λ2 + 20λ + 3)t2 − (λ + 1)]. Using Tate’s algorithm, we find that the singular fibers are generically of the following types: I10 I8 I1 I1 I1 I1 I1 I1 .
(24)
The singular fibers above t = ∞ and t = 0 are visible in the dual graph of Tλ (see Figure 1) as the outer and inner cycles, respectively. The N´eron-Severi group of Xλ /K is the group of algebraic equivalence classes of divisors on Xλ . Shioda [S] showed that the rank rλ of the N´eron-Severi group of Xλ /K is given by X (mj − 1) + 2 + rank(Cλ (K(t)), rλ = where the sum is over the singular fibers, and mj is the number of components of the singular fiber j not meeting the zero section. In particular, he showed that the equivalence classes of the generic fiber, the zero section and the components of the singular fibers not meeting the zero section are linearly independent elements of the N´eron-Severi group modulo torsion. From (24), then, these give 18 linearly independent elements. Also note that (0, 0) is an element of Cλ (K(t)), and one can check that it has infinite order. Therefore rλ ≥ 19. 4. Proof of Theorem 1.1 Given a prime p, the zeta function of a projective variety Y /Fp is ! ∞ X Z(Y /Fp, T ) = exp #Y (Fpn ) · T n /n . n=1
The following proposition is our first step in proving Theorem 1.1. Proposition 4.1. If p is an odd prime, λ ∈ Fp \ {0, −1}, and q = pr , then X #Xλ (Fq ) = 1 + q 2 + 19q + φq (xy(x + 1)(y + 1)(x + λy)). x,y∈Fq
Proof. Recall (see §3) that we have a map ψ : Xλ → P2 ramified above Uλ , that Xλ \ ψ −1 (Uλ ) is a smooth affine surface given by (21) and that ψ −1 (Uλ ) is a union of rational curves whose dual graph is given in Figure 1. Since all the components of Uλ are lines defined over Fp , we can read from Figure 1 that the number of Fq -rational points on ψ −1 (Uλ ) is 9(q + 1) + 15(q − 1) = 24q − 6.
ZETA FUNCTIONS OF AN INFINITE FAMILY OF K3 SURFACES
9
Moreover, from (21) we see that the number of Fq -rational points on Xλ \ψ −1 (Uλ ) with s = 0 is 5q − 7. Therefore, we have that X #Xλ (Fq ) = (φq (xy(x + 1)(y + 1)(x + λy)) + 1) + (24q − 6) − (5q − 7). x,y∈Fq
Recall that if Eλ (λ ∈ Q) has good reduction at p, then the zeta function of Eλ at p is Z(Eλ /Fp , T ) =
(1 − πλ,p T )(1 − π λ,p T ) 1 − a(λ, p)T + pT 2 = . (1 − T )(1 − pT ) (1 − T )(1 − pT )
Then Proposition 4.1 and Theorem 2.1 imply Theorem 4.2. If λ ∈ Q \ {0, −1} and Eλ has good reduction at a prime p, then #Xλ (Fp ) = 1 + p2 + 19p + φp (λ + 1) a(λ, p)2 − p and #Xλ (Fp2 ) = 1 + p4 + 18p2 + a(λ, p)2 − 2p
2
.
(25) (26)
Now let us prove Theorem 1.1. The Weil conjectures tell us that Z(Xλ /Fp , T ) ∈ Q(T ). Indeed, since Xλ is a K3 surface, its zeta function has the form 1 , Z(Xλ /Fp , T ) = (1 − T )(1 − p2 T )Pλ,p (T ) with Pλ,p (T ) =
22 Y (1 − αi T ),
αi ∈ C,
|αi | = p.
i=1
As in [B-S], §12, we will use the N´eron-Severi group of Xλ /Fp to find a large factor of Pλ,p (T ). Recall from §3 that on the elliptic surface Xλ , the generic fiber, zero section and the components of the singular fibers not meeting the zero section are linearly independent elements of the N´eron-Severi group modulo torsion. By (24) these comprise 18 linearly independent elements. As all of these are fixed by the action of Frobenius, we have that 4 Y Pλ,p (T ) = (1 − pT ) · (1 − αi T ). 18
i=1
Hence, for any positive integer r, #Xλ (Fpr ) = 1 + p2r + 18pr + α1r + α2r + α3r + α4r .
(27)
Since rλ ≥ 19, it follows that one of the αi (1 ≤ i ≤ 4) is equal to p times a root of unity. If we denote by ωr a primitive r-th root of unity, then α1 = pωr for some r. √ In what follows, for brevity write πλ,p := π and π λ,p := π, and recall that |π| = |π| = p. Note that (26) and (27) give 4 X i=1
αi2 = (a(λ, p)2 − 2p)2 = π 4 + π 4 + 2p2 .
(28)
10
SCOTT AHLGREN, KEN ONO AND DAVID PENNISTON
We claim that one of the αi is real. To see this, suppose that none of the αi is real. Then α1 is either quadratic or quartic over Z. If it is quartic, then r ∈ {5, 8, 10, 12}. If r = 5 or P P 2 r = 10, then 4i=1 αi2 = −p2 , a contradiction since by (28), αi is the square of an integer. P4 2 2 If r = 8, then i=1 αi = 0, and (28) gives a(λ, p) = 2p, a contradiction since p is odd. P P 2 Finally, if r = 12, then 4i=1 αi2 = 2p2 , again contradicting the fact that αi is the square of an integer. We conclude that α1 is not quartic over Z. Similar arguments show that α1 is not quadratic over Z, and our claim follows. Since one of the αi is real, and the αi satisfy the same quartic polynomial over Z, it follows that two of the αi , say α1 and α2 , are real. Equations (25), (26) and (27) now yield α1 + α2 + α3 + α4 = (1 + φp (λ + 1)) p + φp (λ + 1)(π 2 + π 2 )
(29)
α32 + α42 = π 4 + π 4 .
(30)
(α3 + α4 )2 − (π 4 + π 4 ) = 2α3 α4 ∈ {2p2 , −2p2 }.
(31)
and These imply that We now claim that there exist 1 ≤ i, j ≤ 4, i 6= j, with αi = p and αj = φp (λ + 1)p. We argue by contradiction. If this does not hold, then we have four possible outcomes. Case 1: φp (λ + 1) = 1, α1 = α2 = −p. Then (29) gives α3 + α4 = 4p + π 2 + π 2 . Since |αi | = p = |π 2 |, this implies that α3 = α4 = p, contradicting our assumption. Case 2: φp (λ + 1) = 1, α1 = p, α2 = −p. Then (29) gives α3 + α4 = 2p + π 2 + π 2 , and the left hand side of (31) is 6p2 +4p(π 2 +π 2 ). Hence π 2 +π 2 ∈ {−p, −2p}. If π 2 +π 2 = −p, then a(λ, p)2 = p, a contradiction since a(λ, p) ∈ Z. If π 2 + π 2 = −2p, then π 2 = π 2 = −p, which by (29) gives α3 + α4 = 0, implying that either α3 = ip, α4 = −ip (contradicting (30), since π 4 + π 4 = 2p2 ), or α3 = p, α4 = −p, which contradicts our assumption. Case 3: φp (λ + 1) = −1, α1 = α2 = p. Then (29) gives α3 + α4 = −2p − π 2 − π 2 , which gives a contradiction by the same argument as in Case 2. Case 4: φp (λ + 1) = −1, α1 = α2 = −p. Then (29) gives α3 + α4 = 2p − π 2 − π 2 , and the left hand side of (31) becomes 6p2 − 4p(π 2 + π 2 ), so π 2 + π2 ∈ {p, 2p}. If π 2 + π2 = p, then π 2 = pω6 , which contradicts the fact that π is at most quadratic over Z. So π 2 +π 2 = 2p, which implies that a(λ, p)2 = 4p, also a contradiction. Upon renumbering, then, we have α1 = p, α2 = φp (λ + 1)p. Then (29) becomes α3 + α4 = φp (λ + 1)(π 2 + π 2 ).
(32)
To finish, we just need to show that {α3 , α4 } = {φp (λ + 1)π 2 , φp (λ + 1)π2 }. First suppose that α3 ∈ R. Then α4 ∈ R, and either α3 = α4 (which by (32) immediately gives our result since |π 2 | = |π 2 | = p), or α3 = −α4 , which by (32) gives π 2 + π 2 = 0. This latter implies that a(λ, p)2 = 2p, a contradiction since p is odd.
ZETA FUNCTIONS OF AN INFINITE FAMILY OF K3 SURFACES
11
Finally, suppose that α3 ∈ / R. We have seen that π 2 + π 2 6= 0, so by (32), π 2 ∈ / R and 2 Re(α3 ) = φp (λ + 1)Re(π ), and this gives our result since |α3 | = |π 2 | = p. 5. Proof of Theorem 1.2
P n 2πiz Recall that Xλ is modular if there is a weight 3 cusp form fλ (z) = ∞ n=1 aλ (n)q (q := e throughout) which is an eigenform of the Hecke operators and has the property that L(Xλ , s) :=
Y
∗
−s
Z (Xλ /Fp , p ) =
p
∞ X aλ (n) n=1
ns
.
(33)
Here the product is over the primes p at which Eλ has good reduction, and Z ∗ (Xλ /Fp , T ) denotes the product of the two nontrivial factors of Z(Xλ /Fp , T ). In view of Theorem 1.1 and our formulation above, care must be taken in defining Z ∗ (Xλ /Fp , T ) for the primes p at which Eλ has supersingular reduction. If η(z) is Dedekind’s eta-function η(z) := q
1/24
∞ Y
(1 − q n ),
(34)
n=1
then the four weight 3 newforms we require are: A(z) := η 6 (4z) = q − 6q 5 + 9q 9 + · · · ∈ S3 (Γ0 (16), χ−1 ),
(35)
B(z) := η 2 (z)η(2z)η(4z)η 2 (8z) = q − 2q 2 − 2q 3 + · · · ∈ S3 (Γ0 (8), χ−2 ),
(36)
C(z) := η (2z)η (6z) = q − 3q + 2q + 9q − · · · ∈ S3 (Γ0 (12), χ−3 ),
(37)
D(z) := η (z)η (7z) = q − 3q + 5q − 7q − · · · ∈ S3 (Γ0 (7), χ−7 ).
(38)
3
3
3
3
3
2
7
4
9
7
We begin by considering the “if” direction of Theorem 1.2 (i.e. that Xλ is modular for the seven exceptional values of λ). If λ ∈ Q \ {0, −1}, then the j-invariant of Eλ is 64(λ + 4)3 . λ2 Using the fact that an elliptic curve E/Q has complex multiplication if and only if j(Eλ ) =
(39)
j(E) ∈ {123, 663 , 203 , 0, 2 · 303 , −3 · 1603, −153 , 2553 , −323 , −963 , −9603 , −52803 , −6403203}, (39) implies that Eλ has complex multiplication if and only if λ is one of the seven exceptional values. Arguing as in [B-S], one can show that f1 = B ⊗χ−4 , f8 = A, f1/8 = A⊗χ8 , f−4 = C, f−1/4 = C ⊗ χ−4 , f−64 = D, and f−1/64 = D ⊗ χ−4 . Here f ⊗ χ denotes the χ-twist√of the modular form f , and χD denotes the Kronecker character for the quadratic field Q( D). Now we prove the “only if” direction in Theorem 1.2. If λ ∈ Q \ {0, −1} is not one of these exceptional values, then it suffices to show that L(Xλ , s) cannot be a modular L-function. For these λ, it is well known that the set of primes p at which Eλ has supersingular reduction has density zero. Moreover, if p is a prime of good reduction which is not supersingular for Eλ , then the local Euler factor at p for L(Xλ , s) is 1 . 1 − φp (λ + 1) (a(λ, p)2 − 2p) p−s + p2−2s
(40)
12
SCOTT AHLGREN, KEN ONO AND DAVID PENNISTON
Therefore, for such p the coefficient aλ (p) in L(Xλ , s) is φp (λ + 1)(a(λ, p)2 − 2p). If Xλ is modular, then for every prime ` there is a Galois representation ρ` : Gal(Q/Q) → GL2 (F` ) with the property that Tr(ρ` (Frob(p)) ≡ φp (λ + 1)(a(λ, p)2 − 2p) (mod `)
(41)
for a set of primes with density one. This contradicts a well known theorem due to Ribet which implies that all but finitely many such ρ` are large for eigenforms without complex multiplication (for example see [O-S], Lemma, p. 459). acknowledgements The authors thank Ling Long for an early version of her manuscript [L]. References [B-E-W] [B-S] [B-P-V] [G] [I-S] [K-T] [L] [M] [O-S] [R] [S] [Y]
B. Berndt, R. Evans, and K. Williams, Gauss and Jacobi sums, Canadian Math. Soc. Monographs 21, Wiley Interscience, 1998. F. Beukers and J. Stienstra, On the Picard-Fuchs equation and the formal Brauer group of certain elliptic K3-surfaces, Math. Ann. 271 (1985), 269-304. W. Barth, C. Peters and A. van de Ven, Compact complex surfaces, Springer, Berlin-HeidelbergNew York, 1984. J. Greene, Character sum analogues for hypergeometric and generalized hypergeometric functions over finite fields, Ph.D. thesis, University of Minnesota, 1984. H. Inose and T. Shioda, On singular K3 surfaces, Complex analysis and algebraic geometry (eds. W. Baily and T. Shioda), Cambridge (1977), 119-136. M. Kuwata and J. Top, A singular K3 surface related to sums of consecutive cubes, preprint. L. Long, The modularity conjecture for a class of elliptic modular surfaces, preprint. D. Morrison, On K3 surfaces with large Picard number, Invent. Math. 75 (1984), 105-121. K. Ono and C. Skinner, Fourier coefficients of half-integral weight modular forms modulo `, Annals of Math. 147 (1998), 453-470. K. Ribet, Galois representations attached to eigenforms with Nebentypus, Springer Lecture Notes 601 (1976), 17-51. T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan 24, no. 1 (1972), 20-59. N. Yui, The arithmetic of certain Calabi-Yau varieties over number fields, Arithmetic and geometry of algebraic cycles, Banff (1998).
Department of Mathematics, Colgate University, Hamilton, NY 13346 E-mail address: [email protected] Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706 E-mail address: [email protected] Department of Mathematics, Pennsylvania State University, University Park, PA 16802 E-mail address: [email protected]
