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 {X_{λ}}, for λ ∈ ℚ \ {0, −1}, of K3 surfaces whose function fields are given by

X_{λ} : s^{2} = xy(x + 1)(y + 1)(x + λy)  (1)

in relation to the family of elliptic curves

E_{λ} : y^{2} = (x − 1) (x^{2} − \frac{1}{λ+1}) .  (2)

For convenience, if p is an odd prime, then let ϕ_{p}(x) denote the Legendre symbol (\frac{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.

Theorem 1.1. If λ ∈ ℚ \ {0, −1} and p is an odd prime at which E_{λ} has good reduction, then the local zeta function of X_{λ} at p is

Z(X_{λ}/𝔽_{p}, T) = \frac{1}{(1 − T)(1 − p^{2}T)(1 − pT)^{19}(1 − ϵpT)(1 − ϵπ_{λ,p}^{2}T)(1 − ϵ\overline{π}_{λ,p}^{2}T)},

where π_{λ,p} and \overline{π}_{λ,p} are the eigenvalues of the Frobenius at p on E_{λ}, and ϵ = ϕ_{p}(λ + 1).

In view of Theorem 1.1, it is easy to characterize those λ for which X_{λ} is modular.

Theorem 1.2. If λ ∈ ℚ \ {0, −1}, then the surface X_{λ} 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 X_{8} and X_{1/8}. 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 X_{λ}. 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 ≢ 0, −1 (mod p). If q = p^{r}, then let ϕ_{q} be the extended Legendre symbol on 𝔽_{q} (we will often write ϕ = ϕ_{q} for simplicity). Further, define quantities a(λ, q) and A(λ, q) by

\begin{matrix}a(λ, q) := − ∑_{x∈𝔽_{q}} ϕ_{q} ((x − 1) (x^{2} − \frac{1}{λ+1})) , \\ A(λ, q) := ∑_{x,y∈𝔽_{q}} ϕ_{q}(xy(x + 1)(y + 1)(x + λy)). \end{matrix}  (3)

Theorem 2.1. If λ ∈ ℚ, and λ ≢ 0, −1 (mod p), then A(λ, q) = ϕ_{q}(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 𝔽_{q}, and by ϵ the trivial character on 𝔽_{q} (we agree that ϵ(0) = 0). Given characters χ and ψ, we define the usual Jacobi sum J(χ, ψ) := ∑_{x∈𝔽_{q}} χ(x)ψ(1 − x). Throughout this section, all sums (unless otherwise noted) will run over all elements or over all characters of 𝔽_{q}.

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 ϵ(0) = 0). The second and third are easy consequences of orthogonality.

J(χ, ψ) = χ(−1)J(χ, \overline{χψ}).  (4)
∑_{χ} χ(x) = \begin{cases}q − 1 if x = 1 \\ 0  else. \end{cases}  (5)
ψ(1 − x) = \frac{1}{q−1} ∑_{χ} J(ψ, \overline{χ})χ(x) + \begin{cases}1 if x = 0 \\ 0 else. \end{cases}  (6)

We begin with an easy lemma. If λ ∈ ℚ and λ ≢ 0, −1 (mod p), then define

g(λ) := \frac{1}{q−1} ∑_{χ} J(\overline{χ}, ϕ)J(\overline{χ}, ϕ\overline{χ})χ(\frac{λ}{4(λ+1)}).

Lemma 2.2. If λ ∈ ℚ, λ ≢ 0, −1 (mod p), then −ϕ(2)a(q, λ) = g(λ) + ϕ(\frac{−λ}{λ+1}). Proof. Expanding the Jacobi sums and using (5) gives

g(λ) = \frac{1}{q−1} ∑_{x,y} ϕ(1 − x)ϕ(1 − y) ∑_{χ} \overline{χ}(\frac{4(λ+1)xy(1−y)}{λ}) = ∑_{y≠0,1} ϕ(1 − y)ϕ(1 − \frac{λ}{4(λ+1)y(1−y)}).

Simplifying the last sum (and accounting for the term y = 1), we find that

g(λ) = ∑_{y} ϕ(y)ϕ (4y(1 − y) − \frac{λ}{λ+1}) − ϕ(\frac{−λ}{λ+1}).

The lemma follows after the change of variables y → \frac{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 E_{1}, E_{2}, etc.) until the end.

Expanding g(λ)^{2} as a sum over χ and ψ and making the change ψ → ψ\overline{χ}, we find that

g(λ)^{2} = \frac{1}{(q−1)^{2}} ∑_{χ,ψ} ψ (\frac{λ}{4(λ+1)}) J(\overline{χ}, ϕ)J(\overline{χ}, ϕ\overline{χ})J(\overline{ψ}χ, ϕ)J(\overline{ψ}χ, ϕ\overline{ψ}χ).

Expanding the final Jacobi sum gives

g(λ)^{2} = \frac{1}{(q−1)^{2}} ∑_{ψ,x} ψ(\frac{λ}{4(λ+1)})\overline{ψ}(x)ϕ\overline{ψ}(1 − x) ∑_{χ} J(\overline{χ}, ϕ)J(\overline{ψ}χ, ϕ)J(\overline{χ}, ϕ\overline{χ})χ(x)χ(1 − x). (7)

Notice that (replacing w with y/w in the second step) we have

J(\overline{χ}, ϕ)J(\overline{ψ}χ, ϕ) = ∑_{w,y} \overline{χ}(w)ϕ(1 − w)\overline{ψ}χ(y)ϕ(1 − y) = ∑_{w,y} ϕχ(w)ϕ(w − y)\overline{ψ}(y)ϕ(1 − y).

Rearranging and using (6), this becomes

∑_{\begin{matrix}w,y \\ y≠1 \end{matrix}} ϕχ(w)ϕ(1 − \frac{1−w}{1−y})\overline{ψ}(y) = \frac{1}{q−1} ∑_{\begin{matrix}w,y \\ y≠1 \end{matrix}} ϕχ(w)\overline{ψ}(y) ∑_{A} J(\overline{A}, ϕ)A(\frac{1−w}{1−y}) + ∑_{y≠1} \overline{ψ}(y)

= \frac{1}{q−1} ∑_{A} J(\overline{A}, ϕ)J(ϕχ, A)J(\overline{ψ}, \overline{A}) + ∑_{y≠1} \overline{ψ}(y).

4 SCOTT AHLGREN, KEN ONO AND DAVID PENNISTON

This together with (7) shows that g(λ)^{2} is given by the expression

\frac{1}{(q−1)^{3}} ∑_{ψ,x} ψ(\frac{λ}{4(λ+1)})\overline{ψ}(x)ϕ\overline{ψ}(1 − x) ∑_{A}J(\overline{A}, ϕ)J(\overline{ψ}, \overline{A}) ∑_{χ}J(ϕχ, A)J(\overline{χ}, ϕ\overline{χ})χ(x)χ(1 − x) + E_{1}, (8)

where

E_{1} := \frac{1}{(q−1)^{2}} ∑_{ψ,x} ψ(\frac{λ}{4(λ+1)})\overline{ψ}(x)ϕ\overline{ψ}(1 − x) ∑_{χ} J(\overline{χ}, ϕ\overline{χ})χ(x)χ(1 − x) ∑_{y≠1} \overline{ψ}(y).  (9)

By (7), we may assume that x ≠ 0, 1. Therefore the sum on χ in (8) can be written

ϕ(\frac{x}{1−x}) ∑_{\begin{matrix}w,y \\ w≠0,1 \end{matrix}} A(1 − y)ϕ(w) ∑_{χ} ϕχ (\frac{x(1−x)y}{w(1−w)}) = (q − 1)ϕ(\frac{x}{1−x}) ∑_{w≠1} A (1 − \frac{w(1−w)}{x(1−x)}) ϕ(w).

When w = 1, the summand above equals (q − 1)ϕ(\frac{x}{1−x}). Accounting for this term, making the change of variables w → wx and factoring, the last displayed equation becomes

(q − 1)ϕ(1 − x) ∑_{w} ϕ(w)A(1 − w)A(1 − \frac{wx}{1−x}) − (q − 1)ϕ(\frac{x}{1−x}).

Together with (8) this yields

\begin{matrix}g(λ)^{2} = E_{1} + E_{2} + \\ \frac{1}{(q−1)^{2}} ∑_{ψ,x} ψ(\frac{λ}{4(λ+1)})\overline{ψ}(x)\overline{ψ}(1 − x) ∑_{A} J(\overline{A}, ϕ)J(\overline{ψ}, \overline{A}) ∑_{w} ϕ(w)A(1 − w)A(1 − \frac{wx}{1−x}), \end{matrix}(10)

where

E_{2} := \frac{−1}{(q−1)^{2}} ∑_{ψ,x} ψ(\frac{λ}{4(λ+1)})\overline{ψ}ϕ(x)\overline{ψ}(1 − x) ∑_{A} J(\overline{A}, ϕ)J(\overline{ψ}, \overline{A}).  (11)

Using (6), the sum on A in (10) can be written

∑_{\begin{matrix}w,z \\ w≠1,z≠0 \end{matrix}}ϕ(1 − z)ϕ(w) ∑_{A} J(\overline{ψ}, \overline{A})A(\frac{1}{z}(1 − w)(1 − \frac{wx}{1−x})

= (q − 1) ∑_{\begin{matrix}w,z \\ w≠1,z≠0 \end{matrix}} ϕ(1 − z)ϕ(w)\overline{ψ} (1 − \frac{1}{z}(1 − w)(1 − \frac{wx}{1−x})) − (q − 1) ∑_{z≠0} ϕ(1 − z)ϕ(\frac{1−x}{x}).

Replacing z by (1 − w)/z and simplifying, this becomes

\begin{matrix}(q − 1) ∑_{\begin{matrix}w,z \\ w≠1 \end{matrix}} & ϕ(z)ϕ(z − 1 + w)ϕ(w)\overline{ψ} (1 − z(1 − \frac{wx}{1−x})) + (q − 1)ϕ(\frac{1−x}{x}) \\ = (q − 1) ∑_{\begin{matrix}w,z \\ z≠1 \end{matrix}} ϕ(z)ϕ(z − 1 + w)ϕ(w)\overline{ψ} (1 − z(1 − \frac{wx}{1−x})) + F (ψ, x), \end{matrix} (12)

where F (ψ, x) contains the error arising from the terms w = 1 and z = 1; i.e.

F (ψ, x) = (q − 1)ϕ(\frac{1−x}{x}) − (q − 1) ∑_{z≠0} \overline{ψ} (1 − z(1 − \frac{x}{1−x})) + (q − 1) ∑_{w} \overline{ψ} (\frac{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

(q − 1) ∑_{w,z}ϕ(z)ϕ(−1 + w)ϕ(w)\overline{ψ}(1 − z)\overline{ψ} (1 + \frac{wzx}{1−x})) + F (ψ, x)

= ∑_{w,z} ϕ(z)ϕ(−1 + w)ϕ(w)\overline{ψ}(1 − z) ∑_{χ} J(\overline{χ}, \overline{ψ})χ(\frac{−wzx}{1−x}) + F (ψ, x)

= ∑_{χ} J(\overline{χ}, ϕ)J(\overline{ψ}, \overline{χ})J(\overline{ψ}, ϕχ)χ(\frac{x}{x−1}) + F (ψ, x).

With (10) this yields

g(λ)^{2} = \frac{1}{(q−1)^{2}} ∑_{ψ,χ} ψ(\frac{λ}{4(λ+1)})χ(−1)J(\overline{ψ}χ, \overline{ψ}\overline{χ})J(\overline{ψ}, \overline{χ})J(\overline{χ}, ϕ)J(\overline{ψ}, ϕχ) + E_{1} + E_{2} + E_{3},(14)

where

E_{3} := \frac{1}{(q−1)^{2}} ∑_{ψ,x} ψ(\frac{λ}{4(λ+1)})\overline{ψ}(x)\overline{ψ}(1 − x)F (ψ, x).  (15)

To simplify the main term in (14), we apply (4) to find that

χ(−1)(\overline{ψ}χ, \overline{ψ}\overline{χ})J(\overline{ψ}, \overline{χ}) = J(\overline{ψ}χ, ψ^{2})J(\overline{ψ}, χψ) = ∑_{w,y} \overline{ψ}χ(w)ψ^{2}(1 − w)\overline{ψ}(1 − y)ψχ(y).

Making the change w → \frac{w}{y} and simplifying, this becomes

∑_{\begin{matrix}w,y \\ y≠0 \end{matrix}} \overline{ψ}χ(w)ψ^{2}(y − w)\overline{ψ}(1 − y) = ∑_{\begin{matrix}w,y \\ w≠1 \end{matrix}} \overline{ψ}χ(w)ψ^{2}(y − w)\overline{ψ}(1 − y) − ∑_{w} ψχ(w) + ∑_{y} ψ(1 − y). (16)

For ease of notation, define a function δ by

δ(χ) := \begin{cases}0 if χ ≠ ϵ \\ 1 if χ = ϵ. \end{cases}

Making the change y → 1 − (1 − w)y and simplifying, the expression in (16) becomes

ψ(−1)J(\overline{ψ}χ, ψ)J(\overline{ψ}, \overline{ψ}) − (q − 1)δ(ψχ) + (q − 1)δ(ψ).

It is well-known (see[B-E-W]) that \overline{ψ}(4)J(\overline{ψ}, \overline{ψ}) = J(\overline{ψ}, ϕ) + (q − 1)δ(ψ). This, together with (14) and the last displayed equation, shows that

g(λ)^{2} = \frac{1}{(q−1)^{2}} ∑_{χ} J(\overline{χ}, ϕ) ∑_{ψ} ψ(\frac{−λ}{λ+1})J(\overline{ψ}, ϕ)J(\overline{ψ}χ, ψ)J(\overline{ψ}, ϕχ) + E_{1} + E_{2} + E_{3} + E_{4},(17)

where

E_{4} := \frac{−1}{q−1} ∑_{ψ}ψ(\frac{λ}{4(λ+1)})J(ψ, ϕ)J(\overline{ψ}, ϕ\overline{ψ}) + \frac{1}{q−1} ∑_{χ}J(\overline{χ}, ϕ)J(ϵ, ϕχ) + \frac{1}{q−1} ∑_{χ}J(\overline{χ}, ϕ)J(χ, ϵ)J(ϵ, ϕχ). (18)

The main term in (17) is

\frac{1}{(q−1)^{2}} ∑_{χ} J(\overline{χ}, ϕ) ∑_{\begin{matrix}w,y,z \\ wyz≠0 \end{matrix}} ϕ(1 − w)ϕχ(1 − y)χ(z) ∑_{ψ} \overline{ψ} (\frac{(λ+1)wyz}{−λ(1−z)}) .  (19)

Notice that \frac{z}{1−z} = \frac{−λ}{(λ+1)wy} if and only if z = \frac{\frac{−λ}{λ+1}}{wy− \frac{λ}{λ+1}} . Using this fact, making the changes

w → \frac{wλ}{λ+1}, y → \frac{y}{y−1}, and then applying (6), the main term becomes

\frac{1}{(q−1)} ∑_{χ}J(\overline{χ}, ϕ) ∑_{\begin{matrix}w,y \\ wy≠0 \end{matrix}} ϕ(1 − \frac{wλ}{λ+1})ϕχ(1 − y)\overline{χ}(1 − wy)

= \frac{1}{(q−1)} ∑_{χ} J(\overline{χ}, ϕ) ∑_{\begin{matrix}w,y \\ wy≠0 \end{matrix}} ϕ(1 − \frac{wλ}{λ+1})ϕ\overline{χ}(1 − y)\overline{χ}(1 − \frac{wy}{y−1})

= \frac{1}{(q−1)} ∑_{\begin{matrix}w,y \\ wy≠0 \end{matrix}} ϕ(1 − \frac{wλ}{λ+1})ϕ(1 − y) ∑_{χ} J(\overline{χ}, ϕ)χ(\frac{1}{1−y+wy})

= ∑_{\begin{matrix}w,y \\ w≠0 \end{matrix}} ϕ(1 − \frac{wλ}{λ+1})ϕ(1 − y)ϕ(1 − y(1 − w))ϕ(−y)ϕ(1 − w)

= ∑_{w,y} ϕ(1 − \frac{wλ}{λ+1})ϕ(1 − y)ϕ(1 − y(1 − w))ϕ(−y)ϕ(1 − w) + ϕ(−1).

Taking w to 1 − \frac{w}{y} and then replacing w, y by −w, −y, this becomes

ϕ(1 + λ) ∑_{w,y} ϕ(1 + \frac{wλ}{y})ϕ(1 − y)ϕ(1 − w)ϕ(−w) + ϕ(−1) = ϕ(1 + λ)A(λ) + ϕ(−1).

Together with (17), this yields

g(λ)^{2} = ϕ(1 + λ)A(λ) + E_{1} + E_{2} + E_{3} + E_{4} + ϕ(−1).

A calculation (see below) shows that

E_{1} + E_{2} + E_{3} + E_{4} = q − 1 − ϕ(−1) − 2ϕ(\frac{−λ}{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 E_{1} (which is given by (9)). We have

∑_{χ} J(\overline{χ}, ϕ\overline{χ})χ(x)χ(1 − x) = ∑_{w≠1} ϕ(w) ∑_{χ} \overline{χ} (\frac{w(1−w)}{x(1−x)}) .

The argument of χ equals one when w = x or w = 1 − x. Taking into account the case when x = \frac{1}{2}, we find that

∑_{χ} J(\overline{χ}, ϕ\overline{χ})χ(x)χ(1 − x) = (q − 1) [ϕ(x) + ϕ(1 − x)ϵ(x − \frac{1}{2})] .

Returning to (9), we note that ∑_{y≠1} \overline{ψ}(y) = −1 + (q − 1)δ(ψ). Therefore,

\begin{aligned}E_{1} := \frac{−1}{q−1} & ∑_{ψ} ψ(\frac{λ}{4(λ+1)})J(ϕ\overline{ψ}, ϕ\overline{ψ}) − \frac{1}{q−1} ∑_{\begin{matrix}ψ,x \\ x≠ \frac{1}{2} \end{matrix}} ψ(\frac{λ}{4(λ+1)})\overline{ψ}(x)\overline{ψ}(1 − x) + ∑_{x≠ \frac{1}{2} ,0,1} 1 + ∑_{x} ϕ(x)ϕ(1 − x) \\ & = \frac{−1}{q−1} ∑_{ψ} ψ (\frac{λ}{4(λ+1)}) J(ϕ\overline{ψ}, ϕ\overline{ψ}) − \frac{1}{q−1} ∑_{ψ} ψ (\frac{λ}{4(λ+1)}) J(\overline{ψ}, \overline{ψ}) + q − 3 − ϕ(−1). \end{aligned}

In the second sum we have used the fact that the summand is zero when x = \frac{1}{2}.

For the remaining error terms (which are given by (11), (15), (18)), we use similar arguments to find that

E_{2} = −ϕ(\frac{−λ}{λ+1})g(λ),

E_{3} = 1 + \frac{1}{q−1} ∑_{ψ} ψ (\frac{λ}{4(λ+1)}) J(ϕ\overline{ψ}, ϕ\overline{ψ}) + \frac{1}{q−1} ∑_{ψ} ψ (\frac{λ}{4(λ+1)}) J(\overline{ψ}, \overline{ψ}),

E_{4} = 1 − ϕ(\frac{−λ}{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 𝔽_{p} (p an odd prime). For each λ ∈ K \ {0, −1}, we define the surface X_{λ}/K to be the smooth complete model of the double cover of ℙ^{2} branched over the union U_{λ} of six lines given by

U_{λ} : XY Z(X + λY)(Y + Z)(Z + X) = 0.

To see that X_{λ} is a K3 surface, one can refer to [B-P-V], or check that the elliptic fibration of X_{λ} we give below satisfies the conditions given in [B-S], §4.

We now give a description of the natural map ψ : X_{λ} → ℙ^{2}. Let B_{λ} be the surface obtained in the following way: first blow up ℙ^{2} at the triple points of U_{λ}, then blow up the resulting surface at each of the double points of the total transform of U_{λ}. The total transform T_{λ} ⊂ B_{λ} of U_{λ} 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 U_{λ} and the dual graph of T_{λ} are depicted here (• denotes a component of odd multiplicity, and denotes a component of even multiplicity):

[figure: •; •; •; •; •; •; •; •; •; Figure; 1]

Then X_{λ} is the double cover of B_{λ} branched above the components of T_{λ} having odd multiplicity. Moreover, ψ^{−1}(U_{λ}) is a union of rational curves whose dual graph is exactly that of T_{λ}, and X_{λ} \ ψ^{−1}(U_{λ}) is a smooth affine surface given by the equation

s^{2} = xy(x + 1)(y + 1)(x + λy).  (21)

Now let us show that the X_{λ} are elliptic surfaces over ℙ^{1}. For a fixed λ, consider the family of cubics (with parameter τ) in ℙ^{2} given by

X(Y + Z)(Z + X) − τ(X + λY)Y Z = 0.  (22)

One can check that the generic cubic in this family intersects U_{λ} 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 U_{λ} 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 B_{λ} intersects only components of T_{λ} of even multiplicity, and therefore the family (22) lifts through ψ to X_{λ}.

To see the elliptic fibration explicitly, the change of variables

X = λt^{2}V W, Y = U^{2} + λt^{2}UW, Z = UV

(where τ = t^{2}, reflecting the double cover) transforms (22) into

C_{λ} : V^{2}W + (1 − t^{2})UV W + λt^{2}V W^{2} = U^{3} + λt^{2}U^{2}W.  (23)

The discriminant of C_{λ} is −λ^{3}t^{8} [t^{6} + (8λ − 3)t^{4} + (16λ^{2} + 20λ + 3)t^{2} − (λ + 1)]. Using Tate’s algorithm, we find that the singular fibers are generically of the following types:

I_{10} I_{8} I_{1} I_{1} I_{1} I_{1} I_{1} I_{1}.  (24)

The singular fibers above t = ∞ and t = 0 are visible in the dual graph of T_{λ} (see Figure 1) as the outer and inner cycles, respectively.

The Néron-Severi group of X_{λ}/\overline{K} is the group of algebraic equivalence classes of divisors on X_{λ}. Shioda [S] showed that the rank r_{λ} of the Néron-Severi group of X_{λ}/\overline{K} is given by

r_{λ} = ∑(m_{j} − 1) + 2 + rank(C_{λ}(\overline{K}(t)),

where the sum is over the singular fibers, and m_{j} 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 Néron-Severi group modulo torsion. From (24), then, these give 18 linearly independent elements. Also note that (0, 0) is an element of C_{λ}(\overline{K}(t)), and one can check that it has infinite order. Therefore r_{λ} ≥ 19.

4. Proof of Theorem 1.1

Given a prime p, the zeta function of a projective variety Y/𝔽_{p} is

Z(Y/𝔽_{p}, T) = exp (∑_{n=1}^{∞} #Y (𝔽_{p^{n}}) · T^{n}/n) .

The following proposition is our first step in proving Theorem 1.1.

Proposition 4.1. If p is an odd prime, λ ∈ 𝔽_{p} \ {0, −1}, and q = p^{r}, then

#X_{λ}(𝔽_{q}) = 1 + q^{2} + 19q + ∑_{x,y∈𝔽_{q}} ϕ_{q}(xy(x + 1)(y + 1)(x + λy)).

Proof. Recall (see §3) that we have a map ψ : X_{λ} → ℙ^{2} ramified above U_{λ}, that X_{λ} \ψ^{−1}(U_{λ}) is a smooth affine surface given by (21) and that ψ^{−1}(U_{λ}) is a union of rational curves whose dual graph is given in Figure 1. Since all the components of U_{λ} are lines defined over 𝔽_{p}, we can read from Figure 1 that the number of 𝔽_{q}-rational points on ψ^{−1}(U_{λ}) is

9(q + 1) + 15(q − 1) = 24q − 6.

Moreover, from (21) we see that the number of 𝔽_{q}-rational points on X_{λ} \ψ^{−1}(U_{λ}) with s = 0 is 5q − 7. Therefore, we have that

#X_{λ}(𝔽_{q}) = ∑_{x,y∈𝔽_{q}} (ϕ_{q}(xy(x + 1)(y + 1)(x + λy)) + 1) + (24q − 6) − (5q − 7).

Recall that if E_{λ} (λ ∈ ℚ) has good reduction at p, then the zeta function of E_{λ} at p is

Z(E_{λ}/𝔽_{p}, T) = \frac{1 − a(λ, p)T + pT^{2}}{(1 − T)(1 − pT)} = \frac{(1 − π_{λ,p}T)(1 − \overline{π}_{λ,p}T)}{(1 − T)(1 − pT)}.

Then Proposition 4.1 and Theorem 2.1 imply

Theorem 4.2. If λ ∈ ℚ \ {0, −1} and E_{λ} has good reduction at a prime p, then

#X_{λ}(𝔽_{p}) = 1 + p^{2} + 19p + ϕ_{p}(λ + 1) (a(λ, p)^{2} − p)  (25)

and

#X_{λ}(𝔽_{p^{2}}) = 1 + p^{4} + 18p^{2} + (a(λ, p)^{2} − 2p)^{2} .  (26)

Now let us prove Theorem 1.1. The Weil conjectures tell us that Z(X_{λ}/𝔽_{p}, T) ∈ ℚ(T). Indeed, since X_{λ} is a K3 surface, its zeta function has the form

Z(X_{λ}/𝔽_{p}, T) = \frac{1}{(1 − T)(1 − p^{2}T)P_{λ,p}(T)},

with

P_{λ,p}(T) = ∏_{i=1}^{22}(1 − α_{i}T), α_{i} ∈ ℂ, |α_{i}| = p.

As in [B-S], §12, we will use the Néron-Severi group of X_{λ}/\overline{𝔽}_{p} to find a large factor of P_{λ,p}(T). Recall from §3 that on the elliptic surface X_{λ}, the generic fiber, zero section and the components of the singular fibers not meeting the zero section are linearly independent elements of the Néron-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

P_{λ,p}(T) = (1 − pT)^{18} · ∏_{i=1}^{4}(1 − α_{i}T).

Hence, for any positive integer r,

#X_{λ}(𝔽_{p^{r}}) = 1 + p^{2r} + 18p^{r} + α_{1}^{r} + α_{2}^{r} + α_{3}^{r} + α_{4}^{r}.  (27)

Since r_{λ} ≥ 19, it follows that one of the α_{i} (1 ≤ i ≤ 4) is equal to p times a root of unity. If we denote by ω_{r} a primitive r-th root of unity, then α_{1} = pω_{r} for some r.

In what follows, for brevity write π_{λ,p} := π and \overline{π}_{λ,p} := \overline{π}, and recall that |π| = |\overline{π}| = \sqrt{p}. Note that (26) and (27) give

∑_{i=1}^{4} α_{i}^{2} = (a(λ, p)^{2} − 2p)^{2} = π^{4} + \overline{π}^{4} + 2p^{2}.  (28)

We claim that one of the α_{i} is real. To see this, suppose that none of the α_{i} is real. Then α_{1} is either quadratic or quartic over ℤ. If it is quartic, then r ∈ {5, 8, 10, 12}. If r = 5 or r = 10, then ∑_{i=1}^{4} α_{i}^{2} = −p^{2}, a contradiction since by (28), ∑ α_{i}^{2} is the square of an integer. If r = 8, then ∑_{i=1}^{4} α_{i}^{2} = 0, and (28) gives a(λ, p)^{2} = 2p, a contradiction since p is odd. Finally, if r = 12, then ∑_{i=1}^{4} α_{i}^{2} = 2p^{2}, again contradicting the fact that ∑ α_{i}^{2} is the square of an integer. We conclude that α_{1} is not quartic over ℤ. Similar arguments show that α_{1} is not quadratic over ℤ, and our claim follows.

Since one of the α_{i} is real, and the α_{i} satisfy the same quartic polynomial over ℤ, it follows that two of the α_{i}, say α_{1} and α_{2}, are real. Equations (25), (26) and (27) now yield

α_{1} + α_{2} + α_{3} + α_{4} = (1 + ϕ_{p}(λ + 1)) p + ϕ_{p}(λ + 1)(π^{2} + \overline{π}^{2})  (29)

and

α_{3}^{2} + α_{4}^{2} = π^{4} + \overline{π}^{4}.  (30)

These imply that

(α_{3} + α_{4})^{2} − (π^{4} + \overline{π}^{4}) = 2α_{3}α_{4} ∈ \{2p^{2}, −2p^{2}\}.  (31)

We now claim that there exist 1 ≤ i, j ≤ 4, i ≠ j, with α_{i} = p and α_{j} = ϕ_{p}(λ + 1)p. We argue by contradiction. If this does not hold, then we have four possible outcomes. Case 1: ϕ_{p}(λ + 1) = 1, α_{1} = α_{2} = −p. Then (29) gives

α_{3} + α_{4} = 4p + π^{2} + \overline{π}^{2}.

Since |α_{i}| = p = |π^{2}|, this implies that α_{3} = α_{4} = p, contradicting our assumption. Case 2: ϕ_{p}(λ + 1) = 1, α_{1} = p, α_{2} = −p. Then (29) gives

α_{3} + α_{4} = 2p + π^{2} + \overline{π}^{2},

and the left hand side of (31) is 6p^{2}+4p(π^{2}+\overline{π}^{2}). Hence π^{2}+\overline{π}^{2} ∈ \{−p, −2p\}. If π^{2}+\overline{π}^{2} = −p, then a(λ, p)^{2} = p, a contradiction since a(λ, p) ∈ ℤ. If π^{2} + \overline{π}^{2} = −2p, then π^{2} = \overline{π}^{2} = −p, which by (29) gives α_{3} + α_{4} = 0, implying that either α_{3} = ip, α_{4} = −ip (contradicting (30), since π^{4} + \overline{π}^{4} = 2p^{2}), or α_{3} = p, α_{4} = −p, which contradicts our assumption. Case 3: ϕ_{p}(λ + 1) = −1, α_{1} = α_{2} = p. Then (29) gives

α_{3} + α_{4} = −2p − π^{2} − \overline{π}^{2},

which gives a contradiction by the same argument as in Case 2. Case 4: ϕ_{p}(λ + 1) = −1, α_{1} = α_{2} = −p. Then (29) gives

α_{3} + α_{4} = 2p − π^{2} − \overline{π}^{2},

and the left hand side of (31) becomes 6p^{2} −4p(π^{2} +\overline{π}^{2}), so π^{2} +\overline{π}^{2} ∈ \{p, 2p\}. If π^{2} +\overline{π}^{2} = p, then π^{2} = pω_{6}, which contradicts the fact that π is at most quadratic over ℤ. So π^{2}+\overline{π}^{2} = 2p, which implies that a(λ, p)^{2} = 4p, also a contradiction.

Upon renumbering, then, we have α_{1} = p, α_{2} = ϕ_{p}(λ + 1)p. Then (29) becomes

α_{3} + α_{4} = ϕ_{p}(λ + 1)(π^{2} + \overline{π}^{2}).  (32)

To finish, we just need to show that \{α_{3}, α_{4}\} = \{ϕ_{p}(λ + 1)π^{2}, ϕ_{p}(λ + 1)\overline{π}^{2}\}. First suppose that α_{3} ∈ ℝ. Then α_{4} ∈ ℝ, and either α_{3} = α_{4} (which by (32) immediately gives our result since |π^{2}| = |\overline{π}^{2}| = p), or α_{3} = −α_{4}, which by (32) gives π^{2} + \overline{π}^{2} = 0. This latter implies that a(λ, p)^{2} = 2p, a contradiction since p is odd.

Finally, suppose that α_{3} ∉ ℝ. We have seen that π^{2} + \overline{π}^{2} ≠ 0, so by (32), π^{2} ∉ ℝ and Re(α_{3}) = ϕ_{p}(λ + 1)Re(π^{2}), and this gives our result since |α_{3}| = |π^{2}| = p.

5. Proof of Theorem 1.2

Recall that X_{λ} is modular if there is a weight 3 cusp form f_{λ}(z) = ∑_{n=1}^{∞} a_{λ}(n)q^{n} (q := e^{2πiz} throughout) which is an eigenform of the Hecke operators and has the property that

L(X_{λ}, s) := ∏_{p} Z^{∗}(X_{λ}/𝔽_{p}, p^{−s}) = ∑_{n=1}^{∞} \frac{a_{λ}(n)}{n^{s}}.  (33)

Here the product is over the primes p at which E_{λ} has good reduction, and Z^{∗}(X_{λ}/𝔽_{p}, T) denotes the product of the two nontrivial factors of Z(X_{λ}/𝔽_{p}, T). In view of Theorem 1.1 and our formulation above, care must be taken in defining Z^{∗}(X_{λ}/𝔽_{p}, T) for the primes p at which E_{λ} has supersingular reduction.

If η(z) is Dedekind’s eta-function

η(z) := q^{1/24} ∏_{n=1}^{∞}(1 − q^{n}),  (34)

then the four weight 3 newforms we require are:

\begin{matrix}A(z) := η^{6}(4z) = q − 6q^{5} + 9q^{9} + ⋯ ∈ S_{3}(Γ_{0}(16), χ_{−1}), & (35) \\ B(z) := η^{2}(z)η(2z)η(4z)η^{2}(8z) = q − 2q^{2} − 2q^{3} + ⋯ ∈ S_{3}(Γ_{0}(8), χ_{−2}), & (36) \\ C(z) := η^{3}(2z)η^{3}(6z) = q − 3q^{3} + 2q^{7} + 9q^{9} − ⋯ ∈ S_{3}(Γ_{0}(12), χ_{−3}), & (37) \\ D(z) := η^{3}(z)η^{3}(7z) = q − 3q^{2} + 5q^{4} − 7q^{7} − ⋯ ∈ S_{3}(Γ_{0}(7), χ_{−7}). & (38) \end{matrix}

We begin by considering the “if” direction of Theorem 1.2 (i.e. that X_{λ} is modular for the seven exceptional values of λ). If λ ∈ ℚ \ {0, −1}, then the j-invariant of E_{λ} is

j(E_{λ}) = \frac{64(λ + 4)^{3}}{λ^{2}} .  (39)

Using the fact that an elliptic curve E/ℚ has complex multiplication if and only if

j(E) ∈ {12^{3}, 66^{3}, 20^{3}, 0, 2 · 30^{3}, −3 · 160^{3}, −15^{3}, 255^{3}, −32^{3}, −96^{3}, −960^{3}, −5280^{3}, −640320^{3}},

(39) implies that E_{λ} has complex multiplication if and only if λ is one of the seven exceptional values. Arguing as in [B-S], one can show that f_{1} = B ⊗χ_{−4}, f_{8} = A, f_{1/8} = A⊗χ_{8}, 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 χ_{D} denotes the Kronecker character for the quadratic field ℚ(\sqrt{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(X_{λ}, s) cannot be a modular L-function. For these λ, it is well known that the set of primes p at which E_{λ} has supersingular reduction has density zero. Moreover, if p is a prime of good reduction which is not supersingular for E_{λ}, then the local Euler factor at p for L(X_{λ}, s) is

\frac{1}{1 − ϕ_{p}(λ + 1) (a(λ, p)^{2} − 2p) p^{−s} + p^{2−2s}} .  (40)

Therefore, for such p the coefficient a_{λ}(p) in L(X_{λ}, s) is ϕ_{p}(λ + 1)(a(λ, p)^{2} − 2p). If X_{λ} is modular, then for every prime ℓ there is a Galois representation

ρ_{ℓ} : Gal(\overline{ℚ}/ℚ) → GL_{2}(𝔽_{ℓ})

with the property that

Tr(ρ_{ℓ}(Frob(p)) ≡ ϕ_{p}(λ + 1)(a(λ, p)^{2} − 2p) (mod ℓ)  (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_{ℓ} 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].

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Department of Mathematics, Colgate University, Hamilton, NY 13346

  E-mail address: sahlgren@mail.colgate.edu

Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706

  E-mail address: ono@math.psu.edu

Department of Mathematics, Pennsylvania State University, University Park, PA 16802

  E-mail address: dkp@math.psu.edu