行政院國家科學委員會專題研究計畫 期中進度報告
賈可比式, 模式, 類數, 及多變數 zeta 函數的研究(1/2)
計畫類別： 個別型計畫
計畫編號： NSC94-2115-M-009-012-
執行期間： 94 年 08 月 01 日至 95 年 07 月 31 日
執行單位： 國立交通大學應用數學系(所)
計畫主持人： 楊一帆
報告類型： 精簡報告
報告附件： 出席國際會議研究心得報告及發表論文
處理方式： 本計畫可公開查詢
中 華 民 國 95 年 5 月 15 日
FORMS AND K3 SURFACES
YIFAN YANG AND NORIKO YUI
Abstract. We study differential equations satisfied by bi-modular forms as-sociated to genus zero subgroups of SL2(R) of the form Γ0(N ) or Γ0(N )∗. In
some examples, these differential equations are realized as the Picard–Fuchs differential equations of families of K3 surfaces with large Picard numbers, e.g., 19, 18, 17, 16. Our method rediscovers some of the Lian–Yau examples of “modular relations” involving power series solutions to second and third order differential equations of Fuchsian type in [10, 11].
1. Introduction
Lian and Yau [10, 11] studied arithmetic properties of mirror maps of pencils of certain K3 surfaces, and further, they considered mirror maps of certain families of Calabi–Yau threefolds [12]. Lian and Yau observed in a number of explicit exam-ples a mysterious relationship (now the so-called mirror moonshine phenomenon) between mirror maps and the McKay–Thompson series (Hauptmoduls of one vari-able associated to a genus zero congruence subgroup of SL2(R)) arising from the
Monster. Inspired by the work of Lian and Yau, Verrill–Yui [16] further computed more examples of mirror maps of one-parameter families of lattice polarized K3 surfaces with Picard number 19. The outcome of Verrill–Yui’s calculations sug-gested that the mirror maps themselves are not always Hauptmoduls, but they are commensurable with Hauptmoduls (referred as the modularity of mirror maps). This fact was indeed established by Doran [6] for Mn-lattice polarized K3 surfaces
of Picard number 19 (where Mn = U ⊥ (−E8)2 ⊥ h−2ni). The mirror maps
were calculated via the Picard–Fuchs differential equations of the K3 families in question. Therefore, the determination of the Picard–Fuchs differential equations played the central role in their investigations.
In this paper, we will address the inverse problem of a kind. That is, instead of starting with families of K3 surfaces or families of Calabi–Yau threefolds, we start with modular forms and functions of more than one variable.
More specifically, the main focus our discussions in this paper are on modular forms and functions of two variables. Here is the precise definition.
Date: May 15, 2006.
2000 Mathematics Subject Classification. Primary 11F03, 11F11, 14D05, 11F03, 14J28. Key words and phrases. K3 surfaces, modular forms, modular functions, Picard–Fuchs differ-ential equations, hypergeometric differdiffer-ential equations.
N. Yui was partially supported by Discovery Grant from the Natural Science and Engineering Research Council (NSERC) of Canada. Y. Yang was supported by Grant 93-2115-M-009-014 from the National Science Council (NSC) of Taiwan and by the National Center for Theoretical Sciences (NCTS) of Taiwan.
Definition 1.1. Let H denote the upper half-plane {τ : =τ > 0}, and let H∗ = H ∪ Q ∪ {∞}. Let Γ1 and Γ2 be two subgroups of SL2(R) commensurable with
SL2(Z). We call a function F : H∗× H∗7−→ C of two variables a bi-modular form
of weight (k1, k2) on Γ1× Γ2with character χ if F is meromorphic on H∗× H∗such
that F (γ1τ1, γ2τ2) = χ(γ1, γ2)(c1τ1+ d1)k1(c2τ2+ d2)k2F (τ1, τ2) for all γ1= a1 b1 c1 d1 ∈ Γ1, γ2= a2 b2 c2 d2 ∈ Γ2.
If F is a bi-modular form of weight (0, 0) with trivial character, then we also call F a bi-modular function on Γ1× Γ2.
Notation. We let q1 = e2πiτ1 and q2= e2πiτ2. For a variable t we let Dt denote
the the differential operator t∂t∂.
Remark 1.1. Stienstra and Zagier [15] have a notion of bi-modular forms. Let Γ ⊂ SL2(Z), and let τ, τ∗∈ H. Let k1, k2be integers. A two-variable meromorphic
function F : H × H → C is called a bi-modular form of weight (k1, k2) on Γ if for
any γ =a b
c d
∈ Γ, it satisfies the transformation formula: F (γτ, γτ∗) = (cτ + d)k1(cτ∗+ d)k2F (τ, τ∗). For instance,
F (τ, τ∗) = τ − τ∗
is a bi-modular form for SL2(Z) of weight (−1, −1). Another typical example is
F (τ, τ∗) = E2(τ ) −
1 τ − τ∗
is a bi-modular form of weight (2, 0) for SL2(Z).
Our definition of bi-modular forms coincides with that of Stienstra and Zagier, if we take Γ1= Γ2 and γ1= γ2.
The problems that we will consider here are formulated as follows : Given a bi-modular form F , determine a differential equation it satisfies, and construct a family of K3 surfaces (or degenerations of a family of Calabi–Yau threefolds at some limit points) having the determined differential equation as its Picard–Fuchs differential equation.
In fact, a similar problem was already raised by Lian and Yau in their papers [10, 11]. They discussed the so-called “modular relations” involving power series solutions to second and third order differential equations of Fuchsian type (e.g., hypergeometric differential equations2F1,3F2) and modular forms of weight 4 using
mirror symmetry.
In this paper, we will focus our discussion on bi-modular forms of weight (1, 1). We will determine the differential equations satisfied by bi-modular forms of weight (1, 1) associated to genus zero subgroups of SL2(R), e.g., Γ0(N ) and Γ0(N )∗. Then
the existence and the construction of particular bi-modular forms of weight (1, 1) are discussed, using solutions of some hypergeometric differential equations. Moreover, we determine the differential equations they satisfy. Further, several examples of bi-modular forms and their differential equations are discussed aiming to realize these differential equations as the Picard–Fuchs differential equations of some families
of K3 surfaces (or degenerations of families of Calabi–Yau threefolds) with large Picard numbers 19, 18, 17 and 16.
It should be pointed out that our paper and our results have non-empty inter-sections with the results of Lian and Yau [10, 11]. Indeed, our approach rediscovers some of the examples of Lian and Yau.
2. Differential equations satisfied by bi-modular forms
We will now determine differential equations satisfied by bi-modular forms of weight (1, 1).
Theorem 2.1. Let F (τ1, τ2) be a bi-modular form of weight (1, 1), and let x(τ1, τ2)
and y(τ1, τ2) be non-constant bi-modular functions on Γ1 × Γ2. Then F , as a
function of x and y, satisfy a system of partial differential equations Dx2F + a0DxDyF + a1DxF + a2DyF + a3F = 0,
D2yF + b0DxDyF + b1DxF + b2DyF + b3F = 0,
(2.1)
where ai and bi are algebraic functions of x and y, and can be expressed explicitly
as follows. Suppose that, for each function t among F , x, and y, we let Gt,1= Dq1t t = 1 2πi dt dτ1 , Gt,2= Dq2t t = 1 2πi dt dτ2 . Then we have a0= 2Gy,1Gy,2 Gx,1Gy,2+ Gy,1Gx,2 , b0= 2Gx,1Gx,2 Gx,1Gy,2+ Gy,1Gx,2 , a1= G2 y,2(Dq1Gx,1− 2GF,1Gx,1) − G 2 y,1(Dq2Gx,2− 2GF,2Gx,2) G2 x,1G2y,2− G2y,1G2x,2 , b1= −G2 x,2(Dq1Gx,1− 2GF,1Gx,1) + G 2 x,1(Dq2Gx,2− 2GF,2Gx,2) G2 x,1G2y,2− G2y,1G2x,2 , a2= G2
y,2(Dq1Gy,1− 2GF,1Gy,1) − G
2
y,1(Dq2Gy,2− 2GF,2Gy,2) G2 x,1G2y,2− G2y,1G2x,2 , b2= −G2 x,2(Dq1Gy,1− 2GF,1Gy,1) + G 2 x,1(Dq2Gy,2− 2GF,2Gy,2) G2 x,1G2y,2− G2y,1G2x,2 , a3= − G2 y,2(Dq1GF,1− G 2 F,1) − G 2 y,1(Dq2GF,2− G 2 F,2) G2x,1G2y,2− G2 y,1G 2 x,2 , and b3= − −G2 x,2(Dq1GF,1− G 2 F,1) + G 2 x,1(Dq2GF,2− G 2 F,2) G2 x,1G2y,2− G2y,1G2x,2 .
In order to prove Theorem 2.1, we first need the following lemma, which is an analogue of the classical Ramanujan’s differential equations
DqE2= E2 2− E4 12 = −24 X n∈N n2qn (1 − qn)2, DqE4= E2E4− E6 3 = 240 X n∈N n4qn (1 − qn)2,
DqE6= E2E6− E42 2 = X n∈N n6qn (1 − qn)2 where (2.2) Ek= 1 − 2k Bk X n∈N nk−1qn 1 − qn
are the Eisenstein series of weight k on SL2(Z), where Bkdenotes the k-th Bernoulli
number, e.g., B2= 16, B4= −301 and B6= 421.
Lemma 2.2. We retain the notations of Theorem 2.1. Then (a) Gx,1 and Gy,1are bi-modular forms of weight (2, 0),
(b) Gx,2 and Gy,2 are bi-modular forms of weight (0, 2),
(c) Dq1Gx,1−2GF,1Gx,1, Dq1Gy,1−2GF,1Gy,1and Dq1GF,1−G
2
F,1are bi-modular
forms of weight (4, 0), and
(d) Dq2Gx,2−2GF,2Gx,2, Dq2Gy,2−2GF,2Gy,2and Dq2GF,2−G
2
F,2are bi-modular
forms of weight (0, 4).
Proof. We shall prove (a) and (c); the proof of (b) and (d) is similar.
By assumption, x is a bi-modular function on Γ1× Γ2. That is, for all γ1 =
a1 b1 c1 d1 ∈ Γ1and all γ2= a2 b2 c2 d2 ∈ Γ2, one has x(γ1τ1, γ2τ2) = x(τ1, τ2)
Taking the logarithmic derivatives of the above equation with respect to τ1, we
obtain 1 (c1τ1+ d1)2 ˙ x x(γ1τ1, τ2) = ˙ x x(τ1, τ2), or (2.3) Gx,1(γ1τ1, γ2τ2) = (c1τ1+ d1)2Gx,1(τ1, τ2),
where we let ˙x denote the derivative of the two-variable function x with respect to the first variable. This shows that Gx,1 is a bi-modular form of weight (2, 0) on
Γ1× Γ2with the trivial character. The proof for the case Gy,1 is similar.
Likewise, taking the logarithmetic derivatives of the equation F (γ1τ1, γ2τ2) = χ(γ1, γ2)(c1τ1+ d1)(c2τ2+ d2)F (τ1, τ2)
with respect to τ1, we obtain
1 (c1τ1+ d1)2 ˙ F F(γ1τ1, γ2τ2) = c1 (c1τ1+ d1) +F˙ F(τ1, τ2), or, equivalently (2.4) GF,1(γ1τ1, γ2τ2) = c1(c1τ1+ d1) 2πi + (c1τ1+ d1) 2G F,1(τ1, τ2).
Now, differentiating (2.3) with respect to τ1 again, we obtain
˙ Gx,1 (c1τ1+ d1)2 (γ1τ1, γ2τ2) = 2c1(c1τ1+ d1)Gx,1(τ1, τ2) + (c1τ1+ d1)2G˙x,1(τ1, τ2), or Dq1Gx,1(γ1τ1, γ2τ2) = c1(c1τ1+ d1)3 πi Gx,1(τ1, τ2) + (c1τ1+ d1) 4D q1Gx,1(τ1, τ2).
On the other hand, we also have, by (2.3) and (2.4), GF,1Gx,1(γ1τ1, γ2τ2) = c1(c1τ1+ d1)3 2πi Gx,1(τ1, τ2) + (c1τ1+ d1) 4G F,1Gx,1(τ1, τ2).
From these two equations we see that Dq1Gx,1− 2GF,1Gx,1is a bi-modular form of weight (4, 0) with the trivial character.
Finally, differentiating (2.4) with respect to τ1 and multiplying by (c1τ1+ d1)2
we have Dq1GF,1(γ1τ1, γ2τ2) = c21(c1τ1+ d1)2 (2πi)2 + c1(c1τ1+ d1)3 πi GF,1(τ1, τ2) + (c1τ1+ d1)4Dq1GF,1(τ1, τ2). Combining this with the square of (2.4) we see that Dq1GF,1− G
2
F,1is a bi-modular
form of weight (4, 0) on Γ1× Γ2. This completes the proof of the lemma.
Proof of Theorem 2.1. In light of Lemma 2.2, the functions ak, bkare all bi-modular
functions on Γ1× Γ2, and thus can be expressed as algebraic functions of x and y.
Therefore, it suffices to verify (2.1) as formal identities. By the chain rule we have Dq1F Dq2F =x −1D q1x y −1D q1y x−1Dq2x y −1D q2y DxF DyF . It follows that DxF DyF = F Gx,1Gy,2− Gx,2Gy,1 Gy,2 −Gy,1 −Gx,2 Gx,1 GF,1 GF,2 Writing ∆ = Gx,1Gy,2− Gx,2Gy,1, and ∆x= GF,1Gy,2− GF,2Gy,1, ∆y= −Gx,2GF,1+ Gx,1GF,2, we have (2.5) DxF = F ∆x ∆ , DyF = F ∆y ∆ . Applying the same procedure on DxF again, we obtain
D2xF DyDxF = 1 ∆ Gy,2 −Gy,1 −Gx,2 Gx,1 Dq1(F ∆x/∆) Dq2(F ∆x/∆) = F ∆ Gy,2 −Gy,1 −Gx,2 Gx,1 ∆x ∆ GF,1 GF,2 +Dq1(∆x/∆) Dq2(∆x/∆) . That is, (2.6) D2xF = F∆ 2 x ∆2 + F ∆ Gy,2Dq1 ∆x ∆ − Gy,1Dq2 ∆x ∆ and (2.7) DyDxF = F ∆x∆y ∆2 + F ∆ −Gx,2Dq1 ∆x ∆ + Gx,1Dq2 ∆x ∆ .
We then substitute (2.5), (2.6), and (2.7) into (2.1) and find that (2.1) indeed holds. (The details are tedious, but straightforward calculations. We omit the
3. Bi-modular forms associated to solutions of hypergeometric differential equations
Here we will construct bi-modular forms of weight (1, 1) using solutions of some hypergeometric differential equations. Our main result of this section is the follow-ing theorem.
Theorem 3.1. Let 0 < a < 1 be a positive real number. Let f (t) = 2F1(a, a; 1; t)
be a solution of the hypergeometric differential equation
(3.1) t(1 − t)f00+ [1 − (1 + 2a)t]f0− a2f = 0. Let F (t1, t2) = f (t1)f (t2)(1 − t1)a(1 − t2)a, x = t1+ t2 (t1− 1)(t2− 1) , y = t1t2 (t1+ t2)2 .
Then F is a bi-modular form of weight (1, 1) for Γ1× Γ2. Furthermore, F , as a
function of x and y is a solution of the partial differential equations
(3.2) Dx(Dx− 2Dy)F + x(Dx+ a)(Dx+ 1 − a)F = 0,
and
(3.3) Dy2F − y(2Dy− Dx+ 1)(2Dy− Dx)F = 0,
where Dx= ∂/∂x and Dy= ∂/∂y.
Remark 3.1. Theorem 2.1 of Lian and Yau [11] is essentially the same as our Theorem 3.1, though the formulation and proof are different.
We will present our proof of Theorem 3.1 now. For this, we need one more ingredient, namely, the Schwarzian derivatives.
Lemma 3.2. Let f (t) and f1(t) be two linearly independent solutions of a
differ-ential equation
f00+ p1f0+ p2f = 0.
Set τ := f1(t)/f (t). Then the associated Schwarzian differential equation
2Q dt dτ
2
+ {t, τ } = 0, where {t, τ } is the Schwarzian derivative
{t, τ } = dt 3/dτ3 dt/dτ − 3 2 dt2/dτ2 dt/dτ 2 , satisfies Q = 4p2− 2p 0 1− p 2 1 4 .
Proof. This is standard, and proof can be found, for instance, in Lian and Yau
Proof of Theorem 3.1. Let f1 be another solution of (3.1) linearly independent of
f , and set τ = f1/f . Then a classical identity asserts that
f2= c exp − Z t1 − (1 + 2a)u u(1 − u) du dt dτ = cdt/dτ t(1 − t)2a,
where c is a constant depending on the choice of f1. Thus, letting
q1= e2πif1(t1)/f (t1) and q2= e2πif1(t2)/f (t2),
the function F , with a suitable choice of f1, is in fact
F (t1, t2) =
Dq1t1· Dq2t2 t1t2
1/2 .
We now apply the differential identities in (2.1), which hold for arbitrary F , x, and y. We have Gx,1 := Dq1x x = (1 + t2)Dq1t1 (t1+ t2)(1 − t1) , Gx,2 := Dq2x x = (1 + t1)Dq2t2 (t1+ t2)(1 − t2) , Gy,1:= Dq1y y = (t2− t1)Dq1t1 t1(t1+ t2) , Gy,2:= Dq2y y = (t1− t2)Dq2t2 t2(t1+ t2) , GF,1:= Dq1F F = t1D2q1t1− (Dq1t1) 2 2t1Dq1t1 , GF,2 := Dq2F F = t2D2q2t2− (Dq2t2) 2 2t2Dq2t2 . It follows that a0:= 2Gy,1Gy,2 Gx,1Gy,2+ Gy,1Gx,2 = −2(t1− 1)(t2− 1) t1t2+ 1 = − 2 1 + x, b0:= 2Gx,1Gx,2 Gx,1Gy,2+ Gy,1Gx,2 =2t1t2(t1+ 1)(t2+ 1) (t1− t2)2(t1t2+ 1) = 2y(1 + 2x) (1 + x)(1 − 4y), a1: = G2 y,2(Dq1Gx,1− 2GF,1Gx,1) − G 2 y,1(Dq2Gx,2− 2GF,2Gx,2) G2 x,1G2y,2− G2y,1G2x,2 = t1+ t2 t1t2+ 1 = x 1 + x, b1: = −G2 x,2(Dq1Gx,1− 2GF,1Gx,1) + G 2 x,1(Dq2Gx,2− 2GF,2Gx,2) G2 x,1G2y,2− G2y,1G2x,2 = t1t2(t1+ 1)(t2+ 1) (t1− t2)2(t1t2+ 1) = y(1 + 2x) (1 + x)(1 − 4y), a2:=
G2y,2(Dq1Gy,1− 2GF,1Gy,1) − G
2
y,1(Dq2Gy,2− 2GF,2Gy,2) G2 x,1G 2 y,2− G 2 y,1G 2 x,2 = 0, b2: = −G2 x,2(Dq1Gy,1− 2GF,1Gy,1) + G 2 x,1(Dq2Gy,2− 2GF,2Gy,2) G2 x,1G2y,2− G2y,1G2x,2 = − 2t1t2 (t1− t2)2 = − 2y 1 − 4y.
Moreover, we have a3: = − G2y,2(Dq1GF,1− G 2 F,1) − G 2 y,1(Dq2GF,2− G 2 F,2) G2 x,1G2y,2− G2y,1G2x,2 =(t1− 1)(t2− 1)(t1+ t2)t 2 1˙t42(2 ˙t1 ... t1− 3¨t21) − t22˙t41(2 ˙t2 ... t2− 3¨t22) 4(t1− t2)(t21t22− 1) ˙t41˙t42 , where, for brevity, we let ˙tj, ¨tj,
...
tj denote the derivatives Dqjtj, D
2
qjtj, and D
3 qjtj, respectively. To express a3 in terms of x and y, we note that, by Lemma 3.2,
2 ˙tj ... tj− 3¨t2j = − ˙t 4 j −4a2 tj(1 − tj) − 2 d dtj 1 − (1 + 2a)tj tj(1 − tj) −(1 − (1 + 2a)tj) 2 t2 j(1 − tj)2 ! = −(tj− 1) 2+ 4a(1 − a)t j t2 j(tj− 1)2 ˙t4 j. It follows that a3= a(1 − a) t1+ t2 t1t2+ 1 = a(1 − a)x 1 + x . Likewise, we have b3: = − −G2 x,2(Dq1GF,1− G 2 F,1) + G 2 x,1(Dq2GF,2− G 2 F,2) G2 x,1G2y,2− G2y,1G2x,2 = a(1 − a) t1t2(t1+ t2) (t1− t2)2(t1t2+ 1) = a(1 − a)xy (1 + x)(1 − 4y). Then, by (2.1), the function F , as a function of x and y, satisfies
(3.4) D2xF − 2 1 + xDxDyF + x 1 + xDxF + a(1 − a)x 1 + x F = 0 and D2yF + 2y(1 + 2x) (1 + x)(1 − 4y)DxDyF + y(1 + 2x) (1 + x)(1 − 4y)DxF − 2y 1 − 4yDyF + a(1 − a)xy (1 + x)(1 − 4y)F = 0. (3.5)
Finally, we can deduce the claimed differential equations by taking (3.4) times
(1 + x) and (3.5) times (1 − 4y) minus (3.4) times y, respectively.
4. Examples
Example 4.1. Let j be the elliptic modular j-function, and let E4(q) be the
Eisenstein series of weight 4 on SL2(Z). Set
x = 2 1/j(q1) + 1/j(q2) − 1728/(j(q1)j(q2)) 1 +p(1 − 1728/j(q1))(1 − 1728/j(q2)) , y = 1 j(q1)j(q2)x2 , and F = (E4(q1)E4(q2))1/4.
Then F is a bi-modular form of weight (1, 1) for SL2(Z) × SL2(Z), and it satisfies
the system of partial differential equations:
(1 − 432x)Dx2F − 2DxDyF − 432xDx− 60xF = 0,
We have noticed that this system of differential equation belongs to a general class of partial differential equations which involve solutions of hypergeometric hy-pergeometric differential equations discussed in Theorem 3.1.
Here we will prove the assertion of Example 4.1 using Theorem 3.1.
Proof of Example 4.1. We first make a change of variable x 7→ −¯x/432. For con-venience, we shall denote the new variable ¯x by x again. Thus, we are required to show that the functions
x = −864 1/j(q1) + 1/j(q2) − 1728/(j(q1)j(q2)) 1 +p(1 − 1728/j(q1))(1 − 1728/j(q2)) , y = 432 2 j(q1)j(q2)x2 , and F = (E4(q1)E4(q2))1/4 satisfy (1 + x)Dx2F − 2DxDyF + xDx+ 5 36xF = 0, and
(1 − 4y)D2yF + 4yDxDyF − yD2xF − yDxF − 2yDyF = 0.
For brevity, we let j1denote j(q1) and j2denote j(q2). We now observe that the
function x can be alternatively expressed as x = −864 1/j1+ 1/j2− 1728/(j1j2) 1 − (1 − 1728/j1)(1 − 1728/j2) 1 −p(1 − 1728/j1)(1 − 1728/j2) = 1 2 p (1 − 1728/j1)(1 − 1728/j2) − 1 . Setting t1= p1 − 1728/j1− 1 p1 − 1728/j1+ 1 , t2= p1 − 1728/j2− 1 p1 − 1728/j2+ 1 , we have x = t1+ t2 (t1− 1)(t2− 1) .
Moreover, the functions jk, written in terms of tk, are jk = 432(tk− 1)2/tk for
k = 1, 2. It follows that y = 432 2 j1j2x2 = t1t2 (t1+ t2)2 . In view of Theorem 3.1, setting
t = p1 − 1728/j(q) − 1 p1 − 1728/j(q) + 1
it remains to show that the function f (t) = E4(q)1/4(1 − t)−1/6 is a solution of the
hypergeometric differential equation
t(1 − t)f00+ (1 + 4t/3)f0− 1 36f = 0, or equivalently, that
E4(q)1/4
(1 − t)1/6 = 2F1(1/6, 1/6; 1; t).
This, however, follows from the classical identity E4(q)1/4= 2F1 1 12, 5 12; 1; 1728 j(q)
and Kummer’s transformation formula 1 +√1 − z 2 2a 2F1 a, b; a + b +1 2; z = 2F1 2a, a − b +1 2; a + b + 1 2; √ 1 − z − 1 √ 1 − z + 1 .
This completes the proof of Example 4.1.
Remark 4.1. The functions x and y in Example 4.1 (up to constant multiple) have also appeared in the paper of Lian and Yau [12], Corollary 1.2, as the mirror map of the family of K3 surfaces defined by degree 12 hypersurfaces in the weighted projective space P3[1, 1, 4, 6]. Further, this K3 family is derived from the square of a family of elliptic curves in the weighted projective space P2[1, 2, 3]. (The geometry
behind this phenomenon is the so-called Shoida–Inose structures, which has been studied in detail by Long [13] for one-parameter families of K3 surfaces, and their Picard–Fuchs differential equations.) Lian and Yau [12] proved that the mirror map of the K3 family can be given in terms of the elliptic j-function, and indeed, by the functions x and y (up to constant multiple). We will discuss more examples of families of K3 surfaces, their Picard–Fuchs differential equations and mirror maps in the section 6.
Along the same vein, we obtain more examples of bi-modular forms of weight (1, 1) and bi-modular functions on Γ0(N ) × Γ0(N ) for N = 2, 3, 4.
Theorem 4.1. We retain the notations of Theorem 3.1. Then the solutions of the differential equations (3.2) and (3.3) for the cases a = 1/2, 1/3, 1/4, 1/6 can be expressed in terms of bi-modular forms and bi-modular functions.
(a) For a = 1/2, they are given by
F (q1, q2) = θ4(q1)2θ4(q2)2, t = θ2(q)4/θ3(q)4,
which are modular on Γ0(4) × Γ0(4).
(b) For a = 1/3, they are F (q1, q2) = 1 2(3E2(q 3 1) − E2(q1))1/2(3E2(q32) − E2(q2))1/2, t = −27 η(3τ )12 η(τ )12 ,
which are modular on Γ0(3) × Γ0(3).
(c) For a = 1/4, they are
F (q1, q2) = (2E2(q12) − E2(q1))1/2(2E2(q22) − E2(q2))1/2, t = −64
η(2τ )24 η(τ )24 ,
which are modular are Γ0(2) × Γ0(2).
(d) For a = 1/6, they are given as in Example 4.1. Here
η(τ ) = q1/24Y
n∈N
(1 − qn) is the Dedekind eta-function, and
θ2(q) = q1/4 X n∈Z qn(n+1), θ3(q) = X n∈Z qn2, θ4(q) = X n∈Z (−1)nqn2 are theta-series.
Lemma 4.2. Let Γ be a subgroup of SL2(R) commensurable with SL2(Z). Let
f (τ ) be a modular form of weight 1, and t(τ ) be a non-constant modular function on Γ. Then, setting Gt= Dqt t , Gf = Dqf f , we have D2tf +DqGt− 2GfGt G2 t Dtf − DqGf− G2f G2 t f = 0.
Proof of Theorem 4.1. To prove part (a) we use the well-known identities θ32= 2F1 1 2, 1 2; 1; θ4 2 θ4 3
(see [17] for a proof using Lemma 4.2) and θ43= θ42+ θ44. Applying Theorem 3.1 and observing that
θ23 1 − θ 4 2 θ4 3 1/2 = θ32θ 2 4 θ2 3 = θ42, we thus obtain the claimed differential equation.
For parts (b), we need to show that the function f (τ ) =(3E2(q 3) − E 2(q))1/2 (1 − t)1/3 satisfies t(1 − t)d 2 dt2f + (1 − 5t/3) d dtf − 1 9f = 0, or, equivalently, (4.1) (1 − t)Dt2f −2 3tDtf − 1 9tf = 0.
Let Gt and Gf be defined as in Lemma 4.2. For convenience we also let g =
(3E2(q3) − E2(q))/2. We have Gt= 1 2(3E2(q 3) − E 2(q)) = g and Gf = Dqg 2g − 1 3(1 − t)Dqt = Dqg 2g + t 3(1 − t)g. It follows that DqGt− 2GfGt G2 t = g−2 Dqg − 2 Dqg 2g + t 3(1 − t)g g = − 2t 3(1 − t). Moreover, we can show that (DqGf−G2f)/G
2
t is equal to −t/(9(1−t)) by comparing
enough Fourier coefficients. This establishes (4.1) and hence part (b).
5. More examples
We may also consider groups like Γ0(N )∗× Γ0(N )∗ where Γ0(N )∗ denotes the
group generated by Γ0(N ) and the Atkin–Lehner involution wN =
0 −1
N 0
for some N . (Note that Γ0(N )∗ is contained in the normalizer of Γ0(N ) in SL2(R).)
Also the entire list of N giving rise to genus zero groups Γ0(N )∗ is known (cf.[4]),
and we will be interested in some of those genuz zero groups. We can determine differential equations satisfied by bi-modular forms of weight (1, 1) on Γ0(N )∗×
Γ0(N )∗ for some N (giving rise to genus zero subgroups Γ0(N )∗).
We first prove a generalization of Theorem 3.1.
Theorem 5.1. Let 0 < a, b < 1 be positive real numbers. Let f (t) = 2F1(a, b; 1; t)
be a solution of the hypergeometric differential equation
(5.1) t(1 − t)f00+ [1 − (1 + a + b)t]f0− abf = 0.
Set
F (t1, t2) = f (t1)f (t2)(1 − t1)(a+b)/2(1 − t2)(a+b)/2,
x = t1+ t2− 2, y = (1 − t1)(1 − t2).
Then F , as a function of x and y, satisfies
(5.2) D2xF + 2DxDyF − 1 x + y + 1DxF + x x + y + 1DyF + (2ab − a − b)x 2(x + y + 1) F = 0 and D2yF +2y x2DxDyF + y2 x2(x + y + 1)DxF + y − x − x2 x(x + y + 1)DyF −(a + b)(a + b − 2)(x 2+ x) + (a − b)2xy − (4ab − 2a − 2b)y 4x(x + y + 1) F = 0. (5.3)
Proof. The proof is very similar to that of Theorem 3.1. Let f1be another solution
of the hypergeometric differential equation (5.1), and set τ := f1/f . We find
f2= c exp − Z t1 − (1 + a + b)u u(1 − u) du dt dτ = cdt/dτ t(1 − t)a+b
for some constant c depending on the choice of f1. Thus, setting
q1= e2πif1(t1)/f (t1) and q2= e2πif1(t2)/f (t2),
we have
F (t1, t2) = c0
Dq1t1· Dq2t2 t1t2
1/2
for some constant c0. We now apply the differential identities (2.1). We have, for j = 1, 2, Gx,j := Dqjx x = Dqjtj t1+ t2− 2 , Gy,j:= Dqjy y = − Dqjtj 1 − tj , and GF,j := DqjF F = tjDq2jtj− (Dqjtj) 2 2tjDqjtj .
It follows that the coefficients in (2.1) are a0= 2, b0= 2(1 − t1)(1 − t2) (t1+ t2− 2)2 = 2y x2, a1= − 1 t1t2 = − 1 x + y + 1, b1= (1 − t1)2(1 − t2)2 t1t2(t1+ t2− 2)2 = y 2 x2(x + y + 1), a2= t1+ t2− 2 t1t2 = x x + y + 1, b2= − t2 1+ t1t2+ t22− 2t1− 2t2+ 1 t1t2(t1+ t2− 2) = y − x − x 2 x(x + y + 1). Moreover, we have a3= −(1 − t1) 2(2 ˙t 1 ... t1− 3¨t21) 4(t1− t2) ˙t41 +(1 − t2) 2(2 ˙t 2 ... t2− 3¨t22) 4(t1− t2) ˙t42 −2t1t2− t1− t2 4t2 1t22 × (t1+ t2− 2),
where we, as before, employ the notations ˙tj, ¨tj,
...
tjfor the derivatives Dqjtj, D
2 qjtj, and D3
qjtj, respectively. Now, by Lemma 3.2, we have 2 ˙tj ... tj− 3¨t2j = ˙t 4 j (a − b)2t2 j− (1 − tj)2+ (4ab − 2a − 2b)tj t2 j(1 − tj)2 . It follows that a3= (2ab − a − b)(t1+ t2− 2) 2t1t2 = (2ab − a − b)x x + y + 1 .
A similar calculation shows that b3= −
(a + b)(a + b − 2)(x2+ x) + (a − b)2xy − (4ab − 2a − 2b)y
4x(x + y + 1) .
This proves the claimed result.
Remark 5.1. It should be pointed out that the first identity in our proof of Theorem 5.1 is equivalent to the formula in Proposition 4.4 of Lian and Yau [10].
We now obtain new examples of bi-modular forms of weight (1, 1) on Γ0(N )∗×
Γ0(N )∗ for some N .
Theorem 5.2. When the pairs of numbers (a, b) in Theorem 5.1 are given by (1/12, 5/12), (1/12, 7/12), (1/8, 3/8), (1/8, 5/8), (1/6, 1/3), (1/6, 2/3), (1/4, 1/4) and (1/4, 3/4), the solutions F (t1, t2) of the differential equations (5.2) and (5.3)
are bi-modular forms of weight (1, 1) on Γ0(N )∗×Γ0(N )∗with N = 1, 1, 2, 2, 3, 3, 4, 4,
respectively.
Proof. We shall prove only the cases (a, b) = (1/6, 1/3) and (1/6, 2/3); the other cases can be proved in the same manner.
Let s(τ ) = −27η(3τ ) 12 η(τ )12 , E2(q) = 1 − 24 ∞ X n=1 nqn 1 − qn.
From the proof of Part (b) of Theorem 4.1 we know that f (τ ) =(3E2(q
3) − E 2(q))1/2
as a function of s, is equal to√22F1(1/3, 1/3; 1; s). Now, applying the quadratic transformation formula 2F1(α, β; α − β + 1; x) = (1 − x)−α2F1 α 2, 1 + α 2 − β; α − β + 1; − 4x (1 − x)2
for hypergeometric functions (see, for example [1, Theorem 3.1.1]) with α = β = 1/3, we obtain (3E2(q3) − E2(q))1/2= √ 22F1 1 6, 1 3; 1; − 4s (1 − s)2 .
Observing that the action of the Atkin-Lehner involution w3sends s to 1/s, we find
that the function s/(1 − s)2 is modular on Γ0(3)∗. This proves that F (t1, t2) is a
bi-modular form of weight (1, 1) for Γ0(3)∗× Γ0(3)∗ in the case (a, b) = (1/6, 1/3).
Furthermore, an application of another hypergeometric function identity
2F1(α, β; γ; x) = (1 − x)−α2F1 α, γ − β; γ; x x − 1 yields (3E2(q3) − E2(q))1/2= √ 2 1 − s 1 + s 1/3 2F1 1 6, 2 3; 1; 4s (1 + s)2 .
This corresponds to the case (a, b) = (1/6, 2/3). Again, the function 4s/(1 + s)2 is
modular on Γ0(3)∗. This implies that F (t1, t2) is a bi-modular form of weight (1, 1)
for Γ0(3)∗× Γ0(3)∗ for the case (a, b) = (1/6, 2/3).
Remark 5.2. For the remaining pairs (a, b) in Theorem 5.2, we simply list the exact expressions of F (t1, t2) in terms of modular forms as proofs are similar.
For (a, b) = (1/12, 5/12) and (1/12, 7/12), they are E6(q1)E6(q2) E4(q1)E4(q2) 1/2 , and E8(q1)E8(q2) E6(q1)E6(q2) 1/2 , respectively, where Ek are the Eisenstein series in (2.2).
For (a, b) = (1/8, 3/8) and (1/8, 5/8), they are
2 Y j=1 1 + sj 1 − sj (2E2(qj2) − E2(qj)) 1/2 , and 2 Y j=1 1 − sj 1 + sj (2E2(qj2) − E2(qj)) 1/2 , respectively, where sj= −64η(τj)24/η(τj)24.
For (a, b) = (1/6, 1/3) and (1/6, 2/3), they are
2 Y j=1 1 + sj 1 − sj (3E2(qj3) − E2(qj)) 1/2 , and 2 Y j=1 1 − sj 1 + sj (3E2(qj3) − E2(qj)) 1/2 , respectively, where sj= −27η(3τj)12/η(τj).
For (a, b) = (1/4, 1/4) and (1/4, 3/4), they are
2 Y j=1 2E2(qj2) − E2(qj) 1/2 , and 2 Y j=1 2E2(qj2) − E2(qj) 1/21 − sj 1 + sj , respectively, where sj= θ2(qj)4/θ3(qj)4.
6. Picard–Fuchs differential equations of Familes of K3 surfaces : Part I
One of the motivations of our investigation is to understand the mirror maps of families of K3 surfaces with large Picard nubmers, e.g., 19, 18, 17 or 16. Some examples of such families of K3 surfaces were discussed in Lian–Yau [11], Hosono– Lian–Yau [8] and also in Verrill-Yui [16]. Some of K3 families occured considering degenerations of Calabi–Yau families.
Our goal here is to construct families of K3 surfaces whose Picard–Fuchs dif-ferential equations are given by the difdif-ferential equations satisfied by bi-modular forms we constructed in the earlier sections. In this section, we will look into the families of K3 surfaces appeared in Lian and Yau [10, 11].
Let S be a K3 surface. We recall some general theory about K3 surfaces which are relevant to our discussion. We know that
H2(S, Z) ' (−E8)2⊥ U3
where U is the hyperbolic plane 0 1
1 0
and E8 is the even unimodular negative
definite lattice of rank 8. The Picard group of S, Pic(S), is the group of linear equivalence classes of Cartier divisors on S. Then Pic(S) injects to H2
(X, Z), and the image of Pic(S) is the algebraic cycles in H2
(S, Z). As P ic(S) is torsion-free, it may be regarded as a lattice in H2
(S, Z), called the Picard lattice, and its rank is denoted by ρ(S).
According to Arnold–Dolgachev [5], two K3 surfaces form a mirror pair (S, ˆS) if Pic(S)⊥H2(S,Z) = Pic( ˆS) ⊥ U as lattices
In terms of ranks, a mirror pair (S, ˆS) is related by the identity: 22 − ρ(S) = ρ( ˆS) + 2 ⇔ ρ(S) + ρ( ˆS) = 20.
Example 6.1. We will be interested in mirror pairs of K3 surfaces (S, ˆS) whose Picard lattices are of the form
P ic(S) = U and P ic( ˆS) = U2⊥ (−E8)2.
We go back to our Example 4.1, and discuss geometry behind that example. As-sociated to this example, there is a family of K3 surfaces in the weighted projective 3-space P3[1, 1, 4, 6] with weight (q
1, q2, q3, q4) = (1, 1, 4, 6). There is a mirror pair
of K3 surfaces (S, ˆS). Here we know (cf. Belcastro [3]) that Pic(S) = U so that ρ(S) = 2,
and that S has a mirror partner ˆS whose Picard lattice is given by Pic( ˆS) = U ⊥ (−E8)2 so that ρ( ˆS) = 18.
The mirror K3 family can be defined by a hypersurface in the orbifold ambient space P3[1, 1, 4, 6]/G of degree 12. Here G is the discrete group of symmetry and
can be given explicitly by G = (Z/3Z) × (Z/2Z) = hg1i × hg2i where g1, g2 are
generatoers whose actions are given by:
g1: (Y1, Y2, Y3, Y4) 7→ (ζ3Y1, Y2, ζ3−1Y3, Y4)
(Here ζ3= e2πi/3.) The G-invariant monomials are
Y112, Y212, Y33, Y42, Y16Y26, Y1Y2Y3Y4.
The matrix of exponents is the following 6 × 5 matrix 12 0 0 0 1 0 12 0 0 1 0 0 3 0 1 0 0 0 2 1 6 6 0 0 1 1 1 1 1 1
whose rank is 2. Therefore we may conclude that the typical G-invariant polyno-mials is in 2-parameters, and ˆS can be defined by the following 2-parameter family of hypersurfaces of degree 12
Y112+ Y212+ Y33+ Y42+ λY1Y2Y3Y4+ φY16Y 6 2 = 0
in P3[1, 1, 4, 6]/G with parameters λ and φ.
How do we conmpute the Picard–Fuchs differential equation of this K3 family? Several physics articles are devoted to this question. For instance, Klemm– Lerche–Mayr [9], Hosono–Klemm–Theisen–Yau [7], Lian and Yau [11] determined the Picard–Fuchs differential equation of the Calabi–Yau family using the GKZ hypergeometric system. Also it was noticed (cf. [9], [11]) that the Picard–Fuchs system of this family of K3 surfaces can be realized as the degeneration of the Picard–Fuchs systems of the Calabi–Yau family. The family of Calabi–Yau three-folds is a degree 24 hypersurfaces in P4[1, 1, 2, 8, 12] with h1,1 = 3. The defining
equation for this family is given by
Z124+ Z224+ Z312+ Z43+ Z52− 12ψ0Z16Z26Z36− 6ψ1(Z1Z2Z3)6− ψ2(Z1Z2)12= 0.
Its Picard–Fuchs system is given by
L1 = Θx(Θx− 2Θz) − 12 x(6Θx+ 5)(6Θx+ 1) L2 = Θ2y− y(2Θy− Θz+ 1)(2Θy− Θz) L3 = Θz(Θz− 2Θy) − z(Θz− Θx+ 1)(2Θz− Θx) where x = − 2ψ1 17282ψ6 0 , y = 1 ψ2 2 and z = − ψ2 4ψ2 1
are deformation coordinates.
Now the intersection of this Calabi–Yau hypersurface with the hyperplane Z2−
t Z1= 0 gives rise to a family of K3 surfaces
b0Y1Y2Y3Y4+ b1Y112+ b2Y212+ b3Y33+ b4Y42+ b5Y16Y 6 2 = 0
in P3[1, 1, 4, 6] of degree 12. Taking (b0, b1, b2, b3, b4, b5) = (λ, 1, 1, 1, 1, φ) we obtain
the 2-parameter family of K3 surfaces described above. The Picard–Fuchs system of this K3 family is obtained by taking the limit y = 0 in the Picard–Fuchs system for the Calabi–Yau family:
L1 = Θx(Θx− 2Θz) − 12 x(6Θx+ 5)(6Θx+ 1)
Further, if we intersect this K3 family with the hyperplane Y2− s Y1= 0, we obtain
a family of elliptic curves:
c0W1W2W3+ c1W16+ c2W23+ c3W32= 0
in P2[1, 2, 3], whose Picard–Fuchs equation is given by
L = Θ2x− 12 x(6Θx+ 5)(6Θx+ 1).
Here we describe a relation of the Picard–Fuchs system of the above family of K3 surfaces to the differential equation discussed in Example 4.1.
Remark 6.1. We note that, in view of our proof of Example 4.1, the process of setting z = 0 in the above Picard–Fuchs system {L1, L3} is equivalent to setting
t1 = 0 or t2 = 0 in x and y in Example 4.1. Our Theorem 3.1 then implies that
F (t) = (1 − t)1/6
2F1(1/6, 1/6; 1; t) satisfies
(1 + x)Dx2F + xDxF +
5
36xF = 0
with x = t/(1 − t), or equivalently, (making a change of variable x 7→ −x) x(1 − x)F00+ (1 − 2x)F0− 5
36F = 0 with x = t/(t − 1). That is,
(1 − t)1/62F1 1 6, 1 6; 1; t = 2F1 1 6, 5 6; 1; t t − 1 . This is the special case of the hypergeometric series identity
(1 − t)a2F1(a, b; c; t) = 2F1 a, c − b; c; t t − 1 .
We will discuss more examples of Picard–Fuchs systems of Calabi–Yau threefolds and K3 surfaces, which have already been considered by several people. For in-stance, the articles [7], [8], and [9] obtained the Picard–Fuchs operators for Calabi– Yau hypersurfaces with h1,1 ≤ 3. The next two examples consider Calabi–Yau hypersurfaces with h1,1 > 3, and the paper of Lian and Yau [11] addressed the question of determining the Picard–Fuchs system of the families of K3 surfaces P3[1, 1, 2, 2] of degree 6 and P3[1, 1, 2, 4] of degree 8. Their results are that
(1) there is an elliptic fibration on these K3 surfaces, and the Picard–Fuchs systems of the K3 families can be derived from the Picard–Fuchs system of the elliptic pencils, and that
(2) the solutions of the Picard–Fuchs systems for the K3 families are given by “squares” of those for the elliptic families.
The system of partial differential equations considered by Lian and Yau [11] is L1 = Θx(Θx− 2Θz) − λ x(Θx+12+ ν)(Θx+12− ν)
L2 = Θ2z− z(2Θz− Θx+ 1)(2Θz− Θx)
and an ordinary differential equations L = Θ2x− λ x(Θx+
1
2 + ν)(Θx+ 1 2− ν) where Θx= x∂ x∂ , etc.) and λ, ν are complex numbers.
Also they noted that the K3 families correspond, respecitvely, to the families of Calabi–Yau threefolds P4[1, 1, 2, 4, 4] of degree 12 and P4[1, 1, 2, 4, 8] of degree 16. However, the Picard–Fuchs systems for the Calabi–Yau families are not explicitly determined.
Example 6.2. We now consider a family of K3 surfaces P3[1, 1, 2, 4]. of degree
8. This K3 family is realized as the degeneration of the family of Calabi–Yau hypersurfaces P4[1, 1, 2, 4, 8] of degree 16 and h1,1 = 4. The most generic defining
equation for this family is given by
a0Z1Z2Z3Z4Z5+ a1Z116+ a2Z216+ a3Z38+ a4Z44+ a5Z52+ a6Z32Z4Z5+ a7Z18Z 8 2 = 0
Again the intersection with the hyperplane Z2− t Z1 = 0 gives rise to a family of
K3 surfaces P3[1, 1, 2, 4]:
Y18+ Y28+ Y34+ Y42+ λY1Y2Y3Y4+ φY14Y 4 2 = 0
Let S denote this family of K3 surfaces. Then
P ic(S) = M(1,1),(1,1),0 with ρ(S) = 3.
The mirror family ˆS exists and its Picard lattice is
P ic( ˆS) = E8⊥ D7⊥ U with ρ( ˆS) = 17.
The Picard lattices are determined by Belcastro [3]. The intersection of this family of K3 surfaces with the hyperplane Y2− s Y2= 0 gives rise to the pencil of elliptic
curves
c0W1W2W3+ c1W14+ c2W24+ c3W32= 0
in P2[1, 1, 2] of degree 4. This means that this family of K3 surfaces has the elliptic fibration with section.
Now translate this “inductive” structure to the Picard–Fuchs systems. The Picard–Fuchs system for the K3 family is given by
L1 = Θx(Θx− 2Θz) − 64 x(Θx+21+14)(Θx+12−14)
L2 = Θ2z− z(2Θz− Θx+ 1)(2Θz− Θx)
and the Picard–Fuchs defferential equation of the elliptic family is given by L = Θ2x− 64 x(Θx+ 1 2 + 1 4)(Θx+ 1 2− 1 4)
The same remark as Remark 6.1 is valid for the Picard–Fuchs system {L1, L3}
which corresponds to Theorem 4.1 (b) with a = 1/3.
Example 6.3. We consider a family of K3 surfaces P3[1, 1, 2, 2] of degree 6. This K3 family is realized as the degeneration of the family of Calabi–Yau hypersurfaces P4[1, 1, 2, 4, 4] of degree 12 and h1,1 = 5:
a0Z1Z2Z3Z4Z5+ a1Z112+ a2Z212+ a3Z36+ a + 4Z43+ a5Z53+ a6Z16Z26= 0.
The intersection of this Calabi–Yau hypersurface with the hyperplane Z2− t Z1= 0
gives rise to the family of K3 hypersurfaces P3[1, 1, 2, 4]:
Y16+ Y26+ Y33+ Y43+ λY1Y2Y3Y4+ φY13Y23= 0.
Let S denote this family of K3 surfaces. Then
There is a mirror family of K3 surfaces, ˆS with
Pic( ˆS) = E8⊥ D4⊥ A2⊥ U with ρ( ˆS) = 16.
The Picard lattices are determined by Belcastro [3].
The intersection of this K3 family with the hyperplane Y2− s Y1 = 0 gives rise
to the family of elliptic curves
c0W1W2W3+ c1W13+ c2W23+ c3W33= 0
in P2[1, 1, 1] of degree 3.
The Picard–Fuchs system of this K3 family is
L1 = Θx(Θx− 2Θz) − 27 x(Θx+21+16)(Θx+12−16)
L2 = Θ2z− z(2Θz− Θx+ 1)(2Θz− Θx)
and the Picard–Fuchs differential equation for the elliptic family is given by L = Θ2x− 27 x(Θx+ 1 2 + 1 6)(Θx+ 1 2− 1 6)
We note that the same remark is valid for the Picard–Fuchs system {L1, L3}
corresponding to a = 1/4 in Theorem 4.1(c).
We will summarize the above discussions for the families of K3 surfaces in the following form.
Proposition 6.1. The Picard–Fuchs systems of families of K3 surfaces obtained by Lian and Yau [11] can be reconstructed starting from the bi-modular forms and then finding the differential equations satisfied by them. In other words, the differential equations satisfied by the bi-modular forms are realized as the Picard– Fuchs differential equations of the families of K3 surfaces, establishing, in a sense, the “modularity” of the K3 families.
7. Picard–Fuchs differential equations of families of K3 surfaces: Part II
The purpose of this section is to study (one-parameter) families of K3 surfaces (some of which are realized as degenerations of some families of Calabi–Yau three-folds), whose mirror maps are expressed in terms of Hauptmodules for genus zero subgroups of the form Γ0(N )∗, aiming to identify their Picard–Fuchs systems with
differential equations assocaited to some to bi-modular forms (e.g., in Theorem 5.1). Dolgachev [5] has discussed several examples of families of MN-polarized K3
surfaces corresponding to Γ0(N )∗ for small values of N , e.g., N = 1, 2 and 3.
Lian and Yau [10] have given examples of families of K3 surfaces and their Picard–Fuchs differential equqtions of order 3. The modular groups are genus zero subgroups of the form Γ0(N )∗ where N ranging from 1 to 30. Here we try to
analyze their examples and their method in relation to our results in the section 5. Example 7.1. We start with the hypergeometric equation:
t(1 − t)f00+ [1 − (1 + a + b)t]f0− abf = 0
in Theorem 5.1. Take a = b = 14 and consider a one-parameter deformation of this equation of the form:
t(1 − t)f00+ (1 −3 2t)f
0− 1
16(1 − 4ν
with a deformation parameter ν. This has a unique solution f0(t) near t = 0 with
f0(0) = 1, and a solution f1(t) with f1(t) = f0(t)log t + O(t). The inverse t(q) of the
power series q = exp(f1(t)
f0(t)) = t + O(t
2) defines an invertible holomorphic function
in a disc, and t(q) is the so-called mirror map. Put x(q) = 1
λt(λq) for a given λ.
One of the main results of Lian and Yau [10] is that for any complex numbers λ, ν with λ 6= 0, there is a power series identity:
3F2( 1 2, 1 2+ ν, 1 2 − ν; 1, 1; λ x(q)) 2= x0 2 x2(1 − λ x)
in the common domain of definitions of both sides. As before, x0(q) = Dqx(q).
For instance, take (λ, ν) = (2633,1 3), (2 8,1 4), (2 233,1 6) and (2 6, 0), then these
relations are given below. The mirror maps in these examples are expressed in terms of Hauptmodules of genus zero modular groups of the form Γ0(N )∗(Γ0(1)∗= Γ).
Label Modular Relation Modular Group
I : P∞ n=0 (6n)! (3n)!(n!)3 1 j(q)n !2 = E4(q) Γ II : P∞ n=0 (4n)! (n!)4x2(q)n !2 = x 0 2 2 x2(1−256x) Γ0(2)∗ III : P∞ n=0 (2n)!(3n)! (n!)5 x3(q)n !2 = x 0 2 3 x2 3(1−108x3) Γ0(3) ∗ IV : P∞ n=0 (2n)!3 (n!)6 x4(q)n !2 = x0 24 x2 4(1−64x4) Γ0(4) ∗.
Here j(q), x2(q), x3(q) and x4(q) are Hauptmodules for the genus zero subgroups
Γ, Γ0(2)∗, Γ0(3)∗ and Γ0(4)∗, respectively. Observe that in each modular relation,
the right hand side is a modular form of weight 4 on the corresponding genus zero subgroup.
We know that3F2(12,12+ ν,12− ν; 1, 1; λ x) is a unique solution with the leading
term 1 + O(x) to the differential operator L = Θ3x− λ x(Θx+ 1 2)(Θx+ 1 2 + ν)(Θx+ 1 2− ν).
In these examples, this differential operator is identified with the Picard–Fuchs differential operator for a one-parameter family of K3 surfaces, which are obtained by degenerating Calabi–Yau families. (Cf. Lian and Yau [10], Klemm, Lercher and Myer [9].)
CY family K3 family PF Operator
I X(1, 1, 2, 2, 2)[8] X(1, 1, 1, 3)[6] Θ3− 8x(6Θ + 5)(6Θ + 3)(6Θ + 1)
II X(1, 1, 2, 2, 6)[12] X(1, 1, 1, 1)[4] Θ3− 4x(4Θ + 3)(4Θ + 2)(4Θ + 1)
III X(1, 1, 2, 2, 2, 2)[6, 4] X(1, 1, 1, 1, 1)[3, 2] Θ3− 6x(2Θ + 1)(3Θ + 2)(3Θ + 1)
The K3 families I and II have already been discussed in Lian–Yau [11] (see also Verrill–Yui [16]) in relation to mirror maps. The Picard group of I (resp. II) is given by
(−E8)2⊕ U2⊕ < −4 > (resp. (−E8)2⊕ U2⊕ < −2 >).
The Calabi–Yau family III can be realized as a complete intersection of the two hypersurfaces:
Y16+ Y26+ Y33+ Y43+ Y53+ Y63= 0
Y4
1 + Y24+ Y32+ Y42+ Y52+ Y62= 0
This Calabi–Yau family has h1,2 = 68 and h1,1 = 2. The K3 family is realized as
the fiber space by setting Y1= Z
1/2
1 , Y2= λZ 1/2
1 , and Yi= Zi for i = 3, · · · , 6
where λ ∈ P1 is a parameter. That is, we obtain a family of complete intersection
K3 surfaces: (1 + λ6)Z3 1 + Z33+ Z43+ Z53+ Z63= 0 (1 + λ4)Z2 1 + Z32+ Z42+ Z52+ Z62= 0 X(1, 1, 1, 1, 1)[3, 2].
Question: What is the Picard group of this K3 family?
In the similar manner, the Calabi–Yau family IV can be realized as a complete intersection of the three hypersurfaces:
Y4 1 + Y24+ Y32+ Y42+ Y52+ Y62+ Y72= 0 Z4 1+ Z24+ Z32+ Z42+ Z52+ Z62+ Z72= 0 W4 1 + W24+ W32+ W42+ W52+ W62+ W72= 0
The K3 family is realized as the fiber space by setting Y1= Y 01 2 1 , Y2= λY 01 2 1 and Yi= Yi0 for i = 3, · · · , 7
and similarly for Z1, Z2 and W1, W2 where λ ∈ P1 is a parameter.
This gives rise to the K3 family
(1 + λ4)Y102+ Y302+ Y402+ Y502+ Y602+ Y70= 0
(1 + λ4)Z02
1 + Z302+ Z402+ Z502+ Z602+ Z702 = 0
(1 + λ4)W02
1 + W302+ W402+ W502+ W602+ W702 = 0
Question: What is the Picard group of this K3 family? Here is the summary:
(1) One starts with a Hauptmodule x(= x(q)) for a genus zero subgroup Γ0(N )∗;
(2) then there associate a modular form x r(x)x0 2 of weight 4,
(3) and a power series solution ω0(x) of an order three differential operator;
(4) this differential operator coincides with the Picard–Fuchs differential operator of a one-parameter family of K3 surfaces in weighted projective spaces.
Lian and Yau [10] further considered generalizations of the above phenomenon, constructing many more examples. Given a genus zero subgroup of the form Γ0(N )∗
and a Hauptmodul x(q), constract (by taking a Schwarzian derivative) a modular form E of weight 4 of the formx r(x)x0 2 and a differential operator L whose monodromy has maximal unipotency at x = 0, such that L E1/2= 0. Further, identify L as the
the fundamental period of this manifold. Then it should be subject to the modular relation
ω0(x)2=
x0 2 x r(x)
How do we associate bi-modular forms of weight (1, 1) corresponding to the groups Γ0(N )∗× Γ0(N )∗ in this situation?
Taking the square root of both sides of the modular relation, we obtain that ω0(x)1/2 is a modular form of weight 1 for the group Γ0(N )∗. Take ω0(q1)ω0(q2).
Then this is a bi-modular form for Γ0(N )∗× Γ0(N )∗ of weight (1, 1). Then this
bi-modular form satisfies a differential equation, which may be identified with the Picard–Fuchs differential equation of the K3 family considered above. We summa-rize the above discussion in the following proposition.
Proposition 7.1. The examples I–IV above are related to our Theorem 5.2. In-deed, the connection is established by the identity
2F1 a, b; a + b +1 2; z 2 = 3F2 2a, a + b, 2b; a + b +1 2, 2a + 2b; z . More explicitly, the examples I–IV correspond to the cases (1/12, 5/12), (1/8, 3/8), (1/6, 1/3), and (1/4, 1/4), respectively.
Note that the generalized hypergeometric series 3F2(α1, α2, α3; 1, 1; z) satisfies
the differential equation of the form:
[Θ3z− λ z(Θz+ α1)(Θz+ α2)(Θz+ α3)]f = 0
for some α1, α2, α3∈ Q and λ ∈ Q, 6= 0.
A natural question we may ask now is: Is is possible to construct families of K3 surfaces corresponding to Theorem 5.2 from this observation?
When the order 3 differential equation of this form becomes the symmetric square of an order 2 differential equation, and if the order 2 differential equation is real-ized as the Picard–Fuchs differential equation of a family of elliptic curves, we may be able to construct a family of K3 surfaces using the method of Long [13], espe-cially when the Picard number of the K3 family in question is 19 or 20. In fact, Rodriguez–Villegas [14] has discussed 4 families of K3 surfaces which fall into this class.
However, at the moment, we do not know if there are readily available methods for constructing K3 families starting from differential equations.
Remark 7.1. If we consider the order 4 generalized hypergeomtric series, there are 14 families of Calabi–Yau threefolds whose Picard–Fuchs differential equations are of the form
[Θ4z− λ z(Θz+ α1)(Θz+ α2)(Θz+ α3)(Θz+ α4)]f = 0
for some αi∈ Q and λ ∈ Q, 6= 0. These 14 differential operators have been found in
Almkvist–Zudilin [2] and Villegas [14] found the corresponding families of Calabi– Yau threefolds all in weighted projective spaces with h1,1= 1.
8. Generalizations and open problems
Problem 1. We have determined differential equations satisfied by bi-modular forms of weight (1, 1). The arguments can be generalized to bi-modular forms of any weight (k1, k2), using the result of Yang [17]. However, differerntial equations
satisfied by them are getting too big to display.
Problem 2. A natural generalization is to consider tri-modular forms F (τ1, τ2, τ3)
of weight (k1, k2, k3) on Γ1× Γ2× Γ3.
Examples of this kind should correspond to Picard–Fuchs differential equations of families of Calabi–Yau threefolds, or Picard–Fuchs differential equations of de-generate families of Calabi–Yau fourfolds.
References
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Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, TAIWAN
E-mail address: [email protected]
Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario Canada K7L 3N6
