Differential equations concerned with mirror symmetryof toric K3 hypersurfaces with arithmetic properties

Atsuhira Nagano(University of Tokyo)

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ContentsSection 1 : Introduction 1: Hypergeometric differential equationsin mirror symmetry (10 minutes)

Section 2 : Introduction 2: Hypergeometric differential equationsin number theory (10 minutes)

Section 3 : Toric varieties and a construction of mirror (15 minutes)

Section 4 : Arithmetic properties of differential equation fromtoric K3 hypersurfaces (15 minutes)

Section 5 : The case of mirror quintic 3-folds (10 minutes)

• Please note that we shall omit precise proofs of results. If you haveany questions, please come to me after the talk. I will try to give youa detailed explanation.

• In Section 1 and 2, we will see basic motivations of hypergeometricdifferential equations.

2

• The main part is Section 4. The main results are based on the works

[N, 2013] A theta expression of the Hilbert modular functions for√5 via the periods

of K3 surfaces, Kyoto J. Math..

[Shiga-N, 2016] To the Hilbert class field from the hypergeometric modular func-tion, J. Number Theor..

[N, 2017] Icosahedral invariants and a construction of class fields via periods ofK3 surfaces, Ramanujan J., in press.

[Hashimoto-Ueda-N, preprint] Modular surfaces associated with toric K3 hy-

persurfaces.

The main results of this talk partially appeared at other conferencesand workshops in algebraic geometry or number theory. But, today,the speaker would like to survey the results from the viewpoint ofmirror symmetry.

3

1 Introduction 1: Hypergeometric differential equa-

tions in mirror symmetry

Candelas et al. (1991) studied Calabi-Yau 3-folds and discovered mirrorsymmetry. This is a pioneering work for mirror symmetry. Their resultsare closely related to (generalized) hypergeometric differential equations.

Definition : A Calabi-Yau manifold S is a simply connected Kahlermanifold such that the canonical bundle KS on S is trivial.

A 2-dimensional Calabi-Yau manifold is called a K3 surface.

Remark: For a compact complex curve C, it is well-known that KC istrivial if and only if C is an elliptic curve. So, Calabi-Yau manifolds givecounterpart of elliptic cuves.

Mirror symmetry is formulated as a mysterious relation between thefollowing A-model and B-model.

4

Let V be the generic quintic hypersurface in P4(C) = {(x1 : x2 : x3 :x4 : x5)}:

V :∑

j1+j2+j3+j4+j5=5

aj1,j2,j3.j4,j5xa11 x

a22 x

a33 x

a44 x

a55 = 0.

V is a Calabi-Yau 3 fold and called the A-model. V is parametrized by101 complex parameters.

On the other hand, let us consider a family of hypersurfaces

W (z) : x51 + x52 + x53 + x54 + x55 − 5zx1x2x3x4x5 = 0.

with a complex parameter z. We have the action of

G = {(ζ1 : · · · : ζ5)|ζ5j = 1, ζ1 · · · ζ5 = 1} ≃ (Z/5Z)3

on the above hypersurface in (x1 : · · · : x5) 7→ (ζ1x1 : · · · : ζ5x5). Via aresolution of singularities of W (z)/G, we have a family of Calabi-Yau 3-folds W (z). This is called the B-model, or mirror quintic 3-fold. Thefamily of this 3-fold is often called the Dwork family.

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■ Geometry of A-model

• It is conjectured that smooth rational curves on the A-model V areisolated (Clemens conjecture). We can count the number Nd of ratio-nal curves on V of degree d. We define the virtual number of curvesas

N vertd =

∑k|d

1

k3Nd/k ∈ Q.

– Now, mathematicians avoid the above conjecture. Namely, theyuse the Gromov-Witten invariants to define the virtual num-ber of curves, instead of counting curves.

• Let us define a generating function

F (t) =5

6t3 +

∑d≥1

N vertd edt

for the virtual numbers.

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■ Hodge structure of B-modelFor generic z, the Hodge structure of B-model W (z) is given as follows:

• We have the Hodge decomposition

H3dR(W ) = H3,0(W )⊕H2,1(W )⊕H1,2(W )⊕H0,3(W ),

wheredimH3,0(W ) = 1, dimH2,1(W ) = 1.

• 0 = ω ∈ H3,0(W (z)) gives the unique holomorphic 3-form on W (z)up to a constant factor.

• Taking a basis γ1, · · · , γ4 ofH3(W (z)), we have four period integrals∫γ1

ω, · · ·∫γ4

ω.

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These period integrals

∫γj

ω vary with the parameter z.

In fact, by setting λ = (5z)−5, periods give solutions of the linear ordi-nary differential equation((

λd

dλ

)4

− λ(λd

dλ+

1

5

)(λd

dλ+

2

5

)(λd

dλ+

3

5

)(λd

dλ+

4

5

))u = 0

for the independent variable λ.

• Four periods give a system of basis of the space of solutions of thisequation.

• The above differential equation is often called the Picard-Fuchsequation. This is coming from the Gauss-Manin connection ofthe variation of the Hodge structure of the B-model.

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■ Mirror symmetryBy the way, the above equation coincides with the generalized hy-

pergeometric equation 4E3

(1/5 2/5 3/5 4/5

1 1 1

). So, we have another

basis of the space of solutions:

ψ0(λ) =∞∑n=0

(5n)!

(n!)5λn,

ψ1(λ) = log(λ)ψ0(λ) + · · · ,

ψ2(λ) =1

2log(λ)2ψ0(λ) + · · · ,

ψ3(λ) =1

6log(λ)3ψ0(λ) + · · · .

Note that ψ0(λ) is holomorphic around λ = 0. This is called the gener-

alized hypergeometric series 4F3

(1/5 2/5 3/5 4/51 1 1

;λ).

The other solutions ψ1(λ), ψ2(λ), ψ3(λ) have logarithmic singularities atλ = 0.

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Please recall the generating function

F (t) =5

6t3 +

∑d≥1

N vertd edt

derived from the Gromov-Witten invariants of the A-model.Then, the following highly-nontrivial formula holds:

F(ψ1(λ)

ψ0(λ)

)=

5

2

ψ1(λ)ψ2(λ)− ψ0(λ)ψ3(λ)

ψ0(λ)2

This is one of the most famous result in mirror symmetry.

• It is very difficult to calculate the virtual numbers N vertd of curves

for the A-model V . However, via the mirror symmetry above, theyare calculated explicitly by the right hand side using the periodsψ0(λ), · · · , ψ3(λ) of the B-model.

• This is predicted by physicists. [Givental 1996] and [Lian-Liu-Yau1997] gave mathematical proofs of it.

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2 Introduction 2: Hypergeometric differential equa-

tions in number theory

In the last section, we saw the generalized hypergeometric differential equa-tions for Calabi-Yau 3-folds.

By the way, the simplest hypergeometric equation is the Gauss hyperge-ometric equation. The classical theory due to Gauss, Kronecker, Schwarz,etc. suggests that the Gauss hypergeometric equation can be applied tonumber theory. In this section, we will see that.

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The Gauss hypergeometric equation is given by

E(a, b, c) : λ(1− λ)d2η

dλ2+ (c− (a+ b+ 1)λ)

dη

dλ− abη = 0.

λ = 0, 1,∞ are regular singular points of the differential equation.The Gauss hypergeometric series is given by

2F1(a, b, c;λ) =∞∑n=0

(a, n)(b, n)

(c, n)n!λn.

Here, we used the Pochhammer symbol

(a, n) = a(a+ 1)(a+ 2) · · · (a+ n− 1).

The series gives a solution of the Gauss hypergeometric equation. This isholomorphic at λ = 0.

Since the Gauss hypergeometric equation is of rank 2, the space ofsolutions of that is 2-dimensional vector space.

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In the following argument, we suppose that1

|1− c|,

1

|c− a− b|,

1

|a− b|∈ Z ∪ {∞}

|1− c|+ |c− a− b|+ |a− b| < 1.

• We can take 2 solutions η1(λ) and η2(λ) of the Gauss hypergeometricequations such that

σ : λ 7→ η1(λ)

η2(λ)

gives a (surjective) multivalued analytic mapping

P1(C) → H = {z ∈ C|Im(z) > 0}.

This is called the Schwarz mapping.

• The inverse of the Schwarz mapping defines a holomorphic mapping

σ−1 : H → P1(C).

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• Set

p =1

|1− c|, q =

1

|c− a− b|, r =

1

|a− b|.

The multivalued Schwarz mapping σ defines a monodromy coveringof P1(C) − {0, 1,∞}. Under our assumption, the monodromy groupis the triangle group △(p, q, r).

• Then, σ−1 is invariant under the action of △(p, q, r).

If parameters a, b, c are appropriate, the inverse Schwarz mapping σ−1

on H has very good arithmetic properties.■ An Important Example: Elliptic j-function

If

a =1

12, b =

5

12, c = 1,

then it holds

p =1

|1− c|= ∞, q =

1

|c− a− b|= 2, r =

1

|a− b|= 3,

1

∞+

1

2+

1

3< 1.

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We can see that the inverse σ−1 : H → P1(C) of the Schwarz mappingof 2E1(

112 ,

512 , 1) coincides with the famous elliptic j-function

j(z) (z ∈ H).

• It is a meromorphic function on H satisfying the functional equation

j(z + 1) = j(z), j(−1

z

)= j(z).

– The above functional equation means that j-function is the ellip-tic modular function for the full-modular group SL2(Z).

• It has the Fourier expansion

j(z) =1

q+ 744 + 196884q + 21493760q2 + · · · , (q = e2π

√−1z).

The Fourier coefficients 196884, 21493760, · · · are coming from themonstrous moonshine. This is proved by [Borcherds, 1992], byusing the relation between j(z) and the Weyl-Kac type denominatorformula of the monstrous Lie superalgebra.

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■ Geometry for j-function

0 1

i¥

SL2(Z) ∋(a bc d

)acts z ∈ H: z 7→ az + b

cz + d. The quotient space SL2(Z)\H

is represented by the union of two non-Euclidean triangles with angles

0,π

2,

π

3.

The speaker would like to note that these angles are coming from

|1− c| = 0, |c− a− b| = 1

2, |a− b| = 1

3.

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■ Arithmetic property of j-functionThe j-function has very deep properties from the viewpoint of algebraic

number theory for number fields. Here, the speaker would like to show itvery briefly.

• Let K = Q(√−d) (d ∈ N) be an imaginary quadratic field.

(For example, K = Q(√−1),Q(

√−3),Q(

√−6), · · · .)

• Let OK be the ring of integers of K. OK is generated by

1 and zK =

{√−d (−d ≡ 2, 3(mod4))

(1 +√−d)/2 (−d ≡ 1(mod4)))

over Z. (For example, if K = Q(√−6), OK = Z+ Z

√−6.)

In this talk, we call zK ∈ H a CM-point.The special values j(zK) at CM-points are very important.

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Although the j-function is an analytic transcendental function on H,j(zK) is an algebraic number: j(zK) ∈ Q.

Namely, there exists an irreducible polynomial fK(X) ∈ Q[X] such that

fK(j(zK)) = 0.

This means that the j-function has very good arithmetic property.

For example, the degree of fK is important, because it gives the index[Q(j(zK)) : Q]. In this case, we can determine it from the property of K.

• The ideal class group GK of K is a finite abelian group attached tothe field K. (ex. If K = Q(

√−6), GK = Z/2Z.)

• The order hK ∈ N of GK is called the class number of K. (ex. IfK = Q(

√−6), hK = 2.)

−→ deg(fK) = hK .

Remark 2.1. Ideal class fields and class numbers are very important objectin number theory.

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Such arithmetic properties of the j-function are coming from the follow-ing result.

Theorem (Kronecker’s Jugendtraum)For any imaginary quadratic field K, the special value j(zK) at CM-pointzK generates the absolute class field of K.

In this talk, the speaker would like to omit the precise definition of classfields. But,

• This implies Gal(K(j(zK))/K) ≃ GK .

• This is conjectured by Kronecker in the 19th century and finally solvedby the works in the 20th century (T. Takagi, E. Artin, etc.).

• This gives an essential motivation ofHilbert’s 12th problem, whichis still unsolved.

Anyway, this theorem gives a deep arithmetic property of the j-function.

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Based on the Schwarz mapping of the Gauss hypergeometric equation,Prof. H. Shiga and the speaker gave an expansion of the arithmetic storyof the j-function.

Theorem (Extension of Kronecker’s Jugendtraum using Gauss hypergeo-metric equation, [Shiga-N, 2016])If p = 1/|1−c|, q = 1/|c−a−b|, r = 1/|a−b| are given as one of the follow-ing (up to permutation), special values of the inverse Schwarz mapping forthe Gauss hypergeometric equation 2E1(a, b, c) generate the absolute classfields of appropriate CM fields.

(p, q, r) =(2, 3,∞), (3, 3, 4), (3, 3, 6), (2, 5, 5), (3, 5, 5), (3, 3, 5), (2, 3, 7), (2, 3, 9),

(3, 3, 8), (5, 5, 10), (3, 3, 12), (5, 5, 15), (3, 3, 15), (4, 5, 5), (2, 3, 11).

• CM fields give a natural counterpart of imaginary quadratic fields.

• The above (p, q, r) are coming from the characterization of a certaintype of unit groups of arithmetic triangle groups.

• To prove this theorem, the authors used the theory of Shimuracurves.

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3 Toric varieties and a construction of mirror

• Section 1:Hypergeometric equations are very important in mirror symmetry.

• Section 2:Hypergeometric equations can be applied to number theory.

Expectation: Can we have applications of hypergeometric equation comingfrom the theory of mirror symmetry to number theory ?

−→ If we obtain such a result, we will obtain an explicit and non-trivialrelation between mirror symmetry and number theory.

In this section, we will see a construction of mirror pair of Calabi-Yauvarieties via toric varieties. This construction is closely related to the GKZhypergeometric differential equations.

Remark: The contents of this section may be close to the talk of Prof.Hosono.

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In Rn = {(u1, u2, · · · , un)}, an inequality

a1u1 + a2u2 + · · ·+ anun ≤ 1, (a1, a2, · · · , an) ∈ Zn

defines a half space in Rn.A bounded intersection P of several half spaces gives a polytope in Rn.

If a polytope P satisfies the conditions

(a) every vertex is a point of Zn,

(b) the origin is the unique inner lattice point,

(c) only the vertices are the lattice points on the boundary,

then P is called a reflexive polytope with at most terminal singularities.

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Let P be such a n-dimensional reflexive polytope. We have the n-dimensional toric variety.

Letting P ∩ Zn = {a1, · · · , an+r} (aj = t(ν(1)j , · · · , ν(n)j )) be the lattice

points. We have the Laurent polynomial

S =n+r∑j=0

cjtν(1)j

1 · · · tν(n)jn = 0, (cj ∈ C)

defines a hypersurface in the toric variety.We can prove that this gives a family of (n−1)-dimensional Calabi-Yau

varieties. We call it a toric Calabi-Yau hypersurface.

In this talk, we shall focus on the typical (and interesting) two cases forthe two polytopes

P0 =

1 0 0 0 −10 1 0 0 −10 0 1 0 −10 0 0 1 −1

, P1 =

1 0 0 0 −10 1 0 0 −10 0 1 −1 −2

(columns gives the coordinates of vertices).

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■ Case of P0

Since P0 is 4-dimensional, we will have a family of Calabi-Yau 3-folds.

For P0 =

1 0 0 0 −10 1 0 0 −10 0 1 0 −10 0 0 1 −1

, we have 6 lattice points P0 ∩ Z4:

0000

,

1000

,

0100

,

0010

,

0001

,

−1−1−1−1

∈ R3.

The Laurent polynomial is given by

S : c0t01t

02t

03t

04+c1t

11t

02t

03t

04+c2t

01t

12t

03t

04+c3t

01t

02t

13t

04+c4t

01t

02t

03t

14+c5t

−11 t−1

2 t−13 t−1

4 = 0,

Namely,

S : c0 + c1t1 + c2t2 + c3t3 + c4t4 + c5t−11 t−1

2 t−13 t−1

4 = 0.

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By setting tj =x5j

x1x2x3x4x5, we have the expression

S : c1x51 + c2x

52 + c3x

53 + c4x

54 + c5x

55 + c0x1x2x3x4x5 = 0,

This is essentially equal to the Dwork family appeared in Section 1.

By setting

x =c1t1c0, y =

c2t2c0, z =

c3t3c0, w =

c4t4c0, λ =

c1c2c3c4c5c50

,

S is transformed to another defining equation

S(λ) : xyzw(x+ y + z + w + 1) + λ = 0.

In the following, we will see the meaning of this equation.

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From P0 =

1 0 0 0 −10 1 0 0 −10 0 1 0 −10 0 0 1 −1

, we set P0 =

1 1 1 1 1 10 1 0 0 0 −10 0 1 0 0 −10 0 0 1 0 −10 0 0 0 1 −1

. The

matrix P0 gives a homomorphism Z6 → Z5 over Z.Let L = Ker(P0). We can see that L is generated by the vector

t(−5, 1, 1, 1, 1, 1).

Our parameter λ correspond this vector.

λ =c11c

12c

13c

14c

15

c50.

Remark: This is very closely related to the talk of Prof. Hosono, Section2 “Warm-Up”.

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Such a construction of parameters can be explained in the sense ofsecondary stack.

• By the generators of L = Ker(P0), we can obtain the matrix β =(−5 1 1 1 1 1

).

• From the columns of the matrix of β, we obtain a fan in R1. This fanis called a secondary fan FP0

of the polytope P0.

• The fan FP0gives a stacky fan. The toric stack derived from the

stacky fan is called the secondary stack XP0in the sense of [Diemer-

Katzarkov-Kerr 2016].

Remark 3.1. Secondary stacks are studied by [Diemer-Katzarkov-Kerr2016] for the purpose to study mirror symmetry of Calabi-Yau varieties.We note that secondary stacks are also very closely related to the Laf-forgue stacks due to [Lafforgue 2003]. The coordinates x, y, z, w can beexplained in terms of the Lafforgue stack.

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Remark from the viewpoint of mirror symmetry:From this construction, we can easily obtain the mirror due to [Batyrev

1994].For the polytope P0, the polar dual

P ◦0 = {v ∈ Rn|⟨u, v⟩ ≥ −1, ∀u ∈ P0}.

is given by

P ◦0 =

4 −1 −1 −1 −1−1 4 −1 −1 −1−1 −1 4 −1 −1−1 −1 −1 4 −1

.

By taking lattice points P ◦0∩Zn, we can obtain the corresponding Calabi-

Yau 3-fold S◦.−→ S◦ is equal to the quintic hypersurface in P4(C), namely A-model.

Thus, Calabi-Yau varieties from toric hypersurfaces are very useful tostudy mirror symmetry.

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■ Case of P1

Since P1 is 3-dimensional, we will have a family of Calabi-Yau 2-folds,namely K3 surfaces.

In the case of P1, we have 6 lattice points P1 ∩ Z3:uvw

=

000

,

100

,

010

,

001

,

00−1

,

−1−1−2

∈ R3.

Then, we have

S : c0 + c1t1 + c2t2 + c3t3 + c4t−1 + c5t

−11 t−1

2 t−23 = 0.

Setting

x =c1t1c0, y =

c2t2c0, z =

c3t3c0, λ =

c3c4c20

, µ =c1c2c

23c5

c50

and S is transformed to the defining equation

S(λ, µ) : xyz2(x+ y + z + 1) + λxyz + µ = 0.

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Here, λ, µ give the coordinates of the secondary stacks, as in the case ofP0.

More precisely,

Proposition ([Hashimoto-Ueda-N Preprint]) The secondary stack XP1

is given by a weighted blow up of weight (1, 2) of P(1 : 2 : 5) at one point.Our (λ, µ) gives the coordinates of the maximal dense torus of XP1

.

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Remark from the viewpoint of mirror symmetry:We can consider the mirror of S = S(λ, µ).We can obtain the polar dual P ◦

1 . We have the corresponding family ofK3 surfaces S◦, which are parametrized by 18 complex parameters.

The Dolgachev conjecture [Dolgachev 1996] is a conjecture of mir-ror symmetry for toricK3 hypersufraces. In this case, we can directly checkthat the Dolgachev conjecture for our K3 surfaces holds. Namely, we have

Tr(S) ≃ NS(S◦)⊕(0 11 0

).

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4 Arithmetic properties of differential equation from

toric K3 hypersurfaces

In this section, we will see the toric K3 hypersurfaces

S(λ, µ) : xyz2(x+ y + z + 1) + λxyz + µ = 0.

coming from the polytope P1.

We have the Torelli type theorem of K3 surfaces. Therefore, we canstudy Hodge theoretical properties (periods, Gauss-Manin connections,etc.) of K3 surfaces in detail.

We will see the arithmetic properties of the differential equation comingfrom the periods of S(λ, µ). Please note that the results in this section arebased on the Torelli theorem.

32

The following properties for S(λ, µ) are proved in [N, 2013].

• For S = S(λ, µ), NS(S) = H1,1(S,C) ∩H2(S,Z) defines the Neron-Severi (or Picard) lattice by the canonical cup product. In thiscase, the intersection matrix of this lattice is

E8(−1)⊕ E8(−1)⊕(2 11 −2

)for generic (λ, µ).

• The orthogonal complement Tr(S) of the Neron-Severi lattice in theK3 lattice II3,19 (the even unimodular lattice of signature (3, 19)) iscalled the transcendental lattice. In this case, it is given by

A =

(0 11 0

)⊕

(2 11 −2

).

33

• Since S(λ, µ) is a K3 surface, by the definition, there exists the holo-morphic 2-form ω ∈ H2,0 on S(λ, µ) up to a constant factor.

• For γ ∈ H2(S(λ, µ)), we have the periods

∫γ

ω.

• The period domain is given by the 2-dimensional symmetric space

DA = {ξ ∈ P3(C)|ξAtξ = 0, ξAtξ > 0}.

• Taking an apropriate basis γ1, · · · , γ4, the quotient of 4 periods givesthe (multivalued) period mapping

Φ : (λ, µ) 7→(∫

γ1

ω : · · · :∫γ4

ω)∈ DA.

Torelli’s theorem guarantees that Φ is surjective.

•∫γ1

ω : · · · :∫γ4

ω satisfy a differential equation for the independent

variables (λ, µ) of rank 4. This is coming from the Gauss-Manin con-nection.

34

The differential equation gives a counterpart of hypergeometric equa-tions.Theorem([N, 2013]) This differential equation is given by{(θλ(θλ + 2θµ)− λ(2θλ + 5θµ + 1)(2θλ + 5θµ + 2))u = 0,

(λ2(4θ2λ − 2θλθµ + 5θ2µ)− 8λ3(1 + 3θλ + 5θµ + 2θ2λ + 5θλθµ) + 25µθλ(θλ − 1))u = 0,

where θλ = λ ∂∂λ , θµ = µ ∂

∂µ .

Proof. • Since our family of K3 surfaces is coming from toric varieties,our periods satisfy the GKZ hypergeometric equation.

But, in this case, the GKZ system is of rank 6.

• We have (holomorphic) power series expansion of a periods∞∑

n,m=0

(−1)m(5m+ 2n)!

(m!)3n!(2m+ n)!λnµm.

• We can determine the irreducible subsystem of the GKZ system ofrank 4 whose solutions contain the power series.

35

So, the multivalued mapping

Φ : (λ, µ) 7→(∫

γ1

ω : · · · :∫γ4

ω)∈ DA,

has the following properties

• Φ is equal to the Schwarz mapping of the differential equation in theabove theorem.

• Also, we have a biholomorphic mapping ψ : DA → H×H.

• We have the inverse Schwarz mapping

Φ−1 ◦ ψ−1 : H×H ∋ (z1, z2) 7→ (λ, µ) = (λ(z1, z2), µ(z1, z2)).

for our differential equation.

This means that the parameters (λ, µ), which are closely related to toricvarieties and mirror symmetry, are naturally regarded as functions onH×Hvia the Schwarz mapping of our differential equations.

36

Let us see the arithmetic property of our inverse Schwarz mapping

Φ−1 ◦ ψ−1 : (z1, z2) 7→ (λ, µ) = (λ(z1, z2), µ(z1, z2)).

Let F be the real quadratic field for the smallest discriminant (F = Q(√5))

and K be an imaginary quadratic extension. Due to Shimura, the ring OK

of integer of K defines a CM-point (z1,K , z2,K) ∈ H×H.

Theorem (Arithmetic properties of (λ, µ), [N, 2017])For any CM-field K over F, K∗(λ(z1,K , z2,K), µ(z1,K , z2,K))/K

∗ gives anunramified class field.

• K∗ is the reflex of K. This is also a CM-field.

• We will omit the precise definition of class fields. But, from this, itfollows λ(z1,K , z2,K), µ(z1,K , z2,K) ∈ Q. Moreover, (λ, µ) have fruitfularithmetic properties.

• Anyway, this theorem gives a natural counterpart of Kronecker’s Ju-gendtarum for this toric K3 hypersurfaces.

37

Proof. We can prove it by two steps.

Step 1. Techniques based on differential equation (, which were essential givenin [N, 2013]).

– By applying the theory of holomorphic conformal structure of dif-ferential equation according to T. Sasaki and M. Yoshida, we canprove that (z1, z2) 7→ (λ, µ) gives Hilbert modular functionsfor the minimal discriminant.

– By a precise study of the monodromy group for our differentialequation, we can obtain an expression of (z1, z2) 7→ (λ, µ) by thetheta functions on H×H.

Step 2. Application of the theory of Shimura varieties.

– Theta functions are often compatible with Shimura varieties. Inour case, our theta functions give the canonical model of theShimura variety for a Hilbert modular surface.

– This implies that the special values of our theta functions generatethe corresponding class fields.

38

3-dimensional reflexive polytopes with 4 or 5 vertices are classified:

Q =

1 0 0 −10 1 0 −10 0 1 −1

, P1 =

1 0 0 0 −10 1 0 0 −10 0 1 −1 −2

, P2 =

1 0 0 −1 00 1 0 0 −10 0 1 −1 −1

,

P3 =

1 0 0 0 −10 1 0 −1 −10 0 1 −1 −1

, P4 =

1 0 0 −1 00 1 0 −1 00 0 1 0 −1

, P5 =

1 0 0 0 −10 1 0 0 −10 0 1 −1 −1

.

In this talk, we saw the case P1. For other cases, the Dolgachev conjec-ture also holds ([Hashimoto-Ueda-N, preprint])

For cases Q and P2, the speaker proved similar arithmetic properties:the secondary stacks via the inverse of Schwarz mappings are Q-valued atCM-points, applying the theory of Shimura varieties.

For other cases, the speaker does not have correct proofs. But, it seemsthat the corresponding secondary stacks also have arithmetic properties.(In fact, the cases P3 and P4 are very similar to the case P2 and the caseP5 seems similar to P1.)

The speaker is hoping to obtain a conceptual proof based on toric ge-ometry, instead of an application of Shimura varieties.

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5 The case of mirror quintic 3-folds

In the last of the talk, the speaker would like to go back to mirror quintic3-folds:

S(λ) : xyzw(x+ y + z + w + 1) + λ = 0.

As we saw in Section 3, the parameter λ has very natural meaning fromthe viewpoint of mirror symmetry and toric geometry.

If possible, the speaker would like to obtain arithmetic properties of λ.

Since this family is very famous and important, many mathematiciansstudied this family.

By the definition of the Calabi-Yau varieties, we can take the uniqueholomorphic 3-form ω on S(λ) up to a constant factor.

For four 3-cycles δ1, δ2, δ3, δ4 on S(λ), we have periods∫δ1

ω,

∫δ2

ω,

∫δ3

ω,

∫δ4

ω.

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At this moment, to the best of the speaker’s knowledge, we do not havesimple Torelli type theorem for Calabi-Yau 3-folds.

• [Kato-Usui, 2009] introduced logarithmic period mapping. TheTorelli type theorem for mirror quintic 3-folds was obtained in thiscontext. But, this Torelli type theorem (especially the image of theperiod mapping) seems so complicated.

– Especially, this theory is much more difficult than that for K3surfaces.

Moreover, some results of the monodromy group for S(λ) are known.The monodromy group is a subgroup of GL4(Z).

• [Brav and Thomas, 2014] proved that the monodromy group Γ isgenerated by

T =

1 0 0 00 1 0 10 0 1 00 0 0 1

, U =

1 1 0 00 1 0 05 5 1 00 −5 −1 1

with the relation (UT )5 = I4.

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They showed that the monodromy group Γ is isomorphic to

Z ∗ (Z/5Z),

where ∗ means amalgamated product.

• The monodromy covering of P1(C)−{0, 1,∞} for Γ is biholomorpohicto H. The moduli space for S(λ), in the sense of [Movasati 2015], isgiven by Γ\H.

• However, this monodromy group Γ is not arithmetic group in the senseof Shimura.−→ We cannot apply the theory of Shimura curves or Shimura vari-eties, directly.

Therefore, to study arithmetic properties of mirror quintic 3-fold is muchmore difficult than that of K3 surfaces.

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On the other hand, there are several good evidences.

• [Cohen-Wolfert, 1990], which is a purely number theoretical work,proved that the group Z∗ (Z/5Z) is embedded in the Hilbert modulargroup for the minimal discriminant, which is equal to the monodromygroup for K3 surfaces in Section 4.

– Question: Can we apply the arithmetic properties for our K3surfaces to mirror quintic 3-folds...?

• [Movasati, 2015] discovered periods for S(λ) has some “modular-like” properties. For appropriate δ1, · · · , δ4, setting

τ0 =

∫δ1

ω/∫

δ2

ω, τ1 =

∫δ3

ω/∫

δ2

ω, τ2 =

∫δ4

ω/∫

δ2

ω,

he proved that P3(C) ∋ (τ0 : 1 : τ1 : τ2) 7→ λ is invariant under theaction

τ0 7→ τ0 + 1, τ0 7→τ0

τ2 + 1.

• For the family S(λ) at a particular λ, [Long-Tu-Yui-Zudilin, 2017]calculated its zeta function and showed the “modularity” .

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At this moment, the speaker does not have correct answers about arith-metic properties of mirror quintic 3-folds.

If we can obtain some arithmetic results from toric Calabi-Yau hyper-surfaces and the corresponding differential equations, (if possible, withoutan application of the theory of Shimura varieties,) they must be new.

Then, we can draw some new and non-trivial relation between geometryand number theory from them.

This is the reason why the speaker would like to understand the arith-metic properties of toric Calabi-Yau hypersurfaces.

Thank you very much for your kind attension.

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