Differential equations concerned with mirror symmetry of toric K3 hypersurfaces with arithmetic properties Atsuhira Nagano (University of Tokyo) 1
Contents Section 1 : Introduction 1: Hypergeometric differential equations in mirror symmetry (10 minutes) Section 2 : Introduction 2: Hypergeometric differential equations in number theory (10 minutes) Section 3 : Toric varieties and a construction of mirror (15 minutes) Section 4 : Arithmetic properties of differential equation from toric 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 have any questions, please come to me after the talk. I will try to give you a detailed explanation. In Section 1 and 2, we will see basic motivations of hypergeometric differential 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 function, J. Number Theor.. [N, 2017] Icosahedral invariants and a construction of class fields via periods of K3 surfaces, Ramanujan J., in press. [Hashimoto-Ueda-N, preprint] Modular surfaces associated with toric K3 hypersurfaces. The main results of this talk partially appeared at other conferences and workshops in algebraic geometry or number theory. But, today, the speaker would like to survey the results from the viewpoint of mirror symmetry. 3
1 Introduction 1: Hypergeometric differential equations in mirror symmetry Candelas et al. (1991) studied Calabi-Yau 3-folds and discovered mirror symmetry. This is a pioneering work for mirror symmetry. Their results are closely related to (generalized) hypergeometric differential equations. Definition : A Calabi-Yau manifold S is a simply connected Kähler manifold such that the canonical bundle K S 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 K C is trivial if and only if C is an elliptic curve. So, Calabi-Yau manifolds give counterpart of elliptic cuves. Mirror symmetry is formulated as a mysterious relation between the following A-model and B-model. 4
Let V be the generic quintic hypersurface in P 4 (C) = {(x 1 : x 2 : x 3 : x 4 : x 5 )}: V : a j1,j 2,j 3.j 4,j 5 x a 1 1 xa 2 2 xa 3 3 xa 4 4 xa 5 5 = 0. j 1 +j 2 +j 3 +j 4 +j 5 =5 V is a Calabi-Yau 3 fold and called the A-model. V is parametrized by 101 complex parameters. On the other hand, let us consider a family of hypersurfaces W (z) : x 5 1 + x 5 2 + x 5 3 + x 5 4 + x 5 5 5zx 1 x 2 x 3 x 4 x 5 = 0. with a complex parameter z. We have the action of G = {(ζ 1 : : ζ 5 ) ζ 5 j = 1, ζ 1 ζ 5 = 1} (Z/5Z) 3 on the above hypersurface in (x 1 : : x 5 ) (ζ 1 x 1 : : ζ 5 x 5 ). Via a resolution 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. The family of this 3-fold is often called the Dwork family. 5
Geometry of A-model It is conjectured that smooth rational curves on the A-model V are isolated (Clemens conjecture). We can count the number N d of rational curves on V of degree d. We define the virtual number of curves as N vert d = k d 1 k 3N d/k Q. Now, mathematicians avoid the above conjecture. Namely, they use the Gromov-Witten invariants to define the virtual number of curves, instead of counting curves. Let us define a generating function F (t) = 5 6 t3 + d 1 Nd vert e dt for the virtual numbers. 6
Hodge structure of B-model For generic z, the Hodge structure of B-model W (z) is given as follows: We have the Hodge decomposition H 3 dr(w ) = H 3,0 (W ) H 2,1 (W ) H 1,2 (W ) H 0,3 (W ), where dimh 3,0 (W ) = 1, dimh 2,1 (W ) = 1. 0 ω H 3,0 (W (z)) gives the unique holomorphic 3-form on W (z) up to a constant factor. Taking a basis γ 1,, γ 4 of H 3 (W (z)), we have four period integrals ω, ω. γ 1 γ 4 7
These period integrals ω vary with the parameter z. γ j In fact, by setting λ = (5z) 5, periods give solutions of the linear ordinary differential equation (( λ d ) 4 ( λ λ d dλ dλ + 1 )( λ d 5 dλ + 2 )( λ d 5 dλ + 3 )( λ d 5 dλ + 4 )) u = 0 5 for the independent variable λ. Four periods give a system of basis of the space of solutions of this equation. The above differential equation is often called the Picard-Fuchs equation. This is coming from the Gauss-Manin connection of the variation of the Hodge structure of the B-model. 8
Mirror symmetry By the way, the above equation ( coincides with) the generalized hypergeometric equation 4 E 3. So, we have another 1/5 2/5 3/5 4/5 1 1 1 basis of the space of solutions: ψ 0 (λ) = n=0 (5n)! (n!) 5 λn, ψ 1 (λ) = log(λ)ψ 0 (λ) +, ψ 2 (λ) = 1 2 log(λ)2 ψ 0 (λ) +, ψ 3 (λ) = 1 6 log(λ)3 ψ 0 (λ) +. Note that ψ 0 (λ) is holomorphic around λ = 0. This is called the generalized hypergeometric series 4 F 3 ( 1/5 2/5 3/5 4/5 ) 1 1 1 ; λ. The other solutions ψ 1 (λ), ψ 2 (λ), ψ 3 (λ) have logarithmic singularities at λ = 0. 9
Please recall the generating function F (t) = 5 6 t3 + d 1 Nd vert e dt derived from the Gromov-Witten invariants of the A-model. Then, the following highly-nontrivial formula holds: ( ψ1 (λ) ) F = 5 ψ 1 (λ)ψ 2 (λ) ψ 0 (λ)ψ 3 (λ) ψ 0 (λ) 2 ψ 0 (λ) 2 This is one of the most famous result in mirror symmetry. It is very difficult to calculate the virtual numbers Nd vert of curves for the A-model V. However, via the mirror symmetry above, they are calculated explicitly by the right hand side using the periods ψ 0 (λ),, ψ 3 (λ) of the B-model. This is predicted by physicists. [Givental 1996] and [Lian-Liu-Yau 1997] gave mathematical proofs of it. 10
2 Introduction 2: Hypergeometric differential equations in number theory In the last section, we saw the generalized hypergeometric differential equations for Calabi-Yau 3-folds. By the way, the simplest hypergeometric equation is the Gauss hypergeometric equation. The classical theory due to Gauss, Kronecker, Schwarz, etc. suggests that the Gauss hypergeometric equation can be applied to number theory. In this section, we will see that. 11
The Gauss hypergeometric equation is given by E(a, b, c) : λ(1 λ) d2 η + (c (a + b + 1)λ)dη abη = 0. dλ2 dλ λ = 0, 1, are regular singular points of the differential equation. The Gauss hypergeometric series is given by 2F 1 (a, b, c; λ) = n=0 (a, n)(b, n) λ n. (c, n)n! 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 is holomorphic at λ = 0. Since the Gauss hypergeometric equation is of rank 2, the space of solutions of that is 2-dimensional vector space. 12
In the following argument, we suppose that 1 1 c, 1 c a b, 1 a b Z { } 1 c + c a b + a b < 1. We can take 2 solutions η 1 (λ) and η 2 (λ) of the Gauss hypergeometric equations such that σ : λ η 1(λ) η 2 (λ) gives a (surjective) multivalued analytic mapping P 1 (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 P 1 (C). 13
Set p = 1 1 c, q = 1 c a b, r = 1 a b. The multivalued Schwarz mapping σ defines a monodromy covering of P 1 (C) {0, 1, }. Under our assumption, the monodromy group is 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. 14
We can see that the inverse σ 1 : H P 1 (C) of the Schwarz mapping of 2 E 1 ( 1 12, 5 12, 1) coincides with the famous elliptic j-function j(z) (z H). It is a meromorphic function on H satisfying the functional equation ( 1 ) j(z + 1) = j(z), j = j(z). z The above functional equation means that j-function is the elliptic modular function for the full-modular group SL 2 (Z). It has the Fourier expansion j(z) = 1 q + 744 + 196884q + 21493760q2 +, (q = e 2π 1z ). The Fourier coefficients 196884, 21493760, are coming from the monstrous moonshine. This is proved by [Borcherds, 1992], by using the relation between j(z) and the Weyl-Kac type denominator formula of the monstrous Lie superalgebra. 15
Geometry for j-function i 0 1 ( ) a b SL 2 (Z) acts z H: z az + b c d cz + d. The quotient space SL 2(Z)\H is represented by the union of two non-euclidean triangles with angles π 0, 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. 16
Arithmetic property of j-function The j-function has very deep properties from the viewpoint of algebraic number theory for number fields. Here, the speaker would like to show it very briefly. Let K = Q( d) (d N) be an imaginary quadratic field. (For example, K = Q( 1), Q( 3), Q( 6),.) Let O K be the ring of integers of K. O K is generated by { d ( d 2, 3(mod4)) 1 and z K = (1 + d)/2 ( d 1(mod4))) over Z. (For example, if K = Q( 6), O K = Z + Z 6.) In this talk, we call z K H a CM-point. The special values j(z K ) at CM-points are very important. 17
Although the j-function is an analytic transcendental function on H, j(z K ) is an algebraic number: j(z K ) Q. Namely, there exists an irreducible polynomial f K (X) Q[X] such that f K (j(z K )) = 0. This means that the j-function has very good arithmetic property. For example, the degree of f K is important, because it gives the index [Q(j(z K )) : Q]. In this case, we can determine it from the property of K. The ideal class group G K of K is a finite abelian group attached to the field K. (ex. If K = Q( 6), G K = Z/2Z.) The order h K N of G K is called the class number of K. (ex. If K = Q( 6), h K = 2.) deg(f K ) = h K. Remark 2.1. Ideal class fields and class numbers are very important object in number theory. 18
Such arithmetic properties of the j-function are coming from the following result. Theorem (Kronecker s Jugendtraum) For any imaginary quadratic field K, the special value j(z K ) at CM-point z K generates the absolute class field of K. In this talk, the speaker would like to omit the precise definition of class fields. But, This implies Gal(K(j(z K ))/K) G K. This is conjectured by Kronecker in the 19th century and finally solved by the works in the 20th century (T. Takagi, E. Artin, etc.). This gives an essential motivation of Hilbert s 12th problem, which is still unsolved. Anyway, this theorem gives a deep arithmetic property of the j-function. 19
Based on the Schwarz mapping of the Gauss hypergeometric equation, Prof. H. Shiga and the speaker gave an expansion of the arithmetic story of the j-function. Theorem (Extension of Kronecker s Jugendtraum using Gauss hypergeometric equation, [Shiga-N, 2016]) If p = 1/ 1 c, q = 1/ c a b, r = 1/ a b are given as one of the following (up to permutation), special values of the inverse Schwarz mapping for the Gauss hypergeometric equation 2 E 1 (a, b, c) generate the absolute class fields 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 certain type of unit groups of arithmetic triangle groups. To prove this theorem, the authors used the theory of Shimura curves. 20
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 coming from the theory of mirror symmetry to number theory? If we obtain such a result, we will obtain an explicit and non-trivial relation between mirror symmetry and number theory. In this section, we will see a construction of mirror pair of Calabi-Yau varieties via toric varieties. This construction is closely related to the GKZ hypergeometric differential equations. Remark: The contents of this section may be close to the talk of Prof. Hosono. 21
In R n = {(u 1, u 2,, u n )}, an inequality a 1 u 1 + a 2 u 2 + + a n u n 1, (a 1, a 2,, a n ) Z n defines a half space in R n. A bounded intersection P of several half spaces gives a polytope in R n. If a polytope P satisfies the conditions (a) every vertex is a point of Z n, (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. 22
Let P be such a n-dimensional reflexive polytope. We have the n- dimensional toric variety. Letting P Z n = {a 1,, a n+r } (a j = t (ν (1) j,, ν (n) j )) be the lattice points. We have the Laurent polynomial n+r S = j=0 c j t ν(1) j 1 t ν(n) j n = 0, (c j 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 for the two polytopes 1 0 0 0 1 P 0 = 0 1 0 0 1 1 0 0 0 1 0 0 1 0 1, P 1 = 0 1 0 0 1 0 0 1 1 2 0 0 0 1 1 (columns gives the coordinates of vertices). 23
Case of P 0 Since P 0 is 4-dimensional, we will have a family of Calabi-Yau 3-folds. 1 0 0 0 1 For P 0 = 0 1 0 0 1 0 0 1 0 1, we have 6 lattice points P 0 Z 4 : 0 0 0 1 1 0 1 0 0 0 1 0 0, 0 0, 1 0, 0 1, 0 0, 1 1 R3. 0 0 0 0 1 1 The Laurent polynomial is given by S : c 0 t 0 1t 0 2t 0 3t 0 4+c 1 t 1 1t 0 2t 0 3t 0 4+c 2 t 0 1t 1 2t 0 3t 0 4+c 3 t 0 1t 0 2t 1 3t 0 4+c 4 t 0 1t 0 2t 0 3t 1 4+c 5 t 1 1 t 1 2 t 1 3 t 1 4 = 0, Namely, S : c 0 + c 1 t 1 + c 2 t 2 + c 3 t 3 + c 4 t 4 + c 5 t 1 1 t 1 2 t 1 3 t 1 4 = 0. 24
By setting t j = x 5 j x 1 x 2 x 3 x 4 x 5, we have the expression S : c 1 x 5 1 + c 2 x 5 2 + c 3 x 5 3 + c 4 x 5 4 + c 5 x 5 5 + c 0 x 1 x 2 x 3 x 4 x 5 = 0, This is essentially equal to the Dwork family appeared in Section 1. By setting x = c 1t 1, y = c 2t 2, z = c 3t 3, w = c 4t 4, λ = c 1c 2 c 3 c 4 c 5, c 0 c 0 c 0 c 0 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. c 5 0 25
1 1 1 1 1 1 1 0 0 0 1 From P 0 = 0 1 0 0 1 0 0 1 0 1, we set P 0 1 0 0 0 1 0 = 0 0 1 0 0 1 0 0 0 1 0 1. The 0 0 0 1 1 0 0 0 0 1 1 matrix P 0 gives a homomorphism Z 6 Z 5 over Z. Let L = Ker( P 0 ). We can see that L is generated by the vector t ( 5, 1, 1, 1, 1, 1). Our parameter λ correspond this vector. λ = c1 1c 1 2c 1 3c 1 4c 1 5 c 5. 0 Remark: This is very closely related to the talk of Prof. Hosono, Section 2 Warm-Up. 26
Such a construction of parameters can be explained in the sense of secondary stack. By the generators of L = Ker( P 0 ), we can obtain the matrix ˇβ = ( 5 1 1 1 1 1 ). From the columns of the matrix of ˇβ, we obtain a fan in R 1. This fan is called a secondary fan F P0 of the polytope P 0. The fan F P0 gives a stacky fan. The toric stack derived from the stacky fan is called the secondary stack X P0 in the sense of [Diemer- Katzarkov-Kerr 2016]. Remark 3.1. Secondary stacks are studied by [Diemer-Katzarkov-Kerr 2016] for the purpose to study mirror symmetry of Calabi-Yau varieties. We note that secondary stacks are also very closely related to the Lafforgue stacks due to [Lafforgue 2003]. The coordinates x, y, z, w can be explained in terms of the Lafforgue stack. 27
Remark from the viewpoint of mirror symmetry: From this construction, we can easily obtain the mirror due to [Batyrev 1994]. For the polytope P 0, the polar dual is given by P 0 = {v R n u, v 1, u P 0 }. 4 1 1 1 1 P0 = 1 4 1 1 1 1 1 4 1 1. 1 1 1 4 1 By taking lattice points P 0 Z n, we can obtain the corresponding Calabi- Yau 3-fold S. S is equal to the quintic hypersurface in P 4 (C), namely A-model. Thus, Calabi-Yau varieties from toric hypersurfaces are very useful to study mirror symmetry. 28
Case of P 1 Since P 1 is 3-dimensional, we will have a family of Calabi-Yau 2-folds, namely K3 surfaces. In the case of P 1, we have 6 lattice points P 1 Z 3 : Then, we have u 0 1 0 0 0 1 v = 0, 0, 1, 0, 0, 1 R 3. w 0 0 0 1 1 2 Setting S : c 0 + c 1 t 1 + c 2 t 2 + c 3 t 3 + c 4 t 1 + c 5 t 1 1 t 1 2 t 2 3 = 0. x = c 1t 1 c 0, y = c 2t 2 c 0, z = c 3t 3 c 0, λ = c 3c 4 and S is transformed to the defining equation c 2 0, µ = c 1c 2 c 2 3c 5 c 5 0 S(λ, µ) : xyz 2 (x + y + z + 1) + λxyz + µ = 0. 29
Here, λ, µ give the coordinates of the secondary stacks, as in the case of P 0. More precisely, Proposition ([Hashimoto-Ueda-N Preprint]) The secondary stack X P1 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 X P1. 30
Remark from the viewpoint of mirror symmetry: We can consider the mirror of S = S(λ, µ). We can obtain the polar dual P1. We have the corresponding family of K3 surfaces S, which are parametrized by 18 complex parameters. The Dolgachev conjecture [Dolgachev 1996] is a conjecture of mirror symmetry for toric K3 hypersufraces. In this case, we can directly check that the Dolgachev conjecture for our K3 surfaces holds. Namely, we have ( ) 0 1 Tr(S) NS(S ). 1 0 31
4 Arithmetic properties of differential equation from toric K3 hypersurfaces In this section, we will see the toric K3 hypersurfaces S(λ, µ) : xyz 2 (x + y + z + 1) + λxyz + µ = 0. coming from the polytope P 1. We have the Torelli type theorem of K3 surfaces. Therefore, we can study Hodge theoretical properties (periods, Gauss-Manin connections, etc.) of K3 surfaces in detail. We will see the arithmetic properties of the differential equation coming from the periods of S(λ, µ). Please note that the results in this section are based on the Torelli theorem. 32
The following properties for S(λ, µ) are proved in [N, 2013]. For S = S(λ, µ), NS(S) = H 1,1 (S, C) H 2 (S, Z) defines the Neron- Severi (or Picard) lattice by the canonical cup product. In this case, the intersection matrix of this lattice is ( ) 2 1 E 8 ( 1) E 8 ( 1) 1 2 for generic (λ, µ). The orthogonal complement Tr(S) of the Néron-Severi lattice in the K3 lattice II 3,19 (the even unimodular lattice of signature (3, 19)) is called the transcendental lattice. In this case, it is given by A = ( 0 1 1 0 ) ( 2 1 1 2 ). 33
Since S(λ, µ) is a K3 surface, by the definition, there exists the holomorphic 2-form ω H 2,0 on S(λ, µ) up to a constant factor. For γ H 2 (S(λ, µ)), we have the periods ω. The period domain is given by the 2-dimensional symmetric space D A = {ξ P 3 (C) ξa t ξ = 0, ξa t ξ > 0}. Taking an apropriate basis γ 1,, γ 4, the quotient of 4 periods gives the (multivalued) period mapping ( ) Φ : (λ, µ) ω : : ω D A. γ 1 γ 4 Torelli s theorem guarantees that Φ is surjective. ω : : ω satisfy a differential equation for the independent γ 1 γ 4 variables (λ, µ) of rank 4. This is coming from the Gauss-Manin connection. γ 34
The differential equation gives a counterpart of hypergeometric equations. Theorem([N, 2013]) This differential equation is given by { (θ λ (θ λ + 2θ µ ) λ(2θ λ + 5θ µ + 1)(2θ λ + 5θ µ + 2))u = 0, (λ 2 (4θ 2 λ 2θ λθ µ + 5θ 2 µ) 8λ 3 (1 + 3θ λ + 5θ µ + 2θ 2 λ + 5θ λθ µ ) + 25µθ λ (θ λ 1))u = 0, where θ λ = λ λ, θ µ = 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 ( 1) m (5m + 2n)! (m!) 3 n!(2m + n)! λn µ m. n,m=0 We can determine the irreducible subsystem of the GKZ system of rank 4 whose solutions contain the power series. 35
So, the multivalued mapping ( Φ : (λ, µ) ω : : γ 1 has the following properties γ 4 ω ) D A, Φ is equal to the Schwarz mapping of the differential equation in the above theorem. Also, we have a biholomorphic mapping ψ : D A H H. We have the inverse Schwarz mapping Φ 1 ψ 1 : H H (z 1, z 2 ) (λ, µ) = (λ(z 1, z 2 ), µ(z 1, z 2 )). for our differential equation. This means that the parameters (λ, µ), which are closely related to toric varieties and mirror symmetry, are naturally regarded as functions on H H via the Schwarz mapping of our differential equations. 36
Let us see the arithmetic property of our inverse Schwarz mapping Φ 1 ψ 1 : (z 1, z 2 ) (λ, µ) = (λ(z 1, z 2 ), µ(z 1, z 2 )). 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 O K of integer of K defines a CM-point (z 1,K, z 2,K ) H H. Theorem (Arithmetic properties of (λ, µ), [N, 2017]) For any CM-field K over F, K (λ(z 1,K, z 2,K ), µ(z 1,K, z 2,K ))/K gives an unramified 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, it follows λ(z 1,K, z 2,K ), µ(z 1,K, z 2,K ) Q. Moreover, (λ, µ) have fruitful arithmetic properties. Anyway, this theorem gives a natural counterpart of Kronecker s Jugendtarum for this toric K3 hypersurfaces. 37
Proof. We can prove it by two steps. Step 1. Techniques based on differential equation (, which were essential given in [N, 2013]). By applying the theory of holomorphic conformal structure of differential equation according to T. Sasaki and M. Yoshida, we can prove that (z 1, z 2 ) (λ, µ) gives Hilbert modular functions for the minimal discriminant. By a precise study of the monodromy group for our differential equation, we can obtain an expression of (z 1, z 2 ) (λ, µ) by the theta functions on H H. Step 2. Application of the theory of Shimura varieties. Theta functions are often compatible with Shimura varieties. In our case, our theta functions give the canonical model of the Shimura variety for a Hilbert modular surface. This implies that the special values of our theta functions generate the corresponding class fields. 38
3-dimensional reflexive polytopes with 4 or 5 vertices are classified: 1 0 0 1 1 0 0 0 1 1 0 0 1 0 Q = 0 1 0 1, P 1 = 0 1 0 0 1, P 2 = 0 1 0 0 1, 0 0 1 1 0 0 1 1 2 0 0 1 1 1 1 0 0 0 1 1 0 0 1 0 1 0 0 0 1 P 3 = 0 1 0 1 1, P 4 = 0 1 0 1 0, P 5 = 0 1 0 0 1. 0 0 1 1 1 0 0 1 0 1 0 0 1 1 1 In this talk, we saw the case P 1. For other cases, the Dolgachev conjecture also holds ([Hashimoto-Ueda-N, preprint]) For cases Q and P 2, the speaker proved similar arithmetic properties: the secondary stacks via the inverse of Schwarz mappings are Q-valued at CM-points, applying the theory of Shimura varieties. For other cases, the speaker does not have correct proofs. But, it seems that the corresponding secondary stacks also have arithmetic properties. (In fact, the cases P 3 and P 4 are very similar to the case P 2 and the case P 5 seems similar to P 1.) The speaker is hoping to obtain a conceptual proof based on toric geometry, instead of an application of Shimura varieties. 39
5 The case of mirror quintic 3-folds In the last of the talk, the speaker would like to go back to mirror quintic 3-folds: S(λ) : xyzw(x + y + z + w + 1) + λ = 0. As we saw in Section 3, the parameter λ has very natural meaning from the 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 mathematicians studied this family. By the definition of the Calabi-Yau varieties, we can take the unique holomorphic 3-form ω on S(λ) up to a constant factor. For four 3-cycles δ 1, δ 2, δ 3, δ 4 on S(λ), we have periods ω, ω, ω, ω. δ 1 δ 2 δ 3 δ 4 40
At this moment, to the best of the speaker s knowledge, we do not have simple Torelli type theorem for Calabi-Yau 3-folds. [Kato-Usui, 2009] introduced logarithmic period mapping. The Torelli type theorem for mirror quintic 3-folds was obtained in this context. But, this Torelli type theorem (especially the image of the period mapping) seems so complicated. Especially, this theory is much more difficult than that for K3 surfaces. Moreover, some results of the monodromy group for S(λ) are known. The monodromy group is a subgroup of GL 4 (Z). [Brav and Thomas, 2014] proved that the monodromy group Γ is generated by 1 0 0 0 1 1 0 0 T = 0 1 0 1 0 0 1 0, U = 0 1 0 0 5 5 1 0 0 0 0 1 0 5 1 1 with the relation (UT ) 5 = I 4. 41
They showed that the monodromy group Γ is isomorphic to Z (Z/5Z), where means amalgamated product. The monodromy covering of P 1 (C) {0, 1, } for Γ is biholomorpohic to H. The moduli space for S(λ), in the sense of [Movasati 2015], is given by Γ\H. However, this monodromy group Γ is not arithmetic group in the sense of Shimura. We cannot apply the theory of Shimura curves or Shimura varieties, directly. Therefore, to study arithmetic properties of mirror quintic 3-fold is much more difficult than that of K3 surfaces. 42
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 modular group for the minimal discriminant, which is equal to the monodromy group for K3 surfaces in Section 4. Question: Can we apply the arithmetic properties for our K3 surfaces to mirror quintic 3-folds...? [Movasati, 2015] discovered periods for S(λ) has some modularlike properties. For appropriate δ 1,, δ 4, setting / / / τ 0 = ω ω, τ 1 = ω ω, τ 2 = ω ω, δ 1 δ 2 δ 3 δ 2 δ 4 δ 2 he proved that P 3 (C) (τ 0 : 1 : τ 1 : τ 2 ) λ is invariant under the action τ 0 τ 0 + 1, τ 0 τ 0 τ 2 + 1. For the family S(λ) at a particular λ, [Long-Tu-Yui-Zudilin, 2017] calculated its zeta function and showed the modularity. 43
At this moment, the speaker does not have correct answers about arithmetic properties of mirror quintic 3-folds. If we can obtain some arithmetic results from toric Calabi-Yau hypersurfaces and the corresponding differential equations, (if possible, without an application of the theory of Shimura varieties,) they must be new. Then, we can draw some new and non-trivial relation between geometry and number theory from them. This is the reason why the speaker would like to understand the arithmetic properties of toric Calabi-Yau hypersurfaces. Thank you very much for your kind attension. 44