Differential equations concerned with mirror symmetry of toric K3 hypersurfaces with arithmetic properties. Atsuhira Nagano (University of Tokyo)

Size: px
Start display at page:

Download "Differential equations concerned with mirror symmetry of toric K3 hypersurfaces with arithmetic properties. Atsuhira Nagano (University of Tokyo)"

Transcription

1 Differential equations concerned with mirror symmetry of toric K3 hypersurfaces with arithmetic properties Atsuhira Nagano (University of Tokyo) 1

2 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

3 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

4 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

5 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 x x x 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

6 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

7 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

8 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

9 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/ 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 ) ; λ. The other solutions ψ 1 (λ), ψ 2 (λ), ψ 3 (λ) have logarithmic singularities at λ = 0. 9

10 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

11 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

12 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

13 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

14 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. 14

15 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 q q2 +, (q = e 2π 1z ). The Fourier coefficients , , 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

16 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 =

17 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

18 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

19 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

20 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

21 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

22 In R n = {(u 1, u 2,, u n )}, an inequality a 1 u 1 + a 2 u 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

23 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 P 0 = , P 1 = (columns gives the coordinates of vertices). 23

24 Case of P 0 Since P 0 is 4-dimensional, we will have a family of Calabi-Yau 3-folds For P 0 = , we have 6 lattice points P 0 Z 4 : , 0 0, 1 0, 0 1, 0 0, 1 1 R 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

25 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 c 2 x c 3 x c 4 x c 5 x 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

26 From P 0 = , we set P = The 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

27 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 = ( ). 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

28 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 } P0 = 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

29 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 v = 0, 0, 1, 0, 0, 1 R 3. w 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

30 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

31 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 )

32 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

33 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 = ( ) ( ). 33

34 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

35 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

36 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

37 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

38 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

39 3-dimensional reflexive polytopes with 4 or 5 vertices are classified: Q = , P 1 = , P 2 = , P 3 = , P 4 = , P 5 = 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

40 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

41 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 T = , U = with the relation (UT ) 5 = I 4. 41

42 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

43 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 For the family S(λ) at a particular λ, [Long-Tu-Yui-Zudilin, 2017] calculated its zeta function and showed the modularity. 43

44 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

Overview of classical mirror symmetry

Overview of classical mirror symmetry Overview of classical mirror symmetry David Cox (notes by Paul Hacking) 9/8/09 () Physics (2) Quintic 3-fold (3) Math String theory is a N = 2 superconformal field theory (SCFT) which models elementary

More information

Periods and generating functions in Algebraic Geometry

Periods and generating functions in Algebraic Geometry Periods and generating functions in Algebraic Geometry Hossein Movasati IMPA, Instituto de Matemática Pura e Aplicada, Brazil www.impa.br/ hossein/ Abstract In 1991 Candelas-de la Ossa-Green-Parkes predicted

More information

Hodge structures from differential equations

Hodge structures from differential equations Hodge structures from differential equations Andrew Harder January 4, 2017 These are notes on a talk on the paper Hodge structures from differential equations. The goal is to discuss the method of computation

More information

Mirror Symmetry: Introduction to the B Model

Mirror Symmetry: Introduction to the B Model Mirror Symmetry: Introduction to the B Model Kyler Siegel February 23, 2014 1 Introduction Recall that mirror symmetry predicts the existence of pairs X, ˇX of Calabi-Yau manifolds whose Hodge diamonds

More information

Introduction To K3 Surfaces (Part 2)

Introduction To K3 Surfaces (Part 2) Introduction To K3 Surfaces (Part 2) James Smith Calf 26th May 2005 Abstract In this second introductory talk, we shall take a look at moduli spaces for certain families of K3 surfaces. We introduce the

More information

Monodromy of the Dwork family, following Shepherd-Barron X n+1. P 1 λ. ζ i = 1}/ (µ n+1 ) H.

Monodromy of the Dwork family, following Shepherd-Barron X n+1. P 1 λ. ζ i = 1}/ (µ n+1 ) H. Monodromy of the Dwork family, following Shepherd-Barron 1. The Dwork family. Consider the equation (f λ ) f λ (X 0, X 1,..., X n ) = λ(x n+1 0 + + X n+1 n ) (n + 1)X 0... X n = 0, where λ is a free parameter.

More information

Ursula Whitcher May 2011

Ursula Whitcher May 2011 K3 Surfaces with S 4 Symmetry Ursula Whitcher ursula@math.hmc.edu Harvey Mudd College May 2011 Dagan Karp (HMC) Jacob Lewis (Universität Wien) Daniel Moore (HMC 11) Dmitri Skjorshammer (HMC 11) Ursula

More information

(τ) = q (1 q n ) 24. E 4 (τ) = q q q 3 + = (1 q) 240 (1 q 2 ) (1 q 3 ) (1.1)

(τ) = q (1 q n ) 24. E 4 (τ) = q q q 3 + = (1 q) 240 (1 q 2 ) (1 q 3 ) (1.1) Automorphic forms on O s+2,2 (R) + and generalized Kac-Moody algebras. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 744 752, Birkhäuser, Basel, 1995. Richard E.

More information

Arithmetic Mirror Symmetry

Arithmetic Mirror Symmetry Arithmetic Mirror Symmetry Daqing Wan April 15, 2005 Institute of Mathematics, Chinese Academy of Sciences, Beijing, P.R. China Department of Mathematics, University of California, Irvine, CA 92697-3875

More information

The geometry of Landau-Ginzburg models

The geometry of Landau-Ginzburg models Motivation Toric degeneration Hodge theory CY3s The Geometry of Landau-Ginzburg Models January 19, 2016 Motivation Toric degeneration Hodge theory CY3s Plan of talk 1. Landau-Ginzburg models and mirror

More information

Chern numbers and Hilbert Modular Varieties

Chern numbers and Hilbert Modular Varieties Chern numbers and Hilbert Modular Varieties Dylan Attwell-Duval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point

More information

The kappa function. [ a b. c d

The kappa function. [ a b. c d The kappa function Masanobu KANEKO Masaaki YOSHIDA Abstract: The kappa function is introduced as the function κ satisfying Jκτ)) = λτ), where J and λ are the elliptic modular functions. A Fourier expansion

More information

Crash Course on Toric Geometry

Crash Course on Toric Geometry Crash Course on Toric Geometry Emily Clader RTG Workshop on Mirror Symmetry February 2012 The Kähler cone If X Σ is a simplicial projective toric variety, then A n 1 (X Σ ) R = H 2 (X Σ ; R), so H 1,1

More information

Modular forms and the Hilbert class field

Modular forms and the Hilbert class field Modular forms and the Hilbert class field Vladislav Vladilenov Petkov VIGRE 2009, Department of Mathematics University of Chicago Abstract The current article studies the relation between the j invariant

More information

Homological Mirror Symmetry and VGIT

Homological Mirror Symmetry and VGIT Homological Mirror Symmetry and VGIT University of Vienna January 24, 2013 Attributions Based on joint work with M. Ballard (U. Wisconsin) and Ludmil Katzarkov (U. Miami and U. Vienna). Slides available

More information

GENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY. Sampei Usui

GENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY. Sampei Usui GENERIC TORELLI THEOREM FOR QUINTIC-MIRROR FAMILY Sampei Usui Abstract. This article is a geometric application of polarized logarithmic Hodge theory of Kazuya Kato and Sampei Usui. We prove generic Torelli

More information

Toric Varieties and the Secondary Fan

Toric Varieties and the Secondary Fan Toric Varieties and the Secondary Fan Emily Clader Fall 2011 1 Motivation The Batyrev mirror symmetry construction for Calabi-Yau hypersurfaces goes roughly as follows: Start with an n-dimensional reflexive

More information

arxiv: v3 [math.ag] 26 Jun 2017

arxiv: v3 [math.ag] 26 Jun 2017 CALABI-YAU MANIFOLDS REALIZING SYMPLECTICALLY RIGID MONODROMY TUPLES arxiv:50.07500v [math.ag] 6 Jun 07 CHARLES F. DORAN, ANDREAS MALMENDIER Abstract. We define an iterative construction that produces

More information

Constructing Class invariants

Constructing Class invariants Constructing Class invariants Aristides Kontogeorgis Department of Mathematics University of Athens. Workshop Thales 1-3 July 2015 :Algebraic modeling of topological and computational structures and applications,

More information

Equations for Hilbert modular surfaces

Equations for Hilbert modular surfaces Equations for Hilbert modular surfaces Abhinav Kumar MIT April 24, 2013 Introduction Outline of talk Elliptic curves, moduli spaces, abelian varieties 2/31 Introduction Outline of talk Elliptic curves,

More information

Counting problems in Number Theory and Physics

Counting problems in Number Theory and Physics Counting problems in Number Theory and Physics Hossein Movasati IMPA, Instituto de Matemática Pura e Aplicada, Brazil www.impa.br/ hossein/ Encontro conjunto CBPF-IMPA, 2011 A documentary on string theory

More information

Introduction to Borcherds Forms

Introduction to Borcherds Forms Introduction to Borcherds Forms Montreal-Toronto Workshop in Number Theory September 3, 2010 Main Goal Extend theta lift to construct (meromorphic) modular forms on Sh. var. associated to O(p, 2) with

More information

EXERCISES IN MODULAR FORMS I (MATH 726) (2) Prove that a lattice L is integral if and only if its Gram matrix has integer coefficients.

EXERCISES IN MODULAR FORMS I (MATH 726) (2) Prove that a lattice L is integral if and only if its Gram matrix has integer coefficients. EXERCISES IN MODULAR FORMS I (MATH 726) EYAL GOREN, MCGILL UNIVERSITY, FALL 2007 (1) We define a (full) lattice L in R n to be a discrete subgroup of R n that contains a basis for R n. Prove that L is

More information

Hypergeometric Functions and Hypergeometric Abelian Varieties

Hypergeometric Functions and Hypergeometric Abelian Varieties Hypergeometric Functions and Hypergeometric Abelian Varieties Fang-Ting Tu Louisiana State University September 29th, 2016 BIRS Workshop: Modular Forms in String Theory Fang Ting Tu (LSU) Hypergeometric

More information

The Grothendieck-Katz Conjecture for certain locally symmetric varieties

The Grothendieck-Katz Conjecture for certain locally symmetric varieties The Grothendieck-Katz Conjecture for certain locally symmetric varieties Benson Farb and Mark Kisin August 20, 2008 Abstract Using Margulis results on lattices in semi-simple Lie groups, we prove the Grothendieck-

More information

On values of Modular Forms at Algebraic Points

On values of Modular Forms at Algebraic Points On values of Modular Forms at Algebraic Points Jing Yu National Taiwan University, Taipei, Taiwan August 14, 2010, 18th ICFIDCAA, Macau Hermite-Lindemann-Weierstrass In value distribution theory the exponential

More information

a double cover branched along the smooth quadratic line complex

a double cover branched along the smooth quadratic line complex QUADRATIC LINE COMPLEXES OLIVIER DEBARRE Abstract. In this talk, a quadratic line complex is the intersection, in its Plücker embedding, of the Grassmannian of lines in an 4-dimensional projective space

More information

A SHORT INTRODUCTION TO HILBERT MODULAR SURFACES AND HIRZEBRUCH-ZAGIER DIVISORS

A SHORT INTRODUCTION TO HILBERT MODULAR SURFACES AND HIRZEBRUCH-ZAGIER DIVISORS A SHORT INTRODUCTION TO HILBERT MODULAR SURFACES AND HIRZEBRUCH-ZAGIER DIVISORS STEPHAN EHLEN 1. Modular curves and Heegner Points The modular curve Y (1) = Γ\H with Γ = Γ(1) = SL (Z) classifies the equivalence

More information

Riemannian Curvature Functionals: Lecture III

Riemannian Curvature Functionals: Lecture III Riemannian Curvature Functionals: Lecture III Jeff Viaclovsky Park City Mathematics Institute July 18, 2013 Lecture Outline Today we will discuss the following: Complete the local description of the moduli

More information

arxiv: v1 [math.ag] 29 Jan 2015

arxiv: v1 [math.ag] 29 Jan 2015 CLASSIFICATIONS OF ELLIPTIC FIBRATIONS OF A SINGULAR K3 SURFACE arxiv:50.0484v [math.ag] 29 Jan 205 MARIE JOSÉ BERTIN, ALICE GARBAGNATI, RUTHI HORTSCH, ODILE LECACHEUX, MAKIKO MASE, CECÍLIA SALGADO, AND

More information

Mirror symmetry for G 2 manifolds

Mirror symmetry for G 2 manifolds Mirror symmetry for G 2 manifolds based on [1602.03521] [1701.05202]+[1706.xxxxx] with Michele del Zotto (Stony Brook) 1 Strings, T-duality & Mirror Symmetry 2 Type II String Theories and T-duality Superstring

More information

(www.math.uni-bonn.de/people/harder/manuscripts/buch/), files chap2 to chap

(www.math.uni-bonn.de/people/harder/manuscripts/buch/), files chap2 to chap The basic objects in the cohomology theory of arithmetic groups Günter Harder This is an exposition of the basic notions and concepts which are needed to build up the cohomology theory of arithmetic groups

More information

Topics in Geometry: Mirror Symmetry

Topics in Geometry: Mirror Symmetry MIT OpenCourseWare http://ocw.mit.edu 18.969 Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIRROR SYMMETRY:

More information

The j-function, the golden ratio, and rigid meromorphic cocycles

The j-function, the golden ratio, and rigid meromorphic cocycles The j-function, the golden ratio, and rigid meromorphic cocycles Henri Darmon, McGill University CNTA XV, July 2018 Reminiscences of CNTA 0 The 1987 CNTA in Quebec City was an exciting one for me personally,

More information

Counting curves on a surface

Counting curves on a surface Counting curves on a surface Ragni Piene Centre of Mathematics for Applications and Department of Mathematics, University of Oslo University of Pennsylvania, May 6, 2005 Enumerative geometry Specialization

More information

Mirror symmetry. Mark Gross. July 24, University of Cambridge

Mirror symmetry. Mark Gross. July 24, University of Cambridge University of Cambridge July 24, 2015 : A very brief and biased history. A search for examples of compact Calabi-Yau three-folds by Candelas, Lynker and Schimmrigk (1990) as crepant resolutions of hypersurfaces

More information

What is the Langlands program all about?

What is the Langlands program all about? What is the Langlands program all about? Laurent Lafforgue November 13, 2013 Hua Loo-Keng Distinguished Lecture Academy of Mathematics and Systems Science, Chinese Academy of Sciences This talk is mainly

More information

Congruence sheaves and congruence differential equations Beyond hypergeometric functions

Congruence sheaves and congruence differential equations Beyond hypergeometric functions Congruence sheaves and congruence differential equations Beyond hypergeometric functions Vasily Golyshev Lille, March 6, 204 / 45 Plan of talk Report on joint work in progress with Anton Mellit and Duco

More information

Cohomology jump loci of quasi-projective varieties

Cohomology jump loci of quasi-projective varieties Cohomology jump loci of quasi-projective varieties Botong Wang joint work with Nero Budur University of Notre Dame June 27 2013 Motivation What topological spaces are homeomorphic (or homotopy equivalent)

More information

Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture

Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture Rigidity, locally symmetric varieties and the Grothendieck-Katz Conjecture Benson Farb and Mark Kisin May 8, 2009 Abstract Using Margulis s results on lattices in semisimple Lie groups, we prove the Grothendieck-

More information

CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP

CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP TOSHIHIRO SHODA Abstract. In this paper, we study a compact minimal surface in a 4-dimensional flat

More information

Black Holes and Hurwitz Class Numbers

Black Holes and Hurwitz Class Numbers Black Holes and Hurwitz Class Numbers Shamit Kachru a,1, Arnav Tripathy b a Stanford Institute for Theoretical Physics Stanford University, Palo Alto, CA 94305, USA Email: skachru@stanford.edu b Department

More information

SYMPLECTIC GEOMETRY: LECTURE 5

SYMPLECTIC GEOMETRY: LECTURE 5 SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The

More information

K3 Surfaces and Lattice Theory

K3 Surfaces and Lattice Theory K3 Surfaces and Lattice Theory Ichiro Shimada Hiroshima University 2014 Aug Singapore 1 / 26 Example Consider two surfaces S + and S in C 3 defined by w 2 (G(x, y) ± 5 H(x, y)) = 1, where G(x, y) := 9

More information

Topological and arithmetic intersection numbers attached to real quadratic cycles

Topological and arithmetic intersection numbers attached to real quadratic cycles Topological and arithmetic intersection numbers attached to real quadratic cycles Henri Darmon, McGill University Jan Vonk, McGill University Workshop, IAS, November 8 This is joint work with Jan Vonk

More information

Takao Akahori. z i In this paper, if f is a homogeneous polynomial, the correspondence between the Kodaira-Spencer class and C[z 1,...

Takao Akahori. z i In this paper, if f is a homogeneous polynomial, the correspondence between the Kodaira-Spencer class and C[z 1,... J. Korean Math. Soc. 40 (2003), No. 4, pp. 667 680 HOMOGENEOUS POLYNOMIAL HYPERSURFACE ISOLATED SINGULARITIES Takao Akahori Abstract. The mirror conjecture means originally the deep relation between complex

More information

Gromov-Witten invariants and Algebraic Geometry (II) Jun Li

Gromov-Witten invariants and Algebraic Geometry (II) Jun Li Gromov-Witten invariants and Algebraic Geometry (II) Shanghai Center for Mathematical Sciences and Stanford University GW invariants of quintic Calabi-Yau threefolds Quintic Calabi-Yau threefolds: X =

More information

Kleine AG: Travaux de Shimura

Kleine AG: Travaux de Shimura Kleine AG: Travaux de Shimura Sommer 2018 Programmvorschlag: Felix Gora, Andreas Mihatsch Synopsis This Kleine AG grew from the wish to understand some aspects of Deligne s axiomatic definition of Shimura

More information

Modular-type functions attached to mirror quintic Calabi-Yau varieties 1

Modular-type functions attached to mirror quintic Calabi-Yau varieties 1 Modular-type functions attached to mirror quintic Calabi-Yau varieties Hossein Movasati Instituto de Matemática Pura e Aplicada, IMPA, Estrada Dona Castorina,, 2246-32, Rio de Janeiro, RJ, Brazil, www.impa.br/

More information

Combinatorics and geometry of E 7

Combinatorics and geometry of E 7 Combinatorics and geometry of E 7 Steven Sam University of California, Berkeley September 19, 2012 1/24 Outline Macdonald representations Vinberg representations Root system Weyl group 7 points in P 2

More information

Generalized Tian-Todorov theorems

Generalized Tian-Todorov theorems Generalized Tian-Todorov theorems M.Kontsevich 1 The classical Tian-Todorov theorem Recall the classical Tian-Todorov theorem (see [4],[5]) about the smoothness of the moduli spaces of Calabi-Yau manifolds:

More information

Outline of the Seminar Topics on elliptic curves Saarbrücken,

Outline of the Seminar Topics on elliptic curves Saarbrücken, Outline of the Seminar Topics on elliptic curves Saarbrücken, 11.09.2017 Contents A Number theory and algebraic geometry 2 B Elliptic curves 2 1 Rational points on elliptic curves (Mordell s Theorem) 5

More information

Lecture 1. Toric Varieties: Basics

Lecture 1. Toric Varieties: Basics Lecture 1. Toric Varieties: Basics Taras Panov Lomonosov Moscow State University Summer School Current Developments in Geometry Novosibirsk, 27 August1 September 2018 Taras Panov (Moscow University) Lecture

More information

Topics in Geometry: Mirror Symmetry

Topics in Geometry: Mirror Symmetry MIT OpenCourseWare http://ocw.mit.edu 18.969 Topics in Geometry: Mirror Symmetry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MIRROR SYMMETRY:

More information

PARITY OF THE COEFFICIENTS OF KLEIN S j-function

PARITY OF THE COEFFICIENTS OF KLEIN S j-function PARITY OF THE COEFFICIENTS OF KLEIN S j-function CLAUDIA ALFES Abstract. Klein s j-function is one of the most fundamental modular functions in number theory. However, not much is known about the parity

More information

SMOOTH FINITE ABELIAN UNIFORMIZATIONS OF PROJECTIVE SPACES AND CALABI-YAU ORBIFOLDS

SMOOTH FINITE ABELIAN UNIFORMIZATIONS OF PROJECTIVE SPACES AND CALABI-YAU ORBIFOLDS SMOOTH FINITE ABELIAN UNIFORMIZATIONS OF PROJECTIVE SPACES AND CALABI-YAU ORBIFOLDS A. MUHAMMED ULUDAĞ Dedicated to Mehmet Çiftçi Abstract. We give a classification of smooth complex manifolds with a finite

More information

FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS

FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ

More information

Lecture 4: Examples of automorphic forms on the unitary group U(3)

Lecture 4: Examples of automorphic forms on the unitary group U(3) Lecture 4: Examples of automorphic forms on the unitary group U(3) Lassina Dembélé Department of Mathematics University of Calgary August 9, 2006 Motivation The main goal of this talk is to show how one

More information

arxiv: v3 [math.ag] 25 Sep 2012

arxiv: v3 [math.ag] 25 Sep 2012 NORMAL FUNCTIONS, PICARD-FUCHS EQUATIONS, AND ELLIPTIC FIBRATIONS ON K3 SURFACES XI CHEN, CHARLES DORAN, MATT KERR, AND JAMES D. LEWIS arxiv:1108.2223v3 [math.ag] 25 Sep 2012 Abstract. Using Gauss-Manin

More information

On the BCOV Conjecture

On the BCOV Conjecture Department of Mathematics University of California, Irvine December 14, 2007 Mirror Symmetry The objects to study By Mirror Symmetry, for any CY threefold, there should be another CY threefold X, called

More information

Ramanujan s first letter to Hardy: 5 + = 1 + e 2π 1 + e 4π 1 +

Ramanujan s first letter to Hardy: 5 + = 1 + e 2π 1 + e 4π 1 + Ramanujan s first letter to Hardy: e 2π/ + + 1 = 1 + e 2π 2 2 1 + e 4π 1 + e π/ 1 e π = 2 2 1 + 1 + e 2π 1 + Hardy: [These formulas ] defeated me completely. I had never seen anything in the least like

More information

Homological mirror symmetry via families of Lagrangians

Homological mirror symmetry via families of Lagrangians Homological mirror symmetry via families of Lagrangians String-Math 2018 Mohammed Abouzaid Columbia University June 17, 2018 Mirror symmetry Three facets of mirror symmetry: 1 Enumerative: GW invariants

More information

Magdalena Larfors

Magdalena Larfors Uppsala University, Dept. of Theoretical Physics Based on D. Chialva, U. Danielsson, N. Johansson, M.L. and M. Vonk, hep-th/0710.0620 U. Danielsson, N. Johansson and M.L., hep-th/0612222 2008-01-18 String

More information

The tangent space to an enumerative problem

The tangent space to an enumerative problem The tangent space to an enumerative problem Prakash Belkale Department of Mathematics University of North Carolina at Chapel Hill North Carolina, USA belkale@email.unc.edu ICM, Hyderabad 2010. Enumerative

More information

Abelian Varieties and Complex Tori: A Tale of Correspondence

Abelian Varieties and Complex Tori: A Tale of Correspondence Abelian Varieties and Complex Tori: A Tale of Correspondence Nate Bushek March 12, 2012 Introduction: This is an expository presentation on an area of personal interest, not expertise. I will use results

More information

HYPERKÄHLER MANIFOLDS

HYPERKÄHLER MANIFOLDS HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly

More information

arxiv:alg-geom/ v1 3 Sep 1994

arxiv:alg-geom/ v1 3 Sep 1994 DYNKIN GRAPHS AND TRIANGLE SINGULARITIES Tohsuke Urabe arxiv:alg-geom/9409002v1 3 Sep 1994 Department of Mathematics Tokyo Metropolitan University Minami-Ohsawa 1-1, Hachioji-shi Tokyo 192-03 Japan (E-mail:

More information

Hecke Operators, Zeta Functions and the Satake map

Hecke Operators, Zeta Functions and the Satake map Hecke Operators, Zeta Functions and the Satake map Thomas R. Shemanske December 19, 2003 Abstract Taking advantage of the Satake isomorphism, we define (n + 1) families of Hecke operators t n k (pl ) for

More information

Periods, Galois theory and particle physics

Periods, Galois theory and particle physics Periods, Galois theory and particle physics Francis Brown All Souls College, Oxford Gergen Lectures, 21st-24th March 2016 1 / 29 Reminders We are interested in periods I = γ ω where ω is a regular algebraic

More information

When 2 and 3 are invertible in A, L A is the scheme

When 2 and 3 are invertible in A, L A is the scheme 8 RICHARD HAIN AND MAKOTO MATSUMOTO 4. Moduli Spaces of Elliptic Curves Suppose that r and n are non-negative integers satisfying r + n > 0. Denote the moduli stack over Spec Z of smooth elliptic curves

More information

IN POSITIVE CHARACTERISTICS: 3. Modular varieties with Hecke symmetries. 7. Foliation and a conjecture of Oort

IN POSITIVE CHARACTERISTICS: 3. Modular varieties with Hecke symmetries. 7. Foliation and a conjecture of Oort FINE STRUCTURES OF MODULI SPACES IN POSITIVE CHARACTERISTICS: HECKE SYMMETRIES AND OORT FOLIATION 1. Elliptic curves and their moduli 2. Moduli of abelian varieties 3. Modular varieties with Hecke symmetries

More information

FOURIER COEFFICIENTS OF VECTOR-VALUED MODULAR FORMS OF DIMENSION 2

FOURIER COEFFICIENTS OF VECTOR-VALUED MODULAR FORMS OF DIMENSION 2 FOURIER COEFFICIENTS OF VECTOR-VALUED MODULAR FORMS OF DIMENSION 2 CAMERON FRANC AND GEOFFREY MASON Abstract. We prove the following Theorem. Suppose that F = (f 1, f 2 ) is a 2-dimensional vector-valued

More information

Delzant s Garden. A one-hour tour to symplectic toric geometry

Delzant s Garden. A one-hour tour to symplectic toric geometry Delzant s Garden A one-hour tour to symplectic toric geometry Tour Guide: Zuoqin Wang Travel Plan: The earth America MIT Main building Math. dept. The moon Toric world Symplectic toric Delzant s theorem

More information

Geometry of moduli spaces

Geometry of moduli spaces Geometry of moduli spaces 20. November 2009 1 / 45 (1) Examples: C: compact Riemann surface C = P 1 (C) = C { } (Riemann sphere) E = C / Z + Zτ (torus, elliptic curve) 2 / 45 (2) Theorem (Riemann existence

More information

A TRANSCENDENTAL METHOD IN ALGEBRAIC GEOMETRY

A TRANSCENDENTAL METHOD IN ALGEBRAIC GEOMETRY Actes, Congrès intern, math., 1970. Tome 1, p. 113 à 119. A TRANSCENDENTAL METHOD IN ALGEBRAIC GEOMETRY by PHILLIP A. GRIFFITHS 1. Introduction and an example from curves. It is well known that the basic

More information

An introduction to arithmetic groups. Lizhen Ji CMS, Zhejiang University Hangzhou , China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109

An introduction to arithmetic groups. Lizhen Ji CMS, Zhejiang University Hangzhou , China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109 An introduction to arithmetic groups Lizhen Ji CMS, Zhejiang University Hangzhou 310027, China & Dept of Math, Univ of Michigan Ann Arbor, MI 48109 June 27, 2006 Plan. 1. Examples of arithmetic groups

More information

Rational points on elliptic curves. cycles on modular varieties

Rational points on elliptic curves. cycles on modular varieties Rational points on elliptic curves and cycles on modular varieties Mathematics Colloquium January 2009 TIFR, Mumbai Henri Darmon McGill University http://www.math.mcgill.ca/darmon /slides/slides.html Elliptic

More information

THE UNIT GROUP OF A REAL QUADRATIC FIELD

THE UNIT GROUP OF A REAL QUADRATIC FIELD THE UNIT GROUP OF A REAL QUADRATIC FIELD While the unit group of an imaginary quadratic field is very simple the unit group of a real quadratic field has nontrivial structure Its study involves some geometry

More information

On the Langlands Program

On the Langlands Program On the Langlands Program John Rognes Colloquium talk, May 4th 2018 The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2018 to Robert P. Langlands of the Institute for

More information

Hodge Structures. October 8, A few examples of symmetric spaces

Hodge Structures. October 8, A few examples of symmetric spaces Hodge Structures October 8, 2013 1 A few examples of symmetric spaces The upper half-plane H is the quotient of SL 2 (R) by its maximal compact subgroup SO(2). More generally, Siegel upper-half space H

More information

TORIC REDUCTION AND TROPICAL GEOMETRY A.

TORIC REDUCTION AND TROPICAL GEOMETRY A. Mathematisches Institut, Seminars, (Y. Tschinkel, ed.), p. 109 115 Universität Göttingen, 2004-05 TORIC REDUCTION AND TROPICAL GEOMETRY A. Szenes ME Institute of Mathematics, Geometry Department, Egry

More information

Logarithmic functional and reciprocity laws

Logarithmic functional and reciprocity laws Contemporary Mathematics Volume 00, 1997 Logarithmic functional and reciprocity laws Askold Khovanskii Abstract. In this paper, we give a short survey of results related to the reciprocity laws over the

More information

Class Numbers, Continued Fractions, and the Hilbert Modular Group

Class Numbers, Continued Fractions, and the Hilbert Modular Group Class Numbers, Continued Fractions, and the Hilbert Modular Group Jordan Schettler University of California, Santa Barbara 11/8/2013 Outline 1 Motivation 2 The Hilbert Modular Group 3 Resolution of the

More information

Period Domains. Carlson. June 24, 2010

Period Domains. Carlson. June 24, 2010 Period Domains Carlson June 4, 00 Carlson - Period Domains Period domains are parameter spaces for marked Hodge structures. We call Γ\D the period space, which is a parameter space of isomorphism classes

More information

LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS

LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS WEIMIN CHEN, UMASS, SPRING 07 1. Basic elements of J-holomorphic curve theory Let (M, ω) be a symplectic manifold of dimension 2n, and let J J (M, ω) be

More information

Part II. Riemann Surfaces. Year

Part II. Riemann Surfaces. Year Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 96 Paper 2, Section II 23F State the uniformisation theorem. List without proof the Riemann surfaces which are uniformised

More information

CATEGORICAL ASPECTS OF ALGEBRAIC GEOMETRY IN MIRROR SYMMETRY ABSTRACTS

CATEGORICAL ASPECTS OF ALGEBRAIC GEOMETRY IN MIRROR SYMMETRY ABSTRACTS CATEGORICAL ASPECTS OF ALGEBRAIC GEOMETRY IN MIRROR SYMMETRY Alexei Bondal (Steklov/RIMS) Derived categories of complex-analytic manifolds Alexender Kuznetsov (Steklov) Categorical resolutions of singularities

More information

Cusp forms and the Eichler-Shimura relation

Cusp forms and the Eichler-Shimura relation Cusp forms and the Eichler-Shimura relation September 9, 2013 In the last lecture we observed that the family of modular curves X 0 (N) has a model over the rationals. In this lecture we use this fact

More information

COMPUTING ARITHMETIC PICARD-FUCHS EQUATIONS JEROEN SIJSLING

COMPUTING ARITHMETIC PICARD-FUCHS EQUATIONS JEROEN SIJSLING COMPUTING ARITHMETIC PICARD-FUCHS EQUATIONS JEROEN SIJSLING These are the extended notes for a talk given at the Fields Institute on August 24th, 2011, about my thesis work with Frits Beukers at the Universiteit

More information

p-adic gauge theory in number theory K. Fujiwara Nagoya Tokyo, September 2007

p-adic gauge theory in number theory K. Fujiwara Nagoya Tokyo, September 2007 p-adic gauge theory in number theory K. Fujiwara Nagoya Tokyo, September 2007 Dirichlet s theorem F : totally real field, O F : the integer ring, [F : Q] = d. p: a prime number. Dirichlet s unit theorem:

More information

From K3 Surfaces to Noncongruence Modular Forms. Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives ICERM October 19, 2015

From K3 Surfaces to Noncongruence Modular Forms. Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives ICERM October 19, 2015 From K3 Surfaces to Noncongruence Modular Forms Explicit Methods for Modularity of K3 Surfaces and Other Higher Weight Motives ICERM October 19, 2015 Winnie Li Pennsylvania State University 1 A K3 surface

More information

Borcherds proof of the moonshine conjecture

Borcherds proof of the moonshine conjecture Borcherds proof of the moonshine conjecture pjc, after V. Nikulin Abstract These CSG notes contain a condensed account of a talk by V. Nikulin in the London algebra Colloquium on 24 May 2001. None of the

More information

FANO VARIETIES AND EPW SEXTICS

FANO VARIETIES AND EPW SEXTICS FNO VRIETIES ND EPW SEXTICS OLIVIER DEBRRE bstract. We explore a connection between smooth projective varieties X of dimension n with an ample divisor H such that H n = 10 and K X = (n 2)H and a class

More information

A Motivated Introduction to Modular Forms

A Motivated Introduction to Modular Forms May 3, 2006 Outline of talk: I. Motivating questions II. Ramanujan s τ function III. Theta Series IV. Congruent Number Problem V. My Research Old Questions... What can you say about the coefficients of

More information

Plane quartics and. Dedicated to Professor S. Koizumi for his 70th birthday. by Tetsuji Shioda

Plane quartics and. Dedicated to Professor S. Koizumi for his 70th birthday. by Tetsuji Shioda Plane quartics and Mordell-Weil lattices of type E 7 Dedicated to Professor S. Koizumi for his 70th birthday by Tetsuji Shioda Department of Mathematics, Rikkyo University Nishi-Ikebukuro,Tokyo 171, Japan

More information

Some algebraic number theory and the reciprocity map

Some algebraic number theory and the reciprocity map Some algebraic number theory and the reciprocity map Ervin Thiagalingam September 28, 2015 Motivation In Weinstein s paper, the main problem is to find a rule (reciprocity law) for when an irreducible

More information

Toric Varieties. Madeline Brandt. April 26, 2017

Toric Varieties. Madeline Brandt. April 26, 2017 Toric Varieties Madeline Brandt April 26, 2017 Last week we saw that we can define normal toric varieties from the data of a fan in a lattice. Today I will review this idea, and also explain how they can

More information

Height zeta functions

Height zeta functions Geometry Mathematisches Institut July 19, 2006 Geometric background Let X P n be a smooth variety over C. Its main invariants are: Picard group Pic(X ) and Néron-Severi group NS(X ) Λ eff (X ), Λ ample

More information

FAKE PROJECTIVE SPACES AND FAKE TORI

FAKE PROJECTIVE SPACES AND FAKE TORI FAKE PROJECTIVE SPACES AND FAKE TORI OLIVIER DEBARRE Abstract. Hirzebruch and Kodaira proved in 1957 that when n is odd, any compact Kähler manifold X which is homeomorphic to P n is isomorphic to P n.

More information

Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties arxiv:alg-geom/ v1 30 Jul 1993

Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties arxiv:alg-geom/ v1 30 Jul 1993 Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varieties arxiv:alg-geom/9307010v1 30 Jul 1993 Victor V. Batyrev FB Mathematik, Universität-GH-Essen,

More information