Differential equations concerned with mirror symmetry of toric K3 hypersurfaces with arithmetic properties. Atsuhira Nagano (University of Tokyo)
|
|
- Gavin Farmer
- 5 years ago
- Views:
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 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 informationPeriods 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 informationHodge 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 informationMirror 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 informationIntroduction 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 informationMonodromy 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 informationUrsula 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)
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 informationArithmetic 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 informationThe 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 informationChern 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 informationThe 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 informationCrash 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 informationModular 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 informationHomological 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 informationGENERIC 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 informationToric 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 informationarxiv: 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 informationConstructing 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 informationEquations 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 informationCounting 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 informationIntroduction 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 informationEXERCISES 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 informationHypergeometric 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 informationThe 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 informationOn 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 informationa 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 informationA 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 informationRiemannian 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 informationarxiv: 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 informationMirror 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
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 informationTopics 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 informationThe 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 informationCounting 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 informationMirror 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 informationWhat 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 informationCongruence 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 informationCohomology 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 informationRigidity, 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 informationCONSIDERATION 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 informationBlack 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 informationSYMPLECTIC 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 informationK3 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 informationTopological 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 informationTakao 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 informationGromov-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 informationKleine 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 informationModular-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 informationCombinatorics 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 informationGeneralized 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 informationOutline 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 informationLecture 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 informationTopics 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 informationPARITY 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 informationSMOOTH 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 informationFORMAL 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 informationLecture 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 informationarxiv: 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 informationOn 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 informationRamanujan 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 informationHomological 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 informationMagdalena 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 informationThe 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 informationAbelian 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 informationHYPERKÄHLER 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 informationarxiv: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 informationHecke 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 informationPeriods, 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 informationWhen 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 informationIN 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 informationFOURIER 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 informationDelzant 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 informationGeometry 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 informationA 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 informationAn 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 informationRational 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 informationTHE 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 informationOn 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 informationHodge 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 informationTORIC 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 informationLogarithmic 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 informationClass 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 informationPeriod 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 informationLECTURE 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 informationPart 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 informationCATEGORICAL 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 informationCusp 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 informationCOMPUTING 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 informationp-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 informationFrom 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 informationBorcherds 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 informationFANO 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 informationA 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 informationPlane 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 informationSome 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 informationToric 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 informationHeight 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 informationFAKE 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 informationGeneralized 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