Counting problems in Number Theory and Physicsw3.impa.br/~hossein/talks/talks/IMPA-CBPF-2011.pdf · Counting problems in Number Theory and Physics Hossein Movasati IMPA, Instituto

Counting problems in Number Theory andPhysics

Hossein Movasati

IMPA, Instituto de Matemática Pura e Aplicada, Brazilwww.impa.br/∼hossein/

Encontro conjunto CBPF-IMPA, 2011

A documentary on string theory by Brian Greene:

Counting

Fibonacci numbers:

Fn = Fn−1 + Fn−2, F0 = 0, F1 = 1.

The Fibonacci numbers give the number of pairs of rabbits nmonths after a single pair begins breeding (and newly bornbunnies are assumed to begin breeding when they are twomonths old), as first described by Leonardo of Pisa (also knownas Fibonacci) in his book Liber Abaci (from mathworld).The generating function for Fibonacci numbers:

F =∞∑

n=0

Fnqn =q

1− q − q2

From this we get

Fn =αn − βn

α− β, α, β =

12

(1±√

5).

limn→∞

Fn

Fn−1= lim

n→∞F

1n

n =12

(1 +√

5)

CountingFibonacci numbers:

Fn = Fn−1 + Fn−2, F0 = 0, F1 = 1.

The Fibonacci numbers give the number of pairs of rabbits nmonths after a single pair begins breeding (and newly bornbunnies are assumed to begin breeding when they are twomonths old), as first described by Leonardo of Pisa (also knownas Fibonacci) in his book Liber Abaci (from mathworld).The generating function for Fibonacci numbers:

F =∞∑

n=0

Fnqn =q

1− q − q2

From this we get

Fn =αn − βn

α− β, α, β =

12

(1±√

5).

limn→∞

Fn

Fn−1= lim

n→∞F

1n

n =12

(1 +√

5)

CountingFibonacci numbers:

Fn = Fn−1 + Fn−2, F0 = 0, F1 = 1.

The Fibonacci numbers give the number of pairs of rabbits nmonths after a single pair begins breeding (and newly bornbunnies are assumed to begin breeding when they are twomonths old), as first described by Leonardo of Pisa (also knownas Fibonacci) in his book Liber Abaci (from mathworld).

The generating function for Fibonacci numbers:

F =∞∑

n=0

Fnqn =q

1− q − q2

From this we get

Fn =αn − βn

α− β, α, β =

12

(1±√

5).

limn→∞

Fn

Fn−1= lim

n→∞F

1n

n =12

(1 +√

5)

CountingFibonacci numbers:

Fn = Fn−1 + Fn−2, F0 = 0, F1 = 1.

The Fibonacci numbers give the number of pairs of rabbits nmonths after a single pair begins breeding (and newly bornbunnies are assumed to begin breeding when they are twomonths old), as first described by Leonardo of Pisa (also knownas Fibonacci) in his book Liber Abaci (from mathworld).The generating function for Fibonacci numbers:

F =∞∑

n=0

Fnqn =q

1− q − q2

From this we get

Fn =αn − βn

α− β, α, β =

12

(1±√

5).

limn→∞

Fn

Fn−1= lim

n→∞F

1n

n =12

(1 +√

5)

CountingFibonacci numbers:

Fn = Fn−1 + Fn−2, F0 = 0, F1 = 1.

The Fibonacci numbers give the number of pairs of rabbits nmonths after a single pair begins breeding (and newly bornbunnies are assumed to begin breeding when they are twomonths old), as first described by Leonardo of Pisa (also knownas Fibonacci) in his book Liber Abaci (from mathworld).The generating function for Fibonacci numbers:

F =∞∑

n=0

Fnqn =q

1− q − q2

From this we get

Fn =αn − βn

α− β, α, β =

12

(1±√

5).

limn→∞

Fn

Fn−1= lim

n→∞F

1n

n =12

(1 +√

5)

Eisenstein series

E2k = 1 + (−1)k 4kBk

∑n≥1

σ2k−1(n)qn,

k = 1,2,3, q ∈ C, |q| < 1,

B1 =16, B2 =

130, B3 =

142, . . . ,

σi(n) :=∑d |n

d i ,

Eisenstein series

E2k = 1 + (−1)k 4kBk

∑n≥1

σ2k−1(n)qn,

k = 1,2,3, q ∈ C, |q| < 1,

B1 =16, B2 =

130, B3 =

142, . . . ,

σi(n) :=∑d |n

d i ,

Eisenstein series

E2k = 1 + (−1)k 4kBk

∑n≥1

σ2k−1(n)qn,

k = 1,2,3, q ∈ C, |q| < 1,

B1 =16, B2 =

130, B3 =

142, . . . ,

σi(n) :=∑d |n

d i ,

1. The theory of modular forms over SL(2,Z): homogeneouspolynomials of the ring

C[E4,E6], deg(E4) = 4, deg(E6) = 6.

2. The theory of quasi/differential modular forms overSL(2,Z): homogeneous polynomials of the ring

C[E2,E4,E6], deg(E2) = 2, deg(E4) = 4, deg(E6) = 6

3. In general, a (quasi) modular form over a subgroup ofSL(2,Z) of finite rank is an element in the algebraic closureof C(E2,E4,E6).

1. The theory of modular forms over SL(2,Z): homogeneouspolynomials of the ring

C[E4,E6], deg(E4) = 4, deg(E6) = 6.

2. The theory of quasi/differential modular forms overSL(2,Z): homogeneous polynomials of the ring

C[E2,E4,E6], deg(E2) = 2, deg(E4) = 4, deg(E6) = 6

3. In general, a (quasi) modular form over a subgroup ofSL(2,Z) of finite rank is an element in the algebraic closureof C(E2,E4,E6).

1. The theory of modular forms over SL(2,Z): homogeneouspolynomials of the ring

C[E4,E6], deg(E4) = 4, deg(E6) = 6.

2. The theory of quasi/differential modular forms overSL(2,Z): homogeneous polynomials of the ring

C[E2,E4,E6], deg(E2) = 2, deg(E4) = 4, deg(E6) = 6

3. In general, a (quasi) modular form over a subgroup ofSL(2,Z) of finite rank is an element in the algebraic closureof C(E2,E4,E6).

Monstrous moonshine conjecture,

The j-function

j = 1728E3

4

E34 − E2

6=

q−1 + 744 + 196884q + 21493760q2 + 864299970q3 + · · · .

196884 = 196883 + 1

MacKay 1978: 196883 is the number of dimensions in whichthe Monster group can be most simply represented. J.H.Conway, S.P. Norton 1979: Monstrous moonshine conjectureR. Borcherds 1992: Solved

Monstrous moonshine conjecture,

The j-function

j = 1728E3

4

E34 − E2

6=

q−1 + 744 + 196884q + 21493760q2 + 864299970q3 + · · · .

196884 = 196883 + 1

MacKay 1978: 196883 is the number of dimensions in whichthe Monster group can be most simply represented. J.H.Conway, S.P. Norton 1979: Monstrous moonshine conjectureR. Borcherds 1992: Solved

Monstrous moonshine conjecture,

The j-function

j = 1728E3

4

E34 − E2

6=

q−1 + 744 + 196884q + 21493760q2 + 864299970q3 + · · · .

196884 = 196883 + 1

MacKay 1978: 196883 is the number of dimensions in whichthe Monster group can be most simply represented. J.H.Conway, S.P. Norton 1979: Monstrous moonshine conjectureR. Borcherds 1992: Solved

Monstrous moonshine conjecture,

The j-function

j = 1728E3

4

E34 − E2

6=

q−1 + 744 + 196884q + 21493760q2 + 864299970q3 + · · · .

196884 = 196883 + 1

MacKay 1978: 196883 is the number of dimensions in whichthe Monster group can be most simply represented.

J.H.Conway, S.P. Norton 1979: Monstrous moonshine conjectureR. Borcherds 1992: Solved

Monstrous moonshine conjecture,

The j-function

j = 1728E3

4

E34 − E2

6=

q−1 + 744 + 196884q + 21493760q2 + 864299970q3 + · · · .

196884 = 196883 + 1

MacKay 1978: 196883 is the number of dimensions in whichthe Monster group can be most simply represented. J.H.Conway, S.P. Norton 1979: Monstrous moonshine conjecture

R. Borcherds 1992: Solved

Monstrous moonshine conjecture,

The j-function

j = 1728E3

4

E34 − E2

6=

q−1 + 744 + 196884q + 21493760q2 + 864299970q3 + · · · .

196884 = 196883 + 1

MacKay 1978: 196883 is the number of dimensions in whichthe Monster group can be most simply represented. J.H.Conway, S.P. Norton 1979: Monstrous moonshine conjectureR. Borcherds 1992: Solved

Monster group

I If normal subgroups of a group G are {1} and G then G iscalled a simple group.

I In the classification of all finite simple groups there appear26 sporadic groups. The Monster group M is the largest ofthe sporadic groups.

|M| = 246 ·320 ·59 ·76 ·112 ·133 ·17·19·23·29·31·41·47·59·71

I Dimensions of irreducible representations of M: 1, 196883,

21296876, 842609326, 18538750076, 19360062527, 293553734298, 3879214937598, 36173193327999,

125510727015275, 190292345709543, 222879856734249, 1044868466775133, 1109944460516150,

2374124840062976.

Monster group

I If normal subgroups of a group G are {1} and G then G iscalled a simple group.

I In the classification of all finite simple groups there appear26 sporadic groups. The Monster group M is the largest ofthe sporadic groups.

|M| = 246 ·320 ·59 ·76 ·112 ·133 ·17·19·23·29·31·41·47·59·71

I Dimensions of irreducible representations of M: 1, 196883,

21296876, 842609326, 18538750076, 19360062527, 293553734298, 3879214937598, 36173193327999,

125510727015275, 190292345709543, 222879856734249, 1044868466775133, 1109944460516150,

2374124840062976.

Monster group

I If normal subgroups of a group G are {1} and G then G iscalled a simple group.

I In the classification of all finite simple groups there appear26 sporadic groups. The Monster group M is the largest ofthe sporadic groups.

|M| = 246 ·320 ·59 ·76 ·112 ·133 ·17·19·23·29·31·41·47·59·71

I Dimensions of irreducible representations of M: 1, 196883,

21296876, 842609326, 18538750076, 19360062527, 293553734298, 3879214937598, 36173193327999,

125510727015275, 190292345709543, 222879856734249, 1044868466775133, 1109944460516150,

2374124840062976.

Monster group

I If normal subgroups of a group G are {1} and G then G iscalled a simple group.

I In the classification of all finite simple groups there appear26 sporadic groups. The Monster group M is the largest ofthe sporadic groups.

|M| = 246 ·320 ·59 ·76 ·112 ·133 ·17·19·23·29·31·41·47·59·71

I Dimensions of irreducible representations of M: 1, 196883,

21296876, 842609326, 18538750076, 19360062527, 293553734298, 3879214937598, 36173193327999,

125510727015275, 190292345709543, 222879856734249, 1044868466775133, 1109944460516150,

2374124840062976.

Modularity theorem

An elliptic curve over Z:

E : y2 = 4x3 − a2x − a3,

a2,a3 ∈ Z,∆ := a32 − 27a2

3 6= 0.

Let p be a prime and Np be the number of solutions of Eworking modulo p

ap(E) := p − Np

A version of modularity theorem says that there is modular formof weight 2 associated to some congruence group, namelyf =

∑∞n=0 anqn, such that

ap = ap(E)

for all primes p 6 |∆.

Modularity theorem

An elliptic curve over Z:

E : y2 = 4x3 − a2x − a3,

a2,a3 ∈ Z,∆ := a32 − 27a2

3 6= 0.

Let p be a prime and Np be the number of solutions of Eworking modulo p

ap(E) := p − Np

A version of modularity theorem says that there is modular formof weight 2 associated to some congruence group, namelyf =

∑∞n=0 anqn, such that

ap = ap(E)

for all primes p 6 |∆.

Modularity theorem

An elliptic curve over Z:

E : y2 = 4x3 − a2x − a3,

a2,a3 ∈ Z,∆ := a32 − 27a2

3 6= 0.

Let p be a prime and Np be the number of solutions of Eworking modulo p

ap(E) := p − Np

A version of modularity theorem says that there is modular formof weight 2 associated to some congruence group, namelyf =

∑∞n=0 anqn, such that

ap = ap(E)

for all primes p 6 |∆.

Modularity theorem

An elliptic curve over Z:

E : y2 = 4x3 − a2x − a3,

a2,a3 ∈ Z,∆ := a32 − 27a2

3 6= 0.

Let p be a prime and Np be the number of solutions of Eworking modulo p

ap(E) := p − Np

A version of modularity theorem says that there is modular formof weight 2 associated to some congruence group, namelyf =

∑∞n=0 anqn, such that

ap = ap(E)

for all primes p 6 |∆.

Example:E : y2 + y = x3 − x2

The corresponding modular form is

η(q)2η(q11)12 = q−2q2−q3 +2q4 +q5 +2q6−2q7−2q9−2q10

+q11 − 2q12 + 4q13 + · · · ,

where

η(q) = ∆1

24 = q1

24

∞∏n=1

(1− qn)

is the Dedekind eta function and

∆ =1

1728(E3

4 − E26 ).

Example:E : y2 + y = x3 − x2

The corresponding modular form is

η(q)2η(q11)12 = q−2q2−q3 +2q4 +q5 +2q6−2q7−2q9−2q10

+q11 − 2q12 + 4q13 + · · · ,

where

η(q) = ∆1

24 = q1

24

∞∏n=1

(1− qn)

is the Dedekind eta function and

∆ =1

1728(E3

4 − E26 ).

Example:E : y2 + y = x3 − x2

The corresponding modular form is

η(q)2η(q11)12 = q−2q2−q3 +2q4 +q5 +2q6−2q7−2q9−2q10

+q11 − 2q12 + 4q13 + · · · ,

where

η(q) = ∆1

24 = q1

24

∞∏n=1

(1− qn)

is the Dedekind eta function and

∆ =1

1728(E3

4 − E26 ).

1. Taniyama-Shimura conjecture.

2. A. Weils proved for semistable elliptic curves: This was anessential part of the proof of the Fermat last theorem

3. R. Taylor, C. Breuil, B. Conrad, F.Diamond: Modulartiytheorem

1. Taniyama-Shimura conjecture.2. A. Weils proved for semistable elliptic curves: This was an

essential part of the proof of the Fermat last theorem

3. R. Taylor, C. Breuil, B. Conrad, F.Diamond: Modulartiytheorem

1. Taniyama-Shimura conjecture.2. A. Weils proved for semistable elliptic curves: This was an

essential part of the proof of the Fermat last theorem3. R. Taylor, C. Breuil, B. Conrad, F.Diamond: Modulartiy

theorem

Counting holomorphic maps from curves to anelliptic curve

1. Let E be a complex elliptic curve and let p1, . . . ,p2g−2 bedistinct points of E , where g ≥ 2.

2. The set Xg(d) of equivalence classes of holomorphic mapsφ : C → E of degree d from compact connected smoothcomplex curves C to E , which have only one doubleramification point over each point pi ∈ E and no otherramification points, is finite. By the Hurwitz formula thegenus of C is equal to g.

3. Define

Fg :=∑d≥1

∑[φ]∈Xd (d)

1|Aut (φ) |

qd .

4. R. Dijkgraaf, M. Douglas, D. Zagier, M. Kaneko:

Fg ∈ Q[E2,E4,E6].

Counting holomorphic maps from curves to anelliptic curve

1. Let E be a complex elliptic curve and let p1, . . . ,p2g−2 bedistinct points of E , where g ≥ 2.

2. The set Xg(d) of equivalence classes of holomorphic mapsφ : C → E of degree d from compact connected smoothcomplex curves C to E , which have only one doubleramification point over each point pi ∈ E and no otherramification points, is finite. By the Hurwitz formula thegenus of C is equal to g.

3. Define

Fg :=∑d≥1

∑[φ]∈Xd (d)

1|Aut (φ) |

qd .

4. R. Dijkgraaf, M. Douglas, D. Zagier, M. Kaneko:

Fg ∈ Q[E2,E4,E6].

Counting holomorphic maps from curves to anelliptic curve

1. Let E be a complex elliptic curve and let p1, . . . ,p2g−2 bedistinct points of E , where g ≥ 2.

2. The set Xg(d) of equivalence classes of holomorphic mapsφ : C → E of degree d from compact connected smoothcomplex curves C to E , which have only one doubleramification point over each point pi ∈ E and no otherramification points, is finite. By the Hurwitz formula thegenus of C is equal to g.

3. Define

Fg :=∑d≥1

∑[φ]∈Xd (d)

1|Aut (φ) |

qd .

4. R. Dijkgraaf, M. Douglas, D. Zagier, M. Kaneko:

Fg ∈ Q[E2,E4,E6].

Counting holomorphic maps from curves to anelliptic curve

1. Let E be a complex elliptic curve and let p1, . . . ,p2g−2 bedistinct points of E , where g ≥ 2.

2. The set Xg(d) of equivalence classes of holomorphic mapsφ : C → E of degree d from compact connected smoothcomplex curves C to E , which have only one doubleramification point over each point pi ∈ E and no otherramification points, is finite. By the Hurwitz formula thegenus of C is equal to g.

3. Define

Fg :=∑d≥1

∑[φ]∈Xd (d)

1|Aut (φ) |

qd .

4. R. Dijkgraaf, M. Douglas, D. Zagier, M. Kaneko:

Fg ∈ Q[E2,E4,E6].

Counting holomorphic maps from curves to anelliptic curve

1. Let E be a complex elliptic curve and let p1, . . . ,p2g−2 bedistinct points of E , where g ≥ 2.

2. The set Xg(d) of equivalence classes of holomorphic mapsφ : C → E of degree d from compact connected smoothcomplex curves C to E , which have only one doubleramification point over each point pi ∈ E and no otherramification points, is finite. By the Hurwitz formula thegenus of C is equal to g.

3. Define

Fg :=∑d≥1

∑[φ]∈Xd (d)

1|Aut (φ) |

qd .

4. R. Dijkgraaf, M. Douglas, D. Zagier, M. Kaneko:

Fg ∈ Q[E2,E4,E6].

For instance,

F2(q) =1

103680(10E3

2 − 6E2E4 − 4E6),

F3(q) =1

35831808(−6E6

2 + 15E42 E4 − 12E2

2 E34 + 7E3

4 +

4E32 E6 − 12E2E4E6 + 4E2

6 ).

Number of rational curves on K 3 surfaces

1. K3 surface: simply connected+trivial canonical bundle2. In an (n + g)-dimensional linear system |L| the generic

fiber is of genus n + g.3. Let Nn(g) be the number of geometric genus g curves in|L| passing through g points (so that n is the number ofnodes).

4. Yau-Zaslow (1996), Beauville(1999), Göttsche(1994),Bryan-Leung(1999): For generic (X ,L) we have

∞∑n=0

Nn(g)qn = (−124

∂E2

∂q)g 1728q

E34 − E2

6.

Number of rational curves on K 3 surfaces

1. K3 surface: simply connected+trivial canonical bundle

2. In an (n + g)-dimensional linear system |L| the genericfiber is of genus n + g.

3. Let Nn(g) be the number of geometric genus g curves in|L| passing through g points (so that n is the number ofnodes).

4. Yau-Zaslow (1996), Beauville(1999), Göttsche(1994),Bryan-Leung(1999): For generic (X ,L) we have

∞∑n=0

Nn(g)qn = (−124

∂E2

∂q)g 1728q

E34 − E2

6.

Number of rational curves on K 3 surfaces

1. K3 surface: simply connected+trivial canonical bundle2. In an (n + g)-dimensional linear system |L| the generic

fiber is of genus n + g.

3. Let Nn(g) be the number of geometric genus g curves in|L| passing through g points (so that n is the number ofnodes).

4. Yau-Zaslow (1996), Beauville(1999), Göttsche(1994),Bryan-Leung(1999): For generic (X ,L) we have

∞∑n=0

Nn(g)qn = (−124

∂E2

∂q)g 1728q

E34 − E2

6.

Number of rational curves on K 3 surfaces

1. K3 surface: simply connected+trivial canonical bundle2. In an (n + g)-dimensional linear system |L| the generic

fiber is of genus n + g.3. Let Nn(g) be the number of geometric genus g curves in|L| passing through g points (so that n is the number ofnodes).

4. Yau-Zaslow (1996), Beauville(1999), Göttsche(1994),Bryan-Leung(1999): For generic (X ,L) we have

∞∑n=0

Nn(g)qn = (−124

∂E2

∂q)g 1728q

E34 − E2

6.

Number of rational curves on K 3 surfaces

1. K3 surface: simply connected+trivial canonical bundle2. In an (n + g)-dimensional linear system |L| the generic

fiber is of genus n + g.3. Let Nn(g) be the number of geometric genus g curves in|L| passing through g points (so that n is the number ofnodes).

4. Yau-Zaslow (1996), Beauville(1999), Göttsche(1994),Bryan-Leung(1999): For generic (X ,L) we have

∞∑n=0

Nn(g)qn = (−124

∂E2

∂q)g 1728q

E34 − E2

6.

For the case g = 0 (counting rational curves):

∞∑n=0

Nn(0)qn =1728q

E34 − E2

6= 1 + 24q + 324q2 + 3200q3

+25650q4 + 176256q5 + 1073720q6 + · · ·

(by definition N0(0) = 1). For instance, a smooth quadric X inP3 is K 3 and for such a generic X the number of planes tangentto X in three points is 3200.

For the case g = 0 (counting rational curves):

∞∑n=0

Nn(0)qn =1728q

E34 − E2

6= 1 + 24q + 324q2 + 3200q3

+25650q4 + 176256q5 + 1073720q6 + · · ·

(by definition N0(0) = 1).

For instance, a smooth quadric X inP3 is K 3 and for such a generic X the number of planes tangentto X in three points is 3200.

For the case g = 0 (counting rational curves):

∞∑n=0

Nn(0)qn =1728q

E34 − E2

6= 1 + 24q + 324q2 + 3200q3

+25650q4 + 176256q5 + 1073720q6 + · · ·

(by definition N0(0) = 1). For instance, a smooth quadric X inP3 is K 3 and for such a generic X the number of planes tangentto X in three points is 3200.

Beyond classical modular forms and ellipticcurves?!?

Clemens conjecture:

There exits a finite number of rational curves of a fixed degreein a generic quintic in P4.

Clemens conjecture:

There exits a finite number of rational curves of a fixed degreein a generic quintic in P4.

Candelas, de la Ossa, Green, Parkes (1991), in the frameworkof mirror symmetry calculates a quantity Y called Yukawacoupling:

Y = 5 + 2875q

1− q+ 609250 · 23 q2

1− q2 +

317206375 · 33 q3

1− q3 + · · ·+ ndd3 qd

1− qd + · · ·

They claimed that nd is the number of rational curves of degreed in a generic quintic in P4.

Candelas, de la Ossa, Green, Parkes (1991), in the frameworkof mirror symmetry calculates a quantity Y called Yukawacoupling:

Y = 5 + 2875q

1− q+ 609250 · 23 q2

1− q2 +

317206375 · 33 q3

1− q3 + · · ·+ ndd3 qd

1− qd + · · ·

They claimed that nd is the number of rational curves of degreed in a generic quintic in P4.

Candelas, de la Ossa, Green, Parkes (1991), in the frameworkof mirror symmetry calculates a quantity Y called Yukawacoupling:

Y = 5 + 2875q

1− q+ 609250 · 23 q2

1− q2 +

317206375 · 33 q3

1− q3 + · · ·+ ndd3 qd

1− qd + · · ·

They claimed that nd is the number of rational curves of degreed in a generic quintic in P4.

The main ingredient of the theory of modular forms attached tomirror quintic Calabi-Yau varieties is a particular solution of thedifferential equation Ra1:

t0 = 1t5

(3750t50 + t0t3 − 625t4)

t1 = 1t5

(−390625t60 + 3125t4

0 t1 + 390625t0t4 + t1t3)

t2 = 1t5

(−5859375t70 − 625t5

0 t1 + 6250t40 t2 + 5859375t2

0 t4 + 625t1t4 + 2t2t3)

t3 = 1t5

(−9765625t80 − 625t5

0 t2 + 9375t40 t3 + 9765625t3

0 t4 + 625t2t4 + 3t23 )

t4 = 1t5

(15625t40 t4 + 5t3t4)

t5 = 1t5

(−625t50 t6 + 9375t4

0 t5 + 2t3t5 + 625t4t6)

t6 = 1t5

(9375t40 t6 − 3125t3

0 t5 − 2t2t5 + 3t3t6)

The main ingredient of the theory of modular forms attached tomirror quintic Calabi-Yau varieties is a particular solution of thedifferential equation Ra1:

t0 = 1t5

(3750t50 + t0t3 − 625t4)

t1 = 1t5

(−390625t60 + 3125t4

0 t1 + 390625t0t4 + t1t3)

t2 = 1t5

(−5859375t70 − 625t5

0 t1 + 6250t40 t2 + 5859375t2

0 t4 + 625t1t4 + 2t2t3)

t3 = 1t5

(−9765625t80 − 625t5

0 t2 + 9375t40 t3 + 9765625t3

0 t4 + 625t2t4 + 3t23 )

t4 = 1t5

(15625t40 t4 + 5t3t4)

t5 = 1t5

(−625t50 t6 + 9375t4

0 t5 + 2t3t5 + 625t4t6)

t6 = 1t5

(9375t40 t6 − 3125t3

0 t5 − 2t2t5 + 3t3t6)

q-expansion:

Taket = 5q

∂t∂q

and write each ti as a formal power series in q, ti =∑∞

n=0 ti,nqn

and substitute in the above differential equation. We see that itdetermines all the coefficients ti,n uniquely with the initialvalues:

t0,0 =15, t0,1 = 24, t4,0 = 0, t5,0 6= 0

q-expansion:

Taket = 5q

∂t∂q

and write each ti as a formal power series in q, ti =∑∞

n=0 ti,nqn

and substitute in the above differential equation. We see that itdetermines all the coefficients ti,n uniquely with the initialvalues:

t0,0 =15, t0,1 = 24, t4,0 = 0, t5,0 6= 0

124 t0 = 1

120 + q + 175q2 + 117625q3 + 111784375q4 +

1269581056265 + 160715581780591q6 +218874699262438350q7 + 314179164066791400375q8 +469234842365062637809375q9+722875994952367766020759550q10 + O(q11)

−1750 t1 = 1

30 + 3q + 930q2 + 566375q3 + 526770000q4 +

592132503858q5 + 745012928951258q6 +1010500474677945510q7 + 1446287695614437271000q8 +2155340222852696651995625q9+3314709711759484241245738380q10 + O(q11)

−150 t2 = 7

10 + 107q + 50390q2 + 29007975q3 +

26014527500q4 + 28743493632402q5+35790559257796542q6 + 48205845153859479030q7 +68647453506412345755300q8+101912303698877609329100625q9 +156263153250677320910779548340q10 + O(q11)

−15 t3 = 6

5 + 71q + 188330q2 + 100324275q3 +

86097977000q4 + 93009679497426q5+114266677893238146q6 + 152527823430305901510q7 +215812408812642816943200q8+318839967257572460805706125q9 +487033977592346076373921829980q10 + O(q11)

−t4 =0− 1q1 + 170q2 + 41475q3 + 32183000q4 + 32678171250q5 +38612049889554q6 + 50189141795178390q7 +69660564113425804800q8 + 101431587084669781525125q9

153189681044166218779637500q10 + O(q11)

1125 t5 = −1

125 + 15q + 938q2 + 587805q3 + 525369650q4 +

577718296190q5 + 716515428667010q6 +962043316960737646q7 + 1366589803139580122090q8 +2024744003173189934886225q9+3099476777084481347731347688q10 + O(q11)

t6 = · · ·

ConjectureAll q-expansions of

124

t0 −1

120,−1750

t1 −130,−150

t2 −7

10,−15

t3 −65, −t4,

1125

t5 +1

125, · · ·

have positive integer coefficients.

1125 t5 = −1

125 + 15q + 938q2 + 587805q3 + 525369650q4 +

577718296190q5 + 716515428667010q6 +962043316960737646q7 + 1366589803139580122090q8 +2024744003173189934886225q9+3099476777084481347731347688q10 + O(q11)

t6 = · · ·

ConjectureAll q-expansions of

124

t0 −1

120,−1750

t1 −130,−150

t2 −7

10,−15

t3 −65, −t4,

1125

t5 +1

125, · · ·

have positive integer coefficients.

1. We get the Yukawa coupling calculated by Candelas, de laOssa, Green, Parkes (1991):

−511(t4 − t50 )2

t35

= 5 + 2875q

1− q+ 609250 · 23 q2

1− q2 +

317206375 · 33 q3

1− q3 + · · ·+ ndd3 qd

1− qd + · · ·

2. Using a result of Yamaguchi and Yau (1994) we get alsogenus g topological string partition functions.

1. We get the Yukawa coupling calculated by Candelas, de laOssa, Green, Parkes (1991):

−511(t4 − t50 )2

t35

= 5 + 2875q

1− q+ 609250 · 23 q2

1− q2 +

317206375 · 33 q3

1− q3 + · · ·+ ndd3 qd

1− qd + · · ·

2. Using a result of Yamaguchi and Yau (1994) we get alsogenus g topological string partition functions.

Darboux-Halphen-Ramanujan:

Ra2 :

t1 = t2

1 −1

12 t2t2 = 4t1t2 − 6t3t3 = 6t1t3 − 1

3 t22

t = 12q∂

∂q

Write each ti as a formal power series in q, ti =∑∞

n=0 ti,nqn andsubstitute in the above differential equation. We see that itdetermines all the coefficients ti,n uniquely with the initialvalues:

t1,0 = 1, t1,1 = −24

Darboux-Halphen-Ramanujan:

Ra2 :

t1 = t2

1 −1

12 t2t2 = 4t1t2 − 6t3t3 = 6t1t3 − 1

3 t22

t = 12q∂

∂q

Write each ti as a formal power series in q, ti =∑∞

n=0 ti,nqn andsubstitute in the above differential equation. We see that itdetermines all the coefficients ti,n uniquely with the initialvalues:

t1,0 = 1, t1,1 = −24

In fact we have explicit formulas for ti . They are the well-knownEisenstein series:

ti = aiE2i = ai

(1 + bi

∞∑d=1

d2i−1 qd

1− qd

), i = 1,2,3, (1)

where

(b1,b2,b3) = (−24,240,−504), (a1,a2,a3) = (1,12,8).

Mirror quintic Calabi-Yau varieties:

Let Wψ be the variety obtained by the resolution of singularitiesof the following quotient:

{x ∈ P4 | Q = 0}/G,

Q = x50 + x5

1 + x52 + x5

3 + x54 − 5ψx0x1x2x3x4

where G is the group

G := {(ζ1, ζ2, · · · , ζ5) | ζ5i = 1, ζ1ζ2ζ3ζ4ζ5 = 1}

acting in a canonical way.

Mirror quintic Calabi-Yau varieties:

Let Wψ be the variety obtained by the resolution of singularitiesof the following quotient:

{x ∈ P4 | Q = 0}/G,

Q = x50 + x5

1 + x52 + x5

3 + x54 − 5ψx0x1x2x3x4

where G is the group

G := {(ζ1, ζ2, · · · , ζ5) | ζ5i = 1, ζ1ζ2ζ3ζ4ζ5 = 1}

acting in a canonical way.