Ana Mar a Botero - uni-regensburg.deboa15169/botero... · 2016-06-26 · Singular metrics, convex...

Singular metrics, convex bodies and multiple zeta values

Ana Marıa Botero

ECCO 2016: 5th Encuentro Colombiano de Combinatoria

Universidad de Antioquia and Universidad Nacional sede Medellın

June 21 2016

Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values

The computation of ζ(2)

The Riemann zeta function is defined as the series

ζ(s) :=∞∑n=1

1

ns,

where s ∈ C with Re > 1.Goal: Compute ζ(2) = 1 + 1

22 + 132 + · · ·+ 1

n2 + · · · as a sum of triangle areas.

Start by rewriting (for n ∈ Z>0)

1

n2= −

1

n·e−nx

n

∣∣∣∣∞0

=

∫ ∞0

e−nx

ndx .

Hence, the term 1n2 equals the area

y

x

1n

Figure: 1n2 as the area below an exponential curve


Therefore,

ζ(2) = 1 +1

22+

1

32+ · · ·+

1

n2+ · · ·

can be visualized geometrically as follows

y

x

e−x + e−2x

2+ e−3x

3+ e−4x

4+ e−5x

5+ · · ·

Figure: ζ(2) as a sum of areas of curved triangles


Using the Taylor expansion for the logarithm function

∞∑n=1

e−nx

n= −log(1− e−x ),

we obatin

ζ(2) =∞∑n=1

1

n2=∞∑n=1

∫ ∞0

e−nx

ndx =

∫ ∞0

∞∑n=1

e−nx

ndx

= −∫ ∞

0log(1− e−x )dx .

Hence, ζ(2) equals the area of the region A determined by the curve C

y

x

C : e−y + e−x = 1

for 0 ≤ x < ∞.

A

Figure: ζ(2) as the area of the region A


To compute the area of A, we make the change of variables

(α, β) 7→ (x , y) = φ(α, β) :=

(log

(sin(α+ β)

sin(α)

), log

(sin(α+ β)

sin(β)

))depicted below

y

x

log(2)

log(2)

y

xπ2

π3

π2

π3

B2

B1

φ

Figure: ζ(2) as the area of two triangles B1 and B2

ζ(2) = area(A) =

∫∫A

dx dy =

∫∫B

∣∣∣∣det

(∂φ

∂α,∂φ

∂β

) ∣∣∣∣dα dβ

= area(B) = area(B1) + area(B2) = 2 ·1

2·π

2·π

3=π2

6.


ζ(2) as the volume of a moduli space

Consider A1, the moduli space of elliptic curves. This means that the points of A1 arein bijection with the isomorphism classes of elliptic curves over C. We have

H/ SL2(Z) ' A1

z 7→ C/ (Z⊕ zZ) .

A fundamental domain for this quotient space is

F :=

{z = x + iy ∈ H

∣∣∣− 1

2< x ≤

1

2, x2 + y2 ≥ 1

}.

y

x

F

Figure: Fundamental domain F for SL2(Z) acting on H


Hence, the volume of A1 with respect to the normalized hyperbolic metric dµ can becomputed as

vol (A1) =

∫A1

dµ =1

4π

∫ 1/2

−1/2

∫ ∞√

1−x2

dx ∧ dy

y2=

1

4π

∫ 1/2

−1/2

1√

1− x2dx

=1

4π· arcsin(x)

∣∣∣∣1/2

−1/2

=1

12=

1

2π2ζ(2).

Using the functional equation of the zeta function

π−s/2Γ( s

2

)ζ(s) = π−(1−s)/2Γ

(1− s

2

)ζ(1− s),

the formula of the volume of A1 can be rewritten in the simplified form

vol(A1) = −ζ(−1).


Summary

We have computed the special value ζ(2) by relating it to the area of two triangles.

We have computed the volume of the moduli space A1 and seen it as a specialvalue at a negative integer.

Question: What about universal objects? Can we relate their volume to values of moregeneral Riemann zeta functions?


Countability of the rational numbers

Consider the set of rational numbers

Q :={ a

b

∣∣∣ (a, b) ∈ Z× Z>0 , (a, b) = 1}.

The mediant of two rational numbers ab

and cd

is given by ab⊕ c

d:= a+c

b+d.

01

01

01

01

13

13

12

12

12

23

23

11

11

11

11

32

32

21

21

21

31

31

10

10

10

10

14

25

35

34

43

53

52

41

Figure: A way of enumerating the rationals: The Stern–Brocot tree


The computation of ζMT(2, 2; 2)

The Mordell–Tornheim zeta function is defined as the doubles series

ζMT(s2, s2; s3) :=∞∑

m,n=1

1

ms1ns2 (m + n)s3,

where s1, s2, s3 ∈ C with Re(s1) ≥ Re(s2) ≥ Re(s3) > 1.We aim at computing the value

ζMT(2, 2; 2) :=∞∑

m,n=1

1

m2n2(m + n)2,

We will do this by computing the volume of a convex set.


Consider the concave function φ : R≥0 → R , (a, b) 7→ aba+b

. In the order given by the

Stern–Brocot tree, we assign values on the positive primitive vectors in Z2 dictated by φ.

13

12

23

11

32

21

31

14

25

35

34

43

53

52

41

Figure: The Stern–Brocot tree

12

12

23

23

Figure: Values on primitive rays and the dualpicture


In the limit, we get

· · ·

· · ·

Figure: In the limit


Looking at the complement, we see

Note,

1

22=

1

1212(1 + 1)2

1

62=

1

1222(1 + 2)2

1

122=

1

1232(1 + 3)2

1

302=

1

3222(3 + 2)2

1

702=

1

2252(2 + 5)2


We get

2 · vol (∆) = 1−∑

m,n∈Z> 0(m,n)=1

1

m2n2(m + n)2,

where

√x +√

y = 1(0, 1)

(1, 0)

∆ =

On the other hand, we can calculate

vol (∆) =1

2−∫ 1

0(1−

√x)2dx =

1

2− 16 =

1

3.

Hence,

A :=∑

m,n∈Z> 0(m,n)=1

1

m2n2(m + n)2= 1− 2 ·

1

3=

1

3.

Finally, we end up with

ζMT(2, 2; 2) = ζ(6) · A =π6

945·

1

3=

π6

2835.


ζMT(2, 2; 2) as the volume of the universal object of a moduli space

Consider now the universal elliptic curve

Θi,v

Pv

A1

X1

π

H

Figure: The compactified universal elliptic curve

Remark: The volume of X1 is a numerical invariant defined in terms of arithmeticintersection numbers. And these in turn are defined in terms of a singular metric.Problem: The type of singularities of the metric along the boundary X1 \ X1.


Approach (B. ’15): Encode the singularity type of the metric in a toric b-divisor D.

⇒ Metric residues = volume of a convex set (degree of toric b-divisor).

We havevol(X1

)= L2 − R = L2 − D2 = L2 − 2 · vol(∆),

where L is a distinguished line bundle on X1 which carries a natural metric and

√x +√

y = 1(0, 1)

(1, 0)

∆ =


Summary

1 We have computed the special value ζ(2, 2; 2) by relating it to the volume of aconvex set.

2 We have computed the volume of the universal moduli space X1 by seeing theneeded correction term as the volume of the above convex set.

Question: What about universal moduli spaces of abelian varieties of higher dimension?(Our approach works for 2 but what about 1?)


Muchas gracias!


Ana Mar a Botero - uni-regensburg.deboa15169/botero... · 2016-06-26 · Singular metrics, convex...

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Transcript of Ana Mar a Botero - uni-regensburg.deboa15169/botero... · 2016-06-26 · Singular metrics, convex...