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
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values
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
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values
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
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values
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.
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values
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
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values
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).
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values
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?
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values
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
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values
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.
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values
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
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values
In the limit, we get
· · ·
· · ·
Figure: In the limit
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values
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
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values
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.
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values
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.
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values
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)
∆ =
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values
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?)
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values
Muchas gracias!
Ana Marıa Botero Singular metrics, convex bodies and multiple zeta values