Download Gergen Lecture I

Periods, Galois theory and particle physics
Francis Brown
All Souls College, Oxford
Gergen Lectures,
21st-24th March 2016
1 / 27
Overview
Periods are a countable class of complex numbers defined by
algebraic integrals. They contain all algebraic numbers, and
include constants such as π, elliptic integrals, and values of
zeta functions.
They provide a unifying framework in which one can think
about many problems in mathematics and physics.
Periods give an elementary way to understand, and compute,
subtle concepts in algebraic geometry, e.g., the theory of
motives. They form a bridge between geometry and
arithmetic; pure mathematics and high-energy physics.
The final goal of these lectures is to provide evidence for a
Galois theory of periods. There should be a large
(pro)-algebraic group acting on the space of periods. This is
the first step towards a classification of these numbers.
2 / 27
Overview
Periods are a countable class of complex numbers defined by
algebraic integrals. They contain all algebraic numbers, and
include constants such as π, elliptic integrals, and values of
zeta functions.
They provide a unifying framework in which one can think
about many problems in mathematics and physics.
Periods give an elementary way to understand, and compute,
subtle concepts in algebraic geometry, e.g., the theory of
motives. They form a bridge between geometry and
arithmetic; pure mathematics and high-energy physics.
The final goal of these lectures is to provide evidence for a
Galois theory of periods. There should be a large
(pro)-algebraic group acting on the space of periods. This is
the first step towards a classification of these numbers.
2 / 27
Overview
Periods are a countable class of complex numbers defined by
algebraic integrals. They contain all algebraic numbers, and
include constants such as π, elliptic integrals, and values of
zeta functions.
They provide a unifying framework in which one can think
about many problems in mathematics and physics.
Periods give an elementary way to understand, and compute,
subtle concepts in algebraic geometry, e.g., the theory of
motives. They form a bridge between geometry and
arithmetic; pure mathematics and high-energy physics.
The final goal of these lectures is to provide evidence for a
Galois theory of periods. There should be a large
(pro)-algebraic group acting on the space of periods. This is
the first step towards a classification of these numbers.
2 / 27
Overview
Periods are a countable class of complex numbers defined by
algebraic integrals. They contain all algebraic numbers, and
include constants such as π, elliptic integrals, and values of
zeta functions.
They provide a unifying framework in which one can think
about many problems in mathematics and physics.
Periods give an elementary way to understand, and compute,
subtle concepts in algebraic geometry, e.g., the theory of
motives. They form a bridge between geometry and
arithmetic; pure mathematics and high-energy physics.
The final goal of these lectures is to provide evidence for a
Galois theory of periods. There should be a large
(pro)-algebraic group acting on the space of periods. This is
the first step towards a classification of these numbers.
2 / 27
Overview
Periods are a countable class of complex numbers defined by
algebraic integrals. They contain all algebraic numbers, and
include constants such as π, elliptic integrals, and values of
zeta functions.
They provide a unifying framework in which one can think
about many problems in mathematics and physics.
Periods give an elementary way to understand, and compute,
subtle concepts in algebraic geometry, e.g., the theory of
motives. They form a bridge between geometry and
arithmetic; pure mathematics and high-energy physics.
The final goal of these lectures is to provide evidence for a
Galois theory of periods. There should be a large
(pro)-algebraic group acting on the space of periods. This is
the first step towards a classification of these numbers.
2 / 27
Elementary definition (Kontsevich-Zagier)
A period is a complex number whose real and imaginary parts are
convergent integrals of rational differential forms over domains
defined by polynomial inequalities (all with rational coefficients),
Z
P
I =
dx1 . . . dxn
σ Q
where P, Q, ∈ Q[x1 , . . . , xn ], and σ is a finite union of sets
{f1 , . . . , fN ≥ 0} with fi ∈ Q[x1 , . . . , xn ]. Examples:
√
2=
Z
x 2 ≤2
π=
Z
dxdy
x 2 +y 2 ≤1
,
dx
2
log 2 =
Z
1≤x≤2
dx
x
The first example generalizes: every algebraic number is a period.
3 / 27
Elementary definition (Kontsevich-Zagier)
A period is a complex number whose real and imaginary parts are
convergent integrals of rational differential forms over domains
defined by polynomial inequalities (all with rational coefficients),
Z
P
I =
dx1 . . . dxn
σ Q
where P, Q, ∈ Q[x1 , . . . , xn ], and σ is a finite union of sets
{f1 , . . . , fN ≥ 0} with fi ∈ Q[x1 , . . . , xn ]. Examples:
√
2=
Z
x 2 ≤2
π=
Z
dxdy
x 2 +y 2 ≤1
,
dx
2
log 2 =
Z
1≤x≤2
dx
x
The first example generalizes: every algebraic number is a period.
3 / 27
Ring of periods
A product of periods is still a period, so the set of periods P forms
a ring. It lies between algebraic numbers Q and C:
Q⊆Q⊂P ⊂C
We have greater flexibility if we allow integrals
Z
ω
I =
σ
where ω is a regular n-form on an algebraic variety X of dimension
n, and σ a smooth chain with boundary contained in Z (C), where
Z ⊂ X and ω defined over the rationals. These are periods.
Since P is countable, almost every complex number is not a
period. Conjecturally, numbers such as
Z ∞
Z
e −x
e −x
x
−
dx
e dx or γ =
e=
e −x − 1
x
x≤1
0
are not expected to be periods. One can try to keep enlarging P to
include more general classes of integrals.
4 / 27
Ring of periods
A product of periods is still a period, so the set of periods P forms
a ring. It lies between algebraic numbers Q and C:
Q⊆Q⊂P ⊂C
We have greater flexibility if we allow integrals
Z
ω
I =
σ
where ω is a regular n-form on an algebraic variety X of dimension
n, and σ a smooth chain with boundary contained in Z (C), where
Z ⊂ X and ω defined over the rationals. These are periods.
Since P is countable, almost every complex number is not a
period. Conjecturally, numbers such as
Z ∞
Z
e −x
e −x
x
e dx or γ =
e=
−
dx
e −x − 1
x
x≤1
0
are not expected to be periods. One can try to keep enlarging P to
include more general classes of integrals.
4 / 27
Ring of periods
A product of periods is still a period, so the set of periods P forms
a ring. It lies between algebraic numbers Q and C:
Q⊆Q⊂P ⊂C
We have greater flexibility if we allow integrals
Z
ω
I =
σ
where ω is a regular n-form on an algebraic variety X of dimension
n, and σ a smooth chain with boundary contained in Z (C), where
Z ⊂ X and ω defined over the rationals. These are periods.
Since P is countable, almost every complex number is not a
period. Conjecturally, numbers such as
Z ∞
Z
e −x
e −x
x
−
dx
e dx or γ =
e=
e −x − 1
x
x≤1
0
are not expected to be periods. One can try to keep enlarging P to
include more general classes of integrals.
4 / 27
Relations
The following relations hold between periods.
Additivity:
Z
Z
ω1 + ω2 =
σ
ω1 +
Z
ω2
σ
σ
and likewise, additivity in the domain of integration σ.
Algebraic changes of variables:
Z
Z
ω=
f ∗ω
f −1 (σ)
σ
where f is defined by polynomial equations with rational
coefficients.
Stokes’ theorem:
Z
dω =
σ
Z
ω
∂σ
5 / 27
Relations
The following relations hold between periods.
Additivity:
Z
Z
ω1 + ω2 =
σ
ω1 +
Z
ω2
σ
σ
and likewise, additivity in the domain of integration σ.
Algebraic changes of variables:
Z
Z
ω=
f ∗ω
f −1 (σ)
σ
where f is defined by polynomial equations with rational
coefficients.
Stokes’ theorem:
Z
dω =
σ
Z
ω
∂σ
5 / 27
Relations
The following relations hold between periods.
Additivity:
Z
Z
ω1 + ω2 =
σ
ω1 +
Z
ω2
σ
σ
and likewise, additivity in the domain of integration σ.
Algebraic changes of variables:
Z
Z
ω=
f ∗ω
f −1 (σ)
σ
where f is defined by polynomial equations with rational
coefficients.
Stokes’ theorem:
Z
dω =
σ
Z
ω
∂σ
5 / 27
Relations
The following relations hold between periods.
Additivity:
Z
Z
ω1 + ω2 =
σ
ω1 +
Z
ω2
σ
σ
and likewise, additivity in the domain of integration σ.
Algebraic changes of variables:
Z
Z
ω=
f ∗ω
f −1 (σ)
σ
where f is defined by polynomial equations with rational
coefficients.
Stokes’ theorem:
Z
dω =
σ
Z
ω
∂σ
5 / 27
Kontsevich and Zagier’s conjecture
Folklore conjecture
All identities between periods can be proved using these operations.
Hopelessly difficult! Even in simple examples this can be hard.
There is no known algorithm to determine if two periods are equal.
Example:
π=
Z
∞
−∞
dx
1 + x2
is also
Z
dxdy
x 2 +y 2 ≤1
Note that in practice, periods come with one or more natural
integral representations. We are not necessarily interested in every
possible integral representation.
6 / 27
Kontsevich and Zagier’s conjecture
Folklore conjecture
All identities between periods can be proved using these operations.
Hopelessly difficult! Even in simple examples this can be hard.
There is no known algorithm to determine if two periods are equal.
Example:
π=
Z
∞
−∞
dx
1 + x2
is also
Z
dxdy
x 2 +y 2 ≤1
Note that in practice, periods come with one or more natural
integral representations. We are not necessarily interested in every
possible integral representation.
6 / 27
Kontsevich and Zagier’s conjecture
Folklore conjecture
All identities between periods can be proved using these operations.
Hopelessly difficult! Even in simple examples this can be hard.
There is no known algorithm to determine if two periods are equal.
Example:
π=
Z
∞
−∞
dx
1 + x2
is also
Z
dxdy
x 2 +y 2 ≤1
Note that in practice, periods come with one or more natural
integral representations. We are not necessarily interested in every
possible integral representation.
6 / 27
Where we are heading
Algebraic numbers can be understood through invariants such as
the degree, height, and so on. Galois theory gives a way to study
algebraic numbers via group theory.
Can we do the same for periods?
Unfortunately we run into difficult transcendence questions. Our
final goal will be to set up a working Galois theory for a slightly
modified notion of periods which circumvents these issues. This
theory will produce a way to categorize periods into different types.
For example, the ring P of periods has a filtration called the
weight which is ≥ 0. The periods of weight 0 should be exactly the
algebraic numbers, and π should be a period of weight 2.
These notions are very intuitive: for example, we should not expect
π −1 to be a period, because then it would have weight −2.
7 / 27
Where we are heading
Algebraic numbers can be understood through invariants such as
the degree, height, and so on. Galois theory gives a way to study
algebraic numbers via group theory.
Can we do the same for periods?
Unfortunately we run into difficult transcendence questions. Our
final goal will be to set up a working Galois theory for a slightly
modified notion of periods which circumvents these issues. This
theory will produce a way to categorize periods into different types.
For example, the ring P of periods has a filtration called the
weight which is ≥ 0. The periods of weight 0 should be exactly the
algebraic numbers, and π should be a period of weight 2.
These notions are very intuitive: for example, we should not expect
π −1 to be a period, because then it would have weight −2.
7 / 27
Where we are heading
Algebraic numbers can be understood through invariants such as
the degree, height, and so on. Galois theory gives a way to study
algebraic numbers via group theory.
Can we do the same for periods?
Unfortunately we run into difficult transcendence questions. Our
final goal will be to set up a working Galois theory for a slightly
modified notion of periods which circumvents these issues. This
theory will produce a way to categorize periods into different types.
For example, the ring P of periods has a filtration called the
weight which is ≥ 0. The periods of weight 0 should be exactly the
algebraic numbers, and π should be a period of weight 2.
These notions are very intuitive: for example, we should not expect
π −1 to be a period, because then it would have weight −2.
7 / 27
Where we are heading
Algebraic numbers can be understood through invariants such as
the degree, height, and so on. Galois theory gives a way to study
algebraic numbers via group theory.
Can we do the same for periods?
Unfortunately we run into difficult transcendence questions. Our
final goal will be to set up a working Galois theory for a slightly
modified notion of periods which circumvents these issues. This
theory will produce a way to categorize periods into different types.
For example, the ring P of periods has a filtration called the
weight which is ≥ 0. The periods of weight 0 should be exactly the
algebraic numbers, and π should be a period of weight 2.
These notions are very intuitive: for example, we should not expect
π −1 to be a period, because then it would have weight −2.
7 / 27
Where we are heading
Algebraic numbers can be understood through invariants such as
the degree, height, and so on. Galois theory gives a way to study
algebraic numbers via group theory.
Can we do the same for periods?
Unfortunately we run into difficult transcendence questions. Our
final goal will be to set up a working Galois theory for a slightly
modified notion of periods which circumvents these issues. This
theory will produce a way to categorize periods into different types.
For example, the ring P of periods has a filtration called the
weight which is ≥ 0. The periods of weight 0 should be exactly the
algebraic numbers, and π should be a period of weight 2.
These notions are very intuitive: for example, we should not expect
π −1 to be a period, because then it would have weight −2.
7 / 27
Families of periods
‘A constant is a function that is taking a break’. What is the class
of functions whose values are periods?
Define a family of periods to be an integral
Z
P
dx1 . . . dxn
I =
σ Q
where P, Q ∈ Q[x1 , . . . , xn , t1 , . . . , tm ] now depend on parameters
(t1 , . . . , tm ) ∈ Cm , and σ is a finite union of algebraic sets
{(x1 , . . . , xn ) ∈ Rn : f1 (x), . . . , fN (x) ≥ 0} where fi also depend on
ti , which converges in some region of Cm . Extend it via analytic
continuation to a multi-valued function on some open of Cm .
Example:
log(t) =
Z
1≤x≤t
dx
x
Its analytic continuation is a multi-valued function of t ∈ C× . It is
‘ambiguous’ up to addition of 2πi Z.
8 / 27
Families of periods
‘A constant is a function that is taking a break’. What is the class
of functions whose values are periods?
Define a family of periods to be an integral
Z
P
dx1 . . . dxn
I =
σ Q
where P, Q ∈ Q[x1 , . . . , xn , t1 , . . . , tm ] now depend on parameters
(t1 , . . . , tm ) ∈ Cm , and σ is a finite union of algebraic sets
{(x1 , . . . , xn ) ∈ Rn : f1 (x), . . . , fN (x) ≥ 0} where fi also depend on
ti , which converges in some region of Cm . Extend it via analytic
continuation to a multi-valued function on some open of Cm .
Example:
log(t) =
Z
1≤x≤t
dx
x
Its analytic continuation is a multi-valued function of t ∈ C× . It is
‘ambiguous’ up to addition of 2πi Z.
8 / 27
Families of periods
‘A constant is a function that is taking a break’. What is the class
of functions whose values are periods?
Define a family of periods to be an integral
Z
P
dx1 . . . dxn
I =
σ Q
where P, Q ∈ Q[x1 , . . . , xn , t1 , . . . , tm ] now depend on parameters
(t1 , . . . , tm ) ∈ Cm , and σ is a finite union of algebraic sets
{(x1 , . . . , xn ) ∈ Rn : f1 (x), . . . , fN (x) ≥ 0} where fi also depend on
ti , which converges in some region of Cm . Extend it via analytic
continuation to a multi-valued function on some open of Cm .
Example:
log(t) =
Z
1≤x≤t
dx
x
Its analytic continuation is a multi-valued function of t ∈ C× . It is
‘ambiguous’ up to addition of 2πi Z.
8 / 27
Example 1: Elliptic integrals
The word ‘period’ comes from the classical problem of determining
the periods of planetary orbits, or the period T of a simple
pendulum (time to complete one swing).
ℓ
θ
..
Its equation of motion is given by Newton’s law: θ + Gℓ sin θ = 0.
By integrating, one gets an expression for T as an elliptic integral
Z 1
dx
p
T =4
(1 − x 2 )(1 − ρ2 x 2 )
0
9 / 27
Example 1: Elliptic integrals
The word ‘period’ comes from the classical problem of determining
the periods of planetary orbits, or the period T of a simple
pendulum (time to complete one swing).
ℓ
θ
..
Its equation of motion is given by Newton’s law: θ + Gℓ sin θ = 0.
By integrating, one gets an expression for T as an elliptic integral
Z 1
dx
p
T =4
(1 − x 2 )(1 − ρ2 x 2 )
0
9 / 27
Example 1: Elliptic integrals
The word ‘period’ comes from the classical problem of determining
the periods of planetary orbits, or the period T of a simple
pendulum (time to complete one swing).
ℓ
θ
..
Its equation of motion is given by Newton’s law: θ + Gℓ sin θ = 0.
By integrating, one gets an expression for T as an elliptic integral
Z 1
dx
p
T =4
(1 − x 2 )(1 − ρ2 x 2 )
0
9 / 27
Elliptic integrals
The constant 0 < ρ < 1 depends on the initial conditions. Let’s
take ρ ∈ Q. The solutions to the equation
y 2 = (1 − x 2 )(1 − ρ2 x 2 )
with x, y ∈ C defines a complex torus (elliptic curve E ).
γ
The period can be written in the form
Z
T = ω
γ
where ω = 4dx/y is a holomorphic 1-form on E . Studying line
integrals on curves leads to the theory of Jacobians, Abelian
varieties, and a huge swathe of mathematics.
10 / 27
Elliptic integrals
The constant 0 < ρ < 1 depends on the initial conditions. Let’s
take ρ ∈ Q. The solutions to the equation
y 2 = (1 − x 2 )(1 − ρ2 x 2 )
with x, y ∈ C defines a complex torus (elliptic curve E ).
γ
The period can be written in the form
Z
T = ω
γ
where ω = 4dx/y is a holomorphic 1-form on E . Studying line
integrals on curves leads to the theory of Jacobians, Abelian
varieties, and a huge swathe of mathematics.
10 / 27
Elliptic integrals
The constant 0 < ρ < 1 depends on the initial conditions. Let’s
take ρ ∈ Q. The solutions to the equation
y 2 = (1 − x 2 )(1 − ρ2 x 2 )
with x, y ∈ C defines a complex torus (elliptic curve E ).
γ
The period can be written in the form
Z
T = ω
γ
where ω = 4dx/y is a holomorphic 1-form on E . Studying line
integrals on curves leads to the theory of Jacobians, Abelian
varieties, and a huge swathe of mathematics.
10 / 27
Example 2. Values of the Riemann zeta function
In the 17th century, Mengoli posed the Basel problem: what is
X 1
=?
k2
k≥1
This was solved by Euler in the 1740’s. Write
X 1
ζ(n) =
n≥2.
kn
k≥1
The even zeta values are rational multiples of powers of π:
ζ(2) =
π2
6
,
ζ(4) =
π4
90
,
ζ(6) =
π6
945
,... .
Theorem (Euler)
Let n ≥ 1, and let B2n be the 2nth Bernoulli number. Then
ζ(2n) = −
(2πi )2n B2n
2
(2n)!
11 / 27
Example 2. Values of the Riemann zeta function
In the 17th century, Mengoli posed the Basel problem: what is
X 1
=?
k2
k≥1
This was solved by Euler in the 1740’s. Write
X 1
n≥2.
ζ(n) =
kn
k≥1
The even zeta values are rational multiples of powers of π:
ζ(2) =
π2
6
,
ζ(4) =
π4
90
,
ζ(6) =
π6
945
,... .
Theorem (Euler)
Let n ≥ 1, and let B2n be the 2nth Bernoulli number. Then
ζ(2n) = −
(2πi )2n B2n
2
(2n)!
11 / 27
Example 2. Values of the Riemann zeta function
In the 17th century, Mengoli posed the Basel problem: what is
X 1
=?
k2
k≥1
This was solved by Euler in the 1740’s. Write
X 1
n≥2.
ζ(n) =
kn
k≥1
The even zeta values are rational multiples of powers of π:
ζ(2) =
π2
6
,
ζ(4) =
π4
90
,
ζ(6) =
π6
945
,... .
Theorem (Euler)
Let n ≥ 1, and let B2n be the 2nth Bernoulli number. Then
ζ(2n) = −
(2πi )2n B2n
2
(2n)!
11 / 27
Example 2. Values of the Riemann zeta function
In the 17th century, Mengoli posed the Basel problem: what is
X 1
=?
k2
k≥1
This was solved by Euler in the 1740’s. Write
X 1
n≥2.
ζ(n) =
kn
k≥1
The even zeta values are rational multiples of powers of π:
ζ(2) =
π2
6
,
ζ(4) =
π4
90
,
ζ(6) =
π6
945
,... .
Theorem (Euler)
Let n ≥ 1, and let B2n be the 2nth Bernoulli number. Then
ζ(2n) = −
(2πi )2n B2n
2
(2n)!
11 / 27
Integral representation
Zeta values are periods. If n ≥ 2 then
Z
dt1 dt2
dtn
ζ(n) =
...
tn
0≤t1 ≤t2 ≤···≤tn ≤1 1 − t1 t2
To see this, do a geometric series expansion in t1 :
Z t2 X
Z
X t m+1
dt1
2
t1m dt1 =
=
m+1
0
0≤t1 ≤t2 1 − t1
m≥0
m≥0
and proceed by integrating out t2 , t3 , . . . in turn:
Z
X t m+1 dt2
X
dtn
1
2
...
= ... =
m + 1 t2
tn
(m + 1)n
0≤t2 ≤···≤tn ≤1
m≥0
m≥0
Therefore Euler’s theorem is an identity in the ring of periods.
2
One can prove that ζ(2) = π6 using the standard relations for
periods (tricky). I know of no proof of Euler’s theorem for all
n ≥ 1 along these lines. The standard relations are ill-adapted for
proving infinite families of identities.
12 / 27
Integral representation
Zeta values are periods. If n ≥ 2 then
Z
dt1 dt2
dtn
ζ(n) =
...
tn
0≤t1 ≤t2 ≤···≤tn ≤1 1 − t1 t2
To see this, do a geometric series expansion in t1 :
Z t2 X
Z
X t m+1
dt1
2
t1m dt1 =
=
m+1
0
0≤t1 ≤t2 1 − t1
m≥0
m≥0
and proceed by integrating out t2 , t3 , . . . in turn:
Z
X t m+1 dt2
X
dtn
1
2
...
= ... =
m + 1 t2
tn
(m + 1)n
0≤t2 ≤···≤tn ≤1
m≥0
m≥0
Therefore Euler’s theorem is an identity in the ring of periods.
2
One can prove that ζ(2) = π6 using the standard relations for
periods (tricky). I know of no proof of Euler’s theorem for all
n ≥ 1 along these lines. The standard relations are ill-adapted for
proving infinite families of identities.
12 / 27
Integral representation
Zeta values are periods. If n ≥ 2 then
Z
dt1 dt2
dtn
ζ(n) =
...
tn
0≤t1 ≤t2 ≤···≤tn ≤1 1 − t1 t2
To see this, do a geometric series expansion in t1 :
Z t2 X
Z
X t m+1
dt1
2
t1m dt1 =
=
m+1
0
0≤t1 ≤t2 1 − t1
m≥0
m≥0
and proceed by integrating out t2 , t3 , . . . in turn:
Z
X
X t m+1 dt2
dtn
1
2
...
= ... =
m + 1 t2
tn
(m + 1)n
0≤t2 ≤···≤tn ≤1
m≥0
m≥0
Therefore Euler’s theorem is an identity in the ring of periods.
2
One can prove that ζ(2) = π6 using the standard relations for
periods (tricky). I know of no proof of Euler’s theorem for all
n ≥ 1 along these lines. The standard relations are ill-adapted for
proving infinite families of identities.
12 / 27
Integral representation
Zeta values are periods. If n ≥ 2 then
Z
dt1 dt2
dtn
ζ(n) =
...
tn
0≤t1 ≤t2 ≤···≤tn ≤1 1 − t1 t2
To see this, do a geometric series expansion in t1 :
Z t2 X
Z
X t m+1
dt1
2
t1m dt1 =
=
m+1
0
0≤t1 ≤t2 1 − t1
m≥0
m≥0
and proceed by integrating out t2 , t3 , . . . in turn:
Z
X t m+1 dt2
X
dtn
1
2
...
= ... =
m + 1 t2
tn
(m + 1)n
0≤t2 ≤···≤tn ≤1
m≥0
m≥0
Therefore Euler’s theorem is an identity in the ring of periods.
2
One can prove that ζ(2) = π6 using the standard relations for
periods (tricky). I know of no proof of Euler’s theorem for all
n ≥ 1 along these lines. The standard relations are ill-adapted for
proving infinite families of identities.
12 / 27
Integral representation
Zeta values are periods. If n ≥ 2 then
Z
dt1 dt2
dtn
ζ(n) =
...
tn
0≤t1 ≤t2 ≤···≤tn ≤1 1 − t1 t2
To see this, do a geometric series expansion in t1 :
Z t2 X
Z
X t m+1
dt1
2
t1m dt1 =
=
m+1
0
0≤t1 ≤t2 1 − t1
m≥0
m≥0
and proceed by integrating out t2 , t3 , . . . in turn:
Z
X t m+1 dt2
X
dtn
1
2
...
= ... =
m + 1 t2
tn
(m + 1)n
0≤t2 ≤···≤tn ≤1
m≥0
m≥0
Therefore Euler’s theorem is an identity in the ring of periods.
2
One can prove that ζ(2) = π6 using the standard relations for
periods (tricky). I know of no proof of Euler’s theorem for all
n ≥ 1 along these lines. The standard relations are ill-adapted for
proving infinite families of identities.
12 / 27
Integral representation
Zeta values are periods. If n ≥ 2 then
Z
dt1 dt2
dtn
ζ(n) =
...
tn
0≤t1 ≤t2 ≤···≤tn ≤1 1 − t1 t2
To see this, do a geometric series expansion in t1 :
Z t2 X
Z
X t m+1
dt1
2
t1m dt1 =
=
m+1
0
0≤t1 ≤t2 1 − t1
m≥0
m≥0
and proceed by integrating out t2 , t3 , . . . in turn:
Z
X t m+1 dt2
X
dtn
1
2
...
= ... =
m + 1 t2
tn
(m + 1)n
0≤t2 ≤···≤tn ≤1
m≥0
m≥0
Therefore Euler’s theorem is an identity in the ring of periods.
2
One can prove that ζ(2) = π6 using the standard relations for
periods (tricky). I know of no proof of Euler’s theorem for all
n ≥ 1 along these lines. The standard relations are ill-adapted for
proving infinite families of identities.
12 / 27
What can we say about the odd zeta values?
It is conjectured that π, ζ(3), ζ(5), ζ(7), . . . are algebraically
independent over Q. Think of ζ(2n + 1) as new ‘variables’.
Theorem (Lindemann 1882)
π is transcendental
Apéry (1978)
ζ(3) is irrational.
(Rivoal and Ball-Rival 2000)
Infinitely many odd zeta values ζ(2n + 1) are irrational.
Zudilin proved that at least one of the four numbers
ζ(5), ζ(7), ζ(9), ζ(11) is irrational.
It is still not known whether ζ(5) is irrational, nor is it known if
ζ(3) is a rational multiple of π 3 .
13 / 27
What can we say about the odd zeta values?
It is conjectured that π, ζ(3), ζ(5), ζ(7), . . . are algebraically
independent over Q. Think of ζ(2n + 1) as new ‘variables’.
Theorem (Lindemann 1882)
π is transcendental
Apéry (1978)
ζ(3) is irrational.
(Rivoal and Ball-Rival 2000)
Infinitely many odd zeta values ζ(2n + 1) are irrational.
Zudilin proved that at least one of the four numbers
ζ(5), ζ(7), ζ(9), ζ(11) is irrational.
It is still not known whether ζ(5) is irrational, nor is it known if
ζ(3) is a rational multiple of π 3 .
13 / 27
What can we say about the odd zeta values?
It is conjectured that π, ζ(3), ζ(5), ζ(7), . . . are algebraically
independent over Q. Think of ζ(2n + 1) as new ‘variables’.
Theorem (Lindemann 1882)
π is transcendental
Apéry (1978)
ζ(3) is irrational.
(Rivoal and Ball-Rival 2000)
Infinitely many odd zeta values ζ(2n + 1) are irrational.
Zudilin proved that at least one of the four numbers
ζ(5), ζ(7), ζ(9), ζ(11) is irrational.
It is still not known whether ζ(5) is irrational, nor is it known if
ζ(3) is a rational multiple of π 3 .
13 / 27
Double zeta values
Euler tried to find polynomial relations between zeta values. Let
X
1
ζ(n1 , n2 ) =
n1 n2
k1 k2
0<k1 <k2
where n2 ≥ 2. He proved that for m, n ≥ 2,
X
X
X 1 X 1
X 1
=
+
+
km
ℓn
k m ℓn
k≥1
i.e.,
ℓ≥1
1≤k<ℓ
1≤ℓ<k
1≤k=ℓ
ζ(m)ζ(n) = ζ(m, n) + ζ(n, m) + ζ(m + n).
14 / 27
Multiple zeta values
The natural generalisation are multiple zeta values (MZV)
X
1
nr ≥ 2 .
ζ(n1 , . . . , nr ) =
n1
k1 . . . krnr
0<k1 <...<kr
They occur in: deformation quantization, theory of knot invariants,
Grothendieck-Teichmuller theory, theory of mixed Tate motives,
quantum field theory, string perturbation theory, . . . .
The weight is the quantity n1 + . . . + nr . The depth is r .
MZV’s are periods, via the integral representation:
Z
dt1
dtn
r
ζ(n1 , . . . , nr ) = (−1)
···
t n − ǫn
0≤t1 ≤···≤tn ≤1 t1 − ǫ1
where (ǫ1 , . . . , ǫn ) = 10n1 −1 . . . 10nr −1 is a sequence of 1’s and 0’s.
MZV’s are a first prototype for a general theory of periods!
15 / 27
Multiple zeta values
The natural generalisation are multiple zeta values (MZV)
X
1
nr ≥ 2 .
ζ(n1 , . . . , nr ) =
n1
k1 . . . krnr
0<k1 <...<kr
They occur in: deformation quantization, theory of knot invariants,
Grothendieck-Teichmuller theory, theory of mixed Tate motives,
quantum field theory, string perturbation theory, . . . .
The weight is the quantity n1 + . . . + nr . The depth is r .
MZV’s are periods, via the integral representation:
Z
dt1
dtn
r
ζ(n1 , . . . , nr ) = (−1)
···
t n − ǫn
0≤t1 ≤···≤tn ≤1 t1 − ǫ1
where (ǫ1 , . . . , ǫn ) = 10n1 −1 . . . 10nr −1 is a sequence of 1’s and 0’s.
MZV’s are a first prototype for a general theory of periods!
15 / 27
Multiple zeta values
The natural generalisation are multiple zeta values (MZV)
X
1
nr ≥ 2 .
ζ(n1 , . . . , nr ) =
n1
k1 . . . krnr
0<k1 <...<kr
They occur in: deformation quantization, theory of knot invariants,
Grothendieck-Teichmuller theory, theory of mixed Tate motives,
quantum field theory, string perturbation theory, . . . .
The weight is the quantity n1 + . . . + nr . The depth is r .
MZV’s are periods, via the integral representation:
Z
dtn
dt1
r
···
ζ(n1 , . . . , nr ) = (−1)
t n − ǫn
0≤t1 ≤···≤tn ≤1 t1 − ǫ1
where (ǫ1 , . . . , ǫn ) = 10n1 −1 . . . 10nr −1 is a sequence of 1’s and 0’s.
MZV’s are a first prototype for a general theory of periods!
15 / 27
Multiple zeta values
The natural generalisation are multiple zeta values (MZV)
X
1
nr ≥ 2 .
ζ(n1 , . . . , nr ) =
n1
k1 . . . krnr
0<k1 <...<kr
They occur in: deformation quantization, theory of knot invariants,
Grothendieck-Teichmuller theory, theory of mixed Tate motives,
quantum field theory, string perturbation theory, . . . .
The weight is the quantity n1 + . . . + nr . The depth is r .
MZV’s are periods, via the integral representation:
Z
dtn
dt1
r
···
ζ(n1 , . . . , nr ) = (−1)
t n − ǫn
0≤t1 ≤···≤tn ≤1 t1 − ǫ1
where (ǫ1 , . . . , ǫn ) = 10n1 −1 . . . 10nr −1 is a sequence of 1’s and 0’s.
MZV’s are a first prototype for a general theory of periods!
15 / 27
Relations (i)
Euler’s identity for double zeta values generalises, by decomposing
the domain of summation into cones. For example:
ζ(m1 )ζ(n1 , n2 ) = ζ(m1 , n1 , n2 ) + ζ(n1 , m1 , n2 ) + ζ(n1 , n2 , m1 )
+ ζ(n1 + m1 , n2 ) + ζ(n1 , n2 + m1 ) .
Given any two multiple zeta values ζ(m1 , . . . , mr ) and
ζ(n1 , . . . , ns ), the product can be written explicitly as a linear
combination of multiple zeta values.
This is called the stuffle product. Note that this relation is
homogeneous in the weight, but not in the depth.
Let Z be the Q-vector space generated by multiple zeta values. It
is an algebra. Thus Z ⊂ P is a subring of the ring of periods.
16 / 27
Relations (i)
Euler’s identity for double zeta values generalises, by decomposing
the domain of summation into cones. For example:
ζ(m1 )ζ(n1 , n2 ) = ζ(m1 , n1 , n2 ) + ζ(n1 , m1 , n2 ) + ζ(n1 , n2 , m1 )
+ ζ(n1 + m1 , n2 ) + ζ(n1 , n2 + m1 ) .
Given any two multiple zeta values ζ(m1 , . . . , mr ) and
ζ(n1 , . . . , ns ), the product can be written explicitly as a linear
combination of multiple zeta values.
This is called the stuffle product. Note that this relation is
homogeneous in the weight, but not in the depth.
Let Z be the Q-vector space generated by multiple zeta values. It
is an algebra. Thus Z ⊂ P is a subring of the ring of periods.
16 / 27
Relations (ii)
The shuffle product formula is derived by multiplying together the
integral representations for MZV’s and decomposing the domain of
integration: a product of simplicies is a union of simplicies.
Z
Z
dt1 dt2
ds1 ds2
0≤t1 ≤t2 ≤1 1 − t1 t2
0≤s1 ≤s2 ≤1 1 − s1 s2
is a sum of six terms, including
Z
dt1 ds1 ds2 dt2
= ζ(1, 3) .
0≤t1 ≤s1 ≤s2 ≤t2 ≤1 1 − t1 1 − s1 s2 t2
We deduce the equation
ζ(2)ζ(2) = 4 ζ(1, 3) + 2 ζ(2, 2)
It generalises to an infinite family of relations called the shuffle
relations. They are homogenous in both the weight and depth.
Thus the algebra Z has two multiplication laws.
17 / 27
Relations (ii)
The shuffle product formula is derived by multiplying together the
integral representations for MZV’s and decomposing the domain of
integration: a product of simplicies is a union of simplicies.
Z
Z
dt1 dt2
ds1 ds2
0≤t1 ≤t2 ≤1 1 − t1 t2
0≤s1 ≤s2 ≤1 1 − s1 s2
is a sum of six terms, including
Z
dt1 ds1 ds2 dt2
= ζ(1, 3) .
0≤t1 ≤s1 ≤s2 ≤t2 ≤1 1 − t1 1 − s1 s2 t2
We deduce the equation
ζ(2)ζ(2) = 4 ζ(1, 3) + 2 ζ(2, 2)
It generalises to an infinite family of relations called the shuffle
relations. They are homogenous in both the weight and depth.
Thus the algebra Z has two multiplication laws.
17 / 27
Relations (ii)
The shuffle product formula is derived by multiplying together the
integral representations for MZV’s and decomposing the domain of
integration: a product of simplicies is a union of simplicies.
Z
Z
dt1 dt2
ds1 ds2
0≤t1 ≤t2 ≤1 1 − t1 t2
0≤s1 ≤s2 ≤1 1 − s1 s2
is a sum of six terms, including
Z
dt1 ds1 ds2 dt2
= ζ(1, 3) .
0≤t1 ≤s1 ≤s2 ≤t2 ≤1 1 − t1 1 − s1 s2 t2
We deduce the equation
ζ(2)ζ(2) = 4 ζ(1, 3) + 2 ζ(2, 2)
It generalises to an infinite family of relations called the shuffle
relations. They are homogenous in both the weight and depth.
Thus the algebra Z has two multiplication laws.
17 / 27
Relations (iii)
These two families of relations are not enough. Denote the stuffle
product by ⋆ and shuffle by x . The sums ζ(n1 , . . . , nr −1 , 1) are
infinite. But it turns out that the formal difference:
ζ(1) ⋆ ζ(n1 , . . . , nr ) − ζ(1) x ζ(n1 , . . . , nr )
is finite: all divergent MZV’s drop out. This gives a third, linear,
relation called the regularisation relation, due to M. Hoffmann.
Example: the equation
ζ(1) ⋆ ζ(2) − ζ(1) x ζ(2) = 0
yields the identity, due to Euler, that ζ(3) = ζ(1, 2).
The three sets of equations defined above are called the regularised
double shuffle equations. They are very complex. Conjecturally,
these are the only relations satisfied by MZV’s.
18 / 27
Relations (iii)
These two families of relations are not enough. Denote the stuffle
product by ⋆ and shuffle by x . The sums ζ(n1 , . . . , nr −1 , 1) are
infinite. But it turns out that the formal difference:
ζ(1) ⋆ ζ(n1 , . . . , nr ) − ζ(1) x ζ(n1 , . . . , nr )
is finite: all divergent MZV’s drop out. This gives a third, linear,
relation called the regularisation relation, due to M. Hoffmann.
Example: the equation
ζ(1) ⋆ ζ(2) − ζ(1) x ζ(2) = 0
yields the identity, due to Euler, that ζ(3) = ζ(1, 2).
The three sets of equations defined above are called the regularised
double shuffle equations. They are very complex. Conjecturally,
these are the only relations satisfied by MZV’s.
18 / 27
Relations (iii)
These two families of relations are not enough. Denote the stuffle
product by ⋆ and shuffle by x . The sums ζ(n1 , . . . , nr −1 , 1) are
infinite. But it turns out that the formal difference:
ζ(1) ⋆ ζ(n1 , . . . , nr ) − ζ(1) x ζ(n1 , . . . , nr )
is finite: all divergent MZV’s drop out. This gives a third, linear,
relation called the regularisation relation, due to M. Hoffmann.
Example: the equation
ζ(1) ⋆ ζ(2) − ζ(1) x ζ(2) = 0
yields the identity, due to Euler, that ζ(3) = ζ(1, 2).
The three sets of equations defined above are called the regularised
double shuffle equations. They are very complex. Conjecturally,
these are the only relations satisfied by MZV’s.
18 / 27
Examples
In weight 2 there is a unique MZV, namely ζ(2).
In weight 3 there are two MZV’s, ζ(3) and ζ(1, 2). Euler’s identity
(regularisation) tells us they are the same ζ(3) = ζ(1, 2).
In weight 4 there are four MZV’s, ζ(4), ζ(1, 3), ζ(2, 2) and
ζ(1, 1, 2). We have the equations:
ζ(2)2 = 2ζ(2, 2) + ζ(4)
2
ζ(2)
= 4ζ(1, 3) + 2ζ(2, 2)
ζ(1, 3) + ζ(4) = 2ζ(1, 3) + ζ(2, 2)
2ζ(1, 1, 2) + ζ(2, 2) + ζ(1, 4) = 3ζ(1, 1, 2)
(stuffle)
(shuffle)
(reg.)
(reg.)
It turns out that every MZV in weight four is a rational multiple of
ζ(2)2 . In particular, ζ(4) = 25 ζ(2)2 , which we knew by Euler.
We needed to use every single equation!
19 / 27
Examples
In weight 2 there is a unique MZV, namely ζ(2).
In weight 3 there are two MZV’s, ζ(3) and ζ(1, 2). Euler’s identity
(regularisation) tells us they are the same ζ(3) = ζ(1, 2).
In weight 4 there are four MZV’s, ζ(4), ζ(1, 3), ζ(2, 2) and
ζ(1, 1, 2). We have the equations:
ζ(2)2 = 2ζ(2, 2) + ζ(4)
2
ζ(2)
= 4ζ(1, 3) + 2ζ(2, 2)
ζ(1, 3) + ζ(4) = 2ζ(1, 3) + ζ(2, 2)
2ζ(1, 1, 2) + ζ(2, 2) + ζ(1, 4) = 3ζ(1, 1, 2)
(stuffle)
(shuffle)
(reg.)
(reg.)
It turns out that every MZV in weight four is a rational multiple of
ζ(2)2 . In particular, ζ(4) = 25 ζ(2)2 , which we knew by Euler.
We needed to use every single equation!
19 / 27
Examples
In weight 2 there is a unique MZV, namely ζ(2).
In weight 3 there are two MZV’s, ζ(3) and ζ(1, 2). Euler’s identity
(regularisation) tells us they are the same ζ(3) = ζ(1, 2).
In weight 4 there are four MZV’s, ζ(4), ζ(1, 3), ζ(2, 2) and
ζ(1, 1, 2). We have the equations:
ζ(2)2 = 2ζ(2, 2) + ζ(4)
2
ζ(2)
= 4ζ(1, 3) + 2ζ(2, 2)
ζ(1, 3) + ζ(4) = 2ζ(1, 3) + ζ(2, 2)
2ζ(1, 1, 2) + ζ(2, 2) + ζ(1, 4) = 3ζ(1, 1, 2)
(stuffle)
(shuffle)
(reg.)
(reg.)
It turns out that every MZV in weight four is a rational multiple of
ζ(2)2 . In particular, ζ(4) = 25 ζ(2)2 , which we knew by Euler.
We needed to use every single equation!
19 / 27
Examples
In weight 2 there is a unique MZV, namely ζ(2).
In weight 3 there are two MZV’s, ζ(3) and ζ(1, 2). Euler’s identity
(regularisation) tells us they are the same ζ(3) = ζ(1, 2).
In weight 4 there are four MZV’s, ζ(4), ζ(1, 3), ζ(2, 2) and
ζ(1, 1, 2). We have the equations:
ζ(2)2 = 2ζ(2, 2) + ζ(4)
2
ζ(2)
= 4ζ(1, 3) + 2ζ(2, 2)
ζ(1, 3) + ζ(4) = 2ζ(1, 3) + ζ(2, 2)
2ζ(1, 1, 2) + ζ(2, 2) + ζ(1, 4) = 3ζ(1, 1, 2)
(stuffle)
(shuffle)
(reg.)
(reg.)
It turns out that every MZV in weight four is a rational multiple of
ζ(2)2 . In particular, ζ(4) = 25 ζ(2)2 , which we knew by Euler.
We needed to use every single equation!
19 / 27
Dimensions
There are many other families of relations satisfied by MZV’s:
sum, cyclic, duality, associator, motivic, . . . The regularised double
shuffle are conjectured to imply all others.
Wt
1
2
ζ(2)
3
ζ(3)
4
ζ(2)2
5
ζ(5)
ζ(2)ζ(3)
6
ζ(2)3
ζ(3)2
7
ζ(7)
ζ(2)ζ(5)
ζ(3)ζ(4)
dim
0
1
1
1
2
2
3
8
ζ(2)4
ζ(3)ζ(5)
ζ(3)2 ζ(2)
ζ(3, 5)
4
The dimensions are conjectural. Not all MZV’s can be reduced to
polynomials in single zetas. The first irreducible MZV is ζ(3, 5)
and occurs in weight 8.
Let dk be the dimension in weight k. Zagier conjectured that
dk = dk−2 + dk−3 .
It is not known whether
ζ(5)
ζ(2)ζ(3)
∈
/ Q.
20 / 27
Dimensions
There are many other families of relations satisfied by MZV’s:
sum, cyclic, duality, associator, motivic, . . . The regularised double
shuffle are conjectured to imply all others.
Wt
1
2
ζ(2)
3
ζ(3)
4
ζ(2)2
5
ζ(5)
ζ(2)ζ(3)
6
ζ(2)3
ζ(3)2
7
ζ(7)
ζ(2)ζ(5)
ζ(3)ζ(4)
dim
0
1
1
1
2
2
3
8
ζ(2)4
ζ(3)ζ(5)
ζ(3)2 ζ(2)
ζ(3, 5)
4
The dimensions are conjectural. Not all MZV’s can be reduced to
polynomials in single zetas. The first irreducible MZV is ζ(3, 5)
and occurs in weight 8.
Let dk be the dimension in weight k. Zagier conjectured that
dk = dk−2 + dk−3 .
It is not known whether
ζ(5)
ζ(2)ζ(3)
∈
/ Q.
20 / 27
Dimensions
There are many other families of relations satisfied by MZV’s:
sum, cyclic, duality, associator, motivic, . . . The regularised double
shuffle are conjectured to imply all others.
Wt
1
2
ζ(2)
3
ζ(3)
4
ζ(2)2
5
ζ(5)
ζ(2)ζ(3)
6
ζ(2)3
ζ(3)2
7
ζ(7)
ζ(2)ζ(5)
ζ(3)ζ(4)
dim
0
1
1
1
2
2
3
8
ζ(2)4
ζ(3)ζ(5)
ζ(3)2 ζ(2)
ζ(3, 5)
4
The dimensions are conjectural. Not all MZV’s can be reduced to
polynomials in single zetas. The first irreducible MZV is ζ(3, 5)
and occurs in weight 8.
Let dk be the dimension in weight k. Zagier conjectured that
dk = dk−2 + dk−3 .
It is not known whether
ζ(5)
ζ(2)ζ(3)
∈
/ Q.
20 / 27
Dimensions
There are many other families of relations satisfied by MZV’s:
sum, cyclic, duality, associator, motivic, . . . The regularised double
shuffle are conjectured to imply all others.
Wt
1
2
ζ(2)
3
ζ(3)
4
ζ(2)2
5
ζ(5)
ζ(2)ζ(3)
6
ζ(2)3
ζ(3)2
7
ζ(7)
ζ(2)ζ(5)
ζ(3)ζ(4)
dim
0
1
1
1
2
2
3
8
ζ(2)4
ζ(3)ζ(5)
ζ(3)2 ζ(2)
ζ(3, 5)
4
The dimensions are conjectural. Not all MZV’s can be reduced to
polynomials in single zetas. The first irreducible MZV is ζ(3, 5)
and occurs in weight 8.
Let dk be the dimension in weight k. Zagier conjectured that
dk = dk−2 + dk−3 .
It is not known whether
ζ(5)
ζ(2)ζ(3)
∈
/ Q.
20 / 27
Theorems
Let Zk denote the Q vector space spanned by MZV’s of weight k.
Zagier conjectured that the weight is a grading
M
Z=
Zk ,
k≥0
i.e., no relations between different weights. Let dk be the sequence
satisfying dk = dk−2 + dk−3 , and d0 = 1, d1 = 0, d2 = 1.
Theorem (Goncharov, and independently Terasoma) 2001
dimQ Zk ≤ dk .
The proof uses the theory of mixed Tate motives. They showed by
a geometric construction that multiple zeta values are examples of
periods of mixed Tate motives over Z, denoted PMT (Z) ⊂ P
Z ⊂ PMT (Z) .
By general theory and algebraic K -theory of Z, the periods of the
latter in weight k are of dimension at most dk .
21 / 27
Theorems
Let Zk denote the Q vector space spanned by MZV’s of weight k.
Zagier conjectured that the weight is a grading
M
Z=
Zk ,
k≥0
i.e., no relations between different weights. Let dk be the sequence
satisfying dk = dk−2 + dk−3 , and d0 = 1, d1 = 0, d2 = 1.
Theorem (Goncharov, and independently Terasoma) 2001
dimQ Zk ≤ dk .
The proof uses the theory of mixed Tate motives. They showed by
a geometric construction that multiple zeta values are examples of
periods of mixed Tate motives over Z, denoted PMT (Z) ⊂ P
Z ⊂ PMT (Z) .
By general theory and algebraic K -theory of Z, the periods of the
latter in weight k are of dimension at most dk .
21 / 27
Theorems
Let Zk denote the Q vector space spanned by MZV’s of weight k.
Zagier conjectured that the weight is a grading
M
Z=
Zk ,
k≥0
i.e., no relations between different weights. Let dk be the sequence
satisfying dk = dk−2 + dk−3 , and d0 = 1, d1 = 0, d2 = 1.
Theorem (Goncharov, and independently Terasoma) 2001
dimQ Zk ≤ dk .
The proof uses the theory of mixed Tate motives. They showed by
a geometric construction that multiple zeta values are examples of
periods of mixed Tate motives over Z, denoted PMT (Z) ⊂ P
Z ⊂ PMT (Z) .
By general theory and algebraic K -theory of Z, the periods of the
latter in weight k are of dimension at most dk .
21 / 27
Theorems
Let Zk denote the Q vector space spanned by MZV’s of weight k.
Zagier conjectured that the weight is a grading
M
Z=
Zk ,
k≥0
i.e., no relations between different weights. Let dk be the sequence
satisfying dk = dk−2 + dk−3 , and d0 = 1, d1 = 0, d2 = 1.
Theorem (Goncharov, and independently Terasoma) 2001
dimQ Zk ≤ dk .
The proof uses the theory of mixed Tate motives. They showed by
a geometric construction that multiple zeta values are examples of
periods of mixed Tate motives over Z, denoted PMT (Z) ⊂ P
Z ⊂ PMT (Z) .
By general theory and algebraic K -theory of Z, the periods of the
latter in weight k are of dimension at most dk .
21 / 27
Theorems 2
Theorem (B. 2012) (Hoffman conjecture)
Every multiple zeta value of weight N is a Q-linear combination of
ζ(n1 , . . . , nr ), where n1 + . . . + nr = N and ni ∈ {2, 3}.
The number of MZV’s with arguments 2 or 3 in weight N is
exactly dN . The proof uses the Galois theory of multiple zeta
values. Not known how to prove such a result using relations.
Theorem (B. 2012)
The periods of MT motives over Z are all MZV’s.
Z = PMT (Z)
This gives some clue why MZV’s show up often. They are the
periods of the motives constructed out of the simplest possible
building blocks. Claire Glanois has recently constructed a new
generating family for multiple zeta values using Euler sums.
22 / 27
Theorems 2
Theorem (B. 2012) (Hoffman conjecture)
Every multiple zeta value of weight N is a Q-linear combination of
ζ(n1 , . . . , nr ), where n1 + . . . + nr = N and ni ∈ {2, 3}.
The number of MZV’s with arguments 2 or 3 in weight N is
exactly dN . The proof uses the Galois theory of multiple zeta
values. Not known how to prove such a result using relations.
Theorem (B. 2012)
The periods of MT motives over Z are all MZV’s.
Z = PMT (Z)
This gives some clue why MZV’s show up often. They are the
periods of the motives constructed out of the simplest possible
building blocks. Claire Glanois has recently constructed a new
generating family for multiple zeta values using Euler sums.
22 / 27
Theorems 2
Theorem (B. 2012) (Hoffman conjecture)
Every multiple zeta value of weight N is a Q-linear combination of
ζ(n1 , . . . , nr ), where n1 + . . . + nr = N and ni ∈ {2, 3}.
The number of MZV’s with arguments 2 or 3 in weight N is
exactly dN . The proof uses the Galois theory of multiple zeta
values. Not known how to prove such a result using relations.
Theorem (B. 2012)
The periods of MT motives over Z are all MZV’s.
Z = PMT (Z)
This gives some clue why MZV’s show up often. They are the
periods of the motives constructed out of the simplest possible
building blocks. Claire Glanois has recently constructed a new
generating family for multiple zeta values using Euler sums.
22 / 27
Examples
A multiple zeta value of depth ≤ r can be written as a sum of
elements with at most r threes. Examples:
192
ζ(2, 2, 2, 2) .
5
The general formula for ζ(2n + 1) is known (Zagier):
ζ(8) =
672
528
352
ζ(3, 2, 2) +
ζ(2, 3, 2) +
ζ(2, 2, 3)
151
151
151
There is no known algorithm to write an arbitrary MZV in this
form, but there is an ‘exact-numerical algorithm’. For example, in
ζ(7) =
62
78
48
496
ζ(2, 2, 2, 2)+ ζ(3, 3, 2)+ ζ(3, 2, 3)+ ζ(2, 3, 3)
125
25
25
25
the black coefficients can be determined exactly, the red coefficient
can only be determined numerically.
ζ(3, 5) = −
In general the denominators have large prime factors. Is there a
generating set which reflects the integral structure of MZV’s?
23 / 27
Examples
A multiple zeta value of depth ≤ r can be written as a sum of
elements with at most r threes. Examples:
192
ζ(2, 2, 2, 2) .
5
The general formula for ζ(2n + 1) is known (Zagier):
ζ(8) =
672
528
352
ζ(3, 2, 2) +
ζ(2, 3, 2) +
ζ(2, 2, 3)
151
151
151
There is no known algorithm to write an arbitrary MZV in this
form, but there is an ‘exact-numerical algorithm’. For example, in
ζ(7) =
62
78
48
496
ζ(2, 2, 2, 2)+ ζ(3, 3, 2)+ ζ(3, 2, 3)+ ζ(2, 3, 3)
125
25
25
25
the black coefficients can be determined exactly, the red coefficient
can only be determined numerically.
ζ(3, 5) = −
In general the denominators have large prime factors. Is there a
generating set which reflects the integral structure of MZV’s?
23 / 27
Examples
A multiple zeta value of depth ≤ r can be written as a sum of
elements with at most r threes. Examples:
192
ζ(2, 2, 2, 2) .
5
The general formula for ζ(2n + 1) is known (Zagier):
ζ(8) =
672
528
352
ζ(3, 2, 2) +
ζ(2, 3, 2) +
ζ(2, 2, 3)
151
151
151
There is no known algorithm to write an arbitrary MZV in this
form, but there is an ‘exact-numerical algorithm’. For example, in
ζ(7) =
62
78
48
496
ζ(2, 2, 2, 2)+ ζ(3, 3, 2)+ ζ(3, 2, 3)+ ζ(2, 3, 3)
125
25
25
25
the black coefficients can be determined exactly, the red coefficient
can only be determined numerically.
ζ(3, 5) = −
In general the denominators have large prime factors. Is there a
generating set which reflects the integral structure of MZV’s?
23 / 27
Examples
A multiple zeta value of depth ≤ r can be written as a sum of
elements with at most r threes. Examples:
192
ζ(2, 2, 2, 2) .
5
The general formula for ζ(2n + 1) is known (Zagier):
ζ(8) =
672
528
352
ζ(3, 2, 2) +
ζ(2, 3, 2) +
ζ(2, 2, 3)
151
151
151
There is no known algorithm to write an arbitrary MZV in this
form, but there is an ‘exact-numerical algorithm’. For example, in
ζ(7) =
62
78
48
496
ζ(2, 2, 2, 2)+ ζ(3, 3, 2)+ ζ(3, 2, 3)+ ζ(2, 3, 3)
125
25
25
25
the black coefficients can be determined exactly, the red coefficient
can only be determined numerically.
ζ(3, 5) = −
In general the denominators have large prime factors. Is there a
generating set which reflects the integral structure of MZV’s?
23 / 27
Examples
A multiple zeta value of depth ≤ r can be written as a sum of
elements with at most r threes. Examples:
192
ζ(2, 2, 2, 2) .
5
The general formula for ζ(2n + 1) is known (Zagier):
ζ(8) =
672
528
352
ζ(3, 2, 2) +
ζ(2, 3, 2) +
ζ(2, 2, 3)
151
151
151
There is no known algorithm to write an arbitrary MZV in this
form, but there is an ‘exact-numerical algorithm’. For example, in
ζ(7) =
62
78
48
496
ζ(2, 2, 2, 2)+ ζ(3, 3, 2)+ ζ(3, 2, 3)+ ζ(2, 3, 3)
125
25
25
25
the black coefficients can be determined exactly, the red coefficient
can only be determined numerically.
ζ(3, 5) = −
In general the denominators have large prime factors. Is there a
generating set which reflects the integral structure of MZV’s?
23 / 27
Examples
A multiple zeta value of depth ≤ r can be written as a sum of
elements with at most r threes. Examples:
192
ζ(2, 2, 2, 2) .
5
The general formula for ζ(2n + 1) is known (Zagier):
ζ(8) =
672
528
352
ζ(3, 2, 2) +
ζ(2, 3, 2) +
ζ(2, 2, 3)
151
151
151
There is no known algorithm to write an arbitrary MZV in this
form, but there is an ‘exact-numerical algorithm’. For example, in
ζ(7) =
62
78
48
496
ζ(2, 2, 2, 2)+ ζ(3, 3, 2)+ ζ(3, 2, 3)+ ζ(2, 3, 3)
125
25
25
25
the black coefficients can be determined exactly, the red coefficient
can only be determined numerically.
ζ(3, 5) = −
In general the denominators have large prime factors. Is there a
generating set which reflects the integral structure of MZV’s?
23 / 27
Examples
A multiple zeta value of depth ≤ r can be written as a sum of
elements with at most r threes. Examples:
192
ζ(2, 2, 2, 2) .
5
The general formula for ζ(2n + 1) is known (Zagier):
ζ(8) =
672
528
352
ζ(3, 2, 2) +
ζ(2, 3, 2) +
ζ(2, 2, 3)
151
151
151
There is no known algorithm to write an arbitrary MZV in this
form, but there is an ‘exact-numerical algorithm’. For example, in
ζ(7) =
62
78
48
496
ζ(2, 2, 2, 2)+ ζ(3, 3, 2)+ ζ(3, 2, 3)+ ζ(2, 3, 3)
125
25
25
25
the black coefficients can be determined exactly, the red coefficient
can only be determined numerically.
ζ(3, 5) = −
In general the denominators have large prime factors. Is there a
generating set which reflects the integral structure of MZV’s?
23 / 27
Conjectures
Two of the many remaining conjectures:
Algebraic Conjecture
The space of formal symbols Z (n1 , . . . , nr ) modulo the regularised
double shuffle relations has dimension dk in weight k.
This is (hard) linear algebra. If true, then Z (n1 , . . . , nr ) with
ni ∈ {2, 3} would be a basis (the dimension is bounded below by
dk by the theory on the previous slide). Ecalle has shown that
every MZV can be expressed as a linear combination of
ζ(n1 , . . . , nr ) with ni ≥ 2 using the regularised double shuffle
equations “the elimination of 1’s”. Very complex.
Transcendence Conjecture (‘Period conjecture’)
The elements ζ(n1 , . . . , nr ) with ni = 2, 3 are linearly independent,
and hence a basis for Z.
This conjecture is totally inaccessible at present.
24 / 27
Conjectures
Two of the many remaining conjectures:
Algebraic Conjecture
The space of formal symbols Z (n1 , . . . , nr ) modulo the regularised
double shuffle relations has dimension dk in weight k.
This is (hard) linear algebra. If true, then Z (n1 , . . . , nr ) with
ni ∈ {2, 3} would be a basis (the dimension is bounded below by
dk by the theory on the previous slide). Ecalle has shown that
every MZV can be expressed as a linear combination of
ζ(n1 , . . . , nr ) with ni ≥ 2 using the regularised double shuffle
equations “the elimination of 1’s”. Very complex.
Transcendence Conjecture (‘Period conjecture’)
The elements ζ(n1 , . . . , nr ) with ni = 2, 3 are linearly independent,
and hence a basis for Z.
This conjecture is totally inaccessible at present.
24 / 27
Conjectures
Two of the many remaining conjectures:
Algebraic Conjecture
The space of formal symbols Z (n1 , . . . , nr ) modulo the regularised
double shuffle relations has dimension dk in weight k.
This is (hard) linear algebra. If true, then Z (n1 , . . . , nr ) with
ni ∈ {2, 3} would be a basis (the dimension is bounded below by
dk by the theory on the previous slide). Ecalle has shown that
every MZV can be expressed as a linear combination of
ζ(n1 , . . . , nr ) with ni ≥ 2 using the regularised double shuffle
equations “the elimination of 1’s”. Very complex.
Transcendence Conjecture (‘Period conjecture’)
The elements ζ(n1 , . . . , nr ) with ni = 2, 3 are linearly independent,
and hence a basis for Z.
This conjecture is totally inaccessible at present.
24 / 27
Conjectures
Two of the many remaining conjectures:
Algebraic Conjecture
The space of formal symbols Z (n1 , . . . , nr ) modulo the regularised
double shuffle relations has dimension dk in weight k.
This is (hard) linear algebra. If true, then Z (n1 , . . . , nr ) with
ni ∈ {2, 3} would be a basis (the dimension is bounded below by
dk by the theory on the previous slide). Ecalle has shown that
every MZV can be expressed as a linear combination of
ζ(n1 , . . . , nr ) with ni ≥ 2 using the regularised double shuffle
equations “the elimination of 1’s”. Very complex.
Transcendence Conjecture (‘Period conjecture’)
The elements ζ(n1 , . . . , nr ) with ni = 2, 3 are linearly independent,
and hence a basis for Z.
This conjecture is totally inaccessible at present.
24 / 27
Remarks
The algebraic conjecture has been verified up to weights in
the mid-20’s using the most powerful computer algebra
systems available. (Vermaseren et al.)
Note that even if the algebraic conjecture were known, the
only algorithm for writing an MZV in the Hoffmann basis
would be ‘write down all the regularised double shuffle
equations and solve the system of equations’. This is hugely
inefficient and hopeless in general. A very interesting problem
is to try to solve the double shuffle equations.
The presently-known transcendence results tell us nothing
about the conjectured inequality
dimk Zk ≥ dk .
Indeed, we only know the trivial bound dimk Zk ≥ 1.
25 / 27
Remarks
The algebraic conjecture has been verified up to weights in
the mid-20’s using the most powerful computer algebra
systems available. (Vermaseren et al.)
Note that even if the algebraic conjecture were known, the
only algorithm for writing an MZV in the Hoffmann basis
would be ‘write down all the regularised double shuffle
equations and solve the system of equations’. This is hugely
inefficient and hopeless in general. A very interesting problem
is to try to solve the double shuffle equations.
The presently-known transcendence results tell us nothing
about the conjectured inequality
dimk Zk ≥ dk .
Indeed, we only know the trivial bound dimk Zk ≥ 1.
25 / 27
Remarks
The algebraic conjecture has been verified up to weights in
the mid-20’s using the most powerful computer algebra
systems available. (Vermaseren et al.)
Note that even if the algebraic conjecture were known, the
only algorithm for writing an MZV in the Hoffmann basis
would be ‘write down all the regularised double shuffle
equations and solve the system of equations’. This is hugely
inefficient and hopeless in general. A very interesting problem
is to try to solve the double shuffle equations.
The presently-known transcendence results tell us nothing
about the conjectured inequality
dimk Zk ≥ dk .
Indeed, we only know the trivial bound dimk Zk ≥ 1.
25 / 27
Depth
The depth is not a grading, but a filtration. For example, we had
ζ(1, 2) = ζ(3) .
One can study relations between multiple zeta values ‘modulo
terms of lower depth’. Perversely, this considerably simplifies the
double shuffle relations, but makes the underlying structure much
more complicated. Some rather subtle new phenomena occur
relating to modular forms for SL2 (Z).
For example, in weight 12 one finds for the first time an exotic
relation between double zeta values with odd arguments:
5197
ζ(12)
691
Modulo terms of depth ≤ 1 it becomes a relation
28ζ(3, 9) + 150ζ(5, 7) + 168ζ(7, 5) =
28ζD (3, 9) + 150ζD (5, 7) + 168ζD (7, 5) = 0
where ζD means MZV modulo lower depth.
26 / 27
Depth
The depth is not a grading, but a filtration. For example, we had
ζ(1, 2) = ζ(3) .
One can study relations between multiple zeta values ‘modulo
terms of lower depth’. Perversely, this considerably simplifies the
double shuffle relations, but makes the underlying structure much
more complicated. Some rather subtle new phenomena occur
relating to modular forms for SL2 (Z).
For example, in weight 12 one finds for the first time an exotic
relation between double zeta values with odd arguments:
5197
ζ(12)
691
Modulo terms of depth ≤ 1 it becomes a relation
28ζ(3, 9) + 150ζ(5, 7) + 168ζ(7, 5) =
28ζD (3, 9) + 150ζD (5, 7) + 168ζD (7, 5) = 0
where ζD means MZV modulo lower depth.
26 / 27
Depth
The depth is not a grading, but a filtration. For example, we had
ζ(1, 2) = ζ(3) .
One can study relations between multiple zeta values ‘modulo
terms of lower depth’. Perversely, this considerably simplifies the
double shuffle relations, but makes the underlying structure much
more complicated. Some rather subtle new phenomena occur
relating to modular forms for SL2 (Z).
For example, in weight 12 one finds for the first time an exotic
relation between double zeta values with odd arguments:
5197
ζ(12)
691
Modulo terms of depth ≤ 1 it becomes a relation
28ζ(3, 9) + 150ζ(5, 7) + 168ζ(7, 5) =
28ζD (3, 9) + 150ζD (5, 7) + 168ζD (7, 5) = 0
where ζD means MZV modulo lower depth.
26 / 27
Depth
The depth is not a grading, but a filtration. For example, we had
ζ(1, 2) = ζ(3) .
One can study relations between multiple zeta values ‘modulo
terms of lower depth’. Perversely, this considerably simplifies the
double shuffle relations, but makes the underlying structure much
more complicated. Some rather subtle new phenomena occur
relating to modular forms for SL2 (Z).
For example, in weight 12 one finds for the first time an exotic
relation between double zeta values with odd arguments:
5197
ζ(12)
691
Modulo terms of depth ≤ 1 it becomes a relation
28ζ(3, 9) + 150ζ(5, 7) + 168ζ(7, 5) =
28ζD (3, 9) + 150ζD (5, 7) + 168ζD (7, 5) = 0
where ζD means MZV modulo lower depth.
26 / 27
Periods of a modular form
Zagier, Gangl and Kaneko explained how to deduce the coefficients
in the previous equation from the ratios of the integrals
Z i∞
∆(τ )τ 2k dτ
0
for 0 ≤ k ≤ 5, where ∆(τ ) is the modular discriminant cusp form
of weight 12 for SL2 (Z). If we write q = exp 2πi τ , it has the
explicit Fourier expansion
Y
∆(q) = q
(1 − q n )24 .
n≥0
This is the beginning of a long story which is far from being
completely understood.
The structure of the ring of multiple zeta values, a small subspace
of the ring of all periods, already encodes a huge amount of
information about apparently unrelated mathematical structures.
27 / 27
Periods of a modular form
Zagier, Gangl and Kaneko explained how to deduce the coefficients
in the previous equation from the ratios of the integrals
Z i∞
∆(τ )τ 2k dτ
0
for 0 ≤ k ≤ 5, where ∆(τ ) is the modular discriminant cusp form
of weight 12 for SL2 (Z). If we write q = exp 2πi τ , it has the
explicit Fourier expansion
Y
∆(q) = q
(1 − q n )24 .
n≥0
This is the beginning of a long story which is far from being
completely understood.
The structure of the ring of multiple zeta values, a small subspace
of the ring of all periods, already encodes a huge amount of
information about apparently unrelated mathematical structures.
27 / 27
Periods of a modular form
Zagier, Gangl and Kaneko explained how to deduce the coefficients
in the previous equation from the ratios of the integrals
Z i∞
∆(τ )τ 2k dτ
0
for 0 ≤ k ≤ 5, where ∆(τ ) is the modular discriminant cusp form
of weight 12 for SL2 (Z). If we write q = exp 2πi τ , it has the
explicit Fourier expansion
Y
∆(q) = q
(1 − q n )24 .
n≥0
This is the beginning of a long story which is far from being
completely understood.
The structure of the ring of multiple zeta values, a small subspace
of the ring of all periods, already encodes a huge amount of
information about apparently unrelated mathematical structures.
27 / 27