A combinatorial approach to classical modular forms inspiredby multiple zeta values
Henrik Bachmann - Universität Hamburg

Jahrestagung der Deutschen Mathematiker-Vereinigung 2015
23.09.2015

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta valuesEisenstein series identities

Definition
For even k > 2 the Eisenstein series of weight k is defined by

Gk (τ ) =12X(m,n)∈Z2 \(0,0)

1(2πi)k X
σk−1 (n)q n ,= ζ(k) +k(mτ + n)(k − 1)! n>0where τ ∈ {x + iy ∈ C | y > 0}, q = exp(2πiτ ) and σk−1 (n) =Pd|n dk−1

The Eisenstein series Gk is a modular form of weight k for SL2 (Z).
The spaces of modular forms for SL2 (Z) and their dimensions are well understood.
For even k the Riemann zeta values ζ(k) are known to be rational multiples of π k ,e.g.

ζ(2) =π4π6π8π2, ζ(4) =, ζ(6) =, ζ(8) =.
6909459450

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values.

Eisenstein series identities

With all this knowledge the following Propositon ist absolutely trivial:
Proposition
We have the following identity

G24 (τ ) =7G8 (τ ) .
6

Proof:

G24 and G8 are modular forms of weight 8
The space of weight 8 modular forms has dimension 1.
7

Their Fourier expansion both have ζ(4)2 = 6 ζ(8) as their constant term.

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta valuesEisenstein series identities

With all this knowledge the following Propositon ist absolutely trivial:
Proposition
We have the following identity

G24 (τ ) =7
G8 (τ ) .
6

Proof:

G24 and G8 are modular forms of weight 8
The space of weight 8 modular forms has dimension 1.
7

Their Fourier expansion both have ζ(4)2 = 6 ζ(8) as their constant term.
7

But how do you prove this without knowing modular forms and ζ(4)2 = 6 ζ(8)?

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values

Aim of this talkIn this talk I try to present a purely combinatorial way to prove such relations.
For this we need...
Multiple zeta values (MZV) - multiple version of the Riemann zeta values
Double shuffle relations - a toolbox to prove relations between MZVMultiple Eisenstein series - multiple version of the Eisenstein series

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values

Multiple zeta values

Definition
For natural numbers s1 ≥ 2, s2 , ..., sl ≥ 1, the multiple zeta value (MZV) of weightk = s1 + · · · + sl and length l is defined byζ(s1 , . . . , sl ) :=1.
s1n.
. . nsl ln >n >···>n >0 1X

12l

By MZ k we denote the space spanned by all MZV of weight k and by MZ the spacespanned by all MZV.
The product of two MZV can be expressed as a linear combination of MZV with thesame weight (harmonic product)
MZV can be expressed as iterated integrals. This gives another way (shuffle product)
to express the product of two MZV as a linear combination of MZV.
These two products give a number of Q-relations (double shuffle relations) between
MZV.

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values

Multiple zeta values - harmonic productIn the smallest length the harmonic product reads

X 1 X 1ns 1
ns2
n1 >0 1 n2 >0 2
XX
X
11
1=
s1 s2 +
s1 s2 +
s1 +s2
nnnnnn >n >0 1 2n >n >0 1 2
n =n >0 1

ζ(s1 ) · ζ(s2 ) =122112= ζ(s1 , s2 ) + ζ(s2 , s1 ) + ζ(s1 + s2 ) .

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values

Multiple zeta values - harmonic productIn the smallest length the harmonic product reads

X 1 X 1ns 1
ns2
n1 >0 1 n2 >0 2
XX
X
1
11
=
s1 s2 +
s1 s2 +
s1 +s2
nnnn
nn >n >0 1 2n >n >0 1 2n =n >0 1

ζ(s1 ) · ζ(s2 ) =122112= ζ(s1 , s2 ) + ζ(s2 , s1 ) + ζ(s1 + s2 ) .
For length 1 times length 2 the same argument givesζ(s1 ) · ζ(s2 , s3 ) = ζ(s1 , s2 , s3 ) + ζ(s2 , s1 , s3 ) + ζ(s2 , s3 , s1 )+ ζ(s1 + s2 , s3 ) + ζ(s2 , s1 + s3 ) .

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values

Multiple zeta values - shuffle productMultiple zeta values can also be written as iterated integrals. For exampleZζ(2, 3) =R(t1 )B(t2 ) R(t3 )R(t4 )B(t5 ) ,{z } |{z}1>t1 >t2 >t3 >t4 >t5 >0 |
dt

23dt

with the differential forms R(t) = t and B(t) = 1−t .

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values

Multiple zeta values - shuffle productMultiple zeta values can also be written as iterated integrals. For exampleZζ(2, 3) =R(t1 )B(t2 ) R(t3 )R(t4 )B(t5 ) ,{z } |{z}1>t1 >t2 >t3 >t4 >t5 >0 |23dtdtwith the differential forms R(t) = t and B(t) = 1−t .
Multiplying two such integrals results in the sum of all possible shuffles of the integrants

ZZζ(2) · ζ(3) =R(t1 )B(t2 ) ·1>t1 >t2 >0

R(u1 )R(u2 )B(u3 )1>u1 >u2 >u3 >0Z

Z=... +
1>t1 >t2 >u1 >u2 >u3 >0Henrik Bachmann - Universität Hamburg... + . . .
1>t1 >u1 >t2 >u2 >u3 >0A combinatorial approach to classical modular forms inspired by multiple zeta values

Multiple zeta values - shuffle productSuppose we have two types of cards (red and blue).
MZV correspond to a deck of these cards

Multiplication of MZV corresponds to shuffling two of these decks (+counting
multiplicities)

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values

Multiple zeta values - shuffle product - example

For example the product ζ(2) · ζ(3) can be evaluated asHenrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values

Multiple zeta-values - double shuffle relations

These two representations for the product give a large family of linear relations between MZV.
shuffleharmonic=⇒ 2ζ(2, 3) + 6ζ(4, 1)

double shuffleζ(3, 2) + 3ζ(2, 3) + 6ζ(4, 1) = ζ(2) · ζ(3) = ζ(2, 3) + ζ(3, 2) + ζ(5) .
=ζ(5) .

But there are more relations between MZV. e.g.:ζ(2, 1) = ζ(3).
These follow from the "extended double shuffle relations" where one use the samecombinatorics as above for "ζ(1) · ζ(2)" in a formal setting.

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values

Multiple zeta values - double shuffle relations - example

7

Now we have enough tools to prove ζ(4)2 = 6 ζ(8).
With the harmonic (h) and shuffle (s) product we obtainh

ζ(4) · ζ(4) = 2ζ(4, 4) + ζ(8) ,(1)sζ(4) · ζ(4) = 2ζ(4, 4) + 8ζ(5, 3) + 20ζ(6, 2) + 40ζ(7, 1)hζ(3) · ζ(5) = ζ(3, 5) + ζ(5, 3) + ζ(8) ,(2)(3)sζ(3) · ζ(5) = ζ(3, 5) + 3ζ(4, 4) + 7ζ(5, 3) + 15ζ(6, 2) + 30ζ(7, 1)(4)

From which we deduceζ(4)2 = 2ζ(4, 4) + ζ(8) =1
72((4) − (3)) − ((2) − (1)) + ζ(8) .
32|{z} |{z} 6=0Henrik Bachmann - Universität Hamburg=0A combinatorial approach to classical modular forms inspired by multiple zeta values

Back to Eisenstein series..

7But how can we prove G4 (τ )2 = 6 G8 (τ ) ?
Introduce multiple Eisenstein series.
Show that product of two multiple Eisenstein series can also be express by theharmonic and the shuffle product.
With this one can use the exact same proof as before by replacing ζ with G.

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values

Multiple Eisenstein series

Definition
For s1 , . . . , sl ≥ 2 we define the multiple Eisenstein series of weight k = s1 + · · · + sl
and length l by
0X

Gs1 ,...,sl (τ ) :=λ1 ···λl 0λi ∈Λτ1,λs11 . . . λsl lwhere λi ∈ Zτ + Z are lattice points and the order  on Z + Zτ is given bym1 τ + n1  m2 τ + n2 :⇔ (m1 > m2 ∨ (m1 = m2 ∧ n1 > n2 )) .
It is easy to see that these are holomorphic functions in the upper half plane and that they
fulfill the harmonic product, i.e. as for MZV we have

G2 (τ ) · G3 (τ ) = G2,3 (τ ) + G3,2 (τ ) + G5 (τ ) .

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values

Multiple Eisenstein series - shuffle product?

What about the shuffle product?
Remember for MZV we haveζ(2) · ζ(3) = ζ(3, 2) + 3ζ(2, 3) + 6ζ(4, 1) .
This equation does not make sense for multiple Eisenstein series if we replace ζ by G since
there is no definition of G4,1 .

Question
What is a good definition of Gs1 ,...,sl for s1 ≥ 2, s2 , . . . , sl ≥ 1 such these series also
"fulfill" the shuffle product ?

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values

Multiple Eisenstein series - shuffle regularization

Theorem (B., K. Tasaka 2014)

For all s1 , . . . , sl ≥ 1 there exist shuffle regularized multiple Eisenstein series Gs1 ,...,sl withthe following properties:
They are holomorphic functions on the upper half plane having a Fourier expansion with themultiple zeta values as the constant term.
They "fulfill" the shuffle product.
For integers s1 , . . . , sl ≥ 2 they equal the multiple Eisenstein seriesGs1 ,...,sl (τ ) = Gs1 ,...,sl (τ )and therefore they fulfill the harmonic product in these cases.
Proof sketch: Uses a beautiful connection of the Fourier expansion of multiple Eisensteinseries to the coproduct of formal iterated integrals.

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values

Double shuffle relations for multiple Eisenstein series

The Theorem enables one to use the double shuffle relations for products of multiple
Eisenstein series Gs1 ,...,sl · Gr1 ,...,rm whenever s1 , . . . , sl , r1 , . . . , rm ≥ 2.
Proposition
We have the following identity

G24 (τ ) =7
G8 (τ ) .
6

Alternative proof: Use the double shuffle relations for G4 · G4 and G3 · G5 .

All algebraic relations between Eisenstein series can be proven this way.

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values

Double shuffle relations for multiple Eisenstein series

There are relations between MZV, which can be proven by double shuffle but which are
not true for Eisenstein series.
For example the relation

ζ(6)2 =

715
ζ(12)
691

can be proven by using the double shuffle relations. But this relation is not true for
Eisenstein series, because there are cusp forms in weight 12, i.e. for some c ∈ R

G6 (τ )2 =715
G12 (τ ) + c · ∆ .
691

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values

But why?

"So you study these things just to give alternative
proofs for easy & well-known results?"
No....
In the theory of multiple zeta values modular forms appear in several ways.
There are relations between multiple zeta values which "come from cusp forms".
Understanding the failure of the double shuffle relations for multiple Eisenstein series
explain these relations.
This failure is still not well understood.

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values

Summary

Multiple zeta values (MZV) are multiple version of the Riemann zeta values.

Q-linear relation between these real numbers can be proven by expressing theproduct of two MZV in two different ways. (harmonic & shuffle product)
There also exist multiple version of the classical Eisenstein series given by multiple
Eisenstein series.
Multiple Eisenstein series also fulfill "some but not all" of the double shuffle relations.
This "some but not all" is crucial to understand the modular aspect of MZV, but still not
well understood so far.
Thank you for your attention!

Henrik Bachmann - Universität Hamburg

A combinatorial approach to classical modular forms inspired by multiple zeta values