[Paper Review] On linear relations among totally odd multiple zeta values related to period polynomials

Koji Tasaka|arXiv (Cornell University)|Feb 14, 2014

Advanced Mathematical Identities7 references4 citations

TL;DR

This paper establishes a connection between totally odd multiple zeta values (MZVs) and period polynomials of modular forms, using matrix constructions related to Brown's motivic coaction. It proves that left annihilators of a matrix $ C_{N,4} $ correspond to restricted even period polynomials, yielding a new upper bound for the rank of $ C_{N,4} $, which supports the uneven part of the motivic Broadhurst-Kreimer conjecture at depth 4.

ABSTRACT

We show that there is a relationship between modular forms and totally odd multiple zeta values, by relating the matrix $E_{N,r}$, whose entries are given by the polynomial representations of the Ihara action, with even period polynomials. We also consider the matrix $C_{N,r}$ defined by Brown and give a new upper bound of the rank of $C_{N,4}$. This result gives support to the uneven part of the motivic Broadhurst-Kreimer conjecture of depth 4.

Motivation & Objective

• To investigate the relationship between totally odd multiple zeta values (MZVs) and period polynomials of modular forms.
• To analyze the matrix $ C_{N,r} $, defined by Brown, which encodes $ \mathbb{Q} $-linear relations among totally odd MZVs of weight $ N $ and depth $ r $.
• To provide a new upper bound for the rank of $ C_{N,4} $, supporting the uneven part of the motivic Broadhurst-Kreimer conjecture at depth 4.
• To establish a correspondence between restricted even period polynomials and left annihilators of the matrix $ E_{N,r} $, a factor of $ C_{N,r} $.

Proposed method

• Define the matrix $ C_{N,r} $ using Brown's operator $ D_m $, which preserves the depth filtration of motivic MZVs.
• Introduce the matrix $ E_{N,r} $ as a right factor of $ C_{N,r} $, whose entries are derived from polynomial representations of the Ihara action.
• Construct the space $ \mathbf{W}_{N,r} $ of even, homogeneous polynomials satisfying a generalized version of the Eichler-Shimura relation.
• Establish an injection from $ \mathbf{W}_{N,r} $ to the left annihilators of $ E_{N,r} $, linking modular form theory to MZV relations.
• Use the dimension of $ \mathbf{W}_{N,r} $ to derive a lower bound for $ \dim_\mathbb{Q} \ker E_{N,r} $, and thus an upper bound for $ \operatorname{rank} C_{N,r} $.
• Apply the theory of Lyndon words and shuffle products to verify linear independence of certain MZV bases, confirming the validity of the bound.

Experimental results

Research questions

• RQ1 How are totally odd multiple zeta values related to period polynomials of modular forms?
• RQ2 Can the rank of the matrix $ C_{N,4} $, which encodes $ \mathbb{Q} $-linear relations among totally odd MZVs, be bounded using modular form theory?
• RQ3 Is there a structural correspondence between the left annihilators of $ C_{N,r} $ and restricted even period polynomials for $ r \geq 2 $?
• RQ4 Does the dimension of the space of restricted even period polynomials of degree $ N-2 $ match the dimension of the space of cusp forms for $ \Gamma_1 $?
• RQ5 Can the linear independence of a specific set of Lyndon basis elements in the shuffle algebra be established to confirm the rank bound for $ C_{N,4} $?

Key findings

• The space $ \mathbf{W}_{N,2} $ of restricted even period polynomials of degree $ N-2 $ has dimension equal to the dimension of the space of cusp forms for $ \Gamma_1 $, via the Eichler-Shimura-Manin correspondence.
• There is a one-to-one correspondence between $ \mathbf{W}_{N,2} $ and the left annihilators of $ E_{N,2} = C_{N,2} $, establishing a foundational link for depth 2.
• For $ r \geq 3 $, there is an injection from $ \mathbf{W}_{N,r} $ to the left annihilators of $ E_{N,r} $, generalizing the depth 2 result.
• The dimension of $ \mathbf{W}_{N,4} $ provides a lower bound for $ \dim_\mathbb{Q} \ker E_{N,4} $, which implies an upper bound for $ \operatorname{rank} C_{N,4} $.
• The set $ \Pi_N \cup \bigcup_{(N_1,N_2) \in S_{N,2}} \Pi_{N_1,N_2}^{(1)} \cup \bigcup_{(N_1,N_2) \in L_N^*} \Pi_{N_1,N_2}^{(2)} $ is linearly independent over $ \mathbb{Q} $, confirming the absence of shuffle product overlaps in the basis construction.
• The upper bound for $ \operatorname{rank} C_{N,4} $ derived from $ \dim \mathbf{W}_{N,4} $ supports the uneven part of the motivic Broadhurst-Kreimer conjecture at depth 4.