Rational structures on multiple zeta values

Thesis

Motivated originally by the question of defining a rational canonical associator, we study rational structures associated to multiple zeta values. In particular, we focus on the question of providing an explicit description of the motivic Lie algebra associated to the projective line minus three points via new families of motivic relations among multiple zeta values.

Inspired by results obtained by considering depth-graded multiple zeta values, we attempt a similar approach. We introduce the block filtration on the space of multiple zeta values and show that it agrees with the coradical filtration induced by the motivic coaction. By considering the associated graded Lie algebra of the motivic Lie algebra with respect to this filtration, we obtain an isomorphic Lie algebra bg with canonical representatives for its generators in Q0,e₁>. This provides a possible route to defining canonical generators of the motivic Lie algebra by finding a section of the projection induced by this isomorphism.

We then consider relations among block graded motivic multiple zeta values, finding several new families of relations and providing a complete description of bg in low block degree. We use the motivic coaction to lift these relations to genuine relations among motivic multiple zeta values, providing new families of relations and generalising previously known relations such as those due to Borwen, Bradley, Broadhurst and Lisonek.

Finally, we consider the implication of previously known relations on the p-adic valuation of coeffcients of a rational associator, providing a bound on the growth in terms of weight and block degree. We also present a partial solution to the question of canonical generators via the introduction of an inner product.

Authors: Keilthy, A

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: motivic periods; multiple zeta values; number theory; shuffle algebra
• Subjects: Mathematics