Block graded relations among motivic multiple zeta values

Abstract: Multiple zeta values, originally considered by Euler, generalise the Riemann zeta function to multiple variables.

While values of the Riemann zeta function at odd positive integers are conjectured to be algebraically independent, multiple zeta values satisfy many algebraic and linear relations, even forming a $\mathbb{Q}$-algebra. While families of well understood relations are known, such as the associator relations and double shuffle relations, they only conjecturally span all algebraic relations. Since multiple zeta values arise as the periods of mixed Tate motives, we obtain further algebraic structures, which have been exploited to provide spanning sets by Brown. In this talk we will aim to define a new set of relations, known to be complete in low block degree.

To achieve this, we will first review the necessary algebraic set up, focusing particularly on the motivic Lie algebra associated to the thrice punctured projective line. We then introduce a new filtration on the algebra of (motivic) multiple zeta values, called the block filtration, based on the work of Charlton. By considering the associated graded algebra, we quickly obtain a new family of graded motivic relations, which can be shown to span all algebraic relations in low block degree. We will also touch on some conjectural ungraded "lifts" of these relations, and if we have time, compare to similar approaches using the depth filtration.

algebraic geometrynumber theory

Audience: researchers in the topic

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Rendez-vous on special values and periods

Series comments: The main objective of this conference is to gather together young researchers interested in special values of L-functions and periods. These objects are at the crossroads of many recent important developments in arithmetic geometry, such as Euler systems or the theory of motives. The different talks will portray the variety of viewpoints with which L-functions and periods are studied at present.

Registration is free and mandatory, to get access to the livestream and recording of the talks.

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