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Canonicalizing Zeta Generators: Genus Zero and Genus One.

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This study introduces zeta generators, linking Riemann zeta values to Lie algebra structures in Riemann surfaces. It establishes a canonical isomorphism for motivic multizeta values and resolves ambiguities in genus-one zeta generators.

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Area of Science:

  • Number Theory
  • Algebraic Geometry
  • Mathematical Physics

Background:

  • Zeta generators are derivations linked to odd Riemann zeta values.
  • They act on the Lie algebra of the fundamental group of Riemann surfaces.
  • Genus-zero zeta generators are Ihara derivations related to the Drinfeld associator.

Purpose of the Study:

  • To characterize canonical Lie polynomials and their non-Lie counterparts.
  • To propose a canonical isomorphism mapping motivic multizeta values to the f-alphabet.
  • To resolve ambiguities in genus-one zeta generators using representation theory.

Main Methods:

  • Characterization via the dual space of formal and motivic multizeta values.
  • Establishing canonical Lie polynomials from genus-zero setup to genus-one.
  • Expansion of genus-one zeta generators using Tsunogai's geometric derivations.
  • Imposing a representation-theoretic condition to resolve ambiguities.

Main Results:

  • Canonical polynomials and their non-Lie counterparts are characterized.
  • A canonical isomorphism between motivic multizeta values and the f-alphabet is proposed.
  • Canonical zeta generators in genus one are determined, acting on elliptic associators.
  • Genus-one zeta generators are systematically expanded, yielding explicit high-order computations.
  • Ambiguities in non-geometric parts of genus-one zeta generators are resolved.

Conclusions:

  • The work reveals a tight interplay between genus-zero and genus-one zeta generators.
  • This connection aids in constructing single-valued multiple polylogarithms on the sphere.
  • It links to iterated-Eisenstein-integral representations of modular graph forms.