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Odd zeta motive and linear forms in odd zeta values

Part of: (Co)homology theory Arithmetic problems. Diophantine geometry Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  30 October 2017

Clément Dupont
Clément Dupont*
Affiliation:
Institut Montpelliérain Alexander Grothendieck, CNRS, Univ. Montpellier, France email [email protected]
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Abstract

We study a family of mixed Tate motives over $\mathbb{Z}$ whose periods are linear forms in the zeta values $\unicode[STIX]{x1D701}(n)$. They naturally include the Beukers–Rhin–Viola integrals for $\unicode[STIX]{x1D701}(2)$ and the Ball–Rivoal linear forms in odd zeta values. We give a general integral formula for the coefficients of the linear forms and a geometric interpretation of the vanishing of the coefficients of a given parity. The main underlying result is a geometric construction of a minimal ind-object in the category of mixed Tate motives over $\mathbb{Z}$ which contains all the non-trivial extensions between simple objects. In a joint appendix with Don Zagier, we prove the compatibility between the structure of the motives considered here and the representations of their periods as sums of series.

Keywords

zeta valuesmixed Tate motives

MSC classification

Primary: 11M06: $zeta (s)$ and $L(s, chi)$ 14F42: Motivic cohomology; motivic homotopy theory 14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture)
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 2 , February 2018 , pp. 342 - 379
Copyright
© The Author 2017 

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