Alternating multizeta values in positive characteristic
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Mathematische Zeitschrift
Alternating multizeta values in positive characteristic Ryotaro Harada1 Received: 11 September 2019 / Accepted: 21 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We introduce alternating multizeta values in positive characteristic which are generalizations of Thakur multizeta values. We establish their fundamental properties including non-vanishing, sum-shuffle relations, period interpretation and linear independence which is a direct sum result for these values. Keywords Multizeta values · Non-vanishing · Sum-shuffle relation · Pre-t-motive · Linear independence Mathematics Subject Classification Primary 11M38 · 11J72 · 11J93
Contents 0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1 Notations and definitions . . . . . . . . . . . . . . . . . . . 2 Fundamental properties . . . . . . . . . . . . . . . . . . . 2.1 Non-vanishing property of AMZVs . . . . . . . . . . . 2.2 Sum-shuffle relation for AMZVs . . . . . . . . . . . . 3 Period interpretation of AMZVs . . . . . . . . . . . . . . . 3.1 Review of pre-t-motive . . . . . . . . . . . . . . . . . 3.2 Normalized AMZVs as periods . . . . . . . . . . . . . 4 Linear independence of monomials of AMZVs . . . . . . . 4.1 ABP-criterion . . . . . . . . . . . . . . . . . . . . . . 4.2 MZ property for AMZVs . . . . . . . . . . . . . . . . Appendix: Explicit sum-shuffle relations in a lower depth case References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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This work was supported by JSPS Overseas Challenge Program for Young Researchers, JSPS KAKENHI Grant Number JP18J15278 and National Center for Theoretical Sciences, Taiwan.
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Ryotaro Harada [email protected] National Center for Theoretical Sciences, National Tsing Hua University, No. 101, Kuang-Fu Road Sec. 2, Hsinchu 300, Taiwan
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R. Harada
0 Introduction Classical multizeta values (characteristic 0 case) are originated from the research in Euler’s work [16] in 1776. He introduced them (originally, r = 2 case) as the following infinite series: 1 ζ (s) := s1 sr ∈ R n · 1 · · nr n >···>n r
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where s := (s1 , . . . , sr ) ∈ Nr and s1 > 1.1 Moreover, r is called the depth and ri=1 si is called the weight of the presentation of ζ (s1 , . . . , sr ). In the last quarter century, it got known that they have connection to number theory [15,34], knot theory [24], quantum field theory [8] and so on. It is also
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