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Linear independence of monomials of multizeta values in positive characteristic

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  27 August 2014

Chieh-Yu Chang
Chieh-Yu Chang*
Affiliation:
Department of Mathematics, National Tsing Hua University and National Center for Theoretical Sciences, Hsinchu City 30042, Taiwan, R.O.C. email [email protected]
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Abstract

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In this paper, we study transcendence theory for Thakur multizeta values in positive characteristic. We prove an analogue of the strong form of Goncharov’s conjecture. The same result is also established for Carlitz multiple polylogarithms at algebraic points.

Keywords

multizeta valuestranscendencelinear independencecarlitz multiple polylogarithms

MSC classification

Primary: 11J91: Transcendence theory of other special functions 11J93: Transcendence theory of Drinfel'd and $t$-modules
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 11 , November 2014 , pp. 1789 - 1808
Copyright
© The Author 2014 

References

Anderson, G. W., t-motives, Duke Math. J. 53 (1986), 457–502.CrossRefGoogle Scholar
Anderson, G. W., Brownawell, W. D. and Papanikolas, M. A., Determination of the algebraic relations among special Γ-values in positive characteristic, Ann. of Math. (2) 160 (2004), 237–313.Google Scholar
Anderson, G. W. and Thakur, D. S., Tensor powers of the Carlitz module and zeta values, Ann. of Math. (2) 132 (1990), 159–191.Google Scholar
Anderson, G. W. and Thakur, D. S., Multizeta values for Fq[t], their period interpretation, and relations between them, Int. Math. Res. Not. IMRN 2009 (2009), 2038–2055.Google Scholar
André, Y., Une introduction aux motifs, in Motifs purs, motifs mixtes, périodes, Panoramas et Synthéses, vol. 17 (Société Mathématique de France, Paris, 2004).Google Scholar
Brown, F., Mixed Tate motives over ℤ, Ann. of Math. (2) 175 (2012), 949–976.CrossRefGoogle Scholar
Baker, A. and Wüstholz, G., Logarithmic forms and Diophatine geometry (Cambridge University Press, Cambridge, 2007).Google Scholar
Carlitz, L., On certain functions connected with polynomials in a Galois field, Duke Math. J. 1 (1935), 137–168.Google Scholar
Chang, C.-Y., A note on a refined version of Anderson–Brownawell–Papanikolas criterion, J. Number Theory 129 (2009), 729–738.Google Scholar
Chang, C.-Y. and Papanikolas, M. A., Algebraic independence of periods and logarithms of Drinfeld modules. With an appendix by Brian Conrad, J. Amer. Math. Soc. 25 (2012), 123–150.Google Scholar
Chang, C.-Y. and Yu, J., Determination of algebraic relations among special zeta values in positive characteristic, Adv. Math. 216 (2007), 321–345.Google Scholar
Goncharov, A. B., The double logarithm and Manin’s complex for modular curves, Math. Res. Lett. 4 (1997), 617–636.Google Scholar
Goss, D., Basic structures of function field arithmetic (Springer, Berlin, 1996).CrossRefGoogle Scholar
Lara Rodriguez, J. A., Relations between multizeta values in characteristic p, J. Number Theory 131 (2011), 2081–2099.Google Scholar
Mishiba, Y., On algebraic independence of certain multizeta values in characteristic $p$, Preprint (2014), arXiv:1401.3628 [math.NT].Google Scholar
Papanikolas, M. A., Tannakian duality for Anderson–Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math. 171 (2008), 123–174.Google Scholar
Thakur, D. S., Function field arithmetic (World Scientific Publishing, River Edge, NJ, 2004).Google Scholar
Thakur, D. S., Power sums with applications to multizeta and zeta zero distribution for Fq[t], Finite Fields Appl. 15 (2009), 534–552.Google Scholar
Thakur, D. S., Relations between multizeta values for Fq[T], Int. Math. Res. Notices IMRN 2009 (2009), 2318–2346.Google Scholar
Thakur, D. S., Shuffle relations for function field multizeta values, Int. Math. Res. Not. IMRN 2010 (2010), 1973–1980.Google Scholar
Waldschmidt, M., Multiple polylogarithms: an introduction, in Number theory and discrete mathematics, Chandigarh, 2000, Trends in Mathematics (Birkhäuser, Basel, 2002), 1–12.Google Scholar
Wüstholz, G., Multiplicity estimates on group varieties, Ann. of Math. (2) 129 (1989), 471–500.Google Scholar
Wüstholz, G., Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen, Ann. of Math. (2) 129 (1989), 501–517.Google Scholar
Yu, J., Transcendence and special zeta values in characteristic p, Ann. of Math. (2) 134 (1991), 1–23.Google Scholar
Yu, J., Analytic homomorphisms into Drinfeld modules, Ann. of Math. (2) 145 (1997), 215–233.Google Scholar
Zagier, D., Evaluation of the multiple zeta values 𝜁(2, …, 2, 3, 2, …, 2), Ann. of Math. (2) 175 (2012), 977–1000.Google Scholar