Linear independence of monomials of multizeta values in positive characteristic. (English) Zbl 1306.11058

Chang, Chieh-Yu

Compos. Math. 150, No. 11, 1789-1808 (2014).

Multiple zeta values over a field of finite characteristic have been introduced by D. S. Thakur [Function field arithmetic. River Edge, NJ: World Scientific (2004; Zbl 1061.11001)]. They are analog of the classical MZV in the field of real numbers, going back to L. Euler. The author proves that they form a graded algebra, a result which is conjectured to hold also in the classical setting. This result occurs as a consequence of a similar statement for the values of Carlitz multiple polylogarithms (CMPL) at algebraic points, which the author introduces, generalizing the work of G. W. Anderson and D. S. Thakur [Int. Math. Res. Not. 2009, No. 11, 2038–2055 (2009; Zbl 1183.11052)]. The main strategy of the author is to use the criterion of [G. W. Anderson et al., Ann. Math. (2) 160, No. 1, 237–313 (2004; Zbl 1064.11055)]. The author points out that to obtain the more comprehensive algebraic independence results on multizeta values or nonzero values of CMPL at algebraic points using Anderson \(t\)-motives [G. W. Anderson, Duke Math. J. 53, 457–502 (1986; Zbl 0679.14001)] via M. A. Papanikolas’ theory [Invent. Math. 171, No. 1, 123–174 (2008; Zbl 1235.11074)], as in previous works on Carlitz zeta values [C.-Y. Chang and J. Yu, Adv. Math. 216, No. 1, 321–345 (2007; Zbl 1123.11025)] or Drinfeld logarithms at algebraic points [C.-Y. Chang and M. A. Papanikolas, J. Am. Math. Soc. 25, No. 1, 123–150 (2012; Zbl 1271.11079)], one needs to compute the dimension of relevant \(t\)-motivic Galois groups, still an open question in general.

Reviewer: Michel Waldschmidt (Paris)

MSC:

11J93	Transcendence theory of Drinfel’d and \(t\)-modules
11G09	Drinfel’d modules; higher-dimensional motives, etc.
11J81	Transcendence (general theory)
11J91	Transcendence theory of other special functions
11R58	Arithmetic theory of algebraic function fields

Keywords:

multizeta values; transcendence; linear independence; Carlitz multiple polylogarithms

Citations:

Zbl 1061.11001; Zbl 0679.14001; Zbl 1235.11074; Zbl 1123.11025; Zbl 1271.11079; Zbl 1183.11052; Zbl 1064.11055

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