Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms. (English) Zbl 1235.11074

This paper is a milestone in transcendence theory in characteristic \(p\). The author shows that every algebraic relation between periods of \(t\)-motives is “motivic”, that is, it arises from a relation between \(t\)-motives. This is a characteristic \(p\) version of Grothendieck’s period conjecture.
Previously, D. Brownawell, G. Anderson and the author had shown that any linear relation between periods of a \(t\)-motive is motivic [Ann. Math. (2) 160, No. 1, 237–313 (2004; Zbl 1064.11055)]. Since the category of \(t\)-motives is closed under tensor product, one may expect to deduce results on algebraic independence from this.
To do so, the author develops a Tannakian machinery for \(t\)-motives, loosely based on similar approaches for differential and difference Galois theory. As a consequence he associates to every \(t\)-motive an algebraic group: its Tannakian Galois group (to be compared with the motivic fundamental group or the Mumford-Tate group of a motive). He shows that the transcendence degree of the set of periods of a \(t\)-motive is the dimension of its Tannakian Galois group.
To illustrate the spectacular power of his theorem, the author determines all algebraic relations between Carlitz logarithms of algebraic points.


11J93 Transcendence theory of Drinfel’d and \(t\)-modules
11G09 Drinfel’d modules; higher-dimensional motives, etc.


Zbl 1064.11055
Full Text: DOI arXiv


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