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

Papanikolas, Matthew A.

Invent. Math. 171, No. 1, 123-174 (2008).

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.

Reviewer: Lenny Taelman (Leiden)

MSC:

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

Keywords:

Drinfeld modules; \(t\)-motives; periods; transcendence; Tannakian category

Citations:

Zbl 1064.11055

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