Tannakian duality for Anderson–Drinfeld motives and algebraic independence of Carlitz logarithms

By Matthew A. Papanikolas. 2005

173 citations recorded by the source corpus.

We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7a3f4d95dc1e94bf2b144aadd9f1b8e040b57bc3

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In Computational Theory and Mathematics

1. Bruno Anglès
2. F. Pellarin
3. Floric Tavares-Ribeiro
4. Jing Yu
5. Chieh-Yu Chang

In Geometry and Topology

1. Bruno Anglès
2. Floric Tavares-Ribeiro
3. F. Pellarin
4. Jing Yu
5. Chieh-Yu Chang

Citations in the corpus

• Determination of algebraic relations among special zeta values in positive characteristic (2007, methodology): “ALGEBRAIC RELATIONS AMONG SPECIAL ZETA VALUES 9 Proof. (cf. Proposition 7.1.3 of [ 5 ]).”
• Values of certain L-series in positive characteristic (2011, methodology): “We refer to [17] for a description of the main properties of the functions sCar,Ω, or to the papers [1, 4, 16], where Ω was originally introduced and studied.”
• Linear independence of monomials of multizeta values in positive characteristic (2012, methodology): “nd hence appeal to Theorem 3.4.5 showing Theorem 5.4.3. 4 CHIEH-YU CHANG We mention that the tools of proving algebraic independence using t-motives introduced by Anderson [A86] come from Papanikolas [P08], which can be regarded as a function field analogue of Grothendieck’s periods conjecture. Using”
• Arithmetic of positive characteristic $L$-series values in Tate algebras (2014, result): “not divisible by q−1 and by the characteristic pare algebraically independent over K (see the work of Chieh-Yu Chang and Jing Yu [10], where they use the Tannaka approach introduced by Papanikolas in [17] and the algebraic independence theory of Anderson, Brownawell, Papanikolas in [1]). This result”
• An effective criterion for Eulerian multizeta values in positive characteristic (2014, unknown): “(t−θ)s1+···+s r−1m 1,···,(t−θ)m 1,m 1,...,(t −θ)s −1mr,...,(t −θ)mr,mr o is a k[σ]-basis of M′. We further observe that (t −θ)NM′/σM′= (0) for N ≫0 and hence M′is an Anderson t-motive in the sense of [P08], which is called a dual t-motive in [ABP04]. For such M′, we can identity M′/(σ −1)M′with the ”