Inventiones mathematicae

, Volume 171, Issue 1, pp 123–174 | Cite as

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



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.


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  1. 1.
    Anderson, G.W.: t-motives. Duke Math. J. 53, 457–502 (1986)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Anderson, G.W., Brownawell, W.D., Papanikolas, M.A.: Determination of the algebraic relations among special Γ-values in positive characteristic. Ann. Math. (2) 160, 237–313 (2004)MATHMathSciNetGoogle Scholar
  3. 3.
    Anderson, G.W., Thakur, D.S.: Tensor powers of the Carlitz module and zeta values. Ann. Math. (2) 132, 159–191 (1990)CrossRefMathSciNetGoogle Scholar
  4. 4.
    André, Y.: Différentielles non commutatives et théeorie de Galois différentielle ou aux différences. Ann. Sci. Éc. Norm. Supér., IV. Sér. 34, 685–739 (2001)MATHGoogle Scholar
  5. 5.
    Bertolin, C.: Périodes de 1-motifs et transcendance. J. Number Theory 97, 204–221 (2002)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Beukers, F.: Differential Galois theory. In: From Number Theory to Physics (Les Houches, 1989), pp. 413–439. Springer, Berlin (1992)Google Scholar
  7. 7.
    Böckle, G., Hartl, U.: Uniformizable families of t-motives. Trans. Am. Math. Soc. 359, 3933–3972 (2007)MATHGoogle Scholar
  8. 8.
    Breen, L.: Tannakian categories. In: Motives (Seattle, WA, 1991). Proc. Sympos. Pure Math., vol. 55, part 1, pp. 337–376. Am. Math. Soc., Providence, RI (1994)Google Scholar
  9. 9.
    Brownawell, W.D.: Transcendence in positive characteristic. In: Number Theory (Tiruchirapalli, 1996). Contemp. Math., vol. 210, pp. 317–332. Am. Math. Soc., Providence, RI (1998)Google Scholar
  10. 10.
    Deligne, P.: Catégories tannakiennes. In: The Grothendieck Festschrift, Vol. II. Progr. Math., vol. 87, pp. 111–195. Birkhäuser, Boston, MA (1990)Google Scholar
  11. 11.
    Deligne, P., Milne, J.S., Ogus, A., Shih, K.-Y.: Hodge Cycles, Motives, and Shimura Varieties. Lect. Notes Math., vol. 900. Springer, Berlin (1982)MATHGoogle Scholar
  12. 12.
    Denis, L.: Independence algébrique de logarithmes en caractéristique p. Bull. Austral. Math. Soc. 74, 461–470 (2006)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Fresnel, J., van der Put, M.: Rigid Analytic Geometry and its Applications. Birkhäuser, Boston (2004)MATHGoogle Scholar
  14. 14.
    Goss, D.: Drinfeld modules: cohomology and special functions. In: Motives (Seattle, WA, 1991). Proc. Sympos. Pure Math., vol. 55, part 2, pp. 309–362. Am. Math. Soc., Providence, RI (1994)Google Scholar
  15. 15.
    Goss, D.: Basic Structures of Function Field Arithmetic. Springer, Berlin (1996)MATHGoogle Scholar
  16. 16.
    Hartl, U., Pink, R.: Vector bundles with a Frobenius structure on the punctured unit disc. Compos. Math. 140, 689–716 (2004)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Kedlaya, K.S.: The algebraic closure of the power series field in positive characteristic. Proc. Am. Math. Soc. 129, 3461–3470 (2001)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kolchin, E.R.: Differential Algebra and Algebraic Groups. Academic Press, New York (1973)MATHGoogle Scholar
  19. 19.
    Lang, S.: Algebraic groups over finite fields. Am. J. Math. 78, 555–563 (1956)MATHCrossRefGoogle Scholar
  20. 20.
    Magid, A.R.: Lectures on Differential Galois Theory. Univ. Lect. Ser., vol. 7. Am. Math. Soc., Providence, RI (1994)MATHGoogle Scholar
  21. 21.
    Matsumura, H.: Commutative Algebra, 2nd edn. Benjamin/Cummings Publ., Reading, MA (1980)MATHGoogle Scholar
  22. 22.
    Matsumura, H.: Commutative Ring Theory. Cambridge University Press, Cambridge (1986)MATHGoogle Scholar
  23. 23.
    Pink, R.: Hodge structures for function fields. Preprint (1997)∼pink/Google Scholar
  24. 24.
    van der Put, M.: Galois theory of differential equations, algebraic groups and Lie algebras. J. Symb. Comput. 28, 441–472 (1999)MATHCrossRefGoogle Scholar
  25. 25.
    van der Put, M., Singer, M.F.: Galois Theory of Difference Equations. Lect. Notes Math., vol. 1666. Springer, Berlin (1997)MATHGoogle Scholar
  26. 26.
    van der Put, M., Singer, M.F.: Galois Theory of Linear Differential Equations. Springer, Berlin (2003)MATHGoogle Scholar
  27. 27.
    Ribenboim, P.: Fields: algebraically closed and others. Manuscr. Math. 75, 115–150 (1992)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Sinha, S.K.: Periods of t-motives and transcendence. Duke Math. J. 88, 465–535 (1997)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Taguchi, Y.: The Tate conjecture for t-motives. Proc. Am. Math. Soc. 123, 3285–3287 (1995)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Tamagawa, A.: The Tate conjecture and the semisimplicity conjecture for t-modules. RIMS Kokyuroku 925, 89–94 (1995)MathSciNetGoogle Scholar
  31. 31.
    Thakur, D.S.: Function Field Arithmetic. World Scientific Publishing, River Edge, NJ (2004)MATHGoogle Scholar
  32. 32.
    Waterhouse, W.C.: Introduction to Affine Group Schemes. Springer, New York (1979)MATHGoogle Scholar
  33. 33.
    Yu, J.: Analytic homomorphisms into Drinfeld modules. Ann. Math. (2) 145, 215–233 (1997)MATHCrossRefGoogle Scholar
  34. 34.
    Zariski, O., Samuel, P.: Commutative Algebra, Vol. II. Springer, New York (1975)Google Scholar

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© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

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