Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Algebraic independence of periods and logarithms of Drinfeld modules


Authors: Chieh-Yu Chang and Matthew A. Papanikolas; with an appendix by Brian Conrad
Journal: J. Amer. Math. Soc. 25 (2012), 123-150
MSC (2010): Primary 11J93; Secondary 11G09, 11J89
DOI: https://doi.org/10.1090/S0894-0347-2011-00714-5
Published electronically: August 1, 2011
MathSciNet review: 2833480
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \rho$ be a Drinfeld $ A$-module with generic characteristic defined over an algebraic function field. We prove that all of the algebraic relations among periods, quasi-periods, and logarithms of algebraic points on $ \rho$ are those coming from linear relations induced by endomorphisms of $ \rho$.


References [Enhancements On Off] (What's this?)

  • 1. G. W. Anderson, $ t$-motives, Duke Math. J. 53 (1986), 457-502. MR 850546 (87j:11042)
  • 2. G. W. Anderson, W. D. Brownawell, and M. A. Papanikolas, Determination of the algebraic relations among special $ \Gamma$-values in positive characteristic, Ann. of Math. (2) 160 (2004), 237-313. MR 2119721 (2005m:11140)
  • 3. A. Baker and G. Wüstholz, Logarithmic forms and Diophantine geometry, Cambridge University Press, Cambridge, 2007. MR 2382891 (2009e:11001)
  • 4. W. D. Brownawell, Minimal extensions of algebraic groups and linear independence, J. Number Theory 90 (2001), 239-254. MR 1858075 (2002h:11066)
  • 5. W. D. Brownawell and M. A. Papanikolas, Linear independence of Gamma-values in positive characteristic, J. reine angew. Math. 549 (2002), 91-148. MR 1916653 (2003g:11080)
  • 6. C.-Y. Chang, On periods of the third kind for rank $ 2$ Drinfeld modules, preprint.
  • 7. C.-Y. Chang and M. A. Papanikolas, Algebraic relations among periods and logarithms of rank $ 2$ Drinfeld modules, Amer. J. Math. 133 (2011), 359-391.
  • 8. B. Conrad, O. Gabber, and G. Prasad, Pseudo-reductive groups, New Math. Monographs 17, Cambridge University Press, 2010. MR 2723571
  • 9. S. David and L. Denis, Périodes de modules de Drinfeld ``l'indépendance quadratique en rang II,'' J. Ramanujan Math. Soc. 17 (2002), 65-83. MR 1906421 (2003i:11103)
  • 10. P. Deligne, J. S. Milne, A. Ogus, and K.-Y. Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin, 1982. MR 654325 (84m:14046)
  • 11. V. G. Drinfeld, Elliptic modules, Math. USSR-Sb. 23 (1974), 561-592. MR 0384707 (52:5580)
  • 12. B. Farb and R. K. Dennis, Noncommutative algebra, Springer-Verlag, New York, 1993. MR 1233388 (94j:16001)
  • 13. E.-U. Gekeler, On the de Rham isomorphism for Drinfeld modules, J. reine angew. Math. 401 (1989), 188-208. MR 1018059 (90g:11070)
  • 14. D. Goss, Basic structures of function field arithmetic, Springer-Verlag, Berlin, 1996. MR 1423131 (97i:11062)
  • 15. C. Hardouin, Computation of the Galois groups occurring in M. Papanikolas's study of Carlitz logarithms, arXiv:0906.4429.
  • 16. A.-K. Juschka, The Hodge conjecture for function fields, Diplomarbeit, Universität Münster, 2010.
  • 17. M. A. Papanikolas, Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math. 171 (2008), 123-174. MR 2358057 (2009b:11127)
  • 18. F. Pellarin, Aspects de l'indépendance algébrique en caractéristique non nulle, Séminaire Bourbaki, Vol. 2006/2007, Astérisque No. 317 (2008), Exp. no. 973, viii, 205-242. MR 2487735 (2010c:11086)
  • 19. R. Pink, Hodge structures over function fields, preprint, 1997, http://www.math. ethz.ch/$ \sim$pink/.
  • 20. R. Pink, The Mumford-Tate conjecture for Drinfeld modules, Publ. Res. Inst. Math. Sci. 33 (1997), 393-425. MR 1474696 (98f:11062)
  • 21. G. Prasad and J.-K. Yu, On quasi-reductive group schemes. With an appendix by B. Conrad, J. Algebraic Geom. 15 (2006), 507-549. MR 2219847 (2007c:14047)
  • 22. D. S. Thakur, Function field arithmetic, World Scientific Publishing, River Edge, NJ, 2004. MR 2091265 (2005h:11115)
  • 23. A. Thiery, Indépendance algébrique des périodes et quasi-périodes d'un module de Drinfeld, The arithmetic of function fields (Columbus, OH, 1991), de Gruyter, Berlin (1992), 265-284. MR 1196524 (93j:11034)
  • 24. M. Waldschmidt, Elliptic functions and transcendence, Surveys in Number Theory, Dev. Math. 17 (2008), 143-188. MR 2462949 (2010a:11142)
  • 25. J. Yu, Transcendence and Drinfeld modules, Invent. Math. 83 (1986), 507-517. MR 827364 (87g:11088)
  • 26. J. Yu, On periods and quasi-periods of Drinfeld modules, Compositio Math. 74 (1990), 235-245. MR 1055694 (91i:11089)
  • 27. J. Yu, Analytic homomorphisms into Drinfeld modules, Ann. of Math. (2) 145 (1997), 215-233. MR 1441876 (98c:11054)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 11J93, 11G09, 11J89

Retrieve articles in all journals with MSC (2010): 11J93, 11G09, 11J89


Additional Information

Chieh-Yu Chang
Affiliation: Department of Mathematics, National Tsing Hua University, No. 101, Sec. 2, Kuang Fu Road, Hsinchu City 30042, Taiwan, Republic of China and National Center for Theoretical Sciences, Hsinchu City 30042, Taiwan, Republic of China
Email: cychang@math.cts.nthu.edu.tw

Matthew A. Papanikolas
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843 U.S.A.
Email: map@math.tamu.edu

Brian Conrad
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305 U.S.A.
Email: conrad@math.stanford.edu

DOI: https://doi.org/10.1090/S0894-0347-2011-00714-5
Keywords: Algebraic independence, Drinfeld modules, periods, logarithms
Received by editor(s): May 27, 2010
Received by editor(s) in revised form: May 23, 2011
Published electronically: August 1, 2011
Additional Notes: The first author was supported by an NCTS postdoctoral fellowship.
The second author was supported by NSF Grant DMS-0903838.
The author of the appendix was supported by NSF grant DMS-0917686.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society