Interpolation of hypergeometric ratios in a global field of positive characteristic

Greg W. Anderson[1]

  • [1] University of Minnesota School of Mathematics Minneapolis, MN 55455 (USA)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 5, page 1655-1687
  • ISSN: 0373-0956

Abstract

top
For each global field of positive characteristic we exhibit many examples of two-variable algebraic functions possessing properties consistent with a conjectural refinement of the Stark conjecture in the function field case recently proposed by the author. All the examples are Coleman units. We obtain our results by studying rank one shtukas in which both zero and pole are generic, i. e., shtukas not associated to any Drinfeld module.

How to cite

top

Anderson, Greg W.. "Interpolation of hypergeometric ratios in a global field of positive characteristic." Annales de l’institut Fourier 57.5 (2007): 1655-1687. <http://eudml.org/doc/10274>.

@article{Anderson2007,
abstract = {For each global field of positive characteristic we exhibit many examples of two-variable algebraic functions possessing properties consistent with a conjectural refinement of the Stark conjecture in the function field case recently proposed by the author. All the examples are Coleman units. We obtain our results by studying rank one shtukas in which both zero and pole are generic, i. e., shtukas not associated to any Drinfeld module.},
affiliation = {University of Minnesota School of Mathematics Minneapolis, MN 55455 (USA)},
author = {Anderson, Greg W.},
journal = {Annales de l’institut Fourier},
keywords = {Shtuka; hypergeometric ratio; Coleman unit; Catalan-Drinfeld symbol},
language = {eng},
number = {5},
pages = {1655-1687},
publisher = {Association des Annales de l’institut Fourier},
title = {Interpolation of hypergeometric ratios in a global field of positive characteristic},
url = {http://eudml.org/doc/10274},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Anderson, Greg W.
TI - Interpolation of hypergeometric ratios in a global field of positive characteristic
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 5
SP - 1655
EP - 1687
AB - For each global field of positive characteristic we exhibit many examples of two-variable algebraic functions possessing properties consistent with a conjectural refinement of the Stark conjecture in the function field case recently proposed by the author. All the examples are Coleman units. We obtain our results by studying rank one shtukas in which both zero and pole are generic, i. e., shtukas not associated to any Drinfeld module.
LA - eng
KW - Shtuka; hypergeometric ratio; Coleman unit; Catalan-Drinfeld symbol
UR - http://eudml.org/doc/10274
ER -

References

top
  1. G. W. Anderson, Rank one elliptic A -modules and A -harmonic series, Duke Mathematical Journal 73 (1994), 491-542 Zbl0807.11032MR1262925
  2. G. W. Anderson, A two-variable refinement of the Stark conjecture in the function field case, Compositio Mathematica 142 (2006), 563-615 Zbl1127.11044MR2231193
  3. G. W. Anderson, Interpolation of numbers of Catalan type in a local field of positive characteristic, (to appear) Zbl1133.11041
  4. G. W. Anderson, W. D. Brownawell, M. A. Papanikolas, Determination of the algebraic relations among special Γ -values in positive characteristic, Annals of Mathematics (2) 160 (2004), 237-313 Zbl1064.11055MR2119721
  5. R. F. Coleman, On the Frobenius endomorphisms of Fermat and Artin-Schreier curves, Proceedings of the American Mathematical Society 102 (1988), 463-466 Zbl0666.14014MR928961
  6. V. G. Drinfeld, Commutative subrings of some noncommutative rings, Functional Analysis and Applications 11 (1977), 9-12 Zbl0368.14011MR476732
  7. D. Goss, Basic structures of function field arithmetic, 35 (1996), Springer Verlag, Berlin Zbl0874.11004MR1423131
  8. D. Mumford, An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-deVries equation, and related nonlinear equations, Proceedings of the International Symposium on Algebraic Geometry, Kyoto University, Kyoto, 1977 (1978), 115-153, Kinokuniya Book Store, Tokyo Zbl0423.14007MR578857
  9. J.-P. Serre, Algebraic Groups and Class Fields, 117 (1988), Springer, New York Zbl0703.14001MR918564
  10. J. T. Tate, Number-theoretic background, Proceedings of Symposia in Pure Mathematics 33 (1979), 3-26 Zbl0422.12007MR546607
  11. D. S. Thakur, Gamma functions for function fields and Drinfeld modules, Annals of Mathematics 134 (1991), 25-64 Zbl0734.11036MR1114607
  12. D. S. Thakur, Drinfeld modules and arithmetic in the function fields, International Mathematics Research Notices (1992), 185-197 Zbl0756.11015MR1185833
  13. D. S. Thakur, Shtukas and Jacobi sums, Inventiones Mathematicae 111 (1993), 557-570 Zbl0770.11032MR1202135
  14. D. S. Thakur, Function Field Arithmetic, (2004), World Scientific Publishing Co., River Edge, NJ Zbl1061.11001MR2091265
  15. A. Weil, Basic Number Theory, 144 (1974), Springer-Verlag, New York-Berlin Zbl0326.12001MR427267

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.