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On the field generated by the periods of a Drinfeld module

  • Ernst-Ulrich GekelerEmail author
Article
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Abstract

Generalizing the results of Maurischat in [4], we show that the field \(K_{\infty }(\Lambda )\) of periods of a Drinfeld module \(\phi \) of rank r defined over \(K_{\infty } = \mathbb {F}_{q}((T^{-1}))\) may be arbitrarily large over \(K_{\infty }\). We also show that, in contrast, the residue class degree \(f( K_{\infty }(\Lambda ) | K_{\infty })\) remains bounded by a constant that depends only on r.

Keywords

Drinfeld module Periods Torsion field 

Mathematics Subject Classification

11G09 11R58 11S20 

Notes

References

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    Gekeler, E.-U.: Towers of GL(r)-type of modular curves. J. Reine Angew. Mathematik (to appear).  https://doi.org/10.1515/crelle-2017-0012
  2. 2.
    Gekeler, E.-U.: On Drinfeld modular forms of higher rank II. J. Number Theory (to appear).  https://doi.org/10.1016/j.jnt.2018.11.011
  3. 3.
    Goss, D.: Basic Structures of Function Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35. Springer, Berlin (1996)Google Scholar
  4. 4.
    Maurischat, Andreas: On field extensions given by periods of Drinfeld modules. Arch. Math. (Basel) 113(3), 247–254 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Serre, Jean-Pierre: Corps Locaux. Hermann, Paris (1968)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of SaarlandSaarbrückenGermany

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