Abstract
In this short note, we answer a question raised by M. Papikian on a universal upper bound for the degree of the extension of \(K_\infty \) given by adjoining the periods of a Drinfeld module of rank 2. We show that contrary to the rank 1 case such a universal upper bound does not exist, and the proof generalies to higher rank. Moreover, we give an upper and lower bound for the extension degree depending on the valuations of the defining coefficients of the Drinfeld module. In particular, the lower bound shows the non-existence of a universal upper bound.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Deligne, P., Husemoller, D.: Survey of Drinfeld modules (Arcata, Calif., 1985). In: Current Trends in Arithmetical Algebraic Geometry. Contemporary Mathematics, vol. 67, pp. 25–91. American Mathematical Society, Providence, RI (1987)
Drinfel’d, V.G.: Elliptic modules. Mat. Sb. (N.S.) 94(136), 594–627, 656 (1974)
Drinfel’d, V.G.: Elliptic modules. II. Mat. Sb. (N.S.) 102(144), 182–194, 325 (1977)
Gekeler, E.-U.: The Galois image of twisted Carlitz modules. J. Number Theory 163, 316–330 (2016)
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)
Thakur, D.S.: Function Field Arithmetic. World Scientific Publishing Co., Inc, River Edge, NJ (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Maurischat, A. On field extensions given by periods of Drinfeld modules. Arch. Math. 113, 247–254 (2019). https://doi.org/10.1007/s00013-019-01339-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-019-01339-0