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Newton polygons, successive minima, and different bounds for Drinfeld modules of rank $ 2$


Authors: Imin Chen and Yoonjin Lee
Journal: Proc. Amer. Math. Soc. 141 (2013), 83-91
MSC (2010): Primary 11G09; Secondary 11R58
DOI: https://doi.org/10.1090/S0002-9939-2012-11300-0
Published electronically: May 4, 2012
MathSciNet review: 2988712
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Abstract: Let $ K = \mathbb{F}_q(T)$. For a Drinfeld $ A$-module $ \phi $ of rank $ 2$ defined over $ C_\infty $, there are an associated exponential function $ e_\phi $ and lattice $ \Lambda _\phi $ in $ C_\infty $ given by uniformization over $ C_\infty $. We explicitly determine the Newton polygons of $ e_\phi $ and the successive minima of $ \Lambda _{\phi }$. When $ \phi $ is defined over $ K_\infty $, we give a refinement of Gardeyn's bounds for the action of wild inertia at $ \infty $ on the torsion points of $ \phi $ and a criterion for the lattice field to be unramified over $ K_\infty $. If $ \phi $ is in addition defined over $ K$, we make explicit Gardeyn's bounds for the action of wild inertia at finite primes on the torsion points of $ \phi $, using results of Rosen, and this gives an explicit bound on the degree of the different divisor of division fields of $ \phi $ over $ K$.


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Additional Information

Imin Chen
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: ichen@math.sfu.ca

Yoonjin Lee
Affiliation: Department of Mathematics, Ewha Womans University, Seoul, 120-750, Republic of Korea
Email: yoonjinl@ewha.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-2012-11300-0
Keywords: Drinfeld modules, Newton polygons, exponential functions, minimal bases
Received by editor(s): April 9, 2011
Received by editor(s) in revised form: June 7, 2011
Published electronically: May 4, 2012
Additional Notes: The first-named author was supported by NSERC
The second-named author is the corresponding author and was supported by Priority Research Centers Program through the NRF funded by the Ministry of Education, Science and Technology (2010-0028298) and by the NRF grant funded by the Korea government (MEST) (No. 2011-0015684)
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 American Mathematical Society

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