Newton polygons, successive minima, and different bounds for Drinfeld modules of rank $2$
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- by Imin Chen and Yoonjin Lee PDF
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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 $K$.References
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Additional Information
- Imin Chen
- Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
- MR Author ID: 609304
- Email: ichen@math.sfu.ca
- Yoonjin Lee
- Affiliation: Department of Mathematics, Ewha Womans University, Seoul, 120-750, Republic of Korea
- MR Author ID: 689346
- ORCID: 0000-0001-9510-3691
- Email: yoonjinl@ewha.ac.kr
- 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
- © Copyright 2012 American Mathematical Society
- 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
- MathSciNet review: 2988712