EXPLICIT SURJECTIVITY RESULTS FOR DRINFELD MODULES OF RANK 2

Imin Chen, Yoonjin Lee

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Abstract

Let K = F q (T) and A = F q [T]. Suppose that φ is a Drinfeld A-module of rank 2 over K which does not have complex multiplication. We obtain an explicit upper bound (dependent on φ) on the degree of primes p of K such that the image of the Galois representation on the p-torsion points of φ is not surjective, in the case of q odd. Our results are a Drinfeld module analogue of Serre's explicit large image results for the Galois representations on p-torsion points of elliptic curves (Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331; Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Etudes Sci. Publ. Math. 54 (1981), 323-401.) and are unconditional because the generalized Riemann hypothesis for function fields holds. An explicit isogeny theorem for Drinfeld A-modules of rank 2 over K is also proven.

Original languageEnglish
Pages (from-to)17-45
Number of pages29
JournalNagoya Mathematical Journal
Volume234
DOIs
StatePublished - 1 Jun 2019

Bibliographical note

Funding Information:
Imin Chen is supported by an NSERC Discovery Grant and Yoonjin Lee is a corresponding author and supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827) and also by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2017R1A2B2004574).

Publisher Copyright:
© 2019 Foundation Nagoya Mathematical Journal.

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