Finiteness of an Isogeny Class of Drinfeld Modules—Correction to a Previous Paper
✍ Scribed by Yuichiro Taguchi
- Publisher
- Elsevier Science
- Year
- 1999
- Tongue
- English
- Weight
- 141 KB
- Volume
- 74
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
✦ Synopsis
In this note, we prove the finiteness of an isogeny class of Drinfeld A-modules over a finite extension of the fraction field of A. This contradicts Remark (3.4(ii-1)) of my previous paper [4]. This is because Proposition (3.1) is wrong, on which Section 3 of that paper was based (the wrong point is in the ``because'' of the second to last sentence of the proof). Besides, the property (ii) of Example (2.2) fails, which made use of a result in Section 3 (at present, its (in)validity is not known). The following replaces the whole Section 3 (except for Remarks (3.3) and (3.5)). To have a correct statement along the line of Proposition (3.1), one would need a more delicate analysis of Galois representations on division points of Drinfeld modules.
The strategy of our proof in this note (Section 1) is similar to that of Faltings' [0], but in our case, we have to check that the Galois action on the determinants of certain subgroups of a Drinfeld module is just ``as expected''; to do so, we shall show (Section 2) that a truncated v-divisible group is liftable to a v-divisible group (cf. [1]).
I thank R. Pink and A. Tamagawa for pointing out the error, and apologize to those who may be concerned. I also thank D. Goss for encouragement. Finally, I thank the referee for careful reading and many helpful comments.
1 Notation and convention are basically the same as in the ``semi-simplicity'' paper [3]. Let F be an algebraic function field in one variable over