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On the singular Drinfeld modules of rank 2. (English) Zbl 0765.11027

The theory of Drinfeld modules of rank 2 is extremely similar to that of elliptic curves. So, for instance, as one can reduce an elliptic curve over a number field modulo primes of that number field, so one can also reduce Drinfeld modules modulo primes of global function fields. In particular, one then obtains Drinfeld modules over “finite fields”. Thus, one can then discuss “singular” and “supersingular” Drinfeld modules. In the paper under review, the authors use these concepts to establish analogs of the classical results of Deuring.

MSC:

11G09 Drinfel’d modules; higher-dimensional motives, etc.
11R58 Arithmetic theory of algebraic function fields
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References:

[1] [D-H] Deligne, P., Husemoller, D.: Survey of Drinfeld Modules. Contemp. Math.67, 25–91 (1987)
[2] [G] Gekeler, E.: Zur Arithmetik von Drinfeld Moduln. Math. Ann.262, 167–182 (1983) · Zbl 0536.14028 · doi:10.1007/BF01455309
[3] [H] Hayes, D.: Explicit Class Field Theory in Global Function Fields. In: Rota, G.C. (ed.) Studies in Algebra and Number Theory. New York: Academic Press 1979
[4] [L] Lang, S.: Elliptic Functions. (Grad. Texts Math., vol. 112) Berlin Heidelberg New York: Springer 1987
[5] [Sh] Shimura, G.: Introduction to the Arithmetic Theory of Automorphic Functions. (Publ. Math. Soc. Japan, vol. 11) Tokyo Princeton: University Press 1971 · Zbl 0221.10029
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