Deligne’s reciprocity for function fields. (English) Zbl 0932.11040
In 1982, guided by conjectures on special values of \(L\)-functions, P. Deligne [Lect. Notes Math. 900, 9-100 (1982; Zbl 0537.14006)]proved the reciprocity law for the Euler \(\Gamma\)-function as an application of results on Hodge cycles on abelian varieties. In this paper the author presents an analogue of this reciprocity law in the context of the arithmetic of function fields and the theory of Drinfeld modules. Here the reciprocity law is proved for the geometric \(\Gamma\)-function, based on the Carlitz exponential, and uses Anderson’s theory of characteristic \(p\)-solitons.
The author also obtains a finer version of the reciprocity law, which addresses questions of rationality and is the analogue of one of the conjectures in P. Deligne’s paper [Proc. Symp. Pure Math. 33, No. 2, 313-346 (1979; Zbl 0449.10022)].
The author also obtains a finer version of the reciprocity law, which addresses questions of rationality and is the analogue of one of the conjectures in P. Deligne’s paper [Proc. Symp. Pure Math. 33, No. 2, 313-346 (1979; Zbl 0449.10022)].
Reviewer: Barry Green (Stellenbosh)
MSC:
| 11G09 | Drinfel’d modules; higher-dimensional motives, etc. |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11R58 | Arithmetic theory of algebraic function fields |
Keywords:
reciprocity law; arithmetic of function fields; Drinfeld modules; geometric \(\Gamma\)-function; Carlitz exponential; characteristic \(p\)-solitonsReferences:
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