Algebraic Hecke Characters and Compatible Systems of Abelian Mod P Galois Representations Over Global fields
Algebraic Hecke characters and compatible systems of abelian mod p Galois representations over global fields Gebhard B¨ockle January 28, 2010 Abstract In [Kh1, Kh2, Kh3], C. Khare showed that any strictly compatible systems of semisimple abelian mod p Galois representations of a number field arises from a unique finite set of algebraic Hecke Characters. In this article, we consider a similar problem for arbitrary global fields. We give a definition of Hecke character which in the function field setting is more general than previous definitions by Goss and Gross and define a corresponding notion of compatible system of mod p Galois representations. In this context we present a unified proof of the analog of Khare's result for arbitrary global fields. In a sequel we shall apply this result to compatible systems arising from Drinfeld modular forms, and thereby attach Hecke characters to cuspidal Drinfeld Hecke eigenforms. 1 Introduction In [Kh1, Kh2, Kh3], C. Khare studied compatible systems of abelian mod p Galois rep- resentations of a number field. He showed that the semisimplification of any such arises from a direct sum of algebraic Hecke characters, as was suggested by the framework of motives. On the one hand, this result shows that with minimal assumptions such as only knowing the mod p reductions, one can reconstruct a motive from an abelian compati- ble system. On the other hand, which is also remarkable, the association of the Hecke characters to the compatible systems is based on fairly elementary tools from algebraic number theory. The first such association was based on deep transcendence results by Henniart [He] and Waldschmidt [Wa] following work of Serre [Se].
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