Independence of l in Lafforgue's theorem

Let X be a smooth curve over a finite field of characteristic p, let l be a prime number different from p, and let L be an irreducible lisse l-adic sheaf on X whose determinant is of finite order. By a theorem of Lafforgue, for each prime number l' different from p, there exists an irreducible...

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Bibliographic Details
Published inarXiv.org
Main Author Chin, CheeWhye
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 25.06.2002
Subjects
Fields (mathematics)
Number theory
Polynomials
Prime numbers
Sheaves
Theorems
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Summary:Let X be a smooth curve over a finite field of characteristic p, let l be a prime number different from p, and let L be an irreducible lisse l-adic sheaf on X whose determinant is of finite order. By a theorem of Lafforgue, for each prime number l' different from p, there exists an irreducible lisse l'-adic sheaf L' on X which is compatible with L, in the sense that at every closed point x of X, the characteristic polynomials of Frobenius at x for L and L' are equal. We prove an "independence of l" assertion on the fields of definition of these irreducible l'-adic sheaves L' : namely, that there exists a number field F such that for any prime number l' different from p, the l'-adic sheaf L' above is defined over the completion of F at one of its l'-adic places.
ISSN:2331-8422
DOI:10.48550/arxiv.0206001