Independence of l in Lafforgue's theorem

• Published in: arXiv.org
• Main Author: Chin, CheeWhye
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 25.06.2002
• Subjects: Fields (mathematics); Number theory; Polynomials; Prime numbers; Sheaves; Theorems
• DOI: 10.48550/arxiv.0206001

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.