Independence of ell of Monodromy Groups

Abstract

Let X be a smooth curve over a finite field of characteristic p, let E be a number field, and consider an E-compatible system of lisse sheaves on the curve X. For each place lambda of E not lying over p, the lambda-component of the system is a lisse E_lambda-sheaf on X, whose associated arithmetic monodromy group is an algebraic group over the local field E_lambda. We use Serre's theory of Frobenius tori and Lafforgue's proof of Deligne's conjecture to show that when the E-compatible system is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is ``independent of lambda''. More precisely: after replacing E by a finite extension, there exists a connected split reductive algebraic group G_0 over the number field E such that for every place lambda of E not lying over p, the identity component of the arithmetic monodromy group of the lambda-component of the system is isomorphic to the group G_0 with coefficients extended to the local field E_lambda.

25 pages, AMS-LaTeX; journal version

References in corpus (2)

• Independence of l in Lafforgue's theorem
• Determining a connected split reductive group from its irreducible representations

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• Independence of $\ell$ for the supports in the Decomposition Theorem
• Potential automorphy of $\hat{G}$-local systems
• Arithmetic Fourier transforms over finite fields: generic vanishing, convolution, and equidistribution