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The semisimplicity conjecture for A-motives

Part of: Discontinuous groups and automorphic forms General commutative ring theory Arithmetic algebraic geometry

Published online by Cambridge University Press:  18 March 2010

Nicolas Stalder
Nicolas Stalder*
Affiliation:
Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092 Zürich, Switzerland (email: [email protected])
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Abstract

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We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V𝔭(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G𝔭(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G𝔭(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V𝔭(M). The second states that the connected component of G𝔭(M) is reductive if M is semisimple and has a separable endomorphism algebra.

Keywords

A-motivet-motivesemisimpleGalois representationmonodromy groupsemilinear algebra

MSC classification

Secondary: 11G09: Drinfel'd modules; higher-dimensional motives, etc. 11F80: Galois representations 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 3 , May 2010 , pp. 561 - 598
Copyright
Copyright © Foundation Compositio Mathematica 2010

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