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Rigid local systems and motives of type \(G_{2}\). (English) Zbl 1194.14036
Let \(G_2\) be the smallest exceptional simple linear algebraic group over an algebraic closed field which is not isomorphic to a classical group. Using the middle convolution functor \(MC_{\chi}\) introduced by N. Katz, the authors prove the existence of rigid local systems whose monodromy is dense in \(G_2\). They derive the existence of motives for motivated cycles which have a motivic Galois group of type \(G_2\). Granting Grothendieck’s standard conjectures, the existence of motives with motivic Galois group of type \(G_2\) can be deduced, giving a partial answer to a question of Serre.

MSC:
14G99 Arithmetic problems in algebraic geometry; Diophantine geometry
14C25 Algebraic cycles
14F20 √Čtale and other Grothendieck topologies and (co)homologies
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[10] doi:10.2307/1990970 · Zbl 0764.14012 · doi:10.2307/1990970
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