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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 Étale and other Grothendieck topologies and (co)homologies
##### Keywords:
rigid local systems; motivic Galois group
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##### References:
 [10] doi:10.2307/1990970 · Zbl 0764.14012 · doi:10.2307/1990970
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