The authors construct explicit non-integral Q-Hodge variations with Zariski dense monodromy, prove new S-integral point results on character varieties, and show the Hodge locus can be non-algebraic.
Monodromy of elliptic curve convolution, seven-point sheaves of $G_2$-type and motives of Beauville type
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abstract
We study the Tannakian properties of the category of perverse sheaves on elliptic curves endowed with the convolution product. We establish that for certain sheaves with unipotent local monodromy over seven points the corresponding Tannaka group is isomorphic to $G_2$. This monodromy approach generalizes a result of Katz on the existence of $G_2$-motives in the middle cohomology of deformations of Beauville surfaces.
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Murphy's law in non-abelian Hodge theory
The authors construct explicit non-integral Q-Hodge variations with Zariski dense monodromy, prove new S-integral point results on character varieties, and show the Hodge locus can be non-algebraic.