Monodromy of elliptic curve convolution, seven-point sheaves of $G_2$-type and motives of Beauville type

TL;DR

This paper investigates the Tannakian structure of perverse sheaves on elliptic curves with convolution, revealing that certain sheaves have a Tannaka group isomorphic to G_2, thus connecting to motives of Beauville type.

Contribution

It demonstrates that specific perverse sheaves on elliptic curves have G_2 as their Tannaka group, extending Katz's results on G_2-motives in Beauville surface deformations.

Findings

Tannaka group for certain sheaves is G_2

Generalizes Katz's G_2-motives result

Establishes monodromy properties of sheaves on elliptic curves

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.