  Let $n\geq 1$. The pro-unipotent completion of the pure braid group of $n$
points on a genus 1 surface has been shown to be isomorphic to an explicit
pro-unipotent group with graded Lie algebra using two types of tools: (a)
minimal models (Bezrukavnikov), (b) the choice of a complex structure on the
genus 1 surface, making it into an elliptic curve $E$, and an appropriate flat
connection on the configuration space of $n$ points in $E$ (joint work of the
authors with D. Calaque). Following a suggestion by P. Deligne, we give an
interpretation of this isomorphism in the framework of the Riemann-Hilbert
correspondence, using the total space $E^\#$ of an affine line bundle over $E$,
which identifies with the moduli space of line bundles over $E$ equipped with a
flat connection.
