Iterated period integrals of elliptic curves

Following Enriquez and Brown-Levin, we study "iterated period
integrals" (also called "elliptic analogs of multiple zeta values") on a
once-punctured, complex elliptic curve. They are constructed from a certain
meromorphic Jacobi form $F$, first studied by Kronecker in the 19th century
and rediscovered in the late 1980s by Zagier. Similar to classical period
integrals, iterated period integrals have a Hodge-theoretic interpretation:
they describe the mixed Hodge structure on the pro-unipotent completion of
the fundamental group of the curve.

In this talk, we show that iterated period integrals can be rewritten as
linear combinations of iterated integrals of Eisenstein series (in the
sense of Manin and Brown). Surprisingly, not all linear combinations of
iterated Eisenstein integrals can occur this way: there is an obstruction
coming from cohomology classes of the modular curve $X(1)$. If time
permits, we will also mention how Hain-Matsumoto's theory of "universal
mixed elliptic motives" explains this phenomenon.