Grothendieck's period conjecture predicts the transcendence
degree of the field generated by the periods of a smooth projective
variety (or more generally, a pure motive) over a number field, in terms
of the dimension of its Mumford-Tate group. For mixed motives a similar
conjecture was made by Andre. The upper bound part of Grothendieck's
conjecture was proved in the case of abelian varieties by Deligne (as a
consequence of his "Hodge implies absolute Hodge" theorem for abelian
varieties). The lower bound part of Grothendieck's conjecture is known for
a CM elliptic curve, thanks to a theorem of G. V. Chudnovsky.
This talk is a report on an aspect of a work in progress with Kumar Murty,
in which we use Hodge theoretic methods and Tannakian formalism to study
quadratic and higher periods of a punctured curve. We start by some
background material and motivation. In the end, we prove the upper bound
part of Andre's conjecture for quadratic periods of a punctured elliptic
curve, defined over a subfield of $\mathbb{R}$. The argument is quite
formal, and in fact, applies to any extension of $H^1\otimes H^1$ by
$H^1$.