  In this expository paper, we review the formula of Chowla and Selberg for the
periods of elliptic curves with complex multiplication, and discuss two methods
of proof. One uses Kronecker's limit formula and the other uses the geometry of
a family of abelian varieties. We discuss a generalization of this formula,
which was proposed by Colmez, as well as some explicit Hodge cycles which
appear in the geometric proof.

Number Theory