  We consider various trace formulas for the cubic Schrodinger equation in the
space of infinitely smooth functions subject to periodic boundary conditions.
The formulas relate conventional integrals of motion to the periods of some
Abelian differentials (holomorphic one-forms) on the spectral curve. We show
that the periods of Abelian differentials are global coordinates on the moduli
space of spectral curves. The exterior derivatives of the holomorphic one-forms
are the basic and higher symplectic structures on the phase space. We write
explicitly these symplectic structures in $QP$ coordinates. We compute the
ratio of two symplectic volume elements in the infinite genus limit.
