  We use the theory of cubic structures to give a fixed point Riemann-Roch
formula for the equivariant Euler characteristics of coherent sheaves on
projective flat schemes over Z with a tame action of a finite abelian group.
This formula supports a conjecture concerning the extent to which such
equivariant Euler characteristics may be determined from the restriction of the
sheaf to an infinitesimal neighborhood of the fixed point locus. Our results
are applied to study the module structure of modular forms having Fourier
coefficients in a ring of algebraic integers, as well as the action of diamond
Hecke operators on the Mordell-Weil groups and Tate-Shafarevich groups of
Jacobians of modular curves.
