  We survey the Hirzebruch signature theorem as a special case of the
Atiyah-Singer index theorem. The family version of the Atiyah-Singer index
theorem in the form of the Riemann-Roch-Grothendieck-Quillen (RRGQ) formula is
then applied to the complexified signature operators varying along the
universal family of elliptic curves. The RRGQ formula allows us to determine a
generalized cohomology class on the base of the elliptic fibration that is
known in physics as (a measure of) the local and global anomaly. Combining
several anomalous operators allows us to cancel the local anomaly on a Jacobian
elliptic surface, a construction that is based on the construction of the
Poincar\'e line bundle over an elliptic surface.
