  In previous work, the authors have each introduced methods for studying the
2-line of the p-local Adams-Novikov spectral sequence in terms of the
arithmetic of modular forms. We give the precise relationship between the
congruences of modular forms introduced by the first author with the Q-spectrum
and the f-invariant of the second author. This relationship enables us to
refine the target group of the f-invariant in a way which makes it more
manageable for computations.

Algebraic Topology