2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/174022In 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.17 pages - minor revision - expanded some arguments, and changed some techniques slightlyAlgebraic Topologytext