Congruences between modular forms given by the divided beta family in homotopy theory

TL;DR

This paper characterizes certain lines in the p-local Adams-Novikov spectral sequence using modular forms with explicit congruences, and shows how the spectrum Q(l) detects key families in stable homotopy groups.

Contribution

It provides a new modular forms-based characterization of the Adams-Novikov spectral sequence lines and relates spectrum Q(l) to detection of alpha and beta families.

Findings

Characterization of the 2-line via modular forms and congruences

Reinterpretation of the 1-line in the spectral sequence

Spectrum Q(l) detects alpha and beta families in stable stems

Abstract

We characterize the 2-line of the p-local Adams-Novikov spectral sequence in terms of modular forms satisfying a certain explicit congruence condition for primes p > 3. We give a similar characterization of the 1-line, reinterpreting some earlier work of A. Baker and G. Laures. These results are then used to deduce that, for l a prime which generates the p-adic units, the spectrum Q(l) detects the alpha and beta families in the stable stems.