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

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 $p\ge 5$. 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 $\ell$ a prime which generates ${\mathbb{Z}}_p^{\times }$, the spectrum $Q\left(\ell \right)$ detects the $\alpha$ and $\beta$ families in the stable stems.