#### Volume 13, issue 1 (2009)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
Congruences between modular forms given by the divided $\beta$ family in homotopy theory

### Mark Behrens

Geometry & Topology 13 (2009) 319–357
##### 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 ${ℤ}_{p}^{×}$, the spectrum $Q\left(\ell \right)$ detects the $\alpha$ and $\beta$ families in the stable stems.

##### Keywords
topological modular forms, chromatic homotopy
##### Mathematical Subject Classification 2000
Primary: 55Q45
Secondary: 55Q51, 55N34, 11F33