Geometry & Topology

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

Mark Behrens

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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 p5. 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 a prime which generates p×, the spectrum Q() detects the α and β families in the stable stems.

Article information

Geom. Topol., Volume 13, Number 1 (2009), 319-357.

Received: 3 May 2008
Revised: 13 October 2008
Accepted: 8 October 2008
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55Q45: Stable homotopy of spheres
Secondary: 55Q51: $v_n$-periodicity 55N34: Elliptic cohomology 11F33: Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50]

topological modular forms chromatic homotopy


Behrens, Mark. Congruences between modular forms given by the divided $\beta$ family in homotopy theory. Geom. Topol. 13 (2009), no. 1, 319--357. doi:10.2140/gt.2009.13.319.

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