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

Behrens, Mark

doi:10.2140/gt.2009.13.319

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

Mark Behrens

Geometry & Topology 13 (2009) 319–357

DOI: 10.2140/gt.2009.13.319

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 $p \geq 5$ . 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 $ℓ$ a prime which generates $ℤ_{p}^{\times}$ , the spectrum $Q (ℓ)$ detects the $α$ and $β$ families in the stable stems.

Keywords

topological modular forms, chromatic homotopy

Mathematical Subject Classification 2000

Primary: 55Q45

Secondary: 55Q51, 55N34, 11F33

References

Publication

Received: 3 May 2008

Revised: 13 October 2008

Accepted: 8 October 2008

Preview posted: 5 November 2009

Published: 1 January 2009

Proposed: Paul Goerss

Seconded: Bill Dwyer, Haynes Miller

Authors

Mark Behrens
MIT Department of Mathematics 2-273 77 Massachusetts Ave Cambridge MA 02140 USA
http://www-math.mit.edu/~mbehrens

Volume 13, issue 1 (2009)