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 > 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.

30 pages, submitted. Revision - repaired an integrality error in the use of Eisenstein series

References in corpus (2)

• A modular description of the K(2)-local sphere at the prime 3
• Some root invariants at the prime 2

Cited by in corpus (15)

• The Picard group of topological modular forms via descent theory
• On the homotopy of Q(3) and Q(5) at the prime 2
• On the ring of cooperations for 2-primary connective topological modular forms
• Topological modular forms (aftern Hopkins, Miller, and Lurie)
• Hecke operators on topological modular forms
• Analogs of Dirichlet $L$-functions in chromatic homotopy theory
• beta-family congruences and the f-invariant
• Topological automorphic forms on U(1,1)
• The beta family at the prime two and modular forms of level three
• Constructing explicit K3 spectra
• On the double transfer and the f-invariant
• The Adams-Novikov E_2-term for Behrens' spectrum Q(2) at the prime 3
• The realizability of operations on homotopy groups concentrated in two degrees
• Toda brackets and congruences of modular forms
• On the Bousfield-Kan spectral sequence for $Q(2{)}_{(3)}$