## Geometry & Topology

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

Mark Behrens

#### 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 $p≥5$. 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 $ℤp×$, the spectrum $Q(ℓ)$ detects the $α$ and $β$ families in the stable stems.

#### Article information

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

Dates
Revised: 13 October 2008
Accepted: 8 October 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800182

Digital Object Identifier
doi:10.2140/gt.2009.13.319

Mathematical Reviews number (MathSciNet)
MR2469520

Zentralblatt MATH identifier
1205.55012

#### Citation

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. https://projecteuclid.org/euclid.gt/1513800182

#### References

• A Baker, Hecke operations and the Adams $E\sb 2$–term based on elliptic cohomology, Canad. Math. Bull. 42 (1999) 129–138
• M Behrens, A modular description of the $K(2)$–local sphere at the prime 3, Topology 45 (2006) 343–402
• M Behrens, Buildings, elliptic curves, and the $K(2)$–local sphere, Amer. J. Math. 129 (2007) 1513–1563
• M Behrens, Some root invariants at the prime $2$, Geom. Top. Mono. 10 (2007) 1–40
• M Behrens, G Laures, $\beta$–family congruences and the $f$–invariant\qua Preprint
• M Behrens, T Lawson, Topological automorphic forms\qua to appear in memoirs of the AMS
• M Behrens, T Lawson, Isogenies of elliptic curves and the Morava stabilizer group, J. Pure Appl. Algebra 207 (2006) 37–49
• A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs 47, American Mathematical Society, Providence, RI (1997) With an appendix by M Cole
• H Hida, $p$–adic automorphic forms on Shimura varieties, Springer Monographs in Mathematics, Springer, New York (2004)
• M J Hopkins, J H Smith, Nilpotence and stable homotopy theory. II, Ann. of Math. $(2)$ 148 (1998) 1–49
• J Hornbostel, N Naumann, Beta-elements and divided congruences, Amer. J. Math. 129 (2007) 1377–1402
• N M Katz, $p$–adic properties of modular schemes and modular forms, from: “Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972)”, Springer, Berlin (1973) 69–190. Lecture Notes in Mathematics, Vol. 350
• S S Kudla, From modular forms to automorphic representations, from: “An introduction to the Langlands program (Jerusalem, 2001)”, Birkhäuser, Boston (2003) 133–151
• G Laures, The Topological $q$–Expansion Principle, PhD thesis, M.I.T. (1996)
• G Laures, The topological $q$–expansion principle, Topology 38 (1999) 387–425
• H R Miller, D C Ravenel, W S Wilson, Periodic phenomena in the Adams–Novikov spectral sequence, Ann. Math. $(2)$ 106 (1977) 469–516
• J Morava, Noetherian localisations of categories of cobordism comodules, Ann. of Math. $(2)$ 121 (1985) 1–39
• D C Ravenel, Localization with respect to certain periodic homology theories, Amer. J. Math. 106 (1984) 351–414
• J H Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer, New York (1986)
• W C Waterhouse, Abelian varieties over finite fields, Ann. Sci. École Norm. Sup. $(4)$ 2 (1969) 521–560