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Semistable models for modular curves and power operations for Morava E-theories of height 2. (English) Zbl 1467.55010

This article proves a result about power operations in algebraic topology by means of a new result in arithmetic geometry.
\(E_{\infty}\)-ring spectra in algebraic topology are a refinement of multiplicative cohomology theories that additionally allows for a theory of power operations. For \(H\mathbb{F}_p\) these are given by Dyer-Lashof operations. Lubin-Tate theories are further examples of \(E_{\infty}\)-ring spectra of paramount importance in chromatic homotopy theory and depend on the choice of height and prime. The theory of power operations for Lubin-Tate theories has been developed first by Ando, Hopkins and Strickland and has been brought into a rather definitive abstract form by Rezk.
For computations one needs to bring the theory in a more concrete form though, i.e.provide generators and relations for the ring of power operations and describe explicitly the error terms arising when commuting power operations (the analogue for \(H\mathbb{F}_p\) being the Adem relations). This concrete form is not difficult to obtain at height \(1\), and at height \(2\) has mostly been provided by Rezk and Zhu at the primes \(2\), \(3\) and \(5\). The paper under review extends this to a uniform and complete presentation at height \(2\) and all primes.
The problem has already earlier been translated into arithmetic geometry and the key is to provide good equations for the moduli of degree \(p\) and \(p^2\) subgroups of the universal deformation of a height \(2\) formal group over \(\mathbb{F}_{p^n}\). The author uses the theory of moduli of elliptic curves to do this.
The work of M. Behrens and C. Rezk [Invent. Math. 220, No. 3, 949–1022 (2020; Zbl 1444.55003)] provides a bridge between power operations for Lubin-Tate theories and unstable chromatic homotopy theory and the present work has already successfully been applied to the latter in [Y. Zhu, Proc. Am. Math. Soc. 146, No. 1, 449–458 (2018; Zbl 1422.55032)].

MSC:

55S25 \(K\)-theory operations and generalized cohomology operations in algebraic topology
14H10 Families, moduli of curves (algebraic)
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
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[1] Ahlgren, S., The theta-operator and the divisors of modular forms on genus zero subgroups, Math. Res. Lett., 10, 6, 787-798 (2003), MR 2024734 · Zbl 1149.11308
[2] Ando, M., Isogenies of formal group laws and power operations in the cohomology theories \(E_n\), Duke Math. J., 79, 2, 423-485 (1995), MR 1344767 · Zbl 0862.55004
[3] Ando, M., Power operations in elliptic cohomology and representations of loop groups, Trans. Amer. Math. Soc., 352, 12, 5619-5666 (2000), MR 1637129 · Zbl 0958.55016
[4] Ando, M.; Hopkins, M. J.; Strickland, N. P., The sigma orientation is an \(H_\infty\) map, Amer. J. Math., 126, 2, 247-334 (2004), MR 2045503 · Zbl 1071.55003
[5] Atkin, A. O.L.; Lehner, J., Hecke operators on \(\Gamma_0(m)\), Math. Ann., 185, 134-160 (1970), MR 0268123 · Zbl 0177.34901
[6] Baker, M. H.; González-Jiménez, E.; González, J.; Poonen, B., Finiteness results for modular curves of genus at least 2, Amer. J. Math., 127, 6, 1325-1387 (2005), MR 2183527 · Zbl 1127.11041
[7] Behrens, M.; Rezk, C., The Bousfield-Kuhn functor and topological André-Quillen cohomology, available at · Zbl 1444.55003
[8] Bruinier, J. H.; Kohnen, W.; Ono, K., The arithmetic of the values of modular functions and the divisors of modular forms, Compos. Math., 140, 3, 552-566 (2004), MR 2041768 · Zbl 1060.11019
[9] Bruner, R. R.; May, J. P.; McClure, J. E.; Steinberger, M., \(H_\infty\) Ring Spectra and Their Applications, Lecture Notes in Mathematics, vol. 1176 (1986), Springer-Verlag: Springer-Verlag Berlin, MR 836132 · Zbl 0585.55016
[10] Buzzard, K., Analytic continuation of overconvergent eigenforms, J. Amer. Math. Soc., 16, 1, 29-55 (2003), MR 1937198 · Zbl 1076.11029
[11] Calegari, F., Congruences between modular forms, available at
[12] Choi, D., On values of a modular form on \(\Gamma_0(N)\), Acta Arith., 121, 4, 299-311 (2006), MR 2224397 · Zbl 1149.11022
[13] Cox, D. A., Fermat, class field theory, and complex multiplication, (Primes of the Form \(x^2 + n y^2\). Primes of the Form \(x^2 + n y^2\), Pure and Applied Mathematics (Hoboken) (2013), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. Hoboken, NJ), MR 3236783 · Zbl 1275.11002
[14] Deligne, P.; Rapoport, M., Les schémas de modules de courbes elliptiques, Lecture Notes in Math., vol. 349, 143-316 (1973), Springer: Springer Berlin, MR 0337993 · Zbl 0281.14010
[15] Devinatz, E. S.; Hopkins, M. J.; Smith, J. H., Nilpotence and stable homotopy theory. I, Ann. of Math. (2), 128, 2, 207-241 (1988), MR 960945 · Zbl 0673.55008
[16] Goerss, P., The Adams-Novikov spectral sequence and the homotopy groups of spheres
[17] Goerss, P. G.; Hopkins, M. J., Moduli Spaces of Commutative Ring Spectra, Structured Ring Spectra, London Math. Soc. Lecture Note Ser., vol. 315, 151-200 (2004), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, MR 2125040 · Zbl 1086.55006
[18] Hopkins, M., Complex oriented cohomology theories and the language of stacks, available at
[19] Hopkins, M. J.; Smith, J. H., Nilpotence and stable homotopy theory. II, Ann. of Math. (2), 148, 1, 1-49 (1998), MR 1652975 · Zbl 0924.55010
[20] Huan, Z., Quasi-elliptic cohomology and its power operations, J. Homotopy Relat. Struct., 13, 4, 715-767 (2018), MR 3870771 · Zbl 1453.55006
[21] Katz, N. M., \(p\)-adic properties of modular schemes and modular forms, (Modular Functions of One Variable, III Proc. Internat. Summer School. Modular Functions of One Variable, III Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972. Modular Functions of One Variable, III Proc. Internat. Summer School. Modular Functions of One Variable, III Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972, Lecture Notes in Mathematics, vol. 350 (1973), Springer: Springer Berlin), 69-190, MR 0447119 · Zbl 0271.10033
[22] Katz, N. M.; Mazur, B., Arithmetic Moduli of Elliptic Curves, Annals of Mathematics Studies, vol. 108 (1985), Princeton University Press: Princeton University Press Princeton, NJ, MR 772569 · Zbl 0576.14026
[23] Lazard, M., Sur les groupes de Lie formels à un paramètre, Bull. Soc. Math. France, 83, 251-274 (1955), MR 0073925 · Zbl 0068.25703
[24] Lubin, J.; Serre, J.-P.; Tate, J., Elliptic curves and formal groups, available at · Zbl 0156.04105
[25] Lubin, J.; Tate, J., Formal moduli for one-parameter formal Lie groups, Bull. Soc. Math. France, 94, 49-59 (1966), MR 0238854 · Zbl 0156.04105
[26] Lurie, J., A Survey of Elliptic Cohomology, Algebraic Topology, Abel Symp., vol. 4, 219-277 (2009), Springer: Springer Berlin, MR 2597740 · Zbl 1206.55007
[27] Lurie, J., Chromatic homotopy theory, esp. Lectures 22 and 23, available at
[28] Mahowald, M.; Rezk, C., Topological modular forms of level 3, Special Issue: In honor of Friedrich Hirzebruch. Part 1. Special Issue: In honor of Friedrich Hirzebruch. Part 1, Pure Appl. Math. Q., 5, 2, 853-872 (2009), MR 2508904 · Zbl 1192.55006
[29] Meier, L., Lifting the Hasse invariant mod 2, (version: 2016-01-16)
[30] Miller, H. R.; Ravenel, D. C.; Wilson, W. S., Periodic phenomena in the Adams-Novikov spectral sequence, Ann. of Math. (2), 106, 3, 469-516 (1977), MR 0458423 · Zbl 0374.55022
[31] Milne, J. S., Modular functions and modular forms, available at
[32] Poonen, B., Supersingular elliptic curves and their “functorial” structure over \(F_{p^2}\), (version: 2010-03-22)
[33] Quillen, D., On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc., 75, 1293-1298 (1969), MR 0253350 · Zbl 0199.26705
[34] Ravenel, D. C., Localization with respect to certain periodic homology theories, Amer. J. Math., 106, 2, 351-414 (1984), MR 737778 · Zbl 0586.55003
[35] Ravenel, D. C., Nilpotence and Periodicity in Stable Homotopy Theory, Annals of Mathematics Studies, vol. 128 (1992), Princeton University Press: Princeton University Press Princeton, NJ, Appendix C by Jeff Smith, MR 1192553 · Zbl 0774.55001
[36] Rezk, C., Lectures on power operations, available at
[37] Rezk, C., Power operations for Morava \(E\)-theory of height 2 at the prime 2
[38] Rezk, C., The congruence criterion for power operations in Morava \(E\)-theory, Homology, Homotopy Appl., 11, 2, 327-379 (2009), MR 2591924 · Zbl 1193.55010
[39] Rezk, C., Modular isogeny complexes, Algebr. Geom. Topol., 12, 3, 1373-1403 (2012), MR 2966690 · Zbl 1254.14030
[40] Rezk, C., Power operations in Morava \(E\)-theory: structure and calculations, (Draft) available at
[41] C. Rezk, Correspondence, 2013.; C. Rezk, Correspondence, 2013.
[42] C. Rezk, Correspondence, 2015.; C. Rezk, Correspondence, 2015.
[43] Rezk, C., Rings of power operations for Morava E-theories are Koszul
[44] Schwede, S., Global Homotopy Theory, New Mathematical Monographs, vol. 34 (2018), Cambridge University Press: Cambridge University Press Cambridge, MR 3838307 · Zbl 1451.55001
[45] Silverman, J. H., The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, vol. 106 (2009), Springer: Springer Dordrecht, MR 2514094 · Zbl 1194.11005
[46] Strickland, N. P., Finite subgroups of formal groups, J. Pure Appl. Algebra, 121, 2, 161-208 (1997), MR 1473889 · Zbl 0916.14025
[47] Strickland, N. P., Morava \(E\)-theory of symmetric groups, Topology, 37, 4, 757-779 (1998), MR 1607736 · Zbl 0912.55012
[48] Tate, J. T., \(p\)-divisible groups, (Proc. Conf. Local Fields. Proc. Conf. Local Fields, Driebergen, 1966 (1967), Springer: Springer Berlin), 158-183, MR 0231827 · Zbl 0157.27601
[49] Weinstein, J., Semistable models for modular curves of arbitrary level, Invent. Math., 205, 2, 459-526 (2016), MR 3529120 · Zbl 1357.14034
[50] Zhu, Y., The power operation structure on Morava \(E\)-theory of height 2 at the prime 3, Algebr. Geom. Topol., 14, 2, 953-977 (2014), MR 3160608 · Zbl 1310.55011
[51] Zhu, Y., The Hecke algebra action and the Rezk logarithm on Morava E-theory of height 2, updated at · Zbl 1439.55019
[52] Zhu, Y., Norm coherence for descent of level structures on formal deformations, updated at · Zbl 07195678
[53] Zhu, Y., Morava \(E\)-homology of Bousfield-Kuhn functors on odd-dimensional spheres, Proc. Amer. Math. Soc., 146, 1, 449-458 (2018), MR 3723154 · Zbl 1422.55032
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