Topological modular forms [after Hopkins, Miller, and Lurie]

Goerss, Paul G.

Séminaire Bourbaki : volume 2008/2009 exposés 997-1011 - Avec table par noms d'auteurs de 1848/49 à 2008/09, Astérisque, no. 332 (2010), Exposé no. 1005, 35 p.

MR Zbl

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     author = {Goerss, Paul G.},
     title = {Topological modular forms [after {Hopkins,} {Miller,} and {Lurie]}},
     booktitle = {S\'eminaire Bourbaki : volume 2008/2009 expos\'es 997-1011  - Avec table par noms d'auteurs de 1848/49 \`a 2008/09},
     series = {Ast\'erisque},
     note = {talk:1005},
     pages = {221--255},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {332},
     year = {2010},
     mrnumber = {2648680},
     zbl = {1222.55003},
     language = {en},
     url = {http://archive.numdam.org/item/AST_2010__332__221_0/}
}

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Goerss, Paul G. Topological modular forms [after Hopkins, Miller, and Lurie], dans Séminaire Bourbaki : volume 2008/2009 exposés 997-1011  - Avec table par noms d'auteurs de 1848/49 à 2008/09, Astérisque, no. 332 (2010), Exposé no. 1005, 35 p. http://archive.numdam.org/item/AST_2010__332__221_0/

Bibliographie
Cité par

[1] J. F. Adams - "On the groups $J (X)$ . IV", Topology 5 (1966), p. 21-71. | DOI | MR | Zbl

[2] J. F. Adams, Stable homotopy and generalised homology, University of Chicago Press, 1974. | MR | Zbl

[3] M. Ando - "Isogenics of formal group laws and power operations in the cohomology theories $E_{n}$ ", Duke Math. J. 79 (1995), p. 423-485. | DOI | MR | Zbl

[4] M. Ando, M. J. Hopkins & C. Rezk - "Multiplicative orientations of $K O$ -theory and of the spectrum of topological modular forms", preprint http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf.

[5] M. Ando, M. J. Hopkins & N. P. Strickland - "Elliptic spectra, the Witten genus and the theorem of the cube", Invent. Math. 146 (2001), p. 595-687. | DOI | MR | Zbl

[6] M. Ando, M. J. Hopkins & N. P. Strickland, "The sigma orientation is an $H_{\infty}$ map", Amer. J. Math. 126 (2004), p. 247-334. | DOI | MR | Zbl

[7] T. Bauer - "Computation of the homotopy of the spectrum tmf", in Groups, homotopy and configuration spaces, Geom. Topol. Monogr., vol. 13, Geom. Topol. Publ., Coventry, 2008, p. 11-40. | DOI | MR | Zbl

[8] M. Behrens - "A modular description of the $K (2)$ -local sphere at the prime $3$ ", Topology 45 (2006), p. 343-402. | DOI | MR | Zbl

[9] M. Behrens, "Congruences between modular forms given by the divided $β$ family in homotopy theory", Geom. Topol. 13 (2009), p. 319-357. | DOI | MR | Zbl

[10] M. Behrens & T. Lawson - "Topological automorphic forms", to appear in Mem. A.M.S. | MR | Zbl

[11] R. E. Borcherds - "Automorphic forms on $O_{s + 2, 2} (𝐑)$ and infinite products", Invent. Math. 120 (1995), p. 161-213. | DOI | EuDML | MR | Zbl

[12] A. K. Bousfield & E. M. Friedlander - "Homotopy theory of $Γ$ -spaces, spectra, and bisimplicial sets", in Geometric applications of homotopy theory (Proc. Conf, Evanston, III, 1977), II, Lecture Notes in Math., vol. 658, Springer, 1978, p. 80-130. | DOI | MR | Zbl

[13] R. R. Bruner, J. P. May, J. E. Mcclure & M. Steinberger - $H_{\infty}$ ring spectra and their applications, Lecture Notes in Math., vol. 1176, Springer, 1986. | MR | Zbl

[14] P. Deligne - "Courbes elliptiques : formulaire d'après J. Tate", in Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., vol. 476, Springer, 1975, p. 53-73. | DOI | MR | Zbl

[15] P. Deligne & D. Mumford - "The irreducibility of the space of curves of given genus", Publ. Math. I.H.É.S. 36 (1969), p. 75-109. | DOI | EuDML | Numdam | MR | Zbl

[16] P. Deligne & M. Rapoport - "Les schémas de modules de courbes elliptiques", in Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math., vol. 349, Springer, 1973, p. 143-316. | DOI | MR | Zbl

[17] E. S. Devinatz & M. J. Hopkins - "Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups", Topology 43 (2004), p. 1-47. | DOI | MR | Zbl

[18] E. S. Devinatz, M. J. Hopkins & J. H. Smith - "Nilpotence and stable homotopy theory. I", Ann. of Math. 128 (1988), p. 207-241. | DOI | MR | Zbl

[19] A. D. Elmendorf, I. Kriz, M. A. Mandell & J. P. May - Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs, vol. 47, Amer. Math. Soc., 1997. | MR | Zbl

[20] J. Franke - "On the construction of elliptic cohomology", Math. Nachr. 158 (1992), p. 43-65. | DOI | MR | Zbl

[21] P. G. Goerss, H. W. Henn, M. E. Mahowald & C. Rezk - "A resolution of the $K (2)$ -local sphere at the prime $3$ ", Ann. of Math. 162 (2005), p. 777-822. | DOI | MR | Zbl

[22] P. G. Goerss & M. J. Hopkins - "Moduli spaces of commutative ring spectra", in Structured ring spectra, London Math. Soc. Lecture Note Ser., vol. 315, Cambridge Univ. Press, 2004, p. 151-200. | DOI | MR | Zbl

[23] H. W. Henn, N. Karamanov & M. E. Mahowald - "The homotopy of the $K (2)$ -local Moore spectrum at the prime $3$ revisited", preprint arXiv:0811.0235. | DOI | MR | Zbl

[24] S. Hollander - "Geometric criteria for Landweber exactness", Proc. London Math. Soc. 99 (2009), p. 697-724. | DOI | MR | Zbl

[25] M. J. Hopkins - "Topological modular forms, the Witten genus, and the theorem of the cube", in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994), Birkhäuser, 1995, p. 554-565. | DOI | MR | Zbl

[26] M. J. Hopkins, "Algebraic topology and modular forms", in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Higher Ed. Press, 2002, p. 291-317. | MR | Zbl

[27] M. J. Hopkins & B. H. Gross - "The rigid analytic period mapping, Lubin-Tate space, and stable homotopy theory", Bull. Amer. Math. Soc. (N.S.) 30 (1994), p. 76-86. | DOI | MR | Zbl

[28] M. J. Hopkins & M. E. Mahowald - "From elliptic curves to homotopy theory", preprint http://hopf.math.purdue.edu//Hopkins-Mahowald/eo2homotopy.pdf, 2005. | MR

[29] M. J. Hopkins & J. H. Smith - "Nilpotence and stable homotopy theory. II", Ann. of Math. 148 (1998), p. 1-49. | DOI | MR | Zbl

[30] M. Hovey - "Bousfield localization functors and Hopkins' chromatic splitting conjecture", in The Čech centennial (Boston, MA, 1993), Contemp. Math., vol. 181, Amer. Math. Soc., 1995, p. 225-250. | MR | Zbl

[31] M. Hovey, B. Shipley & J. H. Smith - "Symmetric spectra", J. Amer. Math. Soc. 13 (2000), p. 149-208. | DOI | MR | Zbl

[32] M. Hovey & N. P. Strickland - "Morava $K$ -theories and localisation", Mem. Amer. Math. Soc. 139 (1999). | MR | Zbl

[33] L. Illusie - Complexe cotangent et deformations. I, Lecture Notes in Math., vol. 239, Springer, 1971. | MR | Zbl

[34] J. F. Jardine - "Stable homotopy theory of simplicial presheaves", Canad. J. Math. 39 (1987), p. 733-747. | DOI | MR | Zbl

[35] J. F. Jardine, "Presheaves of symmetric spectra", J. Pure Appl. Algebra 150 (2000), p. 137-154. | DOI | MR | Zbl

[36] N. M. Katz - "Higher congruences between modular forms", Ann. of Math. 101 (1975), p. 332-367. | DOI | MR | Zbl

[37] N. M. Katz & B. Mazur - Arithmetic moduli of elliptic curves, Annals of Math. Studies, vol. 108, Princeton Univ. Press, 1985. | MR | Zbl

[38] P. S. Landweber - "Homological properties of comodules over $M U_{*} (M U)$ and ${BP}_{*} (BP)$ ", Amer. J. Math. 98 (1976), p. 591-610. | MR | Zbl

[39] P. S. Landweber, D. C. Ravenel & R. E. Stong - "Periodic cohomology theories defined by elliptic curves", in The Čech centennial (Boston, MA, 1993), Contemp. Math., vol. 181, Amer. Math. Soc., 1995, p. 317-337. | MR | Zbl

[40] G. Laumon & L. Moret-Bailly - Champs algébriques, Ergebnisse Math. Grenzg., vol. 39, Springer, 2000. | MR | Zbl

[41] G. Laures - " $K (1)$ -local topological modular forms", Invent. Math. 157 (2004), p. 371-403. | DOI | MR | Zbl

[42] M. Lazard - "Sur les groupes de Lie formels à un paramètre", Bull. Soc. Math. France 83 (1955), p. 251-274. | DOI | EuDML | Numdam | MR | Zbl

[43] L. G. J. Lewis, J. P. May, M. Steinberger & J. E. Mcclure - Equivariant stable homotopy theory, Lecture Notes in Math., vol. 1213, Springer, 1986. | MR | Zbl

[44] J. Lubin & J. Tate - "Formal moduli for one-parameter formal Lie groups", Bull. Soc. Math. France 94 (1966), p. 49-59. | DOI | EuDML | Numdam | MR | Zbl

[45] J. Lurie - "Derived algebraic geometry", Ph.D. Thesis, MIT, 2004. | MR

[46] J. Lurie, "Survey article on elliptic cohomology", preprint http://www.math.harvard.edu/~lurie/papers/survey.pdf, 2007. | MR

[47] J. Lurie, Higher topos theory, Annals of Math. Studies, vol. 170, Princeton Univ. Press, 2009. | MR | Zbl

[48] M. E. Mahowald - "On the order of the image of $J$ ", Topology 6 (1967), p. 371-378. | DOI | MR | Zbl

[49] M. E. Mahowald & C. Rezk - "Brown-Comenetz duality and the Adams spectral sequence", Amer. J. Math. 121 (1999), p. 1153-1177. | DOI | MR | Zbl

[50] M. E. Mahowald & C. Rezk, "Topological modular forms of level $3$ ", Pure Appl. Math. Q. 5 (2009), p. 853-872. | DOI | MR | Zbl

[51] M. A. Mandell, J. P. May, S. Schwede & B. Shipley - "Model categories of diagram spectra", Proc. London Math. Soc. 82 (2001), p. 441-512. | DOI | MR | Zbl

[52] W. Messing - The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lecture Notes in Math., vol. 264, Springer, 1972. | MR | Zbl

[53] H. R. Miller, D. C. Ravenel & W. S. Wilson - "Periodic phenomena in the Adams-Novikov spectral sequence", Ann. Math. 106 (1977), p. 469-516. | DOI | MR | Zbl

[54] N. Naumann - "The stack of formal groups in stable homotopy theory", Adv. in Math. 215 (2007), p. 569-600. | DOI | MR | Zbl

[55] D. Quillen - "Elementary proofs of some results of cobordism theory using Steenrod operations", Adv. in Math. 7 (1971), p. 29-56. | DOI | MR | Zbl

[56] D. C. Ravenel - Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press Inc., 1986. | MR | Zbl

[57] D. C. Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals of Math. Studies, vol. 128, Princeton Univ. Press, 1992. | MR | Zbl

[58] C. Rezk - "Lectures on power operations", notes of a course given at MIT in 2006, http://www.math.uiuc.edu/~rezk/power-operation-lectures.dvi.

[59] C. Rezk, "Supplementary notes for Math 512", notes of a course on topological modular forms given at Northwestern University in 2001, http://www.math. uiuc.edu/~rezk/512-spr2001-notes.pdf.

[60] G. Segal - "Elliptic cohomology (after Landweber-Stong, Ochanine, Witten, and others)", Séminaire Bourbaki, vol. 1987/88, exposé n° 695, Astérisque 161-162 (1988), p. 187-201. | EuDML | Numdam | MR | Zbl

[61] J-P. Serre - Algèbre locale. Multiplicités, Cours au Collège de France, 1957-1958, rédigé par Pierre Gabriel. Seconde édition, 1965. Lecture Notes in Math., vol. 11, Springer, 1965. | MR | Zbl

[62] K. Shimomura & X. Wang - "The homotopy groups $π_{*} (L_{2} S^{0})$ at the prime $3$ ", Topology 41 (2002), p. 1183-1198. | DOI | MR | Zbl

[63] K. Shimomura & A. Yabe - "The homotopy groups $π_{*} (L_{2} S^{0})$ ", Topology 34 (1995), p. 261-289. | DOI | MR | Zbl

[64] J. H. Silverman - The arithmetic of elliptic curves, Graduate Texts in Math., vol. 106, Springer, 1986. | MR | Zbl

[65] B. Smithling - "On the moduli stack of commutative, $1$ -parameter formal Lie groups", Ph.D. Thesis, University of Chicago, 2007. | MR

[66] B. Toën & G. Vezzosi - "Homotopical algebraic geometry. II. Geometric stacks and applications", Mem. Amer. Math. Soc. 193 (2008). | MR | Zbl

[67] E. Witten - "Elliptic genera and quantum field theory", Comm. Math. Phys. 109 (1987), p. 525-536. | DOI | MR | Zbl