Power operations in elliptic cohomology and representations of loop groups
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Abstract:
Part I of this paper describes power operations in elliptic cohomology in terms of isogenies of the underlying elliptic curve. Part II discusses a relationship between equivariant elliptic cohomology and representations of loop groups. Part III investigates the representation of theoretic considerations which give rise to the power operations discussed in Part I.References
- J. F. Adams, A variant of E. H. Brown’s representability theorem, Topology 10 (1971), 185–198. MR 283788, DOI 10.1016/0040-9383(71)90003-6
- Matthew Ando, Isogenies of formal group laws and power operations in the cohomology theories $E_n$, Duke Math. J. 79 (1995), no. 2, 423–485. MR 1344767, DOI 10.1215/S0012-7094-95-07911-3
- Andrew Baker, Hecke operators as operations in elliptic cohomology, J. Pure Appl. Algebra 63 (1990), no. 1, 1–11. MR 1037690, DOI 10.1016/0022-4049(90)90052-J
- R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger, $H_\infty$ ring spectra and their applications, Lecture Notes in Mathematics, vol. 1176, Springer-Verlag, Berlin, 1986. MR 836132, DOI 10.1007/BFb0075405
- Lawrence Breen, Fonctions thêta et théorème du cube, Lecture Notes in Mathematics, vol. 980, Springer-Verlag, Berlin, 1983 (French). MR 823233
- J. Horn, Über eine hypergeometrische Funktion zweier Veränderlichen, Monatsh. Math. Phys. 47 (1939), 359–379 (German). MR 91, DOI 10.1007/BF01695508
- Andrew Baker and Urs Würgler, Liftings of formal groups and the Artinian completion of $v_n^{-1}\textrm {BP}$, Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 3, 511–530. MR 1010375, DOI 10.1017/S0305004100068249
- P. Deligne, Courbes elliptiques: formulaire d’après J. Tate, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, pp. 53–73 (French). MR 0387292
- Michel Demazure and Pierre Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local par Michiel Hazewinkel. MR 0302656
- P. Deligne and M. Rapoport, Correction to: “Les schémas de modules de courbes elliptiques” (Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 143–316, Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973), Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, pp. p. 149 (French). MR 0382292
- V. G. Drinfel′d, Elliptic modules, Mat. Sb. (N.S.) 94(136) (1974), 594–627, 656 (Russian). MR 0384707
- I. B. Frenkel, Representations of affine Lie algebras, Hecke modular forms and Korteweg-de Vries type equations, Lie algebras and related topics (New Brunswick, N.J., 1981) Lecture Notes in Math., vol. 933, Springer, Berlin-New York, 1982, pp. 71–110. MR 675108
- V. Ginzburg, M. Kapranov, and E. Vasserot. Elliptic algebras and equivariant elliptic cohomology, 1995. Preprint.
- Ian Grojnowski. Delocalized equivariant elliptic cohomology, 1994. Submitted to MRL.
- M. J. Hopkins and J. R. Hunton, On the structure of spaces representing a Landweber exact cohomology theory, Topology 34 (1995), no. 1, 29–36. MR 1308488, DOI 10.1016/0040-9383(94)E0013-A
- Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel. Generalized groups, characters and complex oriented cohomology theories. Preprint, 1992.
- Mark Hovey and Neil P. Strickland, Morava $K$-theories and localisation, Mem. Amer. Math. Soc. 139 (1999), no. 666, viii+100. MR 1601906, DOI 10.1090/memo/0666
- John Hunton and Paul Turner. The homology of spaces representing exact pairs of homotopy functors, 1998. Preprint.
- Jun-ichi Igusa, On the transformation theory of elliptic functions, Amer. J. Math. 81 (1959), 436–452. MR 104668, DOI 10.2307/2372750
- R. G. Lintz and V. Buonomano, The concept of differential equation in topological spaces and generalized mechanics, J. Reine Angew. Math. 265 (1974), 31–60. MR 334277
- Victor G. Kac, Infinite-dimensional Lie algebras, 2nd ed., Cambridge University Press, Cambridge, 1985. MR 823672
- P. Deligne, Cristaux ordinaires et coordonnées canoniques, Algebraic surfaces (Orsay, 1976–78) Lecture Notes in Math., vol. 868, Springer, Berlin-New York, 1981, pp. 80–137 (French). With the collaboration of L. Illusie; With an appendix by Nicholas M. Katz. MR 638599
- Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR 772569, DOI 10.1515/9781400881710
- V. G. Kac and M. Wakimoto, Branching functions for winding subalgebras and tensor products, Acta Appl. Math. 21 (1990), no. 1-2, 3–39. MR 1085771, DOI 10.1007/BF00053290
- Eduard Looijenga, Root systems and elliptic curves, Invent. Math. 38 (1976/77), no. 1, 17–32. MR 466134, DOI 10.1007/BF01390167
- Peter S. Landweber, Douglas C. Ravenel, and Robert E. Stong, Periodic cohomology theories defined by elliptic curves, The Čech centennial (Boston, MA, 1993) Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 317–337. MR 1320998, DOI 10.1090/conm/181/02040
- Jonathan Lubin and John Tate, Formal complex multiplication in local fields, Ann. of Math. (2) 81 (1965), 380–387. MR 172878, DOI 10.2307/1970622
- Jonathan Lubin, Finite subgroups and isogenies of one-parameter formal Lie groups, Ann. of Math. (2) 85 (1967), 296–302. MR 209287, DOI 10.2307/1970443
- Haynes Miller, The elliptic character and the Witten genus, Algebraic topology (Evanston, IL, 1988) Contemp. Math., vol. 96, Amer. Math. Soc., Providence, RI, 1989, pp. 281–289. MR 1022688, DOI 10.1090/conm/096/1022688
- Jack Morava, Forms of $K$-theory, Math. Z. 201 (1989), no. 3, 401–428. MR 999737, DOI 10.1007/BF01214905
- David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 0282985
- David Mumford, Tata lectures on theta. III, Progress in Mathematics, vol. 97, Birkhäuser Boston, Inc., Boston, MA, 1991. With the collaboration of Madhav Nori and Peter Norman. MR 1116553, DOI 10.1007/978-0-8176-4579-3
- Andrew Pressley and Graeme Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR 900587
- Douglas C. Ravenel and W. Stephen Wilson, The Hopf ring for complex cobordism, J. Pure Appl. Algebra 9 (1976/77), no. 3, 241–280. MR 448337, DOI 10.1016/0022-4049(77)90070-6
- Graeme Segal, Elliptic cohomology (after Landweber-Stong, Ochanine, Witten, and others), Astérisque 161-162 (1988), Exp. No. 695, 4, 187–201 (1989). Séminaire Bourbaki, Vol. 1987/88. MR 992209
- Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
- Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368, DOI 10.1007/978-1-4612-0851-8
- Neil P. Strickland, Finite subgroups of formal groups, J. Pure Appl. Algebra 121 (1997), no. 2, 161–208. MR 1473889, DOI 10.1016/S0022-4049(96)00113-2
- Neil P. Strickland. Formal schemes and formal groups, 1998. Preprint.
- Olga Taussky, An algebraic property of Laplace’s differential equation, Quart. J. Math. Oxford Ser. 10 (1939), 99–103. MR 83, DOI 10.1093/qmath/os-10.1.99
Additional Information
- Matthew Ando
- Affiliation: Department of Mathematics, The University of Virginia, Charlottesville, Virginia 22903
- Address at time of publication: Department of Mathematics, The University of Illinois at Urbana-Champaign, 1409 W. Green St., Urbana, Illinois 61801
- Email: ando@math.jhu.edu
- Received by editor(s): October 23, 1995
- Received by editor(s) in revised form: June 10, 1998
- Published electronically: July 13, 2000
- Additional Notes: Supported by the NSF
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 5619-5666
- MSC (1991): Primary 55N20, 55S25, 55N91, 22E67
- DOI: https://doi.org/10.1090/S0002-9947-00-02412-0
- MathSciNet review: 1637129