Volume 16 (2009)

Download this article
For screen
For printing
Recent Volumes
Volume 1, 1998
Volume 2, 1999
Volume 3, 2000
Volume 4, 2002
Volume 5, 2002
Volume 6, 2003
Volume 7, 2004
Volume 8, 2006
Volume 9, 2006
Volume 10, 2007
Volume 11, 2007
Volume 12, 2007
Volume 13, 2008
Volume 14, 2008
Volume 15, 2008
Volume 16, 2009
Volume 17, 2011
Volume 18, 2012
Volume 19, 2015
The Series
MSP Books and Monographs
About this Series
Editorial Board
Ethics Statement
Author Index
Submission Guidelines
Author Copyright Form
ISSN (electronic): 1464-8997
ISSN (print): 1464-8989
Other MSP Publications

An overview of abelian varieties in homotopy theory

Tyler Lawson

Geometry & Topology Monographs 16 (2009) 179–214

DOI: 10.2140/gtm.2009.16.179

1 J F Adams, Stable homotopy and generalised homology, University of Chicago Press (1974) MR0402720
2 M Ando, M J Hopkins, N P Strickland, Elliptic spectra, the Witten genus and the theorem of the cube, Invent. Math. 146 (2001) 595–687 MR1869850
3 A Baker, Hecke operators as operations in elliptic cohomology, J. Pure Appl. Algebra 63 (1990) 1–11 MR1037690
4 T Bauer, Computation of the homotopy of the spectrum tmf, from: "Groups, homotopy and configuration spaces (Tokyo 2005)" (editors N Iwase, T Kohno, R Levi, D Tamaki, J Wu), Geom. Topol. Monogr. 13 (2008) 11–40
5 M Behrens, Notes on the construction of tmf, Preprint
6 M Behrens, A modular description of the K(2)–local sphere at the prime 3, Topology 45 (2006) 343–402 MR2193339
7 M Behrens, T Lawson, Topological automorphic forms, To appear in Mem. Amer. Math. Soc.
8 M Behrens, T Lawson, Topological automorphic forms on U(1,1) Submitted
9 A K Bousfield, On the homotopy theory of K–local spectra at an odd prime, Amer. J. Math. 107 (1985) 895–932 MR796907
10 P Deligne, Courbes elliptiques: formulaire d'après J. Tate, from: "Modular functions of one variable, IV (Proc Internat. Summer School, Univ. Antwerp, Antwerp, 1972)", Springer (1975) MR0387292
11 P Deligne, M Rapoport, Les schémas de modules de courbes elliptiques, from: "Modular functions of one variable, II (Proc Internat. Summer School, Univ. Antwerp, Antwerp, 1972)", Lecture Notes in Math. 349, Springer (1973) 143–316 MR0337993
12 E S Devinatz, M J Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology 43 (2004) 1–47 MR2030586
13 E S Devinatz, M J Hopkins, J H Smith, Nilpotence and stable homotopy theory I, Ann. of Math. (2) 128 (1988) 207–241 MR960945
14 G Faltings, C L Chai, Degeneration of abelian varieties, Ergebnisse Series 22, Springer (1990) MR1083353 With an appendix by David Mumford
15 J Franke, Uniqueness theorems for certain triangulated categories with an Adams spectral sequence (1996)
16 P G Goerss, (Pre-)sheaves of ring spectra over the moduli stack of formal group laws, from: "Axiomatic, enriched and motivic homotopy theory", NATO Sci. Ser. II Math. Phys. Chem. 131, Kluwer (2004) 101–131 MR2061853
17 P Goerss, H W Henn, M Mahowald, C Rezk, A resolution of the K(2)–local sphere at the prime 3, Ann. of Math. (2) 162 (2005) 777–822 MR2183282
18 P G Goerss, M J Hopkins, Moduli spaces of commutative ring spectra, from: "Structured ring spectra", London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 151–200 MR2125040
19 V Gorbounov, M Mahowald, Formal completion of the Jacobians of plane curves and higher real K–theories, J. Pure Appl. Algebra 145 (2000) 293–308 MR1733427
20 M Harris, R Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies 151, Princeton University Press (2001) MR1876802 With an appendix by Vladimir G Berkovich
21 H W Henn, On finite resolutions of K(n)–local spheres, from: "Elliptic cohomology", London Math. Soc. Lecture Note Ser. 342, Cambridge Univ. Press (2007) 122–169 MR2330511
22 S Hollander, A homotopy theory for stacks, Israel J. Math. 163 (2008) 93–124 MR2391126
23 M J Hopkins, Complex oriented cohomology theories and the language of stacks (1999)
24 M J Hopkins, Algebraic topology and modular forms, from: "Proceedings of the International Congress of Mathematicians, Vol I (Beijing, 2002)", Higher Ed. Press (2002) 291–317 MR1989190
25 M Hovey, N P Strickland, Morava K–theories and localisation, (1999) MR1601906
26 N Katz, Serre–Tate local moduli, from: "Algebraic surfaces (Orsay, 1976–78)", Lecture Notes in Math. 868, Springer (1981) 138–202 MR638600
27 R E Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992) 373–444 MR1124982
28 P S Landweber, Homological properties of comodules over MU*(MU) and BP_*(BP), Amer. J. Math. 98 (1976) 591–610 MR0423332
29 P S Landweber, D C Ravenel, R E Stong, Periodic cohomology theories defined by elliptic curves, from: "The \v Cech centennial (Boston, MA, 1993)", Contemp. Math. 181, Amer. Math. Soc. (1995) 317–337 MR1320998
30 G Laumon, L Moret-Bailly, Champs algébriques, Ergebnisse Series 39, Springer (2000) MR1771927
31 M Mahowald, C Rezk, Topological modular forms of level 3, To appear in Pure and Applied Math. Quarterly
32 J P May, Idempotents and Landweber exactness in brave new algebra, Homology Homotopy Appl. 3 (2001) 355–359 MR1856031
33 W Messing, The crystals associated to Barsotti–Tate groups: with applications to abelian schemes, Lecture Notes in Mathematics 264, Springer (1972) MR0347836
34 J S Milne, Points on Shimura varieties mod p, from: "Automorphic forms, representations and L–functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2", Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc. (1979) 165–184 MR546616
35 J Milnor, On the cobordism ring Ω* and a complex analogue I, Amer. J. Math. 82 (1960) 505–521 MR0119209
36 J Morava, Forms of K–theory, Math. Z. 201 (1989) 401–428 MR999737
37 J Morava, Complex cobordism and algebraic topology, arXiv:0707.3216 (2007)
38 D Mumford, Geometric invariant theory, Ergebnisse Series, Springer (1965) MR0214602
39 N Naumann, The stack of formal groups in stable homotopy theory, Adv. Math. 215 (2007) 569–600 MR2355600
40 D Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969) 1293–1298 MR0253350
41 D C Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics 121, Academic Press (1986) MR860042
42 D C Ravenel, Toward higher chromatic analogs of elliptic cohomology, from: "Elliptic cohomology", London Math. Soc. Lecture Note Ser. 342, Cambridge Univ. Press (2007) 286–305 MR2330518
43 C Rezk, Notes on the Hopkins–Miller theorem, from: "Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997)", Contemp. Math. 220, Amer. Math. Soc. (1998) 313–366 MR1642902
44 C Rezk, Supplementary notes for Math 512 (2001, url = http://www.math.uiuc.edu/ rezk/papers.html)
45 Y B Rudyak, Exactness theorems for the cohomology theories MU, BP and P(n), Mat. Zametki 40 (1986) 115–126, 141 MR864277
46 K Shimomura, X Wang, The homotopy groups π*(L2S0) at the prime 3, Topology 41 (2002) 1183–1198 MR1923218
47 J H Silverman, The arithmetic of elliptic curves, Graduate Texts in Math 106, Springer (1992) MR1329092 Corrected reprint of the 1986 original
48 J Tate, Finite flat group schemes, from: "Modular forms and Fermat's last theorem (Boston, MA, 1995)", Springer (1997) 121–154 MR1638478
49 A Vistoli, Grothendieck topologies, fibered categories and descent theory, from: "Fundamental algebraic geometry", Math. Surveys Monogr. 123, Amer. Math. Soc. (2005) 1–104 MR2223406