The power operation structure on Morava

Zhu, Yifei¹

¹Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA

Algebraic and Geometric Topology, Tome 14 (2014) no. 2, pp. 953-977

Résumé

We give explicit calculations of the algebraic theory of power operations for a specific Morava E–theory spectrum and its K(1)–localization. These power operations arise from the universal degree-3 isogeny of elliptic curves associated to the E–theory.

Détail

DOI : 10.2140/agt.2014.14.953

Classification : 55S12, 55N20, 55N34
Keywords: power operations, elliptic curves, Morava $E$–theory, $K(1)$–localization

Bibliographie

[1] M Ando, Isogenies of formal group laws and power operations in the cohomology theories $E^n$, Duke Math. J. 79 (1995) 423

[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

[3] M H Baker, E González-Jiménez, J González, B Poonen, Finiteness results for modular curves of genus at least $2$, Amer. J. Math. 127 (2005) 1325

[4] J Borger, B Wieland, Plethystic algebra, Advances Math. 194 (2005) 246

[5] R R Bruner, J P May, J E Mcclure, M Steinberger, $H_\infty $ ring spectra and their applications, Lecture Notes in Mathematics 1176, Springer (1986)

[6] A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs 47, Amer. Math. Soc. (1997)

[7] P G Goerss, M J Hopkins, Moduli spaces of commutative ring spectra, from: "Structured ring spectra" (editors A Baker, B Richter), London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 151

[8] M J Hopkins, $K(1)$–local $E_\infty $ ring spectra

[9] M Hovey, $v_n$–elements in ring spectra and applications to bordism theory, Duke Math. J. 88 (1997) 327

[10] D Husemöller, Elliptic curves, Graduate Texts in Mathematics 111, Springer (2004)

[11] N M Katz, B Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies 108, Princeton Univ. Press (1985)

[12] M Mahowald, C Rezk, Topological modular forms of level $3$, Pure Appl. Math. Q. 5 (2009) 853

[13] A Mazel-Gee, E Peterson, N Stapleton, A relative Lubin–Tate theorem via meromorphic formal geometry

[14] C Rezk, Lectures on power operations

[15] C Rezk, Power operations for Morava ${E}$–theory of height $2$ at the prime $2$

[16] C Rezk, The congruence criterion for power operations in Morava $E$–theory, Homology, Homotopy Appl. 11 (2009) 327

[17] J H Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer (2009)

[18] N E Steenrod, Cohomology operations, Annals of Mathematics Studies 50, Princeton Univ. Press (1962)

[19] N P Strickland, Morava $E$–theory of symmetric groups, Topology 37 (1998) 757

[20] V Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci. (2003) 1