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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the homotopy type of the spectrum representing elliptic cohomology
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by Andrew Baker PDF
Proc. Amer. Math. Soc. 107 (1989), 537-548 Request permission

Abstract:

In this paper we analyse the homotopy type at primes $p > 3$ of the ring spectrum $E\ell \ell$ representing a version of elliptic cohomology whose coefficient ring $E\ell {\ell _ * }$ agrees with the ring of modular forms for $S{L_2}(\mathbb {Z})$. For any prime (=maximal) graded ideal $\mathcal {P} \triangleleft E\ell {\ell _*}$ containing the Eisenstein function ${E_{p - 1}}$ as well as $p$, we show that there is a morphism of ring spectra \[ \widehat {E(2)} \to (E\ell \ell )_{\hat {\mathcal {P}}}\] and a corresponding splitting \[ (E\ell \ell )_{\hat {\mathcal {P}}} \simeq \bigvee \limits _i {\Sigma ^{2\theta (i)}}\widehat {E(2)}\] of algebra spectra over $\widehat {E(2)}$ (the ${I_2}$-adic completion of $E(2)$); here $(\;)_{\hat {\mathcal {P}}}$ denotes the $\mathcal {P}$-adic completion of the spectrum $E\ell \ell$. Moreover, there is a multiplicative reduction ${(E\ell \ell /\mathcal {P})^ * }(\;)$ and we similarly show that there is a splitting of $K(2)$ algebra spectra \[ E\ell \ell /\mathcal {P} \simeq \bigvee \limits _i {\Sigma ^{2\theta ’(i)}}K(2).\] In each case the indexing $i$ ranges over a finite set.
References
  • J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1995. Reprint of the 1974 original. MR 1324104
  • 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
  • —, Elliptic cohomology, $p$-adic modular forms and Atkin’s operator ${U_p}$, preprint, 1988. A. Baker and U. Würgler, Liftings of formal group laws and the Artinian completion of $v_n^{ - 1}BP$, preprint, 1988.
  • Jun-ichi Igusa, On the transformation theory of elliptic functions, Amer. J. Math. 81 (1959), 436–452. MR 104668, DOI 10.2307/2372750
  • Jun-ichi Igusa, On the algebraic theory of elliptic modular functions, J. Math. Soc. Japan 20 (1968), 96–106. MR 240103, DOI 10.2969/jmsj/02010096
  • Nicholas M. Katz, $p$-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 69–190. MR 0447119
  • Neal Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1984. MR 766911, DOI 10.1007/978-1-4684-0255-1
  • Peter S. Landweber, Elliptic cohomology and modular forms, Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986) Lecture Notes in Math., vol. 1326, Springer, Berlin, 1988, pp. 55–68. MR 970281, DOI 10.1007/BFb0078038
  • Peter S. Landweber, Supersingular elliptic curves and congruences for Legendre polynomials, Elliptic curves and modular forms in algebraic topology (Princeton, NJ, 1986) Lecture Notes in Math., vol. 1326, Springer, Berlin, 1988, pp. 69–93. MR 970282, DOI 10.1007/BFb0078039
  • Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
  • Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR 860042
  • J.-P. Serre, Cours d’arithmétique, Presses Universitaires de France, Vendôme.
  • Jean-Pierre Serre, Formes modulaires et fonctions zêta $p$-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin, 1973, pp. 191–268 (French). MR 0404145
  • —, Congruences et formes modulaires, Séminaire Bourbaki, Vol 24$^{e}$, no. 416, Lecture Notes in Mathematics 317 (1971/2), 319-38. J. Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York.
  • Nobuaki Yagita, The exact functor theorem for $\textrm {BP}_\ast /I_{n}$-theory, Proc. Japan Acad. 52 (1976), no. 1, 1–3. MR 394631
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 107 (1989), 537-548
  • MSC: Primary 55N22; Secondary 11F11
  • DOI: https://doi.org/10.1090/S0002-9939-1989-0982399-6
  • MathSciNet review: 982399