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The units of a ring spectrum and a logarithmic cohomology operation. (English) Zbl 1106.55002
Let \(R\) be a spectrum with additional structure making it a commutative \(S\)-algebra in the sense of A. K. Elmendorf, I. Kriz, M. A. Mandell and J. P. May [Rings, Modules and Algebras. Mathematical Surveys and Monographs. 47. (Providence), RI: American Mathematical Society (AMS). (1997; Zbl 0894.55001)], \(\text{GL}_1(R)\) the grouplike \(E_\infty\)-space determined by the pullback \[ \begin{tikzcd}GL_1(R) \rar\dar & \Omega^\infty R\dar\\ (\pi_0 R)^\times \rar & \pi_0 R \end{tikzcd} \] and \(\text{gl}_1(R)\) the \((-1)\) connected spectrum with \(\text{GL}_1(R)\) as \(0\)-space. The cohomology theory associated to \(\text{gl}_1(R)\) has the property that \(\text{gl}_1(R)^0 (X) = (R^0(X))^\times\) (and so is referred to as the units of the spectrum \(R\)). For \(n \geq 1\) and \(p\) a prime the Bousfield-Kuhn construction [A. K. Bousfield, Pac. J. Math 129, 1–31 (1987; Zbl 0664.55006), N. Kuhn, Algebraic topology, Proc. Int. Conf., Arcata/Calif. 1986, Lect. Notes Math. 1370, 243–257 (1989; Zbl 0692.55005)] determines a natural map \(\ell_{n,p}: \text{gl}_1(R) \to L_{K(n)}(R)\) where \(L_{K(n)}\) denotes localization with respect to \(K(n)\), the \(n\)-th Morava \(K\)-theory. The main result of the paper under review is a formula for the map \[ \ell_{n,p}: (R^0 X)^\times \to (L_{K(n)} R)^0 (X) \] in the case that \(R\) is equal to \(E\), a Morava \(E\)-theory associated to a height \(n \geq 1\) formal group law over a perfect field of characterisitc \(p\). For these theories power operations can be constructed [M. Ando, Duke Math. J. 79, 423–485 (1995; Zbl 0862.55004)] by showing that for every finite subgroup, \(A\), of \(\Lambda^* = ({\mathbb Q}_p / {\mathbb Z}_p)^n\) there is associated a natural ring homomorphism \[ \Psi_A: E^0 (X) \to D \otimes_{E_0} E^0 (X). \] The author shows that for \(x \in (E^0 X)^\times\) \[ \ell_{n,p}(x) = \sum^\infty_{k=1} (-1)^{k-1} \frac{p^{k-1}}{k} M(x)^k \] where \(M: E^0(X) \to E^0(X)\) is the unique cohomology operation such that \[ 1+p M(x) = \prod^n_{j=0} \Biggl( \prod_{_{\substack{ A \subseteq \Lambda^* [p] \\ | A| = p^j }}} \Psi_A (x) \Biggr)^{(-1)^j (j-1)(j-2)/2}. \] Here \(\Lambda^*[p]\) denotes the kernel of multiplication by \(p\) on \(\Lambda^*\). The author also expresses this formula in terms of certain additive \(E\)-operations which he calls Hecke operators and notes the similarity with expressions arising in the theory of \(L\)-functions.

MSC:
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
55S05 Primary cohomology operations in algebraic topology
55S25 \(K\)-theory operations and generalized cohomology operations in algebraic topology
55P47 Infinite loop spaces
55P60 Localization and completion in homotopy theory
55N34 Elliptic cohomology
11F25 Hecke-Petersson operators, differential operators (one variable)
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