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We show that Lubin-Tate spectra at the prime $2$ are Real oriented and Real Landweber exact. The proof is by application of the Goerss-Hopkins-Miller theorem to algebras with involution. For each height $n$, we compute the entire homotopy fixed point spectral sequence for $E_n$ with its $C_2$-action given by the formal inverse. We study, as the height varies, the Hurewicz images of the stable homotopy groups of spheres in the homotopy of these $C_2$-fixed points.
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We study certain formal group laws equipped with an action of the cyclic group of order a power of $2$. We construct $C_{2^n}$-equivariant Real oriented models of Lubin-Tate spectra $E_h$ at heights $h=2^{n-1}m$ and give explicit formulas of the $C_{2^n}$-action on their coefficient rings. Our construction utilizes equivariant formal group laws associated with the norms of the Real bordism theory $MU_{mathbb{R}}$, and our work examines the height of the formal group laws of the Hill-Hopkins-Ravenel norms of $MU_{mathbb{R}}$.
We take a direct approach to computing the orbits for the action of the automorphism group $mathbb{G}_2$ of the Honda formal group law of height $2$ on the associated Lubin-Tate rings $R_2$. We prove that $(R_2/p)_{mathbb{G}_2} cong mathbb{F}_p$. The result is new for $p=2$ and $p=3$. For primes $pgeq 5$, the result is a consequence of computations of Shimomura and Yabe and has been reproduced by Kohlhaase using different methods.
We completely compute the slice spectral sequence of the $C_4$-spectrum $BP^{((C_4))}langle 2 rangle$. After periodization and $K(4)$-localization, this spectrum is equivalent to a height-4 Lubin-Tate theory $E_4$ with $C_4$-action induced from the Goerss-Hopkins-Miller theorem. In particular, our computation shows that $E_4^{hC_{12}}$ is 384-periodic.
We give a new formula for $p$-typical real topological cyclic homology that refines the fiber sequence formula discovered by Nikolaus and Scholze for $p$-typical topological cyclic homology to one involving genuine $C_2$-spectra. To accomplish this, we give a new definition of the $infty$-category of real $p$-cyclotomic spectra that replaces the usage of genuinely equivariant dihedral spectra with the parametrized Tate construction $(-)^{t_{C_2} mu_p}$ associated to the dihedral group $D_{2p} = mu_p rtimes C_2$. We then define a $p$-typical and $infty$-categorical version of H{o}genhavens $O(2)$-orthogonal cyclotomic spectra, construct a forgetful functor relating the two theories, and show that this functor restricts to an equivalence between full subcategories of appropriately bounded below objects.
Ando, Morava, and Sadofsky showed that the Tate construction for a trivial $mathbb{Z}/p$-action decreases the chromatic height of Johnson-Wilson theory, and Greenlees and Sadofsky proved that the Tate construction for a trivial finite group action vanishes on Morava K-theory. We prove $C_2$-equivariant enrichments of these results using the parametrized Tate construction. The $C_2$-fixed points of our results produce new blueshift and vanishing results for Real Johnson-Wilson theories $ER(n)$ and Real Morava $K$-theories $KR(n)$, respectively, for all $n$. In particular, our blueshift results generalize Greenlees and Mays Tate splitting of $KO$ to all chromatic heights.
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