A modular description of \(ER(2)\) (Q462696)

In this paper the real Johnson--Wilson theory \(ER(2)\) is shown to arise from certain modular curves in an analogous way to how \(TMF\) arises from the moduli stack of smooth elliptic curves. To define the spectrum \(ER(2)\) consider the orientation map \(BP \to E(n)\), for \(E(n)\) the Johnson--Wilson theory of period \(2(2^n-1)\). This map can be refined to a map of equivariant spectra \(BP\mathbb{R} \to E\mathbb{R}(n)\). The underlying spectrum of \(E\mathbb{R}(n)\) is \(E(n)\) and its homotopy fixed points with respect to conjugation is the spectrum \(ER(n)\), which has period \(2^{n+2}(2^n-1)\). In particular, \(ER(1) = KO_{(2)}\), the localisation of real \(K\)-theory at \(p=2\).NEWLINENEWLINEFor a \(2\)-local commutative \(S\)-algebra \(A\), let \(A(\mu_\infty, 2)\) denote the \(S\)-algebra with all roots of unity of order prime to \(2\) adjoined. The main result of this paper is that \(L_{K(2)} ER(2)(\mu_\infty, 2)\) is an algebra over \(TMF\) and NEWLINE\[NEWLINE L_{K(2)} ER(2)(\mu_\infty, 2) \to L_{K(2)} E(2)(\mu_\infty, 2) NEWLINE\]NEWLINE is a \(K(2)\)--local \(C_2\)-Galois extension isomorphic to NEWLINE\[NEWLINE L_{K(2)} TMF_0(3)(\mu_\infty, 2) \to L_{K(2)} TMF_1(3)(\mu_\infty, 2). NEWLINE\]