Remarks on the elliptic cohomology of finite groups (Q1894176)

Elliptic cohomology \(Ell^*\) is a complex oriented cohomology theory whose ring of coefficients coincides with the ring of modular forms over \(\mathbb{Z} [{1\over 6}]\) for the group \(SL_2 (\mathbb{Z})\). It is known that in elliptic cohomology there is a topological \(q\)-expansion map, i.e. a natural transformation \(\lambda: Ell^*(-) \to K^*(-)[{1\over 6}] ((q))\) which preserves complex orientations and is given on the coefficient rings by the ordinary \(q\)-expansion morphism of modular forms. The author studies for a given finite group \(G\) the elliptic character, i.e. the composition \(\chi:Ell^*(BG) @>\lambda>> K^*(BG) [{1\over 6}]((q)) \to \widehat {R(G)} [{1\over 6}] ((q))\) of the \(q\)-expansion map with Atiyah's classical isomorphism of \(K^*(BG)\) with the completed character ring \(\widehat {R(G)}\). Let \(p\geq 5\) be a prime number. Evaluation of characters at elements \(g\) of order \(p^n\) yields a homomorphism of \(p \)-adic completions \(e(g): \widehat {R(G)}_{\widehat p} \to\widehat {\overline{Q_p}}=: \mathbb{C}_p.\) Here \(\mathbb{C}_p\) denotes the ``\(p\)-adic complex numbers'', i.e. the \(p\)-adic completion of the algebraic closure of \(Q_p\). The author shows that for \(x\in Ell^{2k} (BG)\) the power series \(e(g) \chi(x) \in\mathbb{C}_p((q))\) is the \(q\)-expansion of a modular form of weight \(2k\) with coefficients in \(\mathbb{Z}_p [\zeta_{p^n}]\). Moreover he obtains a precise comparison result of his elliptic character with the Hopkins-Kuhn-Ravenel characters associated to suitable completions of \(Ell^*\).