Modular isogeny complexes
The author defines the modular isogeny complex and proves a vanishing result for the cohomology of this complex. Let \(E/S\) be an elliptic scheme over a scheme \(S,\) let \([N\text{-Isog}](E/S)\) be the set of locally free finite commutative \(S\) subgroup schemes \(G\subset E\) of rank \(N\) over \(S\) (cf. [\textit{N. M. Katz} and \textit{B. Mazur}, Arithmetic moduli of elliptic curves. Princeton, New Jersey: Princeton University Press (1985; Zbl 0576.14026)]). For \(S=\mathrm{Spec}A\) let \({\mathcal S}_{N}(E/S)\) be the function ring of \([N\text{-Isog}](E/S).\) Similarly for integers \(N_{1},\dots , N_{q}\), let \([N_{1},\dots , N_{q} \text{-Isog}](E/S)\) denote the set of sequences \({\underline G}= G_{1} \subsetneq G_{2} \dots \subsetneq G_{q}, \) where \(G_{i}\) is a locally free commutative \(S\)-subgroup scheme of \(E\) of rank \(N_{1},\dots, N_{i}.\) For \(S=\mathrm{Spec} A\) denote by \({\mathcal S}_{N_{1},\dots , N_{q}}(E/S)\) the function ring of \([N_{1},\dots , N_{q} \text{-Isog}](E/S).\) The modular \(N\)-isogeny complex \({\mathcal K}_{N}^{\bullet}(E/S)\) of \(A\)-modules is defined in the following way: NEWLINE\[NEWLINE{\mathcal K}_{N}^{q}(E/S)={\prod}_{N_{1},\dots , N_{q}}{\mathcal S}_{N_{1},\dots , N_{q}}(E/S),NEWLINE\]NEWLINE where the product runs over all \(q\)-tuples \((N_{1},\dots , N_{q})\) of integers such that \(N_{1}\cdots N_{q}=N\) and \(N_{i}>1.\) The coboundary map \({\delta}: {\mathcal K}_{N}^{q-1} \rightarrow {\mathcal K}_{N}^{q}\) is defined by the formula: NEWLINE\[NEWLINE{{\delta} f}_{N_{1},\dots , N_{q}} = {\sum}_{i=1}^{q-1} (-1)^i u_{i}(f_{N_{1},\dots, N_iN_{i+1},\dots , N_{q}}),NEWLINE\]NEWLINE where \(u_{i}\) is induced by forgetting the \(i\)-the group in the sequence \({\underline G}= G_{1} \subsetneq G_{2} \dots \subsetneq G_{q} .\) The results have applications to study the power operations in Morava \(E\)-theory at height \(2.\)
scientific article; zbMATH DE number 6068601
scientific article