Full level structures on elliptic curves (Q6603913)

"The main result of this paper is an integral version of the theorem of Scholze according to which the modular curve \(X(p^\infty)^\mathrm{an}\) with infinite level at \(p\) is a perfectoid space over \(\mathbb{Q}_p^\mathrm{cyc}\), see [\textit{P. Scholze}, Ann. Math. (2) 182, No. 3, 945--1066 (2015; Zbl 1345.14031)]. In the process, the author also proves a mixed-characteristic analogue of Kunz's theorem on the regularity of noetherian \(\mathbb{F}_p\)-algebras, see [\textit{E. Kunz}, Am. J. Math. 98, 999--1013 (1976; Zbl 0341.13009)], which is of independent interest. \par Let us explain the statement of the main theorem in more detail, closely following the paper's introduction. Fix \(p > 0\) a prime. Let \(\overline{\mathrm{Ell}}(p^\infty)\) denote the Deligne-Mumford stack over the ring \(\mathbb{Z}[\zeta_{p^\infty}]\) obtained as the inverse limit of the stacks \(\overline{\mathrm{Ell}}(p^n) \to \mathrm{Spec}(\mathbb{Z}[\zeta_{p^n}])\) of generalised elliptic curves with full level-\(p^n\) structure in the sense of Drinfeld. Write \(\mathbb{F}_p[\zeta_{p^\infty}]\) for \(\mathbb{Z}[\zeta_{p^\infty}]/(p)\) and \(\overline{\mathrm{Ell}}(p^\infty)_{p=0}\) for the closed substack of \(\overline{\mathrm{Ell}}(p^\infty)\) on which \(p\) vanishes. The Frobenius map on \(\mathbb{F}_p[\zeta_{p^\infty}]\) is surjective, with kernel generated by the image of \(\pi = (\zeta_{p^2}-1)^{p-1}\) via the reduction map \(\mathbb{Z}[\zeta_{p^\infty}] \to \mathbb{F}_p[\zeta_{p^\infty}]\). The main theorem states that the absolute Frobenius morphism \(\varphi \colon \overline{\mathrm{Ell}}(p^\infty)_{p=0} \to \overline{\mathrm{Ell}}(p^\infty)_{p=0}\) induces an isomorphism between \(\overline{\mathrm{Ell}}(p^\infty)_{p=0}\) and the closed substack \(\overline{\mathrm{Ell}}(p^\infty)_{\pi=0}\) of \(\overline{\mathrm{Ell}}(p^\infty)\). This, in turn, implies that for any étale map \(\mathrm{Spec}(R) \to \overline{\mathrm{Ell}}(p^\infty)\), there is some \(\varpi \in R\), image of \(\pi\), such that \(\varpi^p\) is a unit multiple of \(p\) and the Frobenius map \(R/(\varpi) \to R/(p)\) is an isomorphism. In particular, as remarked by the author, this result is not exactly an integral version of the result of Scholze. It is a statement about the local structure of \(\overline{\mathrm{Ell}}(p^\infty)\) with respect to the étale topology, rather than the analytic topology. The theorem of Scholze still follows from the main result of this paper. \par Let us sketch the main ideas of the proof. The main theorem is proved by working at finite level-\(p^n\) and showing, for \(n \geq 3\), the existence of a unique morphism \(\varTheta \colon \overline{\mathrm{Ell}}(p^n)_{\pi=0} \to \overline{\mathrm{Ell}}(p^{n-1})_{p=0}\) that fits in the commutative diagram\N\[\N\begin{tikzcd} \overline{\mathrm{Ell}}(p^n)_{p=0} \arrow[d, ""\theta""] \arrow[r, ""\varphi""] & \overline{\mathrm{Ell}}(p^n)_{\pi=0} \arrow[d] \arrow[dl, dashed, swap, ""\varTheta""]\\\N\overline{\mathrm{Ell}}(p^{n-1})_{p=0} \arrow[r, ""\varphi""] & \overline{\mathrm{Ell}}(p^{n-1})_{\pi=0}, \end{tikzcd}\N\]\Nwhere the vertical maps are given by ``forgetting'' the level structure and the horizontal maps are given by the absolute Frobenii. Away from the cusps, points of \(\overline{\mathrm{Ell}}(p^n)_{p=0}\) can be thought of as elliptic curves \(E\) endowed with a full level-\(p^n\) structure, given by two \(p^n\)-torsion points \(x, y\), for which the Weil pairing \(e_{p^n}(x, y)\) is a primitive \(p^n\)-th root of unity. If \(e_{p^n}(x, y)\) is also a primitive \(p^{n-1}\)-st root of unity, then \((E, x, y)\) is in fact a point of the closed substack \(\overline{\mathrm{Ell}}(p^n)_{\pi=0}\). In that case, and if moreover \(E\) is ordinary, the subgroup \(S \subseteq E\) generated by \(p^{n-1}x, p^{n-1}y\) is étale of degree \(p\) and \(\varTheta\) can be described by the rule \((E, x, y) \mapsto (E/S, x+S, y+S)\). This gives \(\varTheta\) over the ordinary locus \(\overline{\mathrm{Ell}}(p^n)_{\pi=0}^\mathrm{ord}\). To extend \(\varTheta\) to the supersingular points (and the cusps), the author shows that, owing to the density of the ordinary locus where \(\varTheta\) is already defined, the map \(\theta\) from the diagram descends along the faithfully flat Frobenius map \(\varphi \colon \overline{\mathrm{Ell}}(p^n)_{p=0} \to \overline{\mathrm{Ell}}(p^n)_{\pi=0}\). The flatness of \(\varphi\) is implied by the mixed-characteristic Kunz's theorem, together with the regularity of the stack \(\overline{\mathrm{Ell}}(p^n)\). \par Let us state the aforementioned mixed-characteristic version of Kunz's theorem. Let \(R\) be a noetherian ring with \(\varpi \in R\) a regular element such that \(\varpi^p \mid p\). The following are equivalent:\N\begin{itemize}\N\item[1.] For every maximal ideal \(\mathfrak{m} \subseteq R\) such that \(\varpi \in \mathfrak{m}\), the local ring \(R_\mathfrak{m}\) is regular.\N\item[2.] The Frobenius morphism \(\varphi \colon R/(\varpi) \to R/(\varpi^p)\) is flat.\N\end{itemize}\NLet us say a few words about the proof of this theorem. The implication 1 \(\Rightarrow\) 2 follows by reducing to the case of a complete regular local ring with perfect residue field \(k\) and showing that \(\varphi\) is a pushout of a (flat) lift of Frobenius to some \(W(k)[[ X_1, \ldots, X_n ]]\). The converse follows by reduction to the local case and from a bound, afforded by the flatness of \(\varphi\), on the degree of the Hilbert-Samuel function \(\chi(t) = \ell(R/(\varpi R + \mathfrak{m}^t))\) of \(R/(\varpi)\).\N\NFor the entire collection see [Zbl 1515.14008]."