Stable reduction of \(X_0(p^3)\). With an Appendix by Everett W. Howe.

*(English)*Zbl 1215.11060Let \(n\) be an integer and \(p\) a prime. It is known that if \(n\geq3\) and \(p\geq5\), or if \(n\geq1\) and \(p\geq11\), the modular curve \(X_0(p^n)\) does not have a model with good reduction over the ring of integers of any complete subfield of \({\mathbb C}_p\). However, it may have a stable model, and the problem of finding such stable models has been studied intensively, starting with the work of Deligne and Rapoport.

The main achievement of the present paper is the construction of a stable model for \(X_0(p^3)\), when \(p\geq13\), over the ring of integers of some finite extension of \({\mathbb Q}_p\) (which can be made explicit).

The method is rigid analytic. The notion of (semi)stable model is rephrased into the notion of (semi)stable rigid analytic covering of the given curve. The corresponding reduction can be recovered from such a covering, and similarly the genus. At the supersingular loci the covering pieces are the deformation spaces of height two formal groups with level structure, as studied first by Lubin. For the present purposes, results of Hopkins and Gross on these deformation spaces are needed. On the other hand, the affinoids lying over the ordinary locus must be slightly extended into the supersingular locus; this is related to the theory of canonical subgroups of elliptic curves.

The paper starts with complete and generous accounts of all the tools it requires (rigid analysis, modular curves, deformation spaces of formal groups, canonical subgroups of elliptic curves). It then turns to its proper subject, a very profound study of the rigid geometry of modular curves.

The main achievement of the present paper is the construction of a stable model for \(X_0(p^3)\), when \(p\geq13\), over the ring of integers of some finite extension of \({\mathbb Q}_p\) (which can be made explicit).

The method is rigid analytic. The notion of (semi)stable model is rephrased into the notion of (semi)stable rigid analytic covering of the given curve. The corresponding reduction can be recovered from such a covering, and similarly the genus. At the supersingular loci the covering pieces are the deformation spaces of height two formal groups with level structure, as studied first by Lubin. For the present purposes, results of Hopkins and Gross on these deformation spaces are needed. On the other hand, the affinoids lying over the ordinary locus must be slightly extended into the supersingular locus; this is related to the theory of canonical subgroups of elliptic curves.

The paper starts with complete and generous accounts of all the tools it requires (rigid analysis, modular curves, deformation spaces of formal groups, canonical subgroups of elliptic curves). It then turns to its proper subject, a very profound study of the rigid geometry of modular curves.

Reviewer: Elmar Große-Klönne (Berlin)