Formal Models of G m and Ω

Transcription

1 Formal Models of G m and Ω Drew Moore In this document, we explain the formal models of the rigid analytic spaces G m and Ω (the Drinfeld upper half plane) via pictures. When a formal scheme is depicted, its special fiber will be on the right, and its rigid generic fiber on the left. The colored regions in each match under the specialization map from the generic fiber to the special fiber. When we need to choose p, we choose p =. A Formal Model of G m In this section, we aim to describe the construction of a formal model for G m : that is, a formal scheme X over Spf Z p whose rigid generic fiber X rig is isomorphic to the rigid analytic variety G an m. Figure : The formal scheme P /Z p, with its generic fiber P,an /Q p on the left, and its special fiber P F p on the right. We start by seeing what happens when we remove two points from the formal scheme P Z p.

2 In figure, we ve decomposed the rigid generic fiber (P /Q p ) an of P Z p into a non-admissable cover by X orange, X blue, X red. Specifically, if T is a coordinate centered at [ : ] on P, X orange = { T < } X blue = { T = } X red = { T > } Under the specialization map, every C p -point of X orange (resp. X red ) maps to the F p point P (F p ) (resp. P (F p )). In contrast, the C p points of the circle X blue surject onto G m (F p ). Hence, we see that if we remove the points, in the special fiber from the formal scheme (i.e. inverting a uniformizer at those points in the sheaf of functions on the formal scheme), we remove the points T P (C p ) with norm > or <, which is too much. To fix this, we use blowups. Blowups preserve the rigid fiber, but allow for modification of the the special fiber (and thus also the behavior of the specialization map). In figure, we depict the blowup of P Z p at in the special fiber. The rigid spaces X red, X blue are the same as in figure. In addition, we have X orange = { p < T < } X green = { T = p } X yellow = { T < p } = sp() ( ) glued to sp() = Figure : The formal scheme obtained from P by blowing up a point in the special fiber. Here p =.

3 If we were to remove the (red) point and the (yellow) point from the special fiber, this would remove the sets { T < p } and { T > } from the generic fiber. Hence, blowing up the scheme at improved our situation near. So what we will do is continually blow up the special fiber at the specializations of the points and in the generic fiber. Set Y = P /Z p, and X = P /Z p \ {P, P }, where P and P are the closed points corresponding to the usual F p -points. The special fiber is G m/fp, and generic fiber is the blue circle depicted on page. Let Y n+ be the blowup of Y n at the points P n,, P n,, which are the closed points of the formal scheme corresponding to sp n () and sp n () (here, sp n is the specialization map for the formal model Y n ). Denote by X n+ the space obtained by removing P n+,, P n+,. The generic fiber of X n has C p -points { p n < T < p n } P (C p ). We have natural maps X X. Let X be the limiting object. It is a formal scheme, not of finite type. Its special fiber is an infinite chain (in both directions) of P s intersecting transversally, while its rigid generic fiber is G m. Figure : A formal model X of G an m The Formal Model of the Drinfeld Upper Half Plane Now, we aim to give a formal model for the Drinfeld upper half plane Ω, whose C p points are P (C p ) \ P (Q p ). This has a rigid analytic structure by giving an admissable covering, which is obtained by successively removing increasing numbers of smaller balls around the Q p -rational points of P. In figure is a diagram showing P /Z p again. We also show the preimages of each of the 6 F -rational points of P F. Each of the yellow/orange/red domains are open disks of radius. If we naively/prematurely remove these points from the special fiber, we see that we

4 Figure : The formal projective line. The preimages of the F p - rational points under specialization are depicted. Recall, p =. remove too much from the generic fiber. Hence, just as in the case for the formal model of G m, we will blow up at each of these 6 points Figure : The second step in constructing the formal model of the Drinfeld upper half plane. In figure, we ve depicted the blowup Y of Y = P /Z p at its closed points correspond-

5 ing to the F p -rational points of its special fiber. Depicted in yellow, red, and orange are the F p -rational points of Y = Y /Fp. All Q p points (except ) in the generic fiber are contained in the preimages of the yellow points ( is in the preimage of the red point). The orange regions in the generic fiber are the vanishing cycle annuli which specialize to the nodes. Removing the closed points of Y corresponding to the yellow/red points of the special fiber will remove disks of radius p around each Q p rational point in P = {,,, /, /, /, /, } (later we will define P n ). This is an improvement from before - we ve removed less points which aren t Q p -rational, and we will obtain a formal model if we take a limit of objects obtained by blowing up in this way. Define X to be Y with the yellow points (and red point) removed. Namely, let P n be a set of representatives in P (Q p ) for P (Z/p n Z). Define Y n+ to be the blowup of Y n at the points sp n (Q) for each Q P n (where sp n is the specialization map for Y n ). Define X n+ to be formal scheme obtained by removing the closed points corresponding to the points sp n+ (Q) for Q P n+. Then the formal model for the p-adic upper half plane Ω is the limiting object of X X. Its special fiber will be a tree of P s. Each P will have a node (intersecting another P transversely) at each of its (p + ) F p -rational points. Its rigid generic fiber will have C p -points P (C p ) \ P (Q p ). In the pdf document Drinfeld Upper Halfplane, I ve depicted the generic and special fibers of Y, respectively. Only part of the generic fiber is drawn - namely the unit disk centered at the origin. The region in gray in the special fiber corresponds to the omitted region in the generic fiber. References [] Siegfried Bosch and Werner Lütkebohmert. Formal and rigid geometry. I. Rigid spaces. In: Math. Ann. 9. (99), pp issn: -8. [] J.-F. Boutot and H. Carayol. Uniformisation p-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfeld. In: Astérisque (99). Courbes modulaires et courbes de Shimura (Orsay, 987/988), 7, 8 (99). issn: -79. [] P. Deligne and M. Rapoport. Les schémas de modules de courbes elliptiques. In: Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 97). Springer, Berlin, 97, 6. Lecture Notes in Math., Vol. 9. [] Jean Fresnel and Marius van der Put. Rigid analytic geometry and its applications. Vol. 8. Progress in Mathematics., pp. xii+96. isbn: