Formal Models of Gm and Ω
Drew Moore
In this document, we explain the formal models of the rigid analytic spaces Gm 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 = 5.
1
A Formal Model of Gm
In this section, we aim to describe the construction of a formal model for Gm : that is, a
formal scheme X over Spf Zp whose rigid generic fiber Xrig is isomorphic to the rigid analytic
variety Gan
m.
∞
0
0
1,an
Figure 1: The formal scheme P1/Zp , with its generic fiber P/Q
on the
p
1
left, and its special fiber PFp on the right.
b1 .
We start by seeing what happens when we “remove two points” from the formal scheme P
Zp
1
b1 into a non-admissable
In figure 1, we’ve decomposed the rigid generic fiber (P1/Qp )an of P
Zp
cover by Xorange , Xblue , Xred . Specifically, if T is a coordinate centered at [0 : 1] on P1 ,
Xorange = {|T | < 1}
Xblue = {|T | = 1}
Xred = {|T | > 1}
Under the specialization map, every Cp -point of Xorange (resp. Xred ) maps to the Fp point
0 ∈ P1 (Fp ) (resp. ∞ ∈ P1 (Fp )). In contrast, the Cp points of the “circle” Xblue surject onto
Gm (Fp ).
Hence, we see that if we remove the points 0, ∞ 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 ∈ P1 (Cp ) with norm > 1 or < 1, 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
b1
thus also the behavior of the specialization map). In figure 2, we depict the blowup of P
Zp
at 0 in the special fiber.
The rigid spaces Xred , Xblue are the same as in figure 1. In addition, we have
Xorange = {|p| < |T | < 1}
Xgreen = {|T | = |p|}
Xyellow = {|T | < |p|}
∞
10
0
2 = sp(2)
=
10)
0 glued
to ∞
sp(
2
2
Figure 2: The formal scheme obtained from P1 by blowing up a point
in the special fiber. Here p = 5.
2
0
If we were to remove the (red) point ∞ and the (yellow) point 0 from the special fiber, this
would remove the sets {|T | < |p|} and {|T | > 1} from the generic fiber. Hence, blowing up
the scheme at 0 improved our situation near 0.
So what we will do is continually blow up the special fiber at the specializations of the
points 0 and ∞ in the generic fiber.
b1 , and X1 = P
b1 \ {P0 , P∞ }, where P0 and P∞ are the closed points correSet Y1 = P
/Zp
/Zp
sponding to the usual Fp -points. The special fiber is Gm/Fp , and generic fiber is the blue
“circle” depicted on page 2.
Let Yn+1 be the blowup of Yn at the points Pn,0 , Pn,∞ , which are the closed points of the formal scheme corresponding to spn (0) and spn (∞) (here, spn is the specialization map for the
formal model Yn ). Denote by Xn+1 the space obtained by removing Pn+1,0 , Pn+1,∞ .
The generic fiber of Xn has Cp -points {|p|n < |T | < |p|−n } ⊂ P1 (Cp ). We have natural maps
X1 ,→ X2 ,→ · · · . 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 P1 ’s intersecting transversally, while
its rigid generic fiber is Gm .
Figure 3: A formal model X of Gan
m
2
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 Cp points
are P1 (Cp ) \ P1 (Qp ). This has a rigid analytic structure by giving an admissable covering,
which is obtained by successively removing increasing numbers of smaller balls around the
Qp -rational points of P1 .
b1 again. We also show the preimages of each of the 6
In figure 4 is a diagram showing P
/Zp
F5 -rational points of P1F5 . Each of the yellow/orange/red domains are open disks of radius
1. If we naively/prematurely remove these points from the special fiber, we see that we
3
∞
2
4
3
0
1
3
2
4
1
0
Figure 4: The formal projective line. The preimages of the Fp rational points under specialization are depicted. Recall, p = 5.
remove too much from the generic fiber. Hence, just as in the case for the formal model of
Gm , we will blow up at each of these 6 points.
12
1
10
7
1
5
2
17
13
8
10
3
18
22
5
11
6
0
23
15
1
20
14
16
21
9
4
1
20
19
1
15
24
Figure 5: The second step in constructing the formal model of the
Drinfeld upper half plane.
b1 at its closed points correspondIn figure 5, we’ve depicted the blowup Y2 of Y1 = P
/Zp
4
ing to the Fp -rational points of its special fiber. Depicted in yellow, red, and orange are
the Fp -rational points of Y2 = Y2/Fp . All Qp 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 Y2 corresponding to the yellow/red
points of the special fiber will remove disks of radius |p| around each Qp rational point
in P2 = {0, · · · , 24, 1/5, 1/10, 1/15, 1/20, ∞} (later we will define Pn ). This is an improvement from before - we’ve removed less points which aren’t Qp -rational, and we will obtain
a formal model if we take a limit of objects obtained by blowing up in this way. Define X2
to be Y2 with the yellow points (and red point) removed.
Namely, let Pn be a set of representatives in P1 (Qp ) for P1 (Z/pn Z). Define Yn+1 to be
the blowup of Yn at the points spn (Q) for each Q ∈ Pn (where spn is the specialization
map for Yn ). Define Xn+1 to be formal scheme obtained by removing the closed points
corresponding to the points spn+1 (Q) for Q ∈ Pn+1 . Then the formal model for the p-adic
upper half plane Ω is the limiting object of X1 ,→ X2 ,→ · · · . Its special fiber will be a
tree of P1 ’s. Each P1 will have a node (intersecting another P1 transversely) at each of its
(p + 1) Fp -rational points. Its rigid generic fiber will have Cp -points P1 (Cp ) \ P1 (Qp ).
In the pdf document “Drinfeld Upper Halfplane”, I’ve depicted the generic and special fibers
of Y4 , 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
[1]
Siegfried Bosch and Werner Lütkebohmert. “Formal and rigid geometry. I. Rigid spaces”.
In: Math. Ann. 295.2 (1993), pp. 291–317. issn: 0025-5831.
[2] J.-F. Boutot and H. Carayol. “Uniformisation p-adique des courbes de Shimura: les
théorèmes de Čerednik et de Drinfeld”. In: Astérisque 196-197 (1991). Courbes modulaires et courbes de Shimura (Orsay, 1987/1988), 7, 45–158 (1992). issn: 0303-1179.
[3] 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, 1972). Springer, Berlin, 1973, 143–316. Lecture Notes in Math., Vol. 349.
[4] Jean Fresnel and Marius van der Put. Rigid analytic geometry and its applications.
Vol. 218. Progress in Mathematics. 2004, pp. xii+296. isbn: 0-8176-4206-4.
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