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ANNALES DE L’INSTITUT FOURIER
JEREMY TEITELBAUM
Geometry of an étale covering of the p-
adic upper half plane
Annales de l’institut Fourier, tome 40, no 1 (1990), p. 69-78
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Ann. Inst. Fourier, Grenoble
40, 1 (1990), 69-78
GEOMETRY OF AN ETALE COVERING OF THE p-ADIC
UPPER HALF PLANE
by Jeremy TEITELBAUM (*)
Introduction.
In this paper we describe the rigid geometry of the first layer in the
tower of coverings of the p-adic upper half plane obtained from the division
points of the formal group constructed in [2]. This covering is accessible
because it is abelian and in some sense "tame." Using our results, we are
able to describe the stable special fiber at p of Shimura curves with a very
small amount of level p structure.
Preliminaries.
Let "Hp denote the formal scheme over Zp constructed by Mumford
([4]) and commonly referred to as the p-adic upper half plane. Naively, Hp
is the complement of the Qp-rational points in P1 . We let Hp be the rigid
analytic space associated to Hp.
In [2], Drinfeld shows that Hp is a parameter space for two-
dimensional formal groups with a certain endomorphism structure. As a
(*) This research was supported by an NSF postdoctoral fellowship.
Key-words : p-adic unifbrmization - Shimura curves - Formal groups.
A.M.S. Classification : 11G18 - 11F85.
70 JEREMY TEITELBAUM
result, there is a universal family of formal groups Q over Hp. The sub-
groups obtained as the division points of this family of formal groups yield
a tower of coverings of Hp. The rigid spaces associated to these coverings
are a family of etale coverings of Hp. Our goal in this work is to describe
the simplest of these coverings.
For a detailed description of DrinfekTs universal formal group, we
refer the reader to [8]. We recall here the basic definitions which we will
require.
Let D be the quaternion division algebra over Qp, and let OD be
the maximal order in D. A formal group G of dimension 2 and height
4 over a ring R on which p is nilpotent is called a special, formal OD~
module (abbreviated SFD-module) provided that OD acts on G and, at
each maximal ideal m of 2?, both characters of the residue field of OD occur
in the tangent space to G at m. Over Fp, all STD-modules are isogenous,
so fix one such module $. With these conventions, we can state Drinfeld's
theorem.
THEOREM (Drinfeld). — Up 0 Spf Z^ (over 7.p) represents the
functor which assigns to a ring R on which p is nilpotent the set of
isomorphism classes of triples (-0, G, p) where
1. ^ : Z^/pZp7' —^ R / p is a homomorphism,
2. G is an SFD-module over R,
3. and p : ̂ ^ —> G 0 R / p is a "quasi-isogeny of height zero," which
means that p is an isogeny with a certain normalization condition which
will not be important in our work.
We let (^, Q, P) be the universal triple over Up 0 Spf Z^, and Gn be
the kernel of multiplication by II on Q. This is a finite, flat group scheme of
order p2 over 7^(8) Spf Z^. IfPGL^(Qp) denotes the image in PGI^Qp) of
the elements of GLa(Qp) with determinants of even p-adic order, then the
action of PGL^(Qp) on Hp extends to an action on the universal triple over
T^SpfZ^, and therefore to Gn. Furthermore, the residue field OD/TI = F
acts on Gn^ and this action commutes with the action ofPGL^(Qp). In fact,
GLs(Qp) acts on the universal triple; see [2] p. 109 for details.
Let us fix a map '0o : T-^ —^ Zp7' and consider the fiber of the induced
projection map Up 0 Spf Zp7' -^ Spf Z^. The resulting formal scheme
is isomorphic to Hp 0 Spf T^ viewed as a formal scheme over Zp7' with
structure map -0o- The geometry of the formal scheme obtained in this
GEOMETRY OF AN ETALE COVERING 71
way does not depend on the choice of ^o? s° ̂  the remainder of this
paper we will abuse notation, suppress reference to ^o» and denote by Hp
and Hp respectively the formal and rigid p-adic upper half planes over Zp
(resp. Qp), base-changed up to Z^ (resp Q^7'.) Similarly, we base change
the covering Gn to obtain a finite flat group scheme of order p2 over (our
base-changed) Hp. We let E be the complement of the zero section in this
group scheme, and E be the associated rigid space. This paper is devoted
to describing the rigid geometry of E.
Classification of E as /Ap2_i-torsor.
The action of the endomorphism ring of Q induces an action of Fx
on the covering E, as we mentioned before. The two embeddings
^ .̂ . T _>. T^f InrCV^(7o?^l : * —)> Zp /PZp
induce two different actions of /ip2_i on our fixed (SF-D-module $ over Fp,
and therefore, via the universal isogeny P, two different actions on E. This
allows us to view E as a ^p2_i torsor in two different ways.
DEFINITION 1. — Let
c,(E) € H^(Hp 0 SpfCp,/Zp2_i)
be the class representing the covering E 0 Spf Cp viewed as a ̂ pi-\ torsor
via the embedding
a,: W(F) ̂  Z^ C Cp
induced by Oi .
The following lemma relates the two classes.
LEMMA 2. — c^(E) = pCt+i(E), reading subscripts mod 2 .
Proof. — Changing the choice of o-i twists the /ip2_i action on E by
C ̂  ̂  . n
Our goal now is to determine the classes c» precisely. Let T be the
tree of SL2(Qp). We fix a reduction map r : Hp —> T which is compatible
with the action of SL2(Qp). For one of many detailed descriptions of the
relation between T and Hp, see [8], pp. 648-660.
The following theorem of Drinfeld ([!]) relates the cohomology of Hp
to the tree T.
72 JEREMY TEITELBAUM
THEOREM 3 (Drinfeld [1]). — IfN is an integer prime to p, there
is an isomorphism
9 : H^Hp 0 Spf Cp, ̂ N) -^ C^(T, 1/NZ)
where T is the tree ofSLa and (7^y.(T,ZyWZ) is the group of harmonic
1-cochains on T — that is, the set of functions f on the edges ofT such
that, for each vertex v, f satisfies
^f(e)=0
e^->v
where the sum is over the oriented edges ofT meeting v.
Let us briefly recall how the map 9 of the theorem is constructed.
Suppose Y is a torsor over T-ip 0 Spf Cp. Let U be the admissible open set
in Hp corresponding to a vertex v in T, together with its bounding edges. It
follows from Lemma 2 of [8] that U is a GL2(Qp) translate of the standard
open set
V = {P e Up : l / p < \z(P)\ < p} - ̂  B^(z)
i=l
where Bf(i) denotes the closed ball centered at i of radius r. Therefore
Pic(t7) = 0 and so
Oy=^[r]/(r^2-1-/)
where /Ap2_i acts by multiplication on T, and / is uniquely determined up
to (p2 — 1)^ powers.
Let e = {v,v'} be an (oriented) edge leaving v. To evaluate 9(Y)(e),
choose a coordinate function z on U such that if P e U reduces to v then
\z(P)\ = 1 while if P € U reduces to e then
hi < W\ < 1 .
Then we let
9(Y)(e) = Resedf/f (mod p2 - 1)
where ReSe denotes the rigid analytic "annular residue" computed with
respect to the selected coordinate z (which determines the sign of Res.)
Since PGL^(Qp) acts on E, the class c^(E) is invariant by PGL^(Qp).
As we see in the next lemma, this is a very strong condition.
LEMMA 4. — Suppose Y is a PGL^(^p)-invariant torsor over
Hp (g) Spf Cp, and 9(Y) is the associated harmonic 1-cocycle. Then 9(Y)
satisfies the following conditions:
GEOMETRY OF AN ETALE COVERING 73
1. Q(Y)(e) = 9(Y)(^e) for all 7 € PGL^-(Qp) and all oriented edges e
off.
2. <9(r)(e) = 0 (mod p - 1) on all edges e ofT .
Proof. — The first property is a consequence of the invariance of Y
by PGL^(Qp) and the fact that PGL^(Qp) preserves the orientation of
edges in T. For the second, observe that all the edges e leaving any vertex
v are permuted transitively by PGL^(Qp). Therefore 9(Y)(e) is a constant
x on all edges e leaving v. From the harmonicity condition we obtain
^ 9(Y)(e) = (p + l)x = 0 (mod p2 - 1)
e^v
since there are p 4-1 edges leaving v. This proves the lemma. D
In order to give a precise statement of our theorem, we must invoke
the relationship between orientations on T and the embeddings o-i. As
Drinfeld shows, and we explain in Lemma 14 of [8], the action of II on
the tangent space T to Q allows us to partition the vertices of T into two
classes labeled with the o^. To describe this partition, first decompose T
into (7^-eigenspaces for the action for the quadratic unramified extension
of Zp inside OD' A vertex v is labelled with (TI if T!T^ C pT^1 over the
affinoid reducing to v where T^ is the cr^-eigenspace in T. Vertices of the
two classes alternate in the tree.
Since 3%(E) is PGL^"(Qp)-invariant, and this group permutes the
edges of T transitively, it suffices to specify the value of c^(E) on a single
edge. This is the content of our theorem.
THEOREM 5. — Let e = [ v y V ' } be an edge ofT. Suppose that v is
labeled with Oi. Then
9,(E)(e)=p-l .
Proof. — Notice first of all that, by Lemmas 2 and 4,
^(E)([z;,i;'])=^(E)([z;^])
=^(E)(h',t;])
=^i(E)(h',z;])
and therefore we may assume that v is labeled with o-o. Let U be the affine
open set of Hp corresponding to e. It follows from Lemma 2 of [8] that U
is a PGL^(Qp) translate of the standard open set
p-i
U(l) = {P € Hp : 1/p < KP)| < 1} - U (W u Si/pW)
i=l
74 JEREMY TEITELBAUM
where Br(i) denotes the open disc of radius r centered at i. Therefore the
coordinate ring of U is isomorphic to
J^limWfi
where
Z[^i] [ 1 1R=
(^I-P) [l-^-15!-^-1] •
As we recalled prior to stating this theorem, the tangent space T to Q over
U is free of rank 2, and it carries a grading coming from the action of the
maximal unramified extension ofZp in OD- Write T = RtQ^BRt^. Referring
to [8] p. 656, we see that the II action on T is IKo = z\i\ and II^i = zoto.
With these conventions, the vertex v of T labeled with (TO corresponds to
the region where ZQ is a unit; the vertex v ' labeled with o-i corresponds to
the region where z\ is a unit.
Let n be the cotangent space to Q. Then On = 0/110 is naturally
the cotangent space to Gn. If o/o and 0:1 generate the graded pieces of 0,
then we must have IIc^ = 2^-n. It follows that
(1) On = (R/ZiR)u}o C (R/zoR)^ .
The finite flat group scheme (7n ? together with its action by F is of the type
classified by Raynaud. Applying his classification (see [5], Corollary 1.5.1)
we see that the coordinate B ring of Gn must have the form
B = R[Xo,X,]/(XS - SoX^X? - 6^X0)
where the functions 6i and p / 6 i belong to R^^. In addition, Raynaud shows
that the natural identification of On with I / I 2 (J being the augmentation
ideal) means that
(2) On = (R/6^R)Xo C W6oR)X^ .
Combining (1) and (2), we see that Zi and 6i differ by a unit of R.
We now have enough information to compute the class of E. Indeed,
E is defined over U by the equation
XS2-1 -^i=0.L? ' -0501=
The group Fx acts on XQ through the embedding o-o. Let us write / = <^i.
Then
9o(E)(e) = Res,,d/// (mod p2 - 1) .
Our results above tell us that 6^6\ = pz^^h^z) where h is a unit in R.
This residue is clearly p — 1 . D
GEOMETRY OF AN ETALE COVERING 75
Thanks to this theorem, we can construct E over Cp. Let X be the
non-singular projective curve over Cp defined by the affine equation
(3) y ^ 1 = z - z p .
Let W C X be the admissible open set of points P on X such that
(4) Ip1/^1)] < |Y(P)| < |p-P/(P+i)| .
COROLLARY 6. — Over Cp, E consists ofp—1 isomorphic connected
components. Each such component has a covering by admissible open sets
isomorphic to W. The nerve of this covering is the tree T. ItW\ and W^
are two elements of the covering, and E = W\ H W^, C W\, then E is one
of the boundary annuli otW\ .
Proof. — Let U be the subset of Hp^Spf Cp consisting of one vertex
(say, labeled with o-o)? and its bounding edges. Then by Theorem 5, E
over U is obtained by extracting the p2 — 1 root of a function with order
congruent to p— 1 mod p2 — 1 on each bounding annulus. If z is a coordinate
on (7, then the function f(z) = (z - zP)P~1 clearly meets this condition.
Thus E is defined over U by the equation
Yf-^^-z^Y-1
where ZQ is an appropriate parameter on U. Notice first that this equation
factors, so that E consists of p — 1 connected components, and is built up
out of pieces of the curve in (3) as claimed. It is a simple matter to check
that the subset of W satisfying the inequality (4) has genus (p2 - p)/2.
Thus the reduction of E consists of curves meeting in double points, like
the reduction of T^p, except that the rational curves which appear in the
reduction of T-lp are replaced by the curves of equation (3) .
With somewhat more care one can determine the equations for E over
Zp7', instead of just over Cp. Examining the end of the proof of Theorem 5
we see that over Z^, we can take 8^6\ = p(z - zP)P~1 and therefore E is
defined over U by the equation
y^-i=p(^_^>-i.
From this one can obtain a minimal regular model for E over Z^ by
blowing-up. This is a straightforward computational problem whose so-
lution we omit, although we do point out that the minimal model has no
components of multiplicity one.
Finally, notice that the action of PGL^(Qp) on E extends to an
action of PGL2(Qp). Indeed, choose r € PGL2(Qp) - PGLJ-(Qp). Then
76 JEREMY TEITELBAUM
Ci(r*E) = -Ci(E) since r reverses orientations. It follows that r*E is
isomorphic to E as a rigid space but that the action of /Ap2_i on r*E
is twisted by Frobenius. This could have been deduced, of course, from the
general construction of Drinfeld.
Application to Shimura curves.
Now we examine the implications of Theorem 5 for the geometry of
Shimura curves. Let A be an indefinite quaternion algebra over Q ramified
at p and let L be a maximal order in A. Suppose n > 3 is prime to p. Let
Sn be the scheme representing the functor which associates to a scheme S
the set of isomorphism classes of abelian surfaces over S with an L action
and a level n-structure, and let S^ be the associated p-adic rigid space.
Let p C L be the unique prime ideal above p. Let Sn^ be the covering
of Sn which classifies abelian L surfaces together with level n structure and
level p structure. As before, S^ is the associated j>-adic rigid space.
Let
X^^VA'^A^/A^
where A' is the definite quaternion algebra obtained from A by interchange
of invariants at p and oo. Let Un be the principal congruence subgroup
UnC]^(L^Zi).
l^p
With this notation, we can state (a slightly simplified form of)
Drinfeld's theorem.
THEOREM (Drinfeld [2]). — There are isomorphisms
(5) GL2(Qp)\^p 0 Spf Z^ x Xn ̂  S^
and
(6) GL2(Qp)\ExX,-.%.
Furthermore, as Drinfeld points out, the quotient in (5) is actually
the union of a finite number of components, each of the form
r\^0Spfz^
where F is a Schottky group.
GEOMETRY OF AN ETALE COVERING 77
Combining this with our geometric description of E, we obtain the
following theorem.
THEOREM 8. — Let Sn(T) = r\^p (g) Spf T^ be one of the
components of 5^". Suppose $ = T/F is the intersection graph of <Sri(r).
Then the covering S^ over Sn(r) has a stable model over Cp consisting of
p — 1 components. The reduction of each such component has intersection
graph $, but the vertices correspond to curves with the equation (3) rather
than to rational curves.
For the sake of concreteness, we supply an example. Suppose that A
has discriminant 26 and that p = 2. Then A' has discriminant 13. Choose
an embedding A' 0Zp «—>• Af2(Qp). Let A be a maximal Z[l/2] order in A',
and let
r = {7 e A : nr(^f) = 2^, k even} .
The Shimura curve S^ of level 1 (over Cp) is the quotient
T\Hp 0 Spf Cp .
We are allowed to consider level 1 since A' has no multiplicative torsion.
Since A' has class number 1, it is not hard to check that the special fiber of
5'i consists of two rational curves meeting in 3 points — see [3] or [6]. By
the theorem, the special fiber of Si,p consists of two copies of the elliptic
curve Y3 = z — z2 crossing in three points.
Conclusions.
In conclusion, we mention two questions-related to our subject matter.
The first, rather naturally, is to obtain information on the higher coverings
of the p-adic upper half plane; and, in particular, on the covering obtained
from the p-torsion on DrinfekTs formal group. This is clearly a much harder
problem than the one we have solved, since the higher coverings are not
abelian and are in some sense "wildly ramified."
From a rigid analytic point of view, however, it would also be
interesting to study the class of curves which admit a uniformization by
E. Such curves are a type of generalized Mumford curve, and it would be
worthwhile to extend the p-adic analytic theory of Mumford curves to this
more general setting. In particular, the Jacobians of these curves are semi-
abelian schemes, and it would be interesting to obtain some form of the
Manin-Drinfeld theory ofj?-adic automorphic forms on E.
78 JEREMY TEITELBAUM
BIBLIOGRAPHIE
[1] V.G. DRINFELD, Elliptic modules, Math. USSR Sbomik, 23(4) (1976).
[2] V.G. DRINFELD, Coverings of p-adic symmetric regions, Functional Analysis and
its Applications, 10(2) (1976), 29-40.
[3] A. KURIHARA, On some examples of equations defining Shimura curves and the
Mumford uniformization, J. Fac. Sci. Univ. Tokyo, Sec. IA, 25 (1979).
[4] D. MUMFORD, An analytic construction of degenerating curves over complete local
rings, Compos. Math., 24 (1972), 129-174.
[5] M. RAYNAUD, Schemas en groupes de type (p,... ,p), Bull. Soc. Math. France, 102
(1974).
[6] K. RIBET, Bimodules and abelian surfaces, Technical Report PAM-423, Center for
Pure and Applied Mathematics, University of California, Berkeley, August 1988.
[7] P. SCHNEIDER and U. STUHLER, The cohomology of p-adic symmetric spaces,
preprint, 1988.
[8] J. TEITELBAUM, On Drinfeld's universal formal group over the p-adic upper half
plane, Mathematische Annalen, 284 (1989), 647-674.
Manuscrit recu Ie 5 mai 1989,
revise Ie 18 decembre 1989.
Jeremy TEITELBAUM,
Mathematics Department
University of Michigan
Ann Arbor, Michigan 48109 (U.S.A.).


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