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\textbf{
18.727, Topics in Algebraic Geometry (rigid analytic geometry)\\
Kiran S. Kedlaya, fall 2004 \\
$p$-adic uniformization and Shimura curves
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I'm going to wrap things up by, in a sense, coming full circle; we're
going to discuss $p$-adic uniformization, which ties in closely to the
Tate curve we constructed at the very beginning of the course.

\textbf{References:}
I'm pulling a lot of this out of: J.-F. Boutot et H. Carayol,
Uniformisation $p$-adique des courbes de Shimura: les th\'eor\`emes de
\v{C}erednik et de Drinfeld, \textit{Ast\'erisque} \textbf{196-197}.
I will cite this as [BC].
Thanks to Samit for the reference. Also see [FvdP, Section 5.4] for more
details on how to take quotients of $\PP^1$ to make ``Mumford curves''.

\subsection*{Quaternion algebras}

Can someone suggest a good reference for this?

A few reminders about quaternion algebras: a \emph{quaternion algebra}
over a field $F$ is a central simple algebra of degree 2 over $F$, i.e.,
a division algebra with center $F$ of dimension 4 as an $F$-vector space,
i.e., an element of the Brauer group of $F$ of order 2. (Whether you allow
the exceptional case $M_2(F)$, the ``split quaternion algebra'', is a
matter of convention.)

Over a complete discretely valued field, there is a unique ``ring of 
integers'' in a quaternion algebra: it is the set of elements whose norm
(element times its conjugate, a/k/a the determinant of the multiplication-by-said-element map) has absolute value $\leq 1$. We call this the
\emph{maximal order} of the quaternion algebra. Over a number field $F$,
by a maximal order of a quaternion algebra $D$ 
we mean an $\gotho_F$-subalgebra $\gotho_D$ with $\gotho_D \otimes_{\gotho_F}
F = D$. There exists at least one such, and any two are conjugate.

We say a quaternion algebra $D$ over a number field \emph{splits} at a place
$v$ (a completion, either archimedean or corresponding to a finite prime) 
if when you complete at $v$, you get the split quaternion algebra
rather than an honest one. By class field theory, the number of nonsplit
places is always even.

\subsection*{The Drinfeld upper half-plane}

Throughout this lecture and the next, 
let $K$ be a \emph{discretely valued} complete
nonarchimedean field with \emph{finite} residue field $k = \FF_q$. 
Let $\CC$ be the completed algebraic closure of $K$.
Let $\pi$ be a uniformizer of $\gotho$. See [BC, Part I] for details.

Define the \emph{Drinfeld upper half-plane} $\Omega$ to be the analytic
subspace of
$\PP^1_{\CC}$ obtained by removing the $K$-rational points; those
form a closed subset in the metric topology, so their complement is
admissible and this really makes sense as a rigid analytic space.

Note that the group $\PGL_2(K)$ acts on $\Omega$. The action is not free,
because you can have fixed points over quadratic extensions of $K$.

One may think of $\PP^1(\CC)$ as the set of $\CC$-homothety classes
of $K$-linear maps $K^2 \to \CC$; $\Omega$ corresponds to those classes
represented by injective maps. So you may think of $\Omega$ as
classifying ``rank two $K$-lattices inside $\CC$'', much as the complex
upper half plane classifies rank two $\ZZ$-lattices inside $\CC$.

There is also a natural formal scheme $\widehat{\Omega}$ with generic
fibre $\Omega$; it is most easily described in terms of the \emph{building}
associated to $\PGL_2(K)$, which is the following graph. 
Consider the set of homothety classes of 
$\gotho$-lattices in $K^2$ as the vertices; join two classes
by an edge if you can find representing lattices $L_1, L_2$
with $\pi L_1 \subset L_2 \subset L_1$. The edges out
of each vertex can be identified with $\PP^1_{\FF_q}$; if you imagine the
graph as a CW-complex, its points correspond to homothety classes of
norms on $K^2$. (You should be thinking Berkovich here...)
Namely, if $\bv_1, \bv_2$ is a basis, and I take a point
on the edge joining $\gotho \bv_1 + \gotho \bv_2$ with
$\gotho \bv_1 + \pi \gotho \bv_2$, its corresponding norm is
\[
\|a_1 \bv_1 + a_2 \bv_2 \| = \max\{|a_1|, q^t|a_2|\}
\]
with $t \in [0,1]$.

Note that one has a natural map from $\Omega$ to the building: given
an element of $\Omega$ describing an injective map from $K^2$ to $\CC$,
compose with the norm on $\CC$ to get a norm on $K^2$. Call this map
$\lambda$.

Each vertex $v$ of the building gives us a way to identify $\PP^1_K$ with
the generic fibre of a projective line over $\gotho$; call the latter
$\PP_v$. Let $\widehat{\Omega}_v$ be the formal completion, along the
special fibre, of the complement in $\PP_v$ of the rational points on the
special fibre; the corresponding rigid space is $\lambda^{-1}(v)$.

If $vw$ is an edge corresponding to a pair of lattices $L_1, L_2$
with $\pi L_1 \subset L_2 \subset L_1$, let $\PP_{vw}$ be the blowup of
$\PP_v$ at the point on its special fibre defined by $L_2$; we
have a canonical identification $\PP_{vw} \cong \PP_{vw}$.
Let $\widehat{\Omega}_{vw}$ be the formal completion, along the
special fibre, of the complement in $\PP_{vw}$ of the \emph{nonsingular}
rational points on the special fibre (so leave in the crossing point);
the corresponding rigid space is $\lambda^{-1}(vw)$. We can now paste
the $\widehat{\Omega}_v$ and $\widehat{\Omega}_{vw}$ together to get
a formal scheme $\widehat{\Omega}$ with generic fibre $\Omega$, and
$\PGL_2(K)$ acts on it.

There are various ways to interpret $\widehat{\Omega}$ as a ``moduli
space'' (i.e., it represents some natural functors); see [BC, I.4 and I.5].

Aside: I won't discuss it further here, but there is also a Drinfeld
upper half-space of any dimension $n$. You get it by taking 
$\PP^{n+1}(\CC)$ and removing all $K$-rational hyperplanes (not
just $K$-rational points!). This carries an action of $\PGL_{n+1}(K)$;
I think you can also analogize to other linear groups, but I don't know
where anyone has done that.

\subsection*{Drinfeld's half-plane and formal $\gotho_D$-modules}

Let $D$ be a quaternion algebra with center $K$.
Fix a choice of a quadratic extension $K'$ of $K$ contained in $D$,
and let $\gotho'$ be its ring of integers.

We are interested in \emph{special formal $\gotho_D$-modules} over
a $\gotho$-algebra $B$; such a thing is a formal $\gotho$-module $X$
of dimension 2
(in the sense of last time) plus an action $i: \gotho_D \to \End(X)$
compatible with the $\gotho$-action, such that $\Lie(X)$ becomes 
a free $B \otimes_{\gotho} \gotho'$-module of rank 1.
In particular, we are going to insist that these also have height 4
(I think this is as a $\gotho$-algebra, but I'm not positive).

Anyway, Drinfeld showed 
that there exists a universal deformation of such a formal module
over $\FF_q$ plus some extra data (a ``quasi-isogeny of height 0''
on the reduction), 
which lives on $\widehat{\Omega} \widehat{\otimes}_{\gotho}
\widehat{\gotho^{\unr}}$. 
This is kind of a long story, and it's in the
same spirit as the discussion of Lubin-Tate formal groups from last time
(even though the formal groups are all two-dimensional, the extra endomorphisms
make the situation look like the Lubin-Tate case)
so I'm not going to discuss it further here. See [BC, Section 2] for all
the gory details.

\subsection*{Shimura curves and the upper half plane}

Things like modular curves are obtained complex analytically by taking the
complex upper half plane and quotienting by the action of a discrete group,
like $\PGL_2(\ZZ)$. If you take a group commensurable with $\PGL_2(\ZZ)$
(a \emph{congruence subgroup}), you get a classical 
modular curve; in that case, you have to compactify by adding in the orbits
of $\PP^1_{\QQ}$ (the \emph{cusps}). However, Shimura noticed you can also
take quotients by things like the unit group of a maximal order in a
quaternion algebra (the maximal order is , and get algebraic curves over $\QQ$ (not just
over $\CC$). Moreover,
like the modular curves, which are moduli spaces for elliptic curves
plus some extra structure, Shimura's curves are moduli spaces for 
certain two-dimensional
abelian varieties with extra endomorphisms by that maximal order
in the quaternion algebra (so-called ``false elliptic curves'').

Let me make this more precise before proceeding to the rigid analogue of
this description. Let $\Delta$ be an indefinite quaternion algebra
with center $\QQ$.
(``Indefinite'' means that $\Delta \otimes_\QQ \RR$ is split, so 
is congruent to $M_2(\RR)$ rather than the Hamilton quaternions.)
For $R$ a commutative $\QQ$-algebra,
let $\Delta^*(R)$ denote the group of units in the noncommutative
ring $\Delta \otimes_\QQ R$.
Let $\AAA_f = \ZZ \otimes_{\QQ} \widehat{\ZZ}$ denote the ring
of finite ad\`eles (where $\widehat{\ZZ} = \prod_p \ZZ_p$ is the
profinite completion of $\ZZ$), and let $\calH^{\pm} = \PP^1(\CC) \setminus
\PP^1(\RR)$ be the upper and lower half planes in $\CC$. Then Shimura's
observation is that for any open compact subgroup $U$ of $\Delta^*(\AAA_f)$,
the (left) quotient
\[
\Delta^*(\QQ) \backslash [\calH^{\pm} \times \Delta^*(\AAA_f)/U]
\]
can be naturally identified with the $\CC$-points of a \emph{projective}
algebraic curve $S_U$ over $\QQ$ (look Ma, no cusps!),
and that the curve is a coarse moduli space for a moduli problem
concerning two-dimensional abelian varieties with endomorphisms by
$\gotho_{\Delta}$ and ``level $U$ structure''. 
(And of course if $U$ is ``sufficiently small'', the moduli space is even 
fine.)

\subsection*{Shimura curves and the rigid upper half plane}

Let $p$ be a prime where $\Delta$ does not split, so that
$\Delta_p = \Delta \otimes_{\QQ} \QQ_p$ is an honest quaternion algebra
over $\QQ_p$. 
Then $\Delta_p^*$ has a unique maximal compact subgroup $U^0_p$,
namely the units of the maximal order of $\Delta_p$,
and there is a decreasing sequence of compact subgroups $U^n_p$ given
by units congruent to 1 modulo $p^n$.

Suppose my level structure $U$ looks like $U^n_p$ times some compact
open subgroup $U^p$ of the prime-to-$p$ part of $\Delta^*(\AAA_f)$.
The theorem of Cherednik-Drinfeld (you can spell that first one
``\v{C}erednik'' if you prefer; you can spell that second one ``Drinfel'd''
if you prefer too) says that if you form the rigid analytic quotient
\[
\GL_2(\QQ_p) \backslash [(\widehat{\Omega} \widehat{\otimes} \widehat{\ZZ_p^{\unr}})
\times Z_U],
\]
you get the analytification of $S_U$ again! Here
\[
Z_U = U^p \backslash \overline{\Delta}^*(\AAA_f) / \overline{\Delta}^*(\QQ)
\]
is a double coset space whose quotient by $\overline{\Delta}^*(\QQ_p)$
is finite.

But what the heck is $\overline{\Delta}$? I'll come back to that in a
moment. First, let me point out that (if I read [BC] correctly)
Cherednik proved this originally
by some not-so-enlightening argument; Drinfeld's contribution was to
match up the moduli interpretation of the Shimura curve with the moduli
interpretation of $\widehat{\Omega}$, giving a much more conceptual
proof of the theorem in the bargain.

\subsection*{But what the heck is $\overline{\Delta}$?}

Glad you asked. $\overline{\Delta}$
is the quaternion algebra obtained from $\Delta$
by switching the invariants at $p$ and $\infty$; i.e., $\overline{\Delta}$
looks like $\Delta$ at all finite primes other than $p$, but it splits
at $p$ and is definite (nonsplit in the real place).

Why does that come up? By Tate-Honda (see [BC, Proposition III.2]):
\begin{itemize}
\item
there is a unique isogeny class of two-dimensional abelian
varieties over $\overline{\FF_p}$ equipped with an action of $\gotho_\Delta$;
\item
any such abelian variety $A$ is isogenous to the product of two supersingular
elliptic curves;
\item
the algebra $\End_{\gotho_{\Delta}}(A) \otimes \QQ$ is isomorphic to
$\overline{\Delta}$.
\end{itemize}

The fact that there is this ``switcheroo'' between the quaternion algebra
which acts on the abelian surfaces and the quaternion algebra you use
in forming the rigid analytic quotient may seem like a trifle or even a 
nuisance, but it's actually a wonderful thing!
It underlies certain ``hidden symmetries'' in the theory
of automorphic forms, like the Jacquet-Langlands correspondence.
A certain geometric realization of this correspondence is
a crucial part of Ribet's proof of Serre's epsilon conjecture, and
in particular that the modularity of elliptic curves
implies Fermat's last theorem. (I have now told you every last thing I
know about this. Go talk to David Helm for more information.)

\subsection*{That's all, folks!}

Thanks for attending the course. 
Oh, and if you have corrections marked in your notes, I'd love to get
some by email so I can assemble a more definitive compilation of the notes.

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