\input{preamble.tex}
\topheader{Introduction and motivation}

In this unit, 
we list the various ways that $p$-adic differential equations occur
in modern number theory. We then explain in a bit of detail one
example; it is in fact  of the original examples of
Dwork that prompted the creation of the whole theory.

Since this unit is intended as an overview, it involves sweeping 
statements with proofs not included. Starting with the next unit,
things will be done more carefully.

To keep things better organized, I am creating a central references file
``references.pdf'' on the web site. However, I will attach to each
unit some afternotes, and some exercises; the latter are optional unless
you need a genuine course grade (e.g., if you are an undergraduate).

\section{Where are the $p$-adic differential equations?}

Under what circumstances does the consideration of a differential
equation involving $p$-adic numbers, rather than real or complex
numbers, occur in number theory? Let me count the ways; note that I
may not get to discuss all of these in much detail during the semester,
though I'll try to fit some of them in at the end.

\begin{itemize}

\item
The original circumstance, which will be described later in this lecture,
was Dwork's work on the variation of zeta functions of algebraic varieties
over finite fields. Roughly speaking, solving certain $p$-adic differential
equations can give rise to explicit formulas for number of points on
varieties over finite fields.

In contrast to methods involving \'etale cohomology, methods for studying
zeta functions based on $p$-adic analysis (including also the next item)
lend themselves well to
numerical computation. Interest in computing zeta functions
for varieties where straightforward point-counting is not an option
(e.g., curves over tremendously large prime fields) has been driven
by potential applications in computer science, the principal example
being cryptography based on elliptic or hyperelliptic curves.
(There may also be some interesting applications generated by coding
theory, but this remains to be seen.)

\item
Dwork's work suggested, but did not immediately lead to, a proper
analogue of \'etale cohomology based on $p$-adic analytic techniques.
Such an analogue was eventually developed by Berthelot (based on
work of Monsky and Washnitzer, and also ideas of Grothendieck); it is
called \emph{rigid cohomology} (see the unit notes for the origin of the word
``rigid''). It is not yet a fully functional analogue of \'etale cohomology,
particularly because there are still open problems related to
the construction of a good category of coefficients. These problems are
rather closely related to questions concerning 
$p$-adic differential equations, and in fact some of the results presented
in this course have been (or will be) used for this purpose.

\item
The subject of \emph{$p$-adic Hodge theory} aims to do for the cohomology
of varieties over $p$-adic fields what ordinary Hodge theory does for
the cohomology of varieties over $\CC$, namely abstract away the variety
and enable a better understanding of the cohomology of the variety as 
an object in its own right. In the $p$-adic case, the cohomology in question
is often \'etale cohomology, which carries the structure of a Galois
representation.

The study of such representations, as pioneered by Fontaine, involves a number
of exotic auxiliary rings (``rings of $p$-adic periods'') which serve
their intended purposes but are otherwise a bit mysterious. More recently,
the work of Berger has connected much of the theory to the study of
$p$-adic differential equations; notably, a key result that was originally
intended for use in $p$-adic cohomology 
(the ``$p$-adic local monodromy theorem'') turned out to imply an important
conjecture about Galois representations (Fontaine's conjecture on
potential semistability).

\item
There are some interesting analogies between properties of differential
equations over $\CC$ with meromorphic singularities, and wildly
ramified Galois representations of $p$-adic fields. At some level,
this is suggested by the parallel formulation of the Langlands conjectures
in the number field and function field cases. One can use $p$-adic
differential equations to interpolate between the two situations,
by associating differential equations to Galois representations (as in the
previous item) and then using ``differential invariants'' (irregularity)
to recover
``Galois invariants'' (Artin and Swan conductor).

For representations of the \'etale fundamental group of a variety
over a field of positive characteristic of dimension greater than 1,
it is quite a tough problem to construct meaningful numerical invariants 
from the Galois point of view. Recent work of Abbes and Saito attempts to
do this, but the resulting quantities are quite difficult to calculate.
One can alternatively use $p$-adic differential equations to define
invariants which are somewhat easier to deal with for some purposes; for 
instance, one can define a ``differential Swan conductor'' which is
guaranteed to be an integer, whereas one does not know this for the
Abbes-Saito conductor. (In fact, the two are expected to coincide;
see Liang Xiao's thesis project.)

\end{itemize}

\section{Zeta functions of varieties}

Let us now say something more detailed about the first item in the
previous list. For this, I must recall properties of zeta functions
of algebraic varieties.

For $\lambda$ in some field $K$, let $E_\lambda$ be the elliptic curve
over $K$ defined by the equation 
\[
E_\lambda: y^2 = x(x-1)(x-\lambda)
\]
in the projective plane. Remember that there is one point $O = [0:1:0]$ at
infinity, and that there is a natural group law on $E_\lambda(K)$ under
which $O$ is the origin, and three points add to zero if and only if they
are collinear (or better, if they are the three intersections of 
$E_\lambda$ with some line; this correctly allows for degenerate cases).
\begin{theorem}[Hasse]
Suppose $\lambda$ belongs to a finite field $\FF_q$. Then
$\#E_\lambda(\FF_q) = q+1 - a_q(\lambda)$ where $|a_q(\lambda)| \leq 2\sqrt{q}$.
\end{theorem}
\begin{proof}
See [Sil91, Theorem~V.1.1].
\end{proof}
Hasse's theorem was later vastly generalized as follows,
originally as a set of conjectures by Weil. (Despite no longer 
being conjectural, these are still commonly referred to as
the \emph{Weil conjectures}.)
For $X$ an algebraic variety over $\FF_q$, the \emph{zeta function}
of $X$ is defined as the formal power series
\[
\zeta_X(T) = \exp \left( \sum_{n=1}^\infty \frac{T^n}{n} \#X(\FF_{q^n})\right);
\]
another way to write it, which makes it look more like zeta functions
you've seen before, is 
\[
\zeta_X(T) = \prod_x (1 - T^{\deg(x)})^{-1},
\]
where $x$ runs over Galois orbits of $X(\overline{\FF_q})$, and $\deg$
is the size of the orbit. (If you prefer algebro-geometric terminology:
$x$ runs over closed points of $X$, and $\deg$ is the degree over
$\FF_q$.)
For $X = E_\lambda$, one checks (exercise) that
\[
\zeta_X(T) = \frac{1 - a_q(\lambda)T + q T^2}{(1-T)(1-qT)}.
\]

\begin{theorem}[Dwork, Grothendieck, Deligne, et al]
Let $X$ be an algebraic variety over $\FF_q$. Then $\zeta_X(T)$ represents
a rational function of $T$. Moreover, if $X$ is smooth and proper
of dimension $d$, we can write
\[
\zeta_X(T) = \frac{P_1(T) \cdots P_{2d-1}(T)}{P_0(T) \cdots P_{2d}(T)},
\]
where each $P_i(T)$ has integer coefficients,
satisfies $P_i(0) = 1$, and has all roots in $\CC$
on the circle $|T| = q^{-i/2}$.
\end{theorem}
\begin{proof}
The proof of this theorem is a sufficiently massive undertaking that
even a reference is not reasonable here; instead, we give
[Har77, Appendix~C] as a metareference. 
\end{proof}
It is worth pointing out that
the first complete proof  uses the fact that you can
interpret 
\[
\#X(\FF_{q^n}) = \sum_i (-1)^i \Trace(F^n, H^i_{\et}(X, \QQ_\ell)),
\]
where for any prime $\ell \neq p$, $H^i_{\et}(X, \QQ_\ell)$ is the
$i$-th \'etale cohomology group of $X$ with coefficients in $\QQ_\ell$.

\section{Why $p$-adic differential equations?}

All well and good, but there are several downsides of the interpretation
in terms of \'etale cohomology. One important one is that \'etale cohomology
is not explicitly computable; for instance, it is not
straightforward to describe
\'etale cohomology to a computer
 well enough that the computer can make calculations.
(The main problem is that while one can write down \'etale cocycles,
it is very hard to tell whether or not a cocycle is a coboundary.)

Another important downside is that you don't get extremely good information
about what happens to $\zeta_X$ when you vary $X$. This is where $p$-adic
differential equations enter the picture. It was observed by Dwork that
when you have a family of algebraic varieties defined over $\QQ$, 
the same differential
equations appear when you study variation of complex periods,
and when you study variation of zeta functions over $\FF_p$.

Here is an explicit example due to Dwork. 
Recall that the \emph{hypergeometric
series}
\[
F(a, b; c; z) = \sum_{i=0}^\infty \frac{a(a+1)\cdots(a+i) b(b+1)\cdots(b+i)}{c(c+1)\cdots(c+i) i!} z^i
\]
satisfies the \emph{hypergeometric differential equation}
\[
z(1-z)y'' + (c-(a+b+1)z)y' -aby = 0.
\]
Set in particular
\[
\alpha(z) = F(1/2, 1/2; 1; z);
\]
over $\CC$, $\alpha$ is related to an elliptic integral, for instance,
by the formula
\[
\alpha(\lambda) = \frac{2}{\pi} 
\int_0^{\pi/2} \frac{d\theta}{\sqrt{1- \lambda \sin^2 \theta}}
\qquad (0 < \lambda < 1).
\]
(You can extend this to complex $\lambda$ by being careful about branch
cuts.)
This elliptic integral can be viewed as a period integral for the curve
$E_\lambda$, i.e., you're integrating some meromorphic form on
$E_\lambda$ around some loop (homology class).

Let $p \neq 2$ be an odd prime. We now try
to interpret $\alpha(z)$ as a function of a $p$-adic variable
rather than a complex variable. Beware that this means $z$ can take
\emph{any} value in a field with a norm extending the $p$-adic norm
on $\QQ$, not just $\QQ_p$ itself. (For the moment, you can imagine $z$
running over a completed algebraic closure of $\QQ_p$.)

\begin{lemma}
The series $\alpha(z)$ converges $p$-adically for $|z| < 1$.
\end{lemma}
\begin{proof}
Straightforward.
\end{proof}

Dwork discovered that a closely related function admits 
``analytic continuation''. To explain what the result says,
we define the \emph{Igusa polynomial}
\[
H(z) = \sum_{i=0}^{(p-1)/2} \binom{(p-1)/2}{i}^2 z^i.
\]
Modulo $p$, the roots of $H(z)$ are the values of $\lambda \in 
\overline{\FF_p}$ (which actually all belong to $\FF_{p^2}$, for 
reasons we will not discuss) for which
$E_\lambda$ is a supersingular elliptic curve, i.e., 
$a_q(\lambda) \equiv 0 \pmod{p}$.

Dwork's analytic continuation result then is the following.
\begin{prop}[Dwork]
There exists a series $\xi(z) = \sum_j P_i(z)/H(z)^i$ converging uniformly for
$|z| \leq 1$ and $|H(z)| = 1$, with each $P_i(z) \in \QQ_p[z]$, such that
\[
\xi(z) = (-1)^{(p-1)/2} \frac{\alpha(z)}{\alpha(z^p)} \qquad
(|z| < 1).
\]
\end{prop}
\begin{proof}
See [vdP86, \S 7].
\end{proof}
Note that $\xi$ itself satisfies a differential equation, which I won't
write out just yet. We will see it again later.

For $\lambda \in \FF_q$, let $\ZZ_q$ be the unramified extension of
$\ZZ_p$ with residue field $\FF_q$. Let $[\lambda]$ be the unique
$q$-th root of 1 in $\ZZ_q$ congruent to $\lambda$ mod $p$ (the
\emph{Teichm\"uller lift} of $\lambda$).
\begin{theorem}[Dwork]
If $q = p^a$ and $\lambda \in \FF_q$ is not a root of $H(z)$, then
\[
T^2 - a_q(\lambda) T + q = 
(T - u)(T - q/u),
\]
where
\[
u = \xi([\lambda]) \xi([\lambda]^p) \cdots \xi([\lambda]^{p^{a-1}}).
\]
\end{theorem}
That is, the quantity $u$ is the ``unit root'' of the polynomial
$T^2 - a_q(\lambda)T + q$ occurring (up to reversal) in the
zeta function. 
\begin{proof}
See [vdP86, \S 7].
\end{proof}

\section{A word of caution}

Before we embark on the study of  $p$-adic ordinary
differential equations, a cautionary note is in order, concerning
the rather innocuous-looking differential
equation $y' = y$. Over $\RR$ or $\CC$,
this equation is nonsingular everywhere, and its solutions $y = ce^x$
are defined everywhere.

Over a $p$-adic field, things are quite different. As a power series around
$x=0$, we have
\[
y = c \sum_{n=0}^\infty \frac{x^n}{n!}
\]
and the denominators hurt us rather than helping. In fact, the series only
converges for $|x| < p^{-1/(p-1)}$ (assuming that we are normalizing
$|p| = p^{-1}$). 
For comparison, note that the logarithm
series
\[
\log \frac{1}{1-x} = \sum_{n=1}^\infty \frac{x^n}{n}
\]
converges for $|x| < 1$.

The conclusion to be taken away is that 
there is no fundamental theorem of ordinary differential equations
over the $p$-adics! In fact, 
the hypergeometric differential equation 
in the previous example was somewhat special; the fact that it had a solution in a disc
where it had no singularities was not a foregone conclusion. One of 
Dwork's discoveries is that
this typically happens for differential equations that ``come from 
geometry'', such
as the Picard-Fuchs equations that arise from integrals of algebraic functions
(e.g., elliptic integrals). Another is that one can quantify rather
well the obstruction to solving a $p$-adic differential equation
in a nonsingular disc, using similar techniques to those used to study
obstructions to solving complex differential equations in singular discs.

\section{Notes}

I alluded above to the notion of an analytic function, defined as
a uniform limit of rational functions with poles prescribed to certain
regions. To keep down the background required for the course, 
I will stick throughout to this approach of defining
everything in terms of rings, and not making any attempt to introduce
analytic geometry over a nonarchimedean field. However, it must be
noted that it is much better in the long run to build this theory
in terms of nonarchimedean analytic geometry; for example, it is pretty
hopeless to deal with partial differential equations without doing so.

That said, there are several ways to develop a theory of analytic spaces over
a nonarchimedean field. The traditional method is Tate's theory of
rigid analytic spaces, so-called because one develops everything ``rigidly''
by imitating the theory of schemes in algebraic geometry, but using
rings of convergent power series instead of polynomials.
The canonical foundational reference for rigid geometry
is the book of Bosch, G\"untzer, and Remmert [BGR84], but novices
may find the text of Fresnel and van der Put [FvdP04] 
or the lecture notes of Bosch [Bos05]
more approachable.
A more recent method, which in some ways is more robust, is
Berkovich's theory of nonarchimedean analytic spaces (commonly
called \emph{Berkovich spaces}), as introduced in [Ber90]
and further developed in [Ber93]. For both points of view,
see also the lecture notes of Conrad [Con07].

Dwork's original analysis of the Legendre family of elliptic curves
via the associated hypergeometric equation appears in [Dwo69, \S 8].
The treatment in [vdP86] is more overtly related to $p$-adic cohomology.

\section{Exercises}

Exercises are purely optional unless you are an undergraduate, in which
case please see me at once if you have not done so yet.

\begin{enumerate}
\item
Explain why the Dwork-Grothendieck-Deligne et al theorem implies Hasse's
theorem (this includes verifying the formula for the zeta function of 
$E_\lambda$).
\item
Check that the usual formula 
\[
\liminf_{n \to \infty} |a_n|^{-1/n}
\]
for the radius
of convergence of the Taylor series $\sum_{n=0}^\infty a_n x^n$
still works over a nonarchimedean field. (That is, the series converges
inside that radius, and diverges outside.)
\item
Check that the exponential series has radius of convergence $p^{-1/(p-1)}$.
\item
Show that a power series which converges for $|x| \leq 1$ may have an integral
which only converges for $|x| < 1$, but that its derivative still converges
for $|x| \leq 1$. This is backwards from the archimedean situation.
\end{enumerate}

\end{document}