FOREWORD xi
the non-archimedean metric every disc is a disjoint union of smaller discs. So this
is not a local definition, and there’s no obvious way to globalize it.
The first to overcome this difficulty was Marc Krasner. He made a good theory
of analytic functions of one variable. They were defined on certain subsets of
P1(K) = K which he called “quasiconnected”. For example, the set obtained
by removing from P1(K) a finite set of discs of the form |(ax + b)/(cx + d)| r
is quasiconnected and an analytic function on it is one which is a uniform limit
of rational functions with poles in its complement. Krasner’s theory had little
influence on the later global higher dimensional developments described in this
volume, but was valued and further developed by p-adic analysts doing the theory
of p-adic differential equations, etc. I remember that Dwork was very upset that
there was no mention of Krasner in the introduction to the book [BGR] referred to
by Conrad in this volume, and I think he was right to be.
The earliest steps toward the subject of this book were mine. My motivation
was the isomorphism K∗/qZ

Eq(K), for q K, 0 |q| 1, where Eq is an
elliptic curve over K with invariant j = q−1 +744+196884q+···. I still remember
the thrill and amazement I felt when it occurred to me that the classical formulas
for such an isomorphism over C made sense p-adically when properly
normalized.1
This uniformization of some elliptic curves made me wonder if there might not
be a general theory of p-adic manifolds. Two years later in the fall of 1961, very
much influenced by Grothendieck’s theory of schemes, I was ready to make a serious
attempt to create such a theory. In contrast to the difficult circumstances Berkovich
faced as he developed his theory, my situation could not have been more favorable,
with a good job at Harvard and friends like Serre and Grothendieck to help me. I
recorded my progress in a series of letters to Serre. He wrote that he was keeping
them carefully, but not reading them carefully. But he did find and fix a gap in my
proof of the acyclicity
theorem.2
He was interested at the time in another aspect
of p-adic analysis, namely Dwork’s spectacular proof of the rationality of the zeta
function of algebraic varieties over a finite field, and was developing a theory of
Fredholm determinants in p-adic Banach spaces in order to simplify Dwork’s proof.
In his course that winter Serre discussed the curve Eq, p-adic Banach spaces, and
Dwork’s proof and his own generalization of it to some L-functions. In Serre’s
seminar, Houzel talked on my letters, which Serre had had typed at the IHES for
limited distribution. Ten years later they were published as a paper in Inventiones
math. 12 (1971).
Grothendieck was visiting Harvard at the time I was writing the letters, and
his presence was a great help. By then, in contrast to the bizarre negativism he had
shown in his letter to Serre quoted at the end of Brian Conrad’s introduction, he had
become wildly optimistic, writing, again to Serre, in Oct.’61: “...Sooner or later it
will be necessary to subsume ordinary analytic spaces, rigid analytic spaces, formal
schemes, and maybe schemes themselves into a single kind of structure for which
all the usual theorems hold: Stein spaces, Grauert finiteness, Remmert-Grauert
GAGA, maybe also Rothstein type theorems....”
1The
notes I wrote at the time (fall ’59) are published in the book Elliptic curves, modular
forms and Fermat’s last theorem, International Press, Cambridge MA, 1995. Though I published
nothing earlier, the curve Eq became known thanks to others.
2I
had carelessly thought that the two complexes which are mentioned in the proof of
Lemma 8.5 of my Inventiones 1971 paper are the same. Serre explained to me that they are
not at all the same, but are, in fact, homotopic, so the lemma is OK.
Previous Page Next Page