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.