A tribute to oliver Atkin.
(A tribute to Oliver Atkin, 31 July 1925 to 28 Dec 2008; talk delivered by
Bryan Birch at Microsoft research on 21 Oct 2010.)
I'll be talking about Oliver's work, with a bit of biography thrown in;
there will be more history than mathematical detail. There is plenty to talk about,
so I have to be selective. For the purposes of this talk, one may divide Oliver's
life into periods:-
1925-45 Education; 1945-61 Early research;
1961-70 ATLAS; 1970-2 Emigration and crisis;
1972-85 Renaissance; 1985-... Elliptic curve cryptography exists!
As I saw little of him after he moved to the States, I
will mostly talk his earlier years. Fortunately, Dan Bernstein, Francois Morain
and Winnie li (among others) have already told you much about his final period,
and indeed it is what this conference is about!
Oliver has been served well by Google and Wikipedia; I strongly recommend
those who don't know Oliver's work already
to try googling A O L Atkin and Oliver Atkin; you will find lots of Oliver's
mathematics, including the bits I've just said I won't be talking about.
Tyically, Oliver's work is both beautiful and useful, and one can read
about it without having to remember abstract definitions.
There is indeed little abstraction in sight, though much of it
illuminates abstract concepts.
Let's start at the beginning. Oliver's mathematical education was rather
extraordinary; he went to boarding school when he was 9, and went on
to Winchester, one of the oldest schools in Britain, where he was a scholar
from 1938-42. Winchester has always
been an exceedingly academic school; in particular, the boys with scholarships
live very closely together in "chambers" (there were no individual rooms)
in a beautiful but freezingly cold medieval building. Oliver was bright, but
not as bright as the group two years ahead of him, which included
Freeman Dyson and three others who became Fellows of the Royal Society.
There was a war on, so when he was 17 Oliver went to Cambridge
to take the Mathematical Tripos, a two year course rather than the peacetime
three. Cambridge was an odd place during the war: just a few young students
being taught by elderly (though highly distinguished) teachers.
The syllabus was old fashioned, dominated by classical analysis and applied
mathematics; very little algebra and practically no topology. Oliver probably
learnt about elliptic and modular functions as examples in complex analysis.
He graduated in 1944, and was posted to the coding establishment at Bletchley Park.
[ In Britain, we know of Bletchley as the place where serious computing was born. ]
At Bletchley he joined the Newmanry. I should explain
that this was a group of a couple of dozen very clever people, almost all of them
mathematicians, whose job was to enable others to read German messages that
had been intercepted --- most important, they had to break any codes used by the
German Navy that had not been broken already. They had a backup of
about 200 Wrens, who acted as their input and output devices.
The group was directed by Max Newman, with deputy director Shaun Wylie.
Other members I actually knew later on included Henry Whitehead, David Rees, Michael
Crum, Sandy Green and Peter Hilton. As it happened, most of them were topologists
or algebraists --- just the people who would normally have been lecturing to Oliver
in Cambridge. All of them were (or became) distinguished
mathematicians. Though I came to know some of them very well, I never heard
any of them talk about their experiences in Bletchley. Alan Turing was
the brightest star attached to the group, not strictly a member because his
individual contribution to the enterprise was so great.
So by the time he was 20, Oliver had had more experience of a group of
extremely intelligent people working together than most people ever
see in their lifetimes; one can hardly imagine a more exciting mathematical
environment. Oliver had learnt what mathematics was about, and could see how
important computation could become.
Oliver's next few years were (by comparison!) conventional. In 1945, the war
ended, but not Oliver's National Service, and he spent a couple of years in
the National Physical Laboratory thinking about the shape of aircraft wings.
He went back to Cambridge in 1947, and did research under J E Littlewood;
he took his doctorate in 1952, and went to Durham as a university lecturer.
In 1959 he married Gaynor. He did highly interesting work on congruence
properties of the partition function, and he played the organ at the cathedral.
(Durham is a quite small town, dominated by its magnificent cathedral.)
He might have settled down for life, but as you all know, he didn't.
It's time to talk about mathematics, and in particular
Oliver's work on partitions. I need some standard notation:
$p(n)$ will be the partition function, with generating function
$$ f(q)=\Sum_{n=1}^{\infty}p(n)q^n=\Prod(1-q^n)^{-1}. $$
The inverse of this generating function is $f(q)^{-1}=q^{-1/24}\eta(z)$ where
$$ \eta(z)=q^{1/24}\Prod(1-q^n)
=\Sum_{n=-\infty}^{\infty}exp(2\pi i(6n+1)^2/24)
where of course $q$ is $exp(2\pi i z)$. It is clear from the series expansion
that $\eta$ is a theta function, so a modular form of weight $1/2$.
Ramanujan conjectured that
if $24n-1$ is divisible by $5^a7^b11^c$
then $p(n)$ is divisible by $5^a7^{\beta}11^c$ ;
where $\beta=[(b+2)/2]$.
[[ Actually Ramanujan made this conjecture with $b$ instead of $\beta$;
but it wasn't true!]]
The [corrected] conjecture was proved for the
powers of 5 and 7 by Watson in 1938, and I think Morris Newman dealt with
divisibility by 11 and by 121.
In particular, 5 | $p(5n+4)$ ; 7 | $p(7n+5)$ ; 11 | p(11n+6) .
Oliver's very first paper, written jointly with Peter Swinnerton-Dyer, was a
remarkable one. They proved a lovely conjecture of Freeman Dyson that he had
published in Eureka (the magazine of the local student mathematical society) just
before Oliver arrived back in Cambridge. Define the {\it rank} of a
partition $\pi$ as the size of the largest part, minus the number of parts;
Dyson conjectured that when we sort the partitions of 5n+4 into classes
according to their ranks modulo 5, the classes are all the same size;
and the same should be true mutatis mutandis when we partition $7n+5$;
so we have a really natural reason for the divisibility.
Oliver and Peter succeeded in proving these facts in a highly original paper
involving Ramanujan's mock theta functions, which I think they had to rediscover
for the purpose.
[[ Dyson also made a similar conjecture about $p(11n+6)$ using a
different function, the crank instead of the rank, but he had to wait another
40 years or so before Garvan proved it in 1987! ]]
Congruence properties of the partition function were to be Oliver's main
interest for several years. His thesis completed the proof of Ramanujan's
conjecture; recollect that Watson had dealt with the cases where $24n-1$
is divisible by a power of 5 or of 7,
and Morris Newman had dealt with divisibility by 11 and 121, but not
with higher powers of 11. Oliver had to complete the proof that
if $24n-1$ is divisible by $11^c$, with c\ge3, then so is $p(n)$.
For this, it wasn't really a matter of inventing new methods, but rather of using
existing methods better than anyone else had managed. The methods involved using
various special modular functions, and the modular equations connecting them.
Oliver became expert in the theory; one felt that particular modular
functions were his special friends. But as I'm implying, the theory was
like my garden at home, a beautiful untidy jungle with lots of
different modular forms and functions, connected by various functional equations.
Oliver realised that even
beautiful jungles need to be tamed if others are to enjoy them.
Second, and more important, he realised that there was much more to be proved,
with more general properties than the divisibility properties in Ramanujan's
conjecture, and about the coefficients of
more general functions than the very special function $\eta(z)^{-1}$.
Oliver's idea (he wanted to state it as a conjecture, but of course the first
problem was to obtain the evidence for a really precise statement) was as follows:-
" Let $f(z)=\Sum_{n=1}^{\infty}a(n)q^n$ be a modular function, not necessarily of
positive weight, that vanishes at $i\infty$ and is invariant by $\Gamma_0(N)$,
where $N=q^k$ may as well be a prime power.
Then for each prime $p$ there are congruences like
$$ a(pn)+p^{k-1}a(n/p) = \gamma_pa(n) (modulo N) 44
that take the place of the hecke equalities valis for newforms of positive weight. "
He wanted to know what these congruences looked like, and after Bletchley Park
he thought he could use computers to find out.
His opportunity occurred when the ATLAS laboratory advertised a research
fellowship in 1961. [I should explain that the ATLAS computer laboratory was
attached to the Atomic Energy Research Establishment at Chilton (Harwell) about
twelve miles south of Oxford.] Oliver applied, and of course he was appointed;
his duties were to use one of the world's most advanced computers to do research
in topics of his choice. At ATLAS, he was in his element, though there were those
who didn't like it when he walked down the corridor singing Wagner fortissimo.
I am on record as having spoken about that period already, so i should keep
it short. I arrived in Oxford a year or two later, in 1964; I needed to know about
modular functions because Shimura had told me that they parametrise some elliptic
curves very naturally. I quickly found that the theory of modular functions was
beautiful and powerful, but
the available literature was horrible --- it was hard to tell which facts were well
known and which had never been proved properly, and there were far too many
particular functions. Oliver put me straight. I find it hard to believe that the
Atkin-Lehner paper wasn't published till 1970, since it is the theory I remember
Oliver teaching me four years earlier! That paper was enormously important: it
transformed the theory of congruence forms of positive weight from a recondite
mystery to a simple tool for people to use!
His spiel on "Feasible Computation" was witty and to the point; it should be
required reading for everyone setting out in computational number thery (it is not
on Math Sci Net; but easily found, eg by googling either A O L Atkin or `feasible
computation'). Also while at ATLAS, he and Peter Swinnerton-Dyer wrote the paper on
non-congruence subgroups that Winnie Li has already told you about; obviously, their
conjecture had much in common with `Oliver's idea' described above. An important
methodological point about the A-S-D paper is that they show how the coefficients
of modular forms and functions invariant by a subgroup $G$ of $SL_2(R)$ can be
computed simply from the geometry of the Riemann surface $G\H$, without any
previous knowledge of special functions.
But Oliver's main preoccupation at ATLAS was with the congruences properties of
coefficients of modular forms, particularly $c(n)$ the coefficient of
$j(z)=q^{-1}+\Sum_{n=0}^{\infty}c(n)q^n$ and the partition function $p(n)$
(a coefficient of $\eta(z)^{-1}$). The most natural function to which to apply
his idea stated above is $f(z)=j(z/N)+j({z+1}/N)+...+j({z+N-1}/N)$, leading to
to congruences for $c(n)$ modulo powers of $q$ when $N$ is a power of $q$. He was
able to prove and publish such congruences for primes up to 31; he and O'Brien also
proved similar congruences for $p(n)$ modulo powers of 13.
In 1968-72 there was a revolution in the theory of modular forms, starting with
with the introduction of 2-dimensional Galois representations by Deligne and Serre,
followed by the inception of p-adic modular forms and culminating in the Antwerp
conference of 1972. Serrereconised Oliver's j-function congruences as the prototype
of p-adic modular forms, but the new ideas involving representations depend
critically on favourable geometry, and the theory of modular forms of non-positive
weight has become unfashionable.
After he left ATLAS in 1970, I lost contact with Oliver for a while; in
particular, I knew nothing of the two apparently terrible years,
when his wife Gayner died, and he spent a year apiece in Arizona and at Brown,
with two young children (initially aged 7 and 11) to look after and no familiar
home. I fear too that he lost a great deal of the computer data he had garnered at
ATLAS: much of his ATLAS output would have been on lineprinter, too heavy to carry.
In 1972, he was appointed to the University of Illinois at
Chicago Circle, and found a home in the village of Oak Park, where he remained
for the rest of his life; the villagers of Oak Park seem to have taken this mad
Englishman to their hearts. Even then, I still didn't have any contact with him,
and have found it difficult to list his accomplishments, especially as
after he left ATLAS he became very reluctant to publish a paper unless
someone else wrote it for him. In his romantic essay `a walk through Ramanujan's
garden' Freeman Dyson describes discussions he had with Oliver, particularly
on the subject of his conjectured congruence relations. [The best source for this
essay is in Dyson's Selected Works, which also contains very relevant commentary.]
I guess that these discussions started in 1968 when Oliver was on sabbatical in
Baltimore; I suspect they may have continued during Oliver's time at Brown.
Once he reached Chicago, I know he acted as advisor to Duncan Buell; I suspect
that this is the period when he devised the algorithm for composing quadratic forms
that is described in Cohen's book. A little later on he was working with Winnie Li
on Atkin-Lehner theory, particular as it affected the twists of forms.
In 1976 there was a follow-up meeting to Antwerp in Bonn; I remember Oliver
checking which church in Bonn-Beuel (the part of Bonn the wrong side of the Rhine)
had the best organ, and obtaining permission to play it. He became
interested in the finding and certification of large primes; Francois Morain has
told you something about this; and Francois's zoo of class invariant functions
would have delighted Oliver. When Monstrous
Moonshine was conjectured by Conway and Norton in 1978, Oliver was quick to join
McKay and others in providing evidence. And, being Oliver, I am sure he
helped many others by computations that told them whether what they were trying to
prove was really true! But I wasn't there, so I don't know .
In 1985 Rene Schoof's thesis reached Oliver. Rene had deliberately not tried
to optimise his parameters so as to produce an algorithm that was really practical;
but this sort of application of particular modular functions was what Oliver loved.
He and Elkies quickly turned Schoof's method into a practical algorithm,
and counting the number of points on elliptic curves of large finite fields
became feasible and cryptographically useful. He rightly thought I ought to be
interested, so I was back on his mailing list. For me, it was as if a dormant
volcano had erupted, and from then on Oliver's e-mails (either direct or via
Victor's number theory net) remained a delight, and at least some of Oliver's work
became known to you all.