%%%%%%%%%%%%%%%%%%%%%%%
%% B. J. Birch, Jean-Louis Colliot-Th\'el\`ene, G. K. Sankaran, Miles Reid
%% and Alexei Skorobogatov
%% preface to Swinnerton-Dyer birthday volume, 34 pp.
%% latex2e
%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[12pt,twoside]{article}
\usepackage{amsmath,amsfonts,amssymb}
\textheight215truemm \textwidth 138truemm \pagestyle{myheadings} \date{}
\markboth{\qquad In lieu of Birthday Greetings \hfill}{\hfill In lieu of
Birthday Greetings \qquad}
\newcommand{\signoff}[1]{\par\nopagebreak\bigskip\hfill#1\par\pagebreak[2]\bigskip}
\newcommand{\Q}{{\bf Q}}
\newcommand{\sA}{{\cal A}}
\newcommand{\PP}{{\bf P}}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\GG}{{\bf G}}
\newcommand{\bil}{{\rm bil}}
\title{In lieu of Birthday Greetings}
\author{B. J. Birch \and Jean-Louis Colliot-Th\'el\`ene
\and G. K. Sankaran \and Miles Reid \and Alexei Skorobogatov}
\date{}
\begin{document}
\maketitle
\addcontentsline{toc}{chapter}{B. J. Birch, Jean-Louis
Colliot-Th\'el\`ene, G. K. Sankaran, Miles Reid and Alexei Skorobogatov,
In lieu of Birthday Greetings}
% \tableofcontents
\vskip1.4cm
This is a volume of papers in honour of Peter Swinnerton-Dyer's 75th
birthday; we very much regret that it appears a few months late owing to
the usual kind of publication delays. This preface contains four sections
of reminiscences, attempting the impossible task of outlining Peter's
many-sided contributions to human culture. Section~5 is the editor's
summary of the 12 papers making up the book, and the preface ends with a
bibliographical section of Peter's papers to date.
\vskip1.4cm
\section{Peter's first sixty years in Mathematics \\
by Bryan Birch}
\markright{\hfill Bryan Birch \qquad}
Peter Swinnerton-Dyer wrote his first paper \cite{1} as a young schoolboy
just 60 years ago, under the abbreviated name P. S. Dyer; in it, he gave
a new parametric solution for $x^4+y^4=z^4+t^4$. It is very appropriate
that his first paper was on the arithmetic of surfaces, the theme that
recurs most often in his mathematical work; indeed, for several years he
was almost the only person writing substantial papers on the subject; and
he is still writing papers about the arithmetic of surfaces sixty years
later. Peter went straight from school to Trinity College (National
Service had not quite been introduced); after his BA, he began research
as an analyst, advised by J E Littlewood. At the time, Littlewood's
lectures were fairly abstract, heading towards functional analysis; in
contrast, Peter was advised to work on the very combinatorial,
down-to-earth, theory of the van der Pol equation (the subject of
Littlewood's wartime collaboration with Mary Cartwright), where a
surprising sequence of stable periodic orbits arise completely
unexpectedly from a simple-looking but non-linear ordinary differential
equation. Lurking in the background was the three body problem, together
with ambitions to prove the stability of the solar system, compare
\cite{formstab}.
After a couple of years, Peter was elected to a Trinity Junior Research
Fellowship, and became a full member of the mathematical community (he
never needed to submit a doctoral thesis). In 1954, he was selected for a
Commonwealth Fund Fellowship, and went to Chicago intending to work with
Zygmund; but when he reached Chicago, he met Weil, who converted him to
geometry; I believe that Weil was the person who most influenced Peter's
mathematics. Ever since his year in Chicago, Peter has been an arithmetic
geometer, with unexpected expertise in classical analysis.
Peter returned to Cambridge in 1955. In the 1950s, mathematical life in
Cambridge was vigorous and sociable; everyone collaborated with everyone
else. It was the heyday of the Geometry of Numbers (it was sad that so
much excellent mathematical work was poured into such an unworthy
subject!) and Peter joined in. In particular, he and Eric Barnes (later
Professor at Adelaide) wrote a massive series of papers \cite{barnes} on
the inhomogeneous minima of binary quadratic forms, which completely
settled the problem of which real quadratic fields are norm-Euclidean;
like the van der Pol equation, this is a case where a `discrete'
phenomenon arises from a `continuous' question. He went on to collaborate
with Ian Cassels \cite{cassels}, trying to obtain a similar theory for
products of three linear forms; their work was highly interesting, but
only partially successful, and to this day there has been (I believe) no
further progress on the problem.
I first came into contact with Peter in 1953, when he read my Rouse Ball
essay on the Theory of Games (one of Peter's lesser interests, that does
not show up in his list of publications), and I got to know him well
after he returned from Chicago. Over the next couple of years, we talked
a lot and he taught me to enjoy opera and we wrote two or three pretty
but unimportant papers together; but at that stage, he wanted to be a
geometer, and I was turning towards analytic number theory, under the
influence of Harold Davenport. In my turn, I went to the States with a
Commonwealth Fund Fellowship, and while I was away Peter took a post in
the fledgling Computer laboratory. When I returned, I was excited by the
Tamagawa numbers of linear algebraic groups, one of us (probably Peter)
wondered about algebraic groups that aren't affine, and we set to,
computing elliptic curves.
Those four years, from 1958--62, were probably the best of my life; they
were the most productive, and I married Gina (who had a desk in Peter's
office in the Computer laboratory). We were under no pressure to publish:
we both had Fellowships, and knew we could get another job whenever we
needed one; and we didn't have to worry about anyone else anticipating
our work. In the first phase, we made a frontal assault; for the curves
$E(a,b):y^2=x^3+ax+b$ with $|a|\le20$ and $|b|\le30$ we computed the
Mordell--Weil rank, the 2-part of the Tate--Shafarevich group, and a
substitute $T(E,P)$ for a Tamagawa number $\tau(E)$, namely the product of
$p$-adic densities taken over primes $p\le P$ where $P$ was as high as
the market would stand. Peter did the programming, which he made feasible
by dealing with many curves simultaneously; for good primes the $p$-adic
density was of course $N_p/p$ where $N_p$ was the number of points mod
$p$, and the crude methods of computing $N_p$ for medium-sized $p$ were
nearly as fast for a batch of curves as for a single curve; there was an
even better batch-processing gain in the rank computations. (For the
finitely many `bad' primes one needed so-called fudge factors, which I
seem to remember were part of my job). To our delight, the numbers
$T(E,P)$ increased roughly as $c(E)\log^rP$, where $r$ was the
Mordell--Weil rank of $E$; so we prepared \cite{bsd1} for publication, and
proceeded to the second phase. Here, Davenport and Cassels were very
helpful; urged by their prodding, we realised that, rather than
considering the product $T(E,P)$ as $P$ got large, one should be
considering $L(E,s)^{-1}$ as $s$ tends to 1 (so that $L(E,s)$ should have
a zero of order $r$ at $s=1$). (As Weil remarked to a colleague in
Chicago, `it was time for them to learn some mathematics'.) Hecke had
tamed this Dirichlet series for elliptic curves with complex
multiplication, giving an explicit formula that actually converged at
$s=1$. So we approximated to the Dirichlet series $L(E,1)$, in case $E$
had complex multiplication and Mordell--Weil rank~1; and we got numbers
that really seemed to mean something: after the junk factors had been
scraped off, they seemed to be the order of the Tate--Shafarevich group
divided by the torsion squared. Next, Davenport showed us how to evaluate
$L(E,1)$ explicitly in terms of the Weierstrass $\wp$-function; we
computed some more, and \cite{bsd2}, containing the main B--S-D
conjectures, was the result.
In 1962 I left Cambridge to take a job in Manchester, and our
collaboration became less close; we had expected to write further Notes
in the series `On Elliptic Curves', but they didn't happen. Note III
might have been a plan of Peter's, to test the conjecture for abelian
varieties by starting with products of elliptic curves; this turned into
the thesis of Damerell, which essentially computed critical values of
$L(E^{(3)},s)$, where $E^{(3)}$ is the cube of a curve; the numbers were
interesting but he was not able to interpret them. The intended Note IV
was more important; Nelson Stephens was able to compute the higher
derivatives $L^{(r)}(E,1)$, where $r$ is the Mordell--Weil rank; he was
the first to obtain exact evidence for the conjectured formula, for
elliptic curves of higher rank over the rationals, and indeed his thesis
\cite{st} is where it is first precisely stated. In July 1965, Peter
received a letter from Weil \cite{weil} which set the tone for further
progress in the area. Weil reminded us that our conjectures make sense
only if the relevant functions $L(E,s)$ have functional equations, and
this is likely to be true only if the elliptic curve $E/\Q$ is
parametrised by modular functions invariant by some $\Ga_0(N)$. So we had
better be looking at modular curves! I was in Cambridge on sabbatical for
the next term, so we set to work. Indeed, we worked very hard; on one
occasion we were so engrossed talking mathematics after dinner, on
Trinity Backs, that an unobservant porter locked us out; fortunately, we
were able to regain entry by successfully charging the New Court gate.
Weil's letter led to three developments; first, modular symbols: I am
pretty certain that Peter had the first idea \cite{bjb}, but he was very
busy, so I and my students had to make them work, and Manin \cite{ma}
formalised the concept. Next, the tabulation of elliptic curves of small
conductor (Table~I of \cite{antw3}); this involved many people, starting
with Peter and then me, as described in the introduction to the table.
Finally, a few years later, Heegner points came on the scene.
I was most excited in our work on elliptic curves; but indeed Peter's
interests in this period were exceedingly diverse. He did seminal work on
his earliest love, the arithmetic of surfaces: in \cite{2sp} he found the
first counterexample to the Hasse principle for cubic surfaces (I think he
found this example in 1959, as I reported on it in Boulder). A little
later, he improved a result of Mordell, that the Hasse principle is valid
for the intersection of two quadric hypersurfaces in $\PP^n$ so long as
the dimension $n$ is large enough --- this paper \cite{2q} is of interest
as a very early example of Peter's technique of working out what one can
prove if one assumes various useful but unprovable `facts'; with luck, one
may remove such unwanted hypotheses later. His 1969 paper
\cite{stonybrook} at the Stony Brook conference reviewed what was known,
and contained new material. At last, in 1970, Peter ceased to be a lone
voice crying in the wilderness, when Manin introduced the so-called Manin
obstruction in his lecture at Nice \cite{ma2}, and went on to write his
book on cubic surfaces \cite{ma3}. Also in 1969--70, Colliot-Th\'el\`ene
went to Cambridge to work with Peter; since then the theory has
flourished, as this volume amply testifies.
Meanwhile, Peter remained an analyst; in particular, Noel Lloyd was his
research student between 1969 and 1972. He also became interested in
modular forms for their own sake; with Atkin, he investigated modular
forms on non-congruence subgroups \cite{atkin}. Surprisingly, their
results suggested that the power series of such modular forms should have
good $p$-adic properties (their conjecture was proved long afterwards by
Scholl). Peter corresponded with Serre, and published the basic paper on
the structure of (ordinary) modular forms modulo $p$ in the third volume
of the Antwerp Proceedings \cite{antw3}; this volume was of course the
beginning of the theory of $p$-adic modular forms. Peter made yet another
important contribution in the Computing Laboratory, where he was
responsible for implementing Autocode for Titan.
He worried about the inefficiencies of university governance, and took an
increasing interest in administrative matters. In 1973 he was elected
Master of St Catherine's College, from 1979--81 he was Vice Chancellor,
and from 1983--89 he was Chairman of the University Grants Committee. All
this involved an immense amount of committee work, but miraculously (and
with the help of Harriet, and of Jean-Louis Colliot-Th\'el\`ene) he
remained in touch with mathematics. When he returned to Cambridge in 1989
he resumed full-time research, principally on the arithmetic theory of
surfaces, but also on analysis.
\signoff{Oxford, 10th Dec 2002}
\vskip0.4cm
\section{Peter Swinnerton-Dyer's work on the \\ arithmetic of higher
dimensional varieties \\
by Jean-Louis Colliot-Th\'el\`ene}
\markright{\hfill Jean-Louis Colliot-Th\'el\`ene \qquad}
In parallel to his well-known contributions to elliptic curves, modular
forms, $L$-functions, differential equations, bridge, chess and other
respectable topics, Peter has a lifelong interest in the arithmetic
geometry of some -- at first sight -- rather special varieties: cubic
surfaces and hypersurfaces, complete intersections of two quadrics
defining a variety of dimension $\ge2$, and quartic surfaces.
I happened to spend a year in Cambridge when I started research, and Peter
passed on to me his keen interest in the corresponding diophantine
questions. I am thus happy to report here on Peter's past and ongoing
work on these problems. As will be clear from what follows, Peter, at age
75, is still doing entirely original innovative research.
Much of the progress achieved in arithmetic geometry during the twentieth
century has been concerned with curves. For these, we now have a clear
picture: for genus zero, the Hasse principle holds; for genus one, many
problems remain, but we have the Birch and Swinnerton-Dyer conjecture,
and we hope that the Tate--Shafarevich groups are finite; for genus at
least two, Faltings proved the Mordell conjecture.
In higher dimension the situation is much less clear. For the three types
of varieties mentioned above, one is still grappling with the basic
diophantine questions: How can we decide whether there are rational
points on such a variety? Is there a local-to-global principle, or at
least some substitute for such a principle? What are the density
properties of rational points on such varieties (in the sense of the
Chinese remainder theorem)? Can one ``parame\-trize'' the rational points?
Can one estimate the number of rational points of bounded height?
The time when varieties were classified according to their degree, as in
Mordell's book, is long gone, and one may view the varieties just
mentioned as belonging to some general classes of varieties. One general
class of interest is that of rational varieties (varieties birational to
projective space after a finite extension of the ground field). A wider
class, whose interest has been recognized only in the last ten years, is
that of rationally connected varieties. These are now considered as the
natural higher dimensional analogues of curves of genus zero. Nonsingular
intersections of two quadrics (of dimension $\ge2$) are rational
varieties, hence rationally connected; so are nonsingular cubic surfaces.
Higher dimensional cubic hypersurfaces are rationally connected.
Nonsingular quartic surfaces are not rationally connected, but there are
interesting density questions for rational points on them.
Until 1965, there were two kinds of general results on the arithmetic of
rational varieties. One series of works, going back to the papers of H.
Hasse in the twenties (local-to-global principle for the existence of
rational points on quadrics), was concerned with homogeneous spaces of
connected linear algebraic groups. A very different series of works,
going back to the work of G. H. Hardy and J. E. Littlewood, proved very
precise estimates on the number of points of bounded height (hence in
particular proved existence of rational points) on complete intersections
when the number of variables is considerably larger than the multidegree.
There had also been isolated papers by F. Enriques, Th.\ A. Skolem, B.
Segre, L. J. Mordell, E. S. Selmer, F. Ch\^atelet, J. W. S. Cassels and
M. J. T. Guy. Peter himself made various contributions to the topic in
his early work: he produced the first counterexamples to the Hasse
principle and to weak approximation for cubic surfaces \cite{2sp}, he
extended results of Mordell on the existence of rational points on
complete intersections of two quadrics in higher dimensional projective
space \cite{2q}, and he proved the Hasse principle for cubic surfaces with
special rationality properties of the lines.
Over the years 1965--1970, after some prodding by I. R. Shafarevich, Yu.
I. Manin and V.~A. Iskovskikh looked at this field of research in the
light of Grothendieck's algebraic geometry. They did not solve all the
diophantine problems, but they put some order on them. A typical
illustration was Manin's appeal \cite{ma2} to Grothendieck's Brauer group
to reinterpret most known counterexamples to the Hasse principle,
including Peter's.
I spent the academic year 1969/1970 in Cambridge -- I was hoping to learn
more about concrete diophantine problems, not the kind of arithmetic
geometry I was exposed to in France. Professor Cassels advised me to take
Peter as a research supervisor. I was first taken aback, because,
ignorant as I was, the only thing I knew about Peter was that he had
written a paper entitled ``An application of computing to number theory'',
and I was not too keen on computing. I wanted concrete diophantine
equations, but with abstract theory. I nevertheless asked Peter, and this
was certainly one of the most important moves in my mathematical career.
In those days, Peter was neither a Sir nor a Professor. He was known to
Trinity students as ``The Dean'', whose function I understand was to
preserve moral order among the students. To this he contributed by
serving sherry (``Sweet, medium or dry?'')\ each evening in his small flat
in New Court. Sherry time was the ideal time to ask him for advice,
mathematical or other -- I do not remember Peter as a great addict of
long sessions in the Mathematics Department. Well, at least one could
enjoy his beautifully prepared lectures (the young Frenchman enjoyed the
very clear, classical English as much as the mathematics). Peter was well
known for his wit, and Swinnerton-Dyer quotations and stories abounded.
His students enjoyed his avuncular behaviour -- he was not a thesis
adviser in the classical sense -- and at the same time one vaguely feared
him as the possible mastermind of many things going on in Cambridge. (His
masterminding was later to extend to a wider scene -- I remember Spencer
Bloch being rather impressed by a 1982 newspaper representation of Peter
Swinnerton-Dyer portrayed as King Kong climbing up one of London
University's main buildings.)
One day in April 1970, on Burrell's walk, I asked Peter for a research
topic. He mentioned the question of understanding and generalizing some
work of Fran\c cois Ch\^atelet, who had performed for cubic surfaces of
the shape $y^2-az^2=f(x)$ (with $f(x)$ a polynomial of the third degree)
something which looked like descent for elliptic curves -- Peter also had
handwritten lists of questions on a similar process for diagonal cubic
surfaces.
In July 1970 I went back to France, and learned ``French algebraic
geometry'' with J.-J. Sansuc. He and I discussed \'etale cohomology and
Grothen\-dieck's papers on the Brauer group, but I kept on thinking about
Ch\^atelet surfaces and Peter's questions. In 1976--77, Sansuc and I laid
out the general mechanism of descent, which appeals to principal
homogeneous spaces (so-called torsors) with structure group a torus (as
opposed to the finite commutative group schemes used in the study of
curves of genus one). One aim was to find the right descent varieties on
Ch\^atelet surfaces (and to answer a question of Peter, whether descent
here was a one-shot process, as opposed to what happens for elliptic
curves). The theory was first applied to more amenable varieties, namely
to smooth compactification of tori. As far as Ch\^atelet surfaces are
concerned, there were two advances: In 1978, Sansuc and I realized that
Schinzel's hypothesis (a wild generalization of the twin prime
conjectures) -- also considered much earlier by Bouniakowsky, Dickson,
and Hardy and Littlewood -- would imply statements of the type: the
Brauer--Manin obstruction is the only obstruction to the Hasse principle
for generalized Ch\^atelet surfaces, namely for surfaces of the shape
$y^2-az^2=f(x)$ with $f(x)$ a polynomial of arbitrary degree (over the
rationals). The second advance took place in 1979: following a rather
devious route, D. Coray, J.-J. Sansuc and I found a class of generalized
Ch\^atelet surfaces for which the Brauer--Manin obstruction entirely
accounts for the defect of the Hasse principle.
During the period 1970--1982, Peter was busy with any number of different
projects: the Antwerp tables on elliptic curves \cite{ah}, understanding
Ramanujan congruences for coefficients of modular forms \cite{antw3},
\cite{an}, writing, jointly with B. Mazur, an influential paper \cite{ab}
on the arithmetic of Weil curves and on $p$-adic $L$-functions, proving
(jointly with M. Artin \cite{ai}) the Tate conjecture for K3 surfaces
with a pencil of curves of genus one (a function-theoretic analogue of
the finiteness of the Tate--Shafarevich group), and also writing a number
of papers on differential equations. He also wrote a note on the number
of lattice points on a convex curve \cite{y}, which was followed by
papers of other writers (W. M. Schmidt, E. Bombieri and J. Pila). The
ideas in those papers now play a r\^ole in the search for unconditional
upper bounds for the number of rational points of bounded height (work of
D. R. Heath-Brown).
During that period, Peter also contributed papers on rational varieties:
he gave a proof of Enriques' claim that del Pezzo surfaces of degree 5
always have a rational point \cite{ag}, he wrote a paper with B. Birch
producing further counterexamples to the Hasse principle \cite{aj} and he
wrote a paper on $R$-equivalence on cubic surfaces over finite fields and
local fields \cite{as}. This last paper used techniques specific to cubic
surfaces to prove results which have just been generalized to all
rationally connected varieties by J. Koll\'ar and E. Szab\'o, who use
modern deformation techniques. That paper and a later one \cite{by} on a
related topic exemplify how Peter is not deterred by inspection of a very
high number of special cases.
Indeed it is Peter's general attitude that a combination of cleverness
and brute force is just as powerful as modern cohomological machineries.
As the development of many of his ideas has shown, cohomology often
follows, and sometimes helps. As we say in France, ``l'intendance suit''.
Let me here include a parenthesis on Peter's ideal working set-up.
Sitting at a conference and not listening to a lecture on a rather
abstruse topic seems to be an ideal situation for him to conceive and
write mathematical papers. The outcome, written without a slip of the
pen, is then imposed upon the lesser mortal who will definitely take much
more time to digest the contents than it took Peter to write them.
In 1982, I spent another six months in Cambridge. I did not see Peter too
often, as I was rather actively working on algebraic K-theory, not a
field which attracts his attention. However, shortly before I left
Cambridge, in June 1982, Peter invited me for lunch at high table in
Trinity, and while reminding me how to behave in this respectable
environment, he inadvertently mentioned that he could say something new
on descent varieties attached to Ch\^atelet surfaces -- the topic he had
offered to me as a research topic 12 years earlier. If my memory is
correct, what he did was to sketch how to prove the Hasse principle on
the specific intersections of two quadrics appearing in the descent
process on Ch\^atelet surfaces, the method being a reduction by clever
hyperplane sections to some very special intersections of two quadrics in
4-dimensional projective space. Sansuc and I quickly saw how the descent
mechanism we had developed in 1976--77 could combine with this new
result. This was to develop into a Comptes Rendus note of Sansuc,
Swinnerton-Dyer and myself \cite{at} in 1984, then into a 170 page paper
of the three of us in Crelle three years later \cite{ax}. Among other
results, we obtained a characterisation of rational numbers that are sums
of two squares and a fourth power, and we proved that over a totally
imaginary number field two quadratic forms in at least 9 variables have a
nontrivial common zero (this is the analogue of Meyer's result for one
form in 5 variables). An outcome of the algebraic geometry in our work
was a negative answer (joint work of the three of us with A. Beauville
\cite{av}) to a 1949 problem of Zariski: some varieties are stably
rational but not rational.
Around 1992, the idea to use Schinzel's hypothesis to explore the
validity of the Hasse principle (or of its Brauer--Manin substitute) was
revived independently by J-P. Serre and by Peter \cite{be}. In that
paper, conceived during a lengthy coach trip in Anatolia, Peter
simultaneously started developing something he calls the Legendre
obstruction. In many cases, this obstruction can be shown to be
equivalent to the Brauer--Manin obstruction, but Peter tells me there are
cases where this yields information not reachable by means of the
Brauer--Manin obstruction. In 1988, P. Salberger had obtained a remarkable
result on zero-cycles on conic bundles over the projective line. The
paper involved a mixture of algebraic K-theory and approximation of
polynomials. Peter saw how to get rid of the K-theory and how to isolate
the essence of Salberger's trick, which turned out to be an unconditional
analogue of Schinzel's hypothesis. This was developed in papers of Peter,
in a paper with me \cite{bd} and in a paper with A. N. Skorobogatov and
me \cite{bk}. The motto here is: it is worth exploring results
conditional on Schinzel's hypothesis for rational points, because if one
succeeds, then one may hope to replace Schinzel's hypothesis by
Salberger's trick and prove unconditional results for zero-cycles.
Up until about ten years ago, work in this area was concerned with the
total space of one-parameter families of varieties which were close to
being rational. In 1993 Peter invented a very intricate new method, which
enables one to attack pencils of curves of genus one. In its general
form, the method builds upon two well-known but very hard conjectures,
already mentioned: Schinzel's hypothesis and finiteness of
Tate--Shafarevich groups of elliptic curves. The original paper
\cite{bg}, in Peter's own words, looks like a series of lucky
coincidences and ``rather uninspiring'' explicit computations (not many
of us have the good fortune to come across such series). It already had
striking applications to surfaces which are complete intersections of two
quadrics.
It took several years for Skorobogatov and me to get rid of as many
lucky coincidences as possible (one instance being a brute force
computation which turned out to be Peter's rediscovery of Tate's duality
theorem for abelian varieties over local fields). The outcome was a long
joint paper of the three of us \cite{bl} in 1998. In that paper Peter's
original method is extended beyond rational surfaces: the method can
predict a substitute of the Hasse principle and density results for
rational points on some elliptic surfaces (surfaces with a pencil of
curves of genus one). This came as quite a surprise.
Since 1998, Peter has been developing subtle variants of the method, with
application to some of the simplest unsolved diophantine equations:
systems of two quadratic forms in as low as 5 variables \cite{bg},
\cite{bl}, \cite{bv}, diagonal quartics \cite{br} (hence some K3 surfaces,
whose geometry is known to be far more complicated than that of rational
surfaces); diagonal cubic surfaces and hypersurfaces over the rationals
\cite{bw}. The first two applications assume Schinzel's hypothesis and
finiteness of Tate--Shafarevich groups, but \cite{bw} (on diagonal cubic
surfaces) only assumes the latter finiteness: this theorem of Peter's on
diagonal cubic surfaces, both by the result and by the subtlety of the
proof, is certainly the most spectacular one obtained in the area in the
last ten years. For instance, under the finiteness assumption on
Tate--Shafarevich groups, the local-to-global principle holds for
diagonal cubic forms in at least 5 variables over the rationals.
In 1996, rather wild guesses were made on two different topics: For which
varieties do we expect potential density of rational points? For
varieties over the rationals with a Zariski-dense set of rational points,
what should we expect about the closure of the set of rational points in
the set of real points (question of B. Mazur)? Peter had the idea to call
in bielliptic surfaces to produce unexpected answers to the second
question. Skorobogatov and I elaborated, and applied the mechanism to get
rid of preliminary guesses for the first question. This led to a joint
work between the three of us \cite{bh}. There has been recent
(conjectural) progress on an answer to the first question (work of
complex algebraic geometers). The same bielliptic surfaces were later
used by Skorobogatov (1999) to produce the first ever example of a
surface for which the Brauer--Manin obstruction is not the only
obstruction to the Hasse principle. This has led to further developments
by D. Harari and Skorobogatov (descent under noncommutative groups).
Peter also contributed two papers \cite{bc}, \cite{bo} to a topic which
has seen quite some activity over the last ten years: the behaviour of the
counting function for points of bounded height on Fano varieties. He
pointed out the way to the correct guess for the constant in the standard
conjecture (later important work in this area was done by E. Peyre and
others). The lower bound he obtained (jointly with J. B. Slater
\cite{bo}) for cubic surfaces is still one of the best results in this
area.
The line of investigation Peter started in 1994 with the paper \cite{bg}
is very delicate, and while his 2001 paper on diagonal cubic surfaces
\cite{bw} is quite a feat, I am sure that Peter will produce much more in
this exciting new direction. I am confident that he will keep on being as
generous with his ideas as he has always been and that he will allow some
of us to accompany him along the way.
\signoff{Orsay, the 13th of February, 2003}
\vskip0.4cm
\section{Peter Swinnerton-Dyer: Geometer and \\ politician \\
by G.K. Sankaran}
\markright{\hfill G. K. Sankaran \qquad}
Peter Swinnerton-Dyer's interest in algebraic geometry derives arguably
from its relation to number theory, and from the formative period he
spent with Andr\'e Weil in Chicago in the 1950s, but he has also made
important contributions to geometry over algebraically closed fields.
Probably his most notable technical result of a purely geometric nature
is the proof (described elsewhere in this preface by Jean-Louis
Colliot-Th\'el\`ene) that stable rationality does not imply rationality
\cite{av}. This was, probably, contrary to the expectations of the
majority of algebraic geometers at the time; though, as often happens, it
is hard with hindsight to imagine why anybody ever thought the opposite
was true. The result, published in French in a joint paper with
Beauville, Sansuc and Colliot-Th\'el\`ene, uses a wide range of
techniques from different parts of algebraic geometry: torsors, linear
systems with base points, Prym varieties and singularities of the theta
divisor. It arose, however, out of arithmetic work with Sansuc and
Colliot-Th\'el\`ene. Many of Peter's arithmetic results have a geometric
flavour, especially his work with Bombieri and with Artin; and it is now
appreciated among geometers that arithmetic information can be made to
yield geometrical or topological information (in addition to the
well-known consequences of the Weil Conjectures). Rational and abelian
varieties particularly feature in his work: these topics are represented
in this volume by the papers of Reid and Suzuki and of Sankaran
respectively.
Within algebraic geometry, however, Peter's chief influence has been as
teacher, expositor, supplier of encouragement and enthusiasm, and
\'eminence grise. He recognised, at a time when few in Britain were more
than dimly aware of it, the power of the French school of algebraic
geometry of Weil, Serre and Grothendieck. In the 1970s he encouraged his
then student Miles Reid to visit Paris and learn directly from Deligne.
The flourishing state of British algebraic geometry at the present day
owes much to this development, and to Peter's encouragement and direction
of later students. His Cambridge Part III courses have been a source of
inspiration to many, and his book on abelian varieties and his account of
the basic facts of Hodge theory have been of great service to even more.
Many of Peter's multifarious activities are completely unrepresented in
this book. The purpose of the rest of this note is to allude to some of
them. I am not the best person to write such a note (that would be Peter
Swinnerton-Dyer): I have drawn on my memories of conversations with many
people, among them Carl Baron, Arnaud Beauville, Bryan Birch, B\'ela
Bollob\'as, Jean-Louis Colliot-Th\'el\`ene, James Davenport, Nicholas
Handy, Richard Pinch, Colin Sparrow, Miles Reid, Pelham Wilson, Rachel
Wroth and, above all, Peter Swinnerton-Dyer.
Mathematically the most obvious of Peter's other activities is his
substantial contribution to the theory of differential equations,
including a paper with Dame Mary Cartwright published only in Russian
\cite{ar}. He is still active in differential equations. Readers of the
present volume will have no difficulty in finding more information about
this part of Peter's work. Slightly further afield, Peter was a member of
the computing group in Cambridge in the 1960s, in the days of the
Cambridge University computer TITAN. The original operating system for
this famous machine, known as the Temporary Supervisor, was written by
Peter single-handed, and it worked. He wrote the computer language
Autocode for the same machine, and most Cambridge mathematicians of the
1960s had their first programming instruction in this language. Who could
ask for anything more?
Peter, then Dean of Trinity College, was elected Master of St Catharine's
College in 1973 and remained there for ten years. Littlewood is said to
have greeted the news with Clemenceau's remark on hearing that the pianist
Paderewski was to be Prime Minister of Poland: `Ah, quelle chute!'. But St
Catharine's afforded Peter considerable scope, and by all the numerous
accounts I heard, as a later Fellow of St Catharine's, he was highly
successful. The head of a Cambridge College (of Oxford I cannot speak) is
commonly all but invisible to the students, and in some cases even to the
Fellows. Peter was not: he has never been averse to the company of
students and he was even willing to do College teaching. As he could and
would teach almost any course in the Mathematical Tripos, the task of the
Director of Studies (who is responsible for arranging for the students to
be taught) was occasionally much simplified.
While at St Catharine's he served as Vice Chancellor of the University.
This is now a full-time post held for a long period, but at the time the
Vice Chancellor was chosen from among the heads of the various colleges
and served for two years only. The role of the Chancellor (then, as now,
the Duke of Edinburgh) is purely ceremonial, and the Vice Chancellor is
in effect at the head of the University. It is a job for a skilled
diplomatist. Cambridge University is a highly visible organisation, under
constant and occasionally hostile scrutiny by newspapers and television.
Internal matters can lead to very acrimonious public debate, and in
extreme cases, which are quite common, the Vice Chancellor is expected to
reconcile the factions. During Peter's term of office there was one
especially well-publicised dispute about whether a tenured post should be
awarded to a particular person. It was clearly impossible to satisfy all
parties, but Peter nevertheless managed to bring the matter to a
conclusion without offending anybody further. Who could ask for anything
more?
Peter left St Catharine's to take up a post as Chairman of the University
Grants Commission, a semi-independent Government body which was charged
with deciding how Government funding ought to be apportioned among
different universities. He had already written an influential, and in
some quarters unpopular, report on the structure of the University of
London, and was thus well known to be of a reforming cast of mind. He was
also widely assumed to be in general political sympathy with the
government of the time (otherwise, the reasoning ran, why did they
appoint him?); but this was far from the case. He was nevertheless able
to use his position to defend the reputation of the universities for
financial responsibility, and in particular to establish the principle
that research is a core activity for any university and therefore merits
funding on its own account, independently of teaching. The price to be
paid was investigation by government of the research activities of
universities. Peter is thus often held responsible for, or credited with,
the Research Assessment Exercise, which attempts to grade British
university departments (not individuals) roughly according to the quality
of research that they produce, and then hopes that they will be funded
accordingly. The system is agreed to be imperfect, but it is easier to
think of worse alternatives than better ones.
Peter's first involvement in politics dates from early in his tenure as
Master of St Catharine's. The Member of Parliament for Cambridge resigned
his seat and a by-election had to be held. Among the candidates was a
representative of the Science Fiction Loony Party, whose aim in standing
was to have some fun, and if possible to do better than the extreme
right-wing candidate. Candidates in British parliamentary elections are
required to pay a deposit of a few hundred pounds, returnable if they
receive a certain proportion (then one-eighth) of the votes cast. In this
case there was no prospect of that, so the deposit was, in effect, a fee:
Peter, a wealthy man, paid it. He explained that the candidate ``deserved
every possible support, short of actually voting for him''. Later his own
name was mentioned as a possible parliamentary candidate, on behalf of the
more serious but probably less entertaining Social Democratic Party formed
by Roy Jenkins and other disaffected members of the Labour Party in 1981.
Nothing came of the plan, if it ever existed. The SDP seems,
understandably, to have been unable to believe that all Peter's activities
were the work of one man, and on occasion sent him two copies of the same
letter, one for Swinnerton and one for Dyer.
Peter is a strong Chess player. Even when Vice Chancellor, he used to put
in occasional appearances at the Cambridge University Chess Club, playing
five-minute against undergraduates. The story is that when appointed to a
Trinity research fellowship, he was strongly advised to cut down the time
he spent on Chess; and that his interest in Bridge dates from this time.
He was to become a very strong Bridge player. He was a member of the team
that won the British Gold Cup in 1963, and he acted as non-playing
captain of the Great Britain Ladies' Bridge team.
On leaving UGC (by then renamed UFC) Peter resumed work as a
mathematician as if nothing had happened. He also continued his life of
public service, working on behalf of such diverse institutions as the
World Bank and the Isaac Newton Institute: he is still frequently to be
found at the latter, at least.
Peter's work at UGC/UFC was recognised by the award of a knighthood (a
KBE, to be precise). The editors of this volume tell me that ``how did
Swinnerton-Dyer get his title?'' is a frequently asked question after
seminars in places such as Buenos Aires and Vladivostok: at the risk of
spoiling the fun, here is an explanation.
Peter is a baronet: he is also a knight. A baronet is entitled to call
himself ``Sir'', and when he dies his eldest son, or some other male
relative if he has none, inherits the title. It is only a title: it does
not give him a seat in the House of Lords, and never has. Baronetcies were
invented by King James I, early in the seventeenth century, as a way of
raising money: they were simply sold. Later baronetcies were awarded for
actual achievement, but the oldest ones are purely mercenary affairs.
Since no baronetcies have been created for many years, all current
baronets have inherited their titles rather than earning or buying them. A
knight is also entitled to call himself ``Sir'', but the title dies with
him. Knighthoods, which are still awarded in quite large numbers, are for
specific personal achievements: they are given by the Queen on the
recommendation of the Prime Minister. By the time he was knighted, Peter
was already a baronet, so already entitled to call himself ``Sir Peter''
(not ``Sir Swinnerton-Dyer''). For this reason he is sometimes referred to
as (Sir)$^2$ Peter, although strictly speaking ``Sir'' is idempotent: he
is technically Professor Sir Henry Peter Francis Swinnerton-Dyer, FRS,
Bt., KBE.
\signoff{Bath, 12th Feb 2003}
\vskip0.4cm
\section{Peter Swinnerton-Dyer, man and legend \\
by Miles Reid}
\markright{\hfill Miles Reid \qquad}
I was supervised by Peter as a second year Trinity undergraduate. {From}
then on, I was among the many Cambridge students who were occasionally
invited for sherry at 7:30 pm (before Hall at 8:00 pm). For me and many
other middle-class students of my generation, this provided an education
into hitherto unsuspected areas of culture, such as good quality sherry,
opera, college politics, famous math visitors, the workings of the British
upper classes, etc. Peter is 16th Baronet Swinnerton-Dyer, and his family
was an illustration that the feudal system was still alive and well, in
Shropshire, at least in 1949: he had an elegant clock on the mantlepiece
of his Trinity New Court apartment, with an inscription
\begin{quote}
``Presented to Henry Peter Francis Swinnerton-Dyer Esq by the tenants,
cottagers and employees of the Westhope estate on the occasion of his
coming of age''.
\end{quote}
Peter's legendary status was already well established -- as a sample of
the stories in circulation, when Galois theory was introduced as a Part II
course lectured by Cassels, Peter claimed that the whole course could be
given in 4 hours, and made good his claim one evening between 10 pm and
2 am. Another story about bridge, that I heard from Peter himself: At a
tournament, Peter called over the referee, told him formally that he was
not making an error or oversight, then bid 8 clubs. Although this bid is
impossible, he had calculated that he would lose less going down in it
than allowing his opponents to make their grand slam. He knew the fine
wording of the rules of bridge, and the match referee was forced to accept
the impossible bid, since it was not made by error or oversight; the rules
were subsequently changed to block this obscure loophole.
At that time Peter was Dean of Trinity; the position included disciplinary
control of students. Those caught walking on the grass in College would be
sent to Peter, and would in theory be fined in multiples of $6/8$ (that
is, 6 shillings and 8 pence, a third of a pound). In my case, for a
particularly unpleasant misdemeanour, my sentence was to wash Peter's car.
Peter had an affinity with math students, and would drop in on friends in
the evening to see if there was a conversation going on; I can well
believe that student company was more fun than that of the senior
combination room. He would often join in conversations, or dominate them
-- his predilection for that well-turned phrase certainly had a lasting
effect on my literary pretensions. (For example: Would he send his son to
Eton? ``Certainly, it has advantages both in this life, and in the life
that is to come.'') Or, he would sometimes simply be comfortable among
student friends and nod off to sleep (presumably this mainly happened
after wine in the Combination room following High Table dinner). On one
occasion, we played the board game Diplomacy from after dinner until
breakfast the following morning -- with great cunning and skill, Peter
unexpectedly murdered me treacherously at about 6:00 am. Outside board
games, Peter was extremely generous with friends and colleagues -- many of
us were invited to accompany him on a trip to the opera in London, or on a
car trip to Norway, Paris or Italy, with appropriate stops to appreciate
the great cathedrals and the starred restaurants of the Michelin guide.
As a PhD student I started to get more specific mathematical benefit from
Peter's advice. He helped Jean-Louis Colliot-Th\'el\`ene and me set up a
seminar to study Mumford's little red book, and was always in a position
to illustrate our questions with some example from his own research
experience, although his background in Weil foundations meant that there
was always the added challenge of a language barrier. The subject of my
thesis (the cohomology of the intersection of two quadrics), given to me
by Pierre Deligne, turned out to be closely related to Peter's work with
Bombieri on the cubic 3-fold \cite{o}. Peter was also in the thick of the
action surrounding modular forms at the time of the 1972 Antwerp
conference \cite{antw3}--\cite{ah}.
{From} 1978, when I got married and left Cambridge for Warwick, my contact
with Peter became less frequent. A few years later, Peter married the
distinguished archeologist Harriet Crawford (reader at UCL and author of
3 books in the current Amazon catalogue). Together with every\-one else in
British academia, I was frequently aware, often through the media, of
his activities as Vice Chancellor of Cambridge, as Chairman of the
University Grants Committee, as the person who persuaded the conservative
government of Mrs Thatcher (``We shall not see her like again!'') to
accept research as the main criterion for judging the quality of
universities, and in numerous other capacities. As a member of the British
Great and Good, he chaired any number of committees or public enquiries,
investigating anything from parochial malpractice at British universities
(see
\begin{quote}
http://www.freedomtocare.org/page37.htm),
\end{quote}
to the disastrous storm of
16th October 1987 (this on behalf of the Secretary of State, see
Meteorological Magazine 117, 141--144). I met him, for example, in Japan
on a mission to investigate the state of university libraries.
At about the time Peter retired from the UGC, Warwick University had the
foresight to offer him the position of Honorary Professor. He has visited
us on many occasions in this capacity, both on Vice Chancellor's business
and for mathematical visits, on each occasion giving us the full benefit
of his wit and wisdom (for example, his scathing comments on teaching
assessment in universities: ``The Teaching Quality Assessment was an
extremely tedious farce, bloody silly''). On several occasions he has
given two mathematical lectures on the same day, one in Diophantine
geometry and another in differential equations, before taking us all out
to a very good dinner.
Peter is more active in research than ever at age 75, and in closer
contact with us at Warwick: he has repeated his lecture series ``New
methods for Diophantine equations'' (first given in Arizona in December
2002) as a Warwick M.Sc.\ course, driving over each week and meeting us for
lunch in a Kenilworth pub, at which Peter takes two pints of cider to put
him in good voice for the afternoon lectures.
I close with some Swinnerton-Dyer quotes:
\begin{itemize}
\item To have a computer job rejected by the EDSAC 2 Priorities Committee,
``You had to be both stupid and arrogant -- neither alone would do it.''
\item On meeting Colin Sparrow in King's Parade ``I have been made
Chairman of UGC. Waste of a knighthood!''
\item ``They aren't true, of course, but one believes them at least as
much as one believes the Thirty-Nine Articles of the Church of England.''
\item In Trinity College parlour with Alexei Skorobogatov (in connection
with the dogma of the Orthodox Church): ``In order to become a clergyman
in the Church of England you need to believe only one thing -- that it is
better to be wealthy than poor.''
\end{itemize}
\signoff{Warwick, 21st Feb 2002}
\vskip0.4cm
\section{Editor's preface to the volume \\ by Alexei Skorobogatov}
\markright{\hfill Alexei Skorobogatov \qquad}
The papers in this volume offer a representative slice of the delicately
inter\-twined tissue of analytic, geometric and cohomological methods
used to attack the fundamental questions on rational solutions of
Diophantine equations. A unique feature of the study of rational points
is the enormous variety of methods that interact and contribute to our
understanding of their behaviour: to name but a few, the
Hardy--Littlewood circle method, the geometry of the underlying complex
algebraic varieties, arithmetic and geo\-metry over finite and $p$-adic
fields, harmonic analysis, Manin's use of the Brauer--Grothendieck group
to define a systematic obstruction to the Hasse principle, the theory of
universal torsors of Colliot-Th\'el\`ene--Sansuc, and the analysis of
Shafarevich--Tate groups. It is no exaggeration to say that pioneering
work of Peter Swinnerton-Dyer was an early example of many of these
techniques, and a source of inspiration for others. The contents of this
volume, that we now describe, reflect this vast influence.
\subsection*{Analytic number theory}
The paper by {\bf Enrico Bombieri} and {\bf Paula B. Cohen} ``{\em An
elementary approach to effective diophantine approximation on $\GG_m$}''
concerns approximations of high order roots of algebraic numbers, with
applications to Diophantine approximation in a number field by a finitely
generated multiplicative subgroup. Such results can be obtained from the
theory of linear forms in logarithms, whereas Bombieri's new approach is
based on the Thue--Siegel--Roth theorem. The main improvement comes from
a new zero lemma that is simpler than the lemma of Dyson employed up to
now. The results sharpen Liouville's inequality for $r$th roots of an
algebraic number $a$. More precisely, the authors obtain a lower bound
for the distance $|a^{1/r} -\ga|$, where $\ga$ is an algebraic number,
and $|\cdot|$ a non-Archimedean absolute value.
{\bf Roger Heath-Brown's} paper ``{\em Linear relations amongst sums of
two squares}'' is an inspiring example of what analytic methods can do
for the study of rational points. The main result of the paper is an
asymptotic formula for the number of integral points of prescribed height
on a class of intersections of two quadratic forms in six variables. This
formula accounts for possible failures of weak approximation. The result
is a significant advance in the state of knowledge on density of rational
points, for existing methods (such as the circle method) provide
asymptotic formulas given by the product of local densities. Heath-Brown
determines the additional factor that reflects the failure of weak
approximation --- a conclusion that was hitherto inaccessible. Such a
result should provide a stimulus to establish analogous conclusions for a
broader range of examples. The proof involves descent to an intersection
of quadratic forms, to which analytic methods can be applied. The analysis
here is delicate, and motivated by earlier work of Hooley and Daniel.
\subsection*{Diophantine equations}
{\bf Andrew Bremner's} short note ``{\em A Diophantine system}'' finds
infinitely many nontrivial $\Q$-rational points on the complete
intersection surface given by
\[
x_1^k+x_2^k+x_3^k=y_1^k+y_2^k+y_3^k \quad\hbox{for $k=2,3,4$.}
\]
Trivial solutions to this system, with the second triple a permutation of
the first, are of no interest, but only one nontrivial rational solution
was previously known. The proof is the observation that the hyperplane
section $x_1+x_2+y_1+y_2=0$ gives an elliptic curve of rank 1.
In ``{\em Valeurs d'un polyn\^ome \`a une variable repr\'esent\'ees par
une norme}'', {\bf Jean-Louis Colliot-Th\'el\`ene}, {\bf David Harari}
and {\bf Alexei Skorobogatov} consider the Diophantine equation
$P(t)=N_{K/k}(z)$, where $P(t)$ is a polynomial and $N_{K/k}(z)$ the norm
form defined by a finite field extension $K/k$. The paper builds on
previous work by Heath-Brown and Skorobogatov, who combined the circle
method and descent to prove results on rational solutions of this
equation for $P(t)$ a product of two linear factors and $k=\Q$. It
studies in detail the Brauer group of a smooth and proper model of the
variety given by $P(t)=N_{K/k}(z)$, with $k$ an arbitrary field, and
calculates it explicitly under some additional assumptions. On the other
hand, when $k=\Q$ and $P(t)$ is a product of arbitrary powers of two
linear factors, the Brauer--Manin obstruction is proved to be the only
obstruction to the Hasse principle and to weak approximation. This leads
to some new cases of the Hasse principle.
The consensus among experts seems to be that the failure of the Hasse
principle for rational surfaces can be characterised in terms of the
Brauer--Manin obstruction (this is far from being settled; possibly the
closely related problem for zero-cycles of degree~1 has more chances of
success). Recent work of Skorobogatov shows that this fails for some
bielliptic surfaces; the paper of {\bf Laura Basile and Alexei
Skorobogatov} ``{\em On the Hasse principle for bielliptic surfaces}''
explores this area, providing positive and negative results as testing
ground for a future overall conjecture.
In his contribution ``{\em On the obstructions to the Hasse principle}'',
{\bf Per Salberger} gives a new proof of the main theorem of the descent
theory of Colliot-Th\'el\`ene and Sansuc. Surprisingly, this new approach
avoids an explicit computation of the Poitou--Tate pairing at the crucial
point of the proof, relying instead on standard functoriality properties
of \'etale cohomology. One of the results was obtained independently by
Colliot-Th\'el\`ene and Swinnerton-Dyer, following Salberger's innovative
1988 paper. It is interesting to note that whereas Colliot-Th\'el\`ene
and Swinnerton-Dyer extended Salberger's original method, in the present
paper Salberger uses for the first time Colliot-Th\'el\`ene and Sansuc's
universal torsors to prove results about zero-cycles. This demonstrates
in a striking way that universal torsors are well adapted not only for
rational points, but also for zero-cycles. This approach may eventually
advance our understanding of the following question of
Colliot-Th\'el\`ene: is the Brauer--Manin obstruction to the existence of
a zero-cycle of degree~1 the only obstruction, if we assume the existence
of such cycles everywhere locally?
\subsection*{Shafarevich--Tate groups}
{\bf Neil Dummigan}, {\bf William Stein} and {\bf Mark Watkins'} paper
``{\em Constructing elements in Shafarevich--Tate groups of modular
motives}'' gives a criterion for the existence of nontrivial elements of
certain Shafarevich--Tate groups. Their methods build upon Cremona and
Mazur's notion of ``visibility'', but in the context of motives rather
than abelian varieties. The motives considered are attached to modular
forms on $\Ga_0(N)$ of weight~$>2$. Examples are found in which the
Beilinson--Bloch conjectures imply the existence of nontrivial elements of
these Shafarevich--Tate groups. Modular symbols and Tamagawa numbers are
used to compute nontrivial conjectural lower bounds for the orders of the
Shafarevich--Tate groups of modular motives of low level and weight
$\le12$.
{\bf Tom Fisher's} paper ``{\em A counterexample to a conjecture of
Selmer}'' answers the following question. Let $K$ be a number field
containing a primitive cube root of unity, and $E$ an elliptic curve over
$K$ having complex multiplication by $\sqrt{-3}$. Is the kernel of this
complex multiplication on the Shafarevich--Tate group of $E$ over $K$ of
square order? The answer is positive if $E$ is defined over a subfield
$k\subset K$ such that $[K:k]=2$, $K=k(\sqrt{-3})$, assuming that the
Shafarevich--Tate group of $E$ over $k$ is finite. Examples show that
without this assumption the answer can be negative. These results play an
important r\^ole in the new method for proving the Hasse principle for
pencils of curves of genus 1, first used by Heath-Brown and then artfully
employed by Swinnerton-Dyer in his recent paper on the Hasse principle
for diagonal cubic forms.
In ``{\em On Shafarevich--Tate groups of Fermat jacobians}'', {\bf
William Mc\-Callum} and {\bf Pavlos Tzermias} find all the points on the
Fermat curve of degree 19 with quadratic residue field; these turn out to
be the points previously described by Gross and Rohrlich. The result
about rational points is an application of the following result about the
Shafarevich--Tate groups. For an odd prime $p$, let $F$ be a quotient of
the $p$th Fermat curve by $\mu_p$, and let $J$ be the jacobian of $F$.
Then $J$ has complex multiplication by the ring of integers of the
cyclotomic field $K=\Q(\zeta_p)$. The authors prove that in certain cases
there are nontrivial elements of order exactly $(1-\zeta_p)^3$ in the
Shafarevich--Tate group of $J$ over $K$.
\subsection*{Zagier's conjectures} In his paper ``{\em Kronecker double
series and the dilogarithm}'', {\bf Andrey Levin} gives an explicit
expression for the value of a certain Kronecker double series at a point
of complex multiplication as a sum of dilogarithms whose arguments are
values of some modular unit of higher level. This result can be
interpreted in the spirit of Zagier's conjecture. The special value of
the Kronecker double series is equal to the value of the partial zeta
function of an ideal class for an order in an imaginary quadratic field.
The values of the modular unit mentioned above belong to the ray class
field corresponding to this order. This gives an explicit formula for the
value of a partial zeta function at $s=2$ as a combination of
dilogarithms of algebraic numbers.
\subsection*{Complex algebraic geometry}
In ``{\em Cascades of projections from log del Pezzo surfaces}'', {\bf
Miles Reid} and {\bf Kaori Suzuki} weave a fantasy around the fascinating
old algebraic geo\-metric construction (del Pezzo, 1890) of the blowup of
$\PP^2$ in $d\le8$ general points and its anti\-canonical embedding.
Some natural families of del Pezzo surfaces with quotient singularities
are organized in `cascades' of projections, similar to the way that the
classic nonsingular del Pezzo surfaces are obtained by successive
projections from the del Pezzo surface of degree~9 in $\PP^9$ (in other
words, $\PP^2$ in its anticanonical embedding). Apart from their geometric
beauty, these examples illustrate the technique of `unprojection', a good
working substitute for an as yet missing structure theory of Gorenstein
rings of small codimension, and a possible tool to eventually construct
one. The authors also sketch a program for the study of singular Fano
3-folds of index $\ge2$ according to their Hilbert series, modelled on the
2-dimensional case.
{\bf Gregory Sankaran} studies the bilevel structures on abelian surfaces
first introduced by Mukai. Given a $(1,t)$-polarized abelian surface $A$,
a bilevel structure on $A$ consists of a (canonical) level structure on
$A$ and a (canonical) level structure on the dual variety $\widehat{A}$,
which also carries a natural $(1,t)$-polarization. The corresponding
moduli problem gives rise to a Siegel modular threefold $\sA_t^{\bil}$.
Mukai proved the rationality of these moduli spaces for $t=2,3$ and
$5$. He also related them to the symmetry groups of the Platonic solids
and to projective threefolds with many nodes. In ``{\em Abelian surfaces
with odd bilevel structure}'' Sankaran proves that $\sA_t^{\bil}$ is of
general type for odd $t\ge17$. A result of Borisov says that
$\sA_t^{\bil}$ is of general type for all but finitely many $t$.
Borisov's method, however, gives no explicit bound.
\signoff{Imperial College, Mon 24th Feb 2003}
\vskip0.4cm
\medskip
\noindent
Bryan Birch, \\
Mathematical Institute, \\
24-29 St Giles', \\
Oxford, OX1 3LB, UK \\
e-mail: birch@maths.ox.ac.uk
\medskip
\noindent
Jean-Louis Colliot-Th\'el\`ene, \\
C.N.R.S., U.M.R. 8628, \\
Math\'ematiques, B\^atiment 425, \\
Universit\'e de Paris-Sud, \\
F-91405 Orsay, France \\
e-mail: colliot@math.u-psud.fr
\medskip
\noindent
Miles Reid,\\
Math Inst., Univ. of Warwick,\\
Coventry CV4 7AL, England\\
e-mail: miles@maths.warwick.ac.uk \\
web: www.maths.warwick.ac.uk/$\!\sim$miles
\medskip
\noindent
G.K. Sankaran, \\
Department of Mathematical Sciences, \\
University of Bath, \\
Bath BA2 7AY, England \\
e-mail: gks@maths.bath.ac.uk
\medskip
\noindent
Alexei Skorobogatov, \\
Department of Mathematics, \\
Imperial College London, \\
South Kensington Campus, \\
London SW7 2AZ, England \\
e-mail: a.skorobogatov@ic.ac.uk
\cleardoublepage
\def\refname{Peter Swinnerton-Dyer's mathematical papers to date}
\begin{thebibliography}{99}
\addcontentsline{toc}{chapter}{Peter Swinnerton-Dyer's mathematical
papers}
\markright{\hfill Bibliography \qquad}
\bibitem{1} P. S. Dyer, A solution of $A^4+B^4=C^4+D^4$, J. London Math.
Soc. {\bf18} (1943) 2--4
\bibitem{a} H. P. F. Swinnerton-Dyer, On a conjecture of Hardy and
Littlewood, J. London Math. Soc. {\bf27} (1952) 16--21
\bibitem{b} H. P. F. Swinnerton-Dyer, A solution of
$A^5+B^5+C^5=D^5+E^5+F^5$, Proc. Cambridge Phil. Soc. {\bf48} (1952)
516--518
\bibitem{c} H. P. F. Swinnerton-Dyer, Extremal lattices of convex bodies,
Proc. Cambridge Phil. Soc. {\bf49} (1953) 161--162
\bibitem{barnes} E. S. Barnes and H. P. F. Swinnerton-Dyer, The
inhomogeneous minima of binary quadratic forms. I, Acta Math. {\bf87}
(1952) 259--323. II, same J. {\bf88} (1952) 279--316. III, same J.
{\bf92} (1954) 199--234
\bibitem{d} H. P. F. Swinnerton-Dyer, Inhomogeneous lattices, Proc.
Cambridge Phil. Soc. {\bf50} (1954) 20--25
\bibitem{e} A. O. L. Atkin and P. Swinnerton-Dyer, Some properties of
partitions, Proc. London Math. Soc. (3) {\bf4} (1954) 84--106
\bibitem{cassels} J. W. S. Cassels and H. P. F. Swinnerton-Dyer, On the
product of three homogeneous linear forms and the indefinite ternary
quadratic forms, Phil. Trans. Roy. Soc. London. Ser. A. {\bf248} (1955)
73--96
\bibitem{g} H. Davenport and H. P. F. Swinnerton-Dyer, Products of
inhomogeneous linear forms, Proc. London Math. Soc. (3) {\bf5} (1955)
474--499
\bibitem{h} B. J. Birch and H. P. F. Swinnerton-Dyer, On the inhomogeneous
minimum of the product of $n$ linear forms, Mathematika {\bf3} (1956)
25--39
\bibitem{i} H. P. F. Swinnerton-Dyer, On an extremal problem, Proc. London
Math. Soc. (3) {\bf7} (1957) 568--583
\bibitem{j} K. Rogers and H. P. F. Swinnerton-Dyer, The geometry of
numbers over algebraic number fields, Trans. Amer. Math. Soc. {\bf88}
(1958) 227--242
\bibitem{k} B. J. Birch and H. P. F. Swinnerton-Dyer, Note on a problem of
Chowla, Acta Arith. {\bf5} (1959) 417--423
\bibitem{l} D. W. Barron and H. P. F. Swinnerton-Dyer, Solution of
simultaneous linear equations using a magnetic-tape store, Comput. J.
{\bf3} (1960/1961) 28--33
\bibitem{2sp} H. P. F. Swinnerton-Dyer, Two special cubic surfaces,
Mathematika {\bf9} (1962) 54--56
\bibitem{m} H. T. Croft and H. P. F. Swinnerton-Dyer, On the Steinhaus
billiard table problem, Proc. Cambridge Phil. Soc. {\bf59} (1963) 37--41
\bibitem{bsd1} B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic
curves. I, J. reine angew. Math. {\bf212} (1963) 7--25
\bibitem{bsd2} B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic
curves. II, J. reine angew. Math. {\bf218} (1965) 79--108
\bibitem{2q} H. P. F. Swinnerton-Dyer, Rational zeros of two quadratic
forms, Acta Arith. {\bf9} (1964) 261--270
\bibitem{formstab} H. P. F. Swinnerton-Dyer, On the formal stability of
the solar system, Proc. London Math. Soc. (3) {\bf14a} (1965) 265--287
\bibitem{n} H. P. F. Swinnerton-Dyer, The zeta function of a cubic surface
over a finite field, Proc. Cambridge Phil. Soc. {\bf63} (1967) 55--71
\bibitem{o} E. Bombieri and H. P. F. Swinnerton-Dyer, On the local zeta
function of a cubic threefold, Ann. Scuola Norm. Sup. Pisa (3) {\bf21}
(1967) 1--29
\bibitem{p} H. P. F. Swinnerton-Dyer, An application of computing to class
field theory, in Algebraic Number Theory (Brighton 1965), Thompson,
Washington, D.C. (1967), pp.~280--291
\bibitem{q} H. P. F. Swinnerton-Dyer, $A\sp{4}+B\sp{4}=C\sp{4}+D\sp{4}$
revisited, J. London Math. Soc. {\bf43} (1968) 149--151
\bibitem{r} P. Swinnerton-Dyer, The conjectures of Birch and
Swinnerton-Dyer, and of Tate, in Proc. Conf. Local Fields (Driebergen
1966), Springer, Berlin (1967), pp.~132--157
\bibitem{s} P. Swinnerton-Dyer, The use of computers in the theory of
numbers, in Proc. Sympos. Appl. Math., Vol. XIX, Amer. Math. Soc.,
Providence, R.I. (1967), pp.~111--116
\bibitem{t} F. K. C. Rankin and H. P. F. Swinnerton-Dyer, On the zeros of
Eisenstein series, Bull. London Math. Soc. {\bf2} (1970) 169--170
\bibitem{u} H. P. F. Swinnerton-Dyer, On a problem of Littlewood
concerning Riccati's equation, Proc. Cambridge Phil. Soc. {\bf65} (1969)
651--662
\bibitem{v} H. P. F. Swinnerton-Dyer, The birationality of cubic surfaces
over a given field, Michigan Math. J. {\bf17} (1970) 289--295
\bibitem{w} H. P. F. Swinnerton-Dyer, On the product of three homogeneous
linear forms, Acta Arith. {\bf18} (1971) 371--385
\bibitem{x} H. P. F. Swinnerton-Dyer, The products of three and of four
linear forms, in Computers in number theory (Oxford 1969), A. O. L. Atkin
and B. J. Birch (eds.), Academic Press, London-New York (1971),
pp.~231--236
\bibitem{atkin} A. O. L. Atkin and H. P. F. Swinnerton-Dyer, Modular
forms on non\-congruence subgroups, in Combinatorics (UCLA 1968), Proc.
Sympos. Pure Math., Vol. XIX, Amer. Math. Soc., Providence, R.I. (1971),
pp.~1--25
\bibitem{y} H. P. F. Swinnerton-Dyer, The number of lattice points on a
convex curve, J. Number Theory {\bf6} (1974) 128--135
\bibitem{stonybrook} H. P. F. Swinnerton-Dyer, Applications of algebraic
geometry to number theory, in Number Theory (Stony Brook 1969), Proc.
Sympos. Pure Math., Vol. XX, Amer. Math. Soc., Providence,
R.I. (1971), pp.~1--52
\bibitem{z} H. P. F. Swinnerton-Dyer, Applications of computers to the
geometry of numbers, in Computers in algebra and number theory (New York
1970), Proc. Sympos. Appl. Math., SIAM-AMS Proc., Vol. IV, Amer. Math.
Soc., Providence, R.I. (1971), pp.~55--62
\bibitem{aa} H. P. F. Swinnerton-Dyer, An enumeration of all varieties of
degree $4$, Amer. J. Math. {\bf95} (1973) 403--418
\bibitem{ab} B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil curves,
Invent. Math. {\bf25} (1974) 1--61
\bibitem{ac} Mary L. Cartwright and H. P. F. Swinnerton-Dyer, Boundedness
theorems for some second order differential equations. I, Collection of
articles dedicated to the memory of Tadeusz Wa\.zewski, III, Ann. Polon.
Math. {\bf29} (1974) 233--258
\bibitem{ad} H. P. F. Swinnerton-Dyer, Almost-conservative second-order
differential equations, Math. Proc. Cambridge Phil. Soc. {\bf77} (1975)
159--169
\bibitem{ae} H. P. F. Swinnerton-Dyer, Analytic theory of abelian
varieties, London Mathematical Society Lecture Note Series, No. 14.
Cambridge University Press, London-New York, 1974. viii+90 pp.
\bibitem{af} H. P. F. Swinnerton-Dyer, An outline of Hodge theory, in
Algebraic geometry (Oslo 1970), Wolters-Noordhoff, Groningen (1972),
pp.~277--286
\bibitem{ag} H. P. F. Swinnerton-Dyer, Rational points on del Pezzo
surfaces of degree $5$, in Algebraic geometry (Oslo 1970),
Wolters-Noordhoff, Groningen (1972), pp.~287--290
\bibitem{antw3} H. P. F. Swinnerton-Dyer, On $l$-adic representations and
congruences for coefficients of modular forms, in Modular functions of one
variable, III (Antwerp 1972), Lecture Notes in Math., Vol. 350, Springer,
Berlin, 1973, pp.~1--55. Correction in \cite{antw4}, p. 149
\bibitem{antw4} H. P. F. Swinnerton-Dyer and B. J. Birch, Elliptic curves
and modular functions, in Modular functions of one variable, IV (Antwerp
1972), Lecture Notes in Math., Vol. 476, Springer, Berlin (1975),
pp.~2--32
\bibitem{ah} H. P. F. Swinnerton-Dyer, N. M. Stephens, James Davenport,
J. V\'elu, F. B. Coghlan, A. O. L. Atkin and D. J. Tingley, Numerical
tables on elliptic curves, in Modular functions of one variable, IV
(Antwerp 1972), Lecture Notes in Math., Vol. 476, Springer, Berlin
(1975), pp.~74--144
\bibitem{ai} M. Artin and H. P. F. Swinnerton-Dyer, The Shafarevich--Tate
conjecture for pencils of elliptic curves on K3 surfaces, Invent. Math.
{\bf20} (1973) 249--266
\bibitem{aj} B. J. Birch and H. P. F. Swinnerton-Dyer, The Hasse problem
for rational surfaces, in Collection of articles dedicated to Helmut Hasse
on his seventy-fifth birthday. III, J. reine angew. Math. {\bf274/275}
(1975) 164--174
\bibitem{ak} Peter Swinnerton-Dyer, The Hopf bifurcation theorem in three
dimensions, Math. Proc. Cambridge Phil. Soc. {\bf82} (1977) 469--483
\bibitem{al} M. L. Cartwright and H. P. F. Swinnerton-Dyer, The
boundedness of solutions of systems of differential equations, in
Differential equations (Keszthely 1974), Colloq. Math. Soc. J\'anos
Bolyai, Vol. 15, North-Holland, Amsterdam (1977), pp.~121--130
\bibitem{am} H. P. F. Swinnerton-Dyer, Arithmetic groups, in Discrete
groups and auto\-morphic functions (Cambridge, 1975), Academic Press,
London (1977), pp.~377--401
\bibitem{an} H. P. F. Swinnerton-Dyer, On $l$-adic representations and
congruences for coefficients of modular forms. II, in Modular functions of
one variable, V (Bonn 1976), Lecture Notes in Math., Vol. 601, Springer,
Berlin (1977), pp.~63--90
\bibitem{ao} Peter Swinnerton-Dyer, Small parameter theory: the method of
averaging, Proc. London Math. Soc. (3) {\bf34} (1977) 385--420
\bibitem{ap} Peter Swinnerton-Dyer, The Royal Society and its impact on the
intellectual and cultural life of Britain, Jbuch. Heidelberger Akad. Wiss.
1979 (1980) 136--143
\bibitem{aq} H. P. F. Swinnerton-Dyer, The method of averaging for some
almost-conservative differential equations, J. London Math. Soc. (2)
{\bf22} (1980) 534--542
\bibitem{ar} M. L. Kartra\u\i t and H. P. F. Svinnerton-Da\u\i er,
Boundedness theorems for some second-order differential equations. IV,
Differentsial'nye Uravneniya {\bf14} (1978) 1941--1979 and 2106 = Differ.
Equations {\bf14} (1979) 1378--1406
\bibitem{as} H. P. F. Swinnerton-Dyer, Universal equivalence for cubic
surfaces over finite and local fields, in Severi centenary symposium (Rome
1979), Symposia Mathematica, Vol. XXIV, Academic Press, London-New York
(1981), pp.~111--143
\bibitem{at} Jean-Louis Colliot-Th\'el\`ene, Jean-Jacques Sansuc and Peter
Swin\-nerton-Dyer, Intersections de deux quadriques et surfaces de
Ch\^atelet, C. R. Acad. Sci. Paris S\'er. I Math. {\bf298} (1984) 377--380
\bibitem{au} H. P. F. Swinnerton-Dyer, The basic Lorenz list and Sparrow's
conjecture~A, J. London Math. Soc. (2) {\bf29} (1984) 509--520
\bibitem{av} Arnaud Beauville, Jean-Louis Colliot-Th\'el\`ene,
Jean-Jacques Sansuc et Peter Swinnerton-Dyer, Vari\'et\'es stablement
rationnelles non ration\-nelles, Ann. of Math. (2) {\bf121} (1985)
283--318
\bibitem{aw} H. P. F. Swinnerton-Dyer, The field of definition of the
N\'eron-Severi group, in Studies in pure mathematics in memory of Paul
Tur\'an, Birkh\"auser, Basel (1983), pp.~719--731
\bibitem{ax} Jean-Louis Colliot-Th\'el\`ene, Jean-Jacques Sansuc and Peter
Swin\-nerton-Dyer, Intersections of two quadrics and Ch\^atelet surfaces.
I, J. reine angew. Math. {\bf373} (1987) 37--107. II, same J., {\bf374}
(1987) 72--168
\bibitem{az} H. P. F. Swinnerton-Dyer, Congruence properties of $\tau(n)$,
in Ramanujan revisited (Urbana-Champaign 1987), Academic Press, Boston
(1988), pp.~289--311
\bibitem{ba} R. G. E. Pinch and H. P. F. Swinnerton-Dyer, Arithmetic of
diagonal quartic surfaces. I, in $L$-functions and arithmetic (Durham
1989), London Math. Soc. Lecture Note Ser., 153, Cambridge Univ. Press,
Cambridge (1991), pp.~317--338
\bibitem{bb} Peter Swinnerton-Dyer, The Brauer group of cubic surfaces,
Math. Proc. Cambridge Phil. Soc. {\bf113} (1993) 449--460
\bibitem{bc} Peter Swinnerton-Dyer, Counting rational points on cubic
surfaces, in Classification of algebraic varieties (L'Aquila 1992),
Contemp. Math., 162, Amer. Math. Soc., Providence, RI (1994), pp.~371--379
\bibitem{bd} Jean-Louis Colliot-Th\'el\`ene and Peter Swinnerton-Dyer,
Hasse principle and weak approximation for pencils of Severi-Brauer and
similar varieties, J. reine angew. Math. {\bf453} (1994) 49--112
\bibitem{be} Peter Swinnerton-Dyer, Rational points on pencils of conics
and on pencils of quadrics, J. London Math. Soc. (2) {\bf50} (1994)
231--242
\bibitem{bf} H. P. F. Swinnerton-Dyer and C. T. Sparrow, The Falkner--Skan
equation. I, The creation of strange invariant sets, J. Differential
Equations {\bf119} (1995) 336--394
\bibitem{bg} Peter Swinnerton-Dyer, Rational points on certain
intersections of two quadrics, in Abelian varieties (Egloffstein 1993),
de Gruyter, Berlin (1995), pp.~273--292
\bibitem{bh} J.-L. Colliot-Th\'el\`ene, A. N. Skorobogatov and Peter
Swinnerton-Dyer, Double fibres and double covers: paucity of rational
points, Acta Arith. {\bf79} (1997) 113--135
\bibitem{bi} Peter Swinnerton-Dyer, Diophantine equations: the geometric
approach, Jahresber. deutsch. Math.-Verein. {\bf98} (1996) 146--164
\bibitem{bj} Peter Swinnerton-Dyer, Brauer--Manin obstructions on some Del
Pezzo surfaces, Math. Proc. Cambridge Phil. Soc. {\bf125} (1999) 193--198
\bibitem{bk} J.-L. Colliot-Th\'el\`ene, A. N. Skorobogatov and Peter
Swinnerton-Dyer, Rational points and zero-cycles on fibred varieties:
Schinzel's hypothesis and Salberger's device, J. reine angew. Math.
{\bf495} (1998) 1--28
\bibitem{bl} J.-L. Colliot-Th\'el\`ene, A. N. Skorobogatov and Peter
Swinnerton-Dyer, Hasse principle for pencils of curves of genus one whose
Jacobians have rational $2$-division points, Invent. Math. {\bf134} (1998)
579--650
\bibitem{bm} Peter Swinnerton-Dyer, A stability theorem for unsymmetric
Li\'enard equations, Dynam. Stability Systems {\bf14} (1999) 93--94
\bibitem{bn} Peter Swinnerton-Dyer, Some applications of Schinzel's
hypothesis to Diophantine equations, in Number theory in progress, Vol. 1
(Zakopane-Ko\'scielisko 1997), de Gruyter, Berlin (1999), pp.~503--530
\bibitem{bo} John B. Slater and Peter Swinnerton-Dyer, Counting points on
cubic surfaces. I, in Nombre et r\'epartition de points de hauteur
born\'ee (Paris 1996), Ast\'erisque No. 251 (1998), pp.~1--12
\bibitem{bp} D. F. Coray, D. J. Lewis, N. I. Shepherd-Barron and Peter
Swinnerton-Dyer, Cubic threefolds with six double points, in Number theory
in progress, Vol. 1 (Zakopane-Ko\'scielisko 1997), de Gruyter, Berlin
(1999), pp.~63--74
\bibitem{bq} Peter Swinnerton-Dyer, Rational points on some pencils of
conics with 6 singular fibres, Ann. Fac. Sci. Toulouse Math. (6) {\bf8}
(1999) 331--341
\bibitem{br} Peter Swinnerton-Dyer, Arithmetic of diagonal quartic
surfaces. II, Proc. London Math. Soc. (3) {\bf80} (2000) 513--544, and
Corrigenda, same J. {\bf85} (2002) 564
\bibitem{bs} Peter Swinnerton-Dyer, A note on Liapunov's method, Dyn.
Stab. Syst. {\bf15} (2000) 3--10
\bibitem{bt} H. P. F. Swinnerton-Dyer, A brief guide to algebraic number
theory, London Mathematical Society Student Texts, 50. Cambridge
University Press, Cambridge, 2001
\bibitem{bu} Peter Swinnerton-Dyer, Bounds for trajectories of the Lorenz
equations: an illustration of how to choose Liapunov functions, Phys.
Lett. A {\bf281} (2001) 161--167
\bibitem{bv} A. O. Bender and Peter Swinnerton-Dyer, Solubility of certain
pencils of curves of genus 1, and of the intersection of two quadrics in
${\bf P}^4$, Proc. London Math. Soc. (3) {\bf83} (2001) 299--329
\bibitem{bw} Peter Swinnerton-Dyer, The solubility of diagonal cubic
surfaces, Ann. Sci. \'Ecole Norm. Sup. (4) {\bf34} (2001) 891--912
\bibitem{bx} Peter Swinnerton-Dyer, The invariant algebraic surfaces of
the Lorenz system, Math. Proc. Cambridge Phil. Soc. {\bf132} (2002)
385--393
\bibitem{by} Peter Swinnerton-Dyer, Weak approximation and $R$-equivalence
on cubic surfaces, in Rational points on algebraic varieties, Progr.
Math., 199, Birkh\"auser, Basel (2001), pp.~357--404
\bibitem{bz} C. Sparrow and H. P. F. Swinnerton-Dyer, The Falkner--Skan
equation. II, Dynamics and the bifurcations of $P$- and $Q$-orbits, J.
Differential Equations {\bf183} (2002) 1--55
\end{thebibliography}
\def\refname{Other references}
\begin{thebibliography}{99}
\makeatletter
\setcounter{\@listctr}{88}
\makeatother
\bibitem{bjb} B. J. Birch, Elliptic curves over $\Q$: A progress report,
in Number Theory (Stony Brook 1969), Proc. Sympos. Pure Math., Vol. XX,
Amer. Math. Soc., Providence, R.I. (1971), pp.~396--400
\bibitem{ma2} Yu. I. Manin, Le groupe de Brauer--Grothendieck en
g\'eom\'etrie diophantienne, in Actes du Congr\`es International des
Math\'ematiciens (Nice 1970), Gauthier-Villars, Paris (1971), Tome 1,
pp.~401--411
\bibitem{ma} Yu. I. Manin, Parabolic points and zeta functions of
modular curves, Izv. Akad. Nauk SSSR Ser. Mat. {\bf36} (1972) 19--66
\bibitem{ma3} Yu. I. Manin, Cubic forms: algebra, geometry, arithmetic,
North-Holland 1974
\bibitem{st} N. M. Stephens, The diophantine equation $X^3+Y^3=DZ^3$ and
the conjectures of Birch and Swinnerton-Dyer, J. reine angew. Math.
{\bf231} (1968) 121--162
\bibitem{weil} Andr\'e Weil, Letter to Peter Swinnerton-Dyer dated
24/7/1965
\end{thebibliography}
\end{document}