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Preface to the Second Edition

The second edition builds on the first in several ways. There are three new chapters
which survey recent directions and extensions of the theory, and there are two new
appendices. Then there are numerous additions to the original text. For example, a
very elementary addition is another parametrization which the author learned from
2
3
Don Zagier y = x − 3αx + 2β of the basic cubic equation. This parametrization
is useful for a detailed description of elliptic curves over the real numbers.
The three new chapters are Chapters 18, 19, and 20. Chapter 18, on Fermat’s Last
Theorem, is designed to point out which material in the earlier chapters is relevant
as background for reading Wiles’ paper on the subject together with further devel-
opments by Taylor and Diamond. The statement which we call the modular curve
conjecture has a long history associated with Shimura, Taniyama, and Weil over the
last fifty years. Its relation to Fermat, starting with the clever observation of Frey
ending in the complete proof by Ribet with many contributions of Serre, was already
mentioned in the first edition. The proof for a broad class of curves by Wiles was suf-
ficient to establish Fermat’s last theorem. Chapter 18 is an introduction to the papers
on the modular curve conjecture and some indication of the proof.
Chapter 19 is an introduction to K3 surfaces and the higher dimensional Calabi–
Yau manifolds. One of the motivations for producing the second edition was the
utility of the first edition for people considering examples of fibrings of three dimen-
sional Calabi–Yau varieties. Abelian varieties form one class of generalizations of
elliptic curves to higher dimensions, and K3 surfaces and general Calabi–Yau mani-
folds constitute a second class.
Chapter 20 is an extension of earlier material on families of elliptic curves where
the family itself is considered as a higher dimensional variety fibered by elliptic
curves. The first two cases are one dimensional parameter spaces where the family is
two dimensional, hence a surface two dimensional surface parameter spaces where
the family is three dimensional. There is the question of, given a surface or a three
dimensional variety, does it admit a fibration by elliptic curves with a finite number
of exceptional singular fibres. This question can be taken as the point of departure
for the Enriques classification of surfaces.