I would like to think that Enrico Bombieri and Walter Gubler intended for the title of their new book Heights in Diophantine Geometry to be a pun: the book deals primarily with the study of certain functions which are known as heights as well as with the great achievements that they have allowed mathematicians to reach. Diophantine geometry is the study of integer and rational solutions to algebraic equations, and the study of the questions can be thought of as lying at the crossroads of algebraic geometry and number theory.
Bombieri and Gubler have written an excellent introduction to some exciting mathematics, assuming very little background and containing appendices summarizing Everything You Ever Needed To Know About Algebraic Geometry, Ramification Theory, and Minkowski's Geometry Of Numbers, so that the book should be accessible to just about any mathematician. It is also written with an excellent combination of clarity and rigor, with the authors highlighting which parts can be skipped on a first reading and which parts are particularly important for later material. The book also contains a glossary of notation, a good index, and a nice bibliography collecting many of the primary sources in this field.
The authors begin by introducing the concept of the heights of points on curves, the most elementary versions of which can be thought of as generalizations of absolute values. They then go on to prove some of the properties of heights and show how they can be used to study the diophantine geometry of subvarieties of split tori. Some of the applications of these results include the result of Amoroso and Dvornicich giving an absolute lower bound for the height of algebraic numbers other than roots of unity in abelian extensions of the rational numbers as well as discussing the question of the unit equation, which asks how often the numbers x and 1–x are both units and the pair (x,1–x) lies in a finitely generated subgroup of K2 for various number fields K. The book then goes on to discuss Roth's Theorem, whose easiest form bounds the number of rational numbers y = p/q such that |α - p/q| < 1/q2+ε for any algebraic number α and any ε>0.
The next section of the book discusses heights related to abelian varieties. Starting from the very definition of abelian varieties as objects that have both geometric and algebraic structures, the authors introduce the ideas behind elliptic curves, Jacobians of curves, the theorems of the square and cube, and the idea of heights on abelian varieties such as Neron-Tate heights. This allows the authors to devote a full chapter each to the proofs of the Mordell-Weil Theorem (which says that the group of rational points on an abelian variety is finitely generated) and Falting's Theorem (which says that curves of genus two or more defined over number fields have only a finite number of rational points), as well as a chapter on the abc-conjecture. The last two chapters give introductions to Nevanlinna theory as well as the Vojta conjectures. These final chapters show some of the beautiful and exciting mathematics that can be found by starting with questions of rational points on curves, and truly what heights they can reach.
Darren Glass is assistant professor of mathematics at Gettysburg College. He can be reached at firstname.lastname@example.org.