Diophantine geometry is (in this reviewer's highly biased opinion) one of the most beautiful and exciting areas of mathematics. At its heart is a seemingly simple question: take a polynomial in several variables and ask if there are any solutions to this polynomial in integers. From there, one starts to ask how many solutions there are and if there are infinitely many, or if any exist over the rational numbers or a given number field or finite< /FONT > field.
Not surprisingly, as the questions get more difficult and technical, so do the tools needed to solve them. One tool which people working in this area have found quite fruitful in recent times is the idea of a logarithmic form, which is a finite sum of terms of the form Λ = α + log(β) where α and β are algebraic numbers. The typical question one might ask is whether we can bound |Λ| based on the height of α and β.
Two of the pioneers in the use of logarithmic forms to solve diophantine questions are Alan Baker and Gisbert Wüstholz , and they have written a book entitled Logarithmic Forms and Diophantine Geometry which was recently published in Cambridge University Press's series of "New Mathematical Monographs."
The book is a slim volume which assumes quite a bit of knowledge about algebraic geometry, algebraic number theory, and Lie groups, and which also forces the reader to place faith in the authors as they dive right into the mathematics and rarely look forward to preview where they are going or backward to summarize what has been covered.
Quite a bit of material is covered in less than 200 pages: the opening chapter summarizes many results in transcendental number theory, using both classical and modern techniques to prove a variety of results that will be familiar to most number theorists. From there they define logarithmic forms and show several of their properties and relationships with questions of transcendence before moving onto more general Diophantine questions, properties of Lie groups, and multiplicity estimates.
The apex of the book comes with a chapter on the analytic subgroup theorem, which says (roughly) that an analytic subgroup of an algebraic group variety has only trivial algebraic points. This theorem has many nice applications, giving new proofs as well as strengthenings of classical results such as Lindemann's theorem and Baker's Theorem in a more general setting. In this chapter, Baker and Wustholz give a full proof and motivation of the analytic subgroup theorem, as well as explaining these applications in depth. Two final chapters consider quantitative versions of the theory of logarithmic forms which had been dealt with in a qualitative manner through most of the text, as well as considering questions of effectiveness in Diophantine geometry.
In order to cover this much ground, the authors feel no hesitation to refer the reader to other sources for details of some proofs, and as mentioned before they do not give strong motivations for most of the material. Because of this, the book is very concise and to the point, which may be a pro or a con depending on your needs and interests. However, the authors do a nice job of explaining the technical material, and for the right audience this is certainly the right book.
Darren Glass is an Assistant Professor of Mathematics at Gettysburg College. He can be reached at firstname.lastname@example.org.
Preface. 1. Transcendence origins; 2. Logarithmic forms; 3. Diophantine problems; 4. Commutative algebraic groups; 5. Multiplicity estimates; 6. The analytic subgroup theorem; 7. The quantitative theory; 8. Further aspects of Diophantine geometry; Bibliography; Index.