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Function Field Arithmetic

World Scientific
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Quite early in the development of the theory of algebraic numbers, mathematicians noticed that there were deep formal analogies between that theory and the theory of algebraic functions (with complex coefficients, at first) of one variable. Dedekind and Weber, for example, wrote a book dedicated to pursuing exactly this idea, recasting the theory of algebraic functions in entirely algebraic terms, highlighting the similarity with the theory of algebraic number fields.

Early in the 20th century, it became clear that this analogy is especially strong when one studies function fields in finite characteristic of transcendence degree one, that is, finite extensions of the field Fp(t) of rational functions with coefficients in the field with p elements. These are the "Function Fields" with which this book deals.

The analogy can be, and has been, read in several ways. In a famous letter to his sister, André Weil highlighted the idea that function fields might be easier than number fields. For example, a natural analogue of the Riemann Hypothesis can be proved in the function field case. Similarly, the cyclotomic fields that are obtained by adjoining roots of unity in the number field case can be thought of as analogous to the fields we get by enlarging the field of constants from Fp to Fpr, since every element of a finite field is a root of unity. This analogy was the seed of "Iwasawa Theory."

Going the other way, however, can be even more interesting. We can try to study, in the context of function fields, structures that are analogous to certain interesting structures in the number field case. This was the motivation for the development of "Drinfeld Modules", which (with their generalizations) are the main topic of Thakur's book. Drinfeld Modules are classified in part by their rank. If the rank is one, they are a kind of characteristic p version of the multiplicative group, leading to a whole new notion, for example, of what a "root of unity" is. In the case of rank two, they are a little like elliptic curves. In higher rank they are stranger, with no obvious number field analogue. The result is a theory that is rich, complex, and fascinating.

Dinesh Thakur is one of the leaders in the study of Drinfeld Modules and related themes. He describes the book as "just a tour of some topics I enjoyed learning and working on", which makes it seem lighter and less thorough than it actually is, but does give a little bit of the flavor. It is dense with mathematics, but there is also motivation and discussion. The overall feeling is that of a very helpful survey of a very interesting field.

Unfortunately, the publisher has not provided the author with a good copy-editor. There is far too much fractured English, most often things that could easily have been fixed with a little time and a blue pencil. This is particularly the case in the Preface; as things get more technical, the very formal nature of the language helps minimize the problem. This is a real pity. The publisher has done the author a disservice that may limit his readership, or at least reduce the readers' enjoyment of a very nice monograph.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College. His main research interests are in history of mathematics and number theory.

Date Received: 
Saturday, January 1, 2005
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Dinesh S. Thakur
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Fernando Q. Gouvêa
* Number Fields and Function Fields
* Drinfeld Modules
* Explicit Class Field Theory
* Gauss Sums and Gamma Functions
* Zeta Functions
* Higher Rank Theory
* Higher Dimensions and Geometric Tools
* Applications to Gauss Sums, Gamma and Zeta Values
* Diophantine Approximation
* Transcendence Results
* Automata and Algebraicity: Applications

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Sunday, October 16, 2005