New Mathematical Monographs 7. Cambridge: Cambridge University Press (ISBN 0-521-83360-4/hbk). xii, 320 p. $ 99.00; £ 55.00; $ 79.00/e-book (2007).
Hilbert asked (in his so-called “tenth problem”) for an effective procedure to decide whether or not a given Diophantine equation has a solution in integers. We now understand “effective” as “computable on a Turing machine”. Since the spectacular negative solution of Hilbert’s Tenth Problem by Davis, Putnam, Matijasevich and Robinson, much interesting research has happened in this field at the interface of logic and number theory, most of which is concerned with the analogous problem for rings of integers in number fields, for the field of rational numbers, and more general global fields, including function fields. The results and proofs have required more and more advanced number theory and arithmetic geometry, in recent years centered around the arithmetic of elliptic curves. This book is the first research monograph that is completely devoted to these new techniques. The author should be congratulated on her effort to write a notationally and conceptually consistent book with results and methods that were previously scattered throughout the literature. The main topics discussed are the analogues of Hilbert’s Tenth Problem for rings of $S$-integers in number fields and function fields (both with $S$ finite and infinite), for global function fields, and relations to Mazur’s conjecture. The book should be on the desk of every researcher in the field.