Capacity theory on algebraic curves.

*(English)*Zbl 0679.14012
Lecture Notes in Mathematics, 1378. Berlin etc.: Springer-Verlag. 437 p. DM 69.00 (1989).

In the theory of functions of one complex variable, capacity is a measure of size for compact sets. This concept of capacity plays an essential role in the study of harmonic functions, equilibrium distribution, and general potential theory within the theory of one complex variable. On the other hand, it provides significant applications to algebraic number theory, too. By a classical theorem of M. Fekete and G. Szegö [cf. Math. Z. 63, 158-172 (1955; Zbl 0066.270)], the distribution of complete conjugate sets of algebraic integers with respect to a compact set in the complex plane can be measured by the logarithmic capacity of such a set. - The naturally arising question of whether there is an analogous theorem for arbitrary number fields (rather than \({\mathbb{Q}})\) and more general sets at the finite primes has been partially answered by D. Cantor (1967, 1980) who defined a generalized capacity measure for adelic sets in the projective line, and proved, together with P. Roquette in 1984, a generalization of the Fekete-Szegö theorem, and a general existence theorem on rational points in affine varieties defined over arbitrary number fields.

In the present monograph, the author now extends capacity theory to arbitrary curves (of any genus) with an arbitrary global field as ground field. This pretentious enterprise requires to lay fairly new foundations, as on curves of higher genus (instead of \({\mathbb{P}}^ 1)\) one can only lean upon local methods. The author’s far-reaching work to found a generalized capacity theory for algebraic curves and to apply it to arithmetical problems is based upon an ingeniuos combination of the classical methods, on the one hand, and several essentially new ingredients, such as local parametizations by power series, a certain global maximum modulus principle, an existence theorem for algebraic functions with near-prescribed divisors, the Deligne-Mumford theorem on stable reduction, and Néron’s canonical local height pairing on curves, on the other hand.

After a very enlightening and profound introduction to the classical background, D. Cantor’s pioneering work, and the author’s goals and results in this monograph, section \(1\) is devoted to some basic techniques needed in the sequel. In section \(2\) several constructions for canonical distance functions on algebraic curves are given. They form a crucial, and qualitatively new ingredient for defining capacity measures and Green functions on curves. The developed capacity theory is, for the present, a local one, and respectively dependent on the assumption of whether the ground field is archimedean or non-archimedean. The difference consists in the methodical approach, whereas the basic properties and results of the generalized (local) capacity theory for curves turn out to be the same. All this is comprehensively developed in section \(3\) (“Local capacity theory - Archimedean case”) and section \(4\) (“Local capacity - nonarchimedean case”), which form the crucial technical part of the monograph. - \(Section\quad 5\) deals with the global capacity theory for curves, i.e., for adelic sets in complete smooth curves over global fields. This provides a suitable generalization of D. Cantor’s theory for \({\mathbb{P}}^ 1\), where the results are, indeed, adapted from that special case, nevertheless heavily based on the author’s new general local approach. A number of examples of capacities and Green functions for (archimedean and nonarchimedean) local sets, as well as methods for the computation of the global capacity illustrate and varify the usefulness of this extended theory of capacity.

Finally, in the concluding section \(6\) the author discusses important arithmetical applications of his generalized capacity theory. After reproving the theorems of Fekete and Fekete-Szegö in their classical form, just for the convenience of the reader, and in order to prepare him for the following generalizations, the author provides a sagacious construction of rational functions on curves over a global field, which admit special properties needed for an adelic generalization of those classical theorems. This is certainly the ultimate goal, and the most significant result of the whole impressing treatise. The following proofs of the adelic generalizations of the theorems of Fekete and Fekete- Szegö are then rather formal consequences of the foregoing deep results. The book concludes with some examples showing that for the proofs of the Fekete theorem and the Fekete-Szegö theorem really different concepts of capacity are needed, and therefore generally justified and useful.

Altogether, the author has not only written a very deep-going research monograph, but also - at the same time - an excellent textbook on classical and generalized capacity theory. All the proofs are carefully and detailedly carried out, and the ubiquitous motivating discussions and examples make this very advanced topic a real pleasure to the reader. A list of indications and conjectures concerning the possibility of strengthening, vaster applying, and extending (to higher dimensions) capacity theory serves as a good hint to current research.

In the present monograph, the author now extends capacity theory to arbitrary curves (of any genus) with an arbitrary global field as ground field. This pretentious enterprise requires to lay fairly new foundations, as on curves of higher genus (instead of \({\mathbb{P}}^ 1)\) one can only lean upon local methods. The author’s far-reaching work to found a generalized capacity theory for algebraic curves and to apply it to arithmetical problems is based upon an ingeniuos combination of the classical methods, on the one hand, and several essentially new ingredients, such as local parametizations by power series, a certain global maximum modulus principle, an existence theorem for algebraic functions with near-prescribed divisors, the Deligne-Mumford theorem on stable reduction, and Néron’s canonical local height pairing on curves, on the other hand.

After a very enlightening and profound introduction to the classical background, D. Cantor’s pioneering work, and the author’s goals and results in this monograph, section \(1\) is devoted to some basic techniques needed in the sequel. In section \(2\) several constructions for canonical distance functions on algebraic curves are given. They form a crucial, and qualitatively new ingredient for defining capacity measures and Green functions on curves. The developed capacity theory is, for the present, a local one, and respectively dependent on the assumption of whether the ground field is archimedean or non-archimedean. The difference consists in the methodical approach, whereas the basic properties and results of the generalized (local) capacity theory for curves turn out to be the same. All this is comprehensively developed in section \(3\) (“Local capacity theory - Archimedean case”) and section \(4\) (“Local capacity - nonarchimedean case”), which form the crucial technical part of the monograph. - \(Section\quad 5\) deals with the global capacity theory for curves, i.e., for adelic sets in complete smooth curves over global fields. This provides a suitable generalization of D. Cantor’s theory for \({\mathbb{P}}^ 1\), where the results are, indeed, adapted from that special case, nevertheless heavily based on the author’s new general local approach. A number of examples of capacities and Green functions for (archimedean and nonarchimedean) local sets, as well as methods for the computation of the global capacity illustrate and varify the usefulness of this extended theory of capacity.

Finally, in the concluding section \(6\) the author discusses important arithmetical applications of his generalized capacity theory. After reproving the theorems of Fekete and Fekete-Szegö in their classical form, just for the convenience of the reader, and in order to prepare him for the following generalizations, the author provides a sagacious construction of rational functions on curves over a global field, which admit special properties needed for an adelic generalization of those classical theorems. This is certainly the ultimate goal, and the most significant result of the whole impressing treatise. The following proofs of the adelic generalizations of the theorems of Fekete and Fekete- Szegö are then rather formal consequences of the foregoing deep results. The book concludes with some examples showing that for the proofs of the Fekete theorem and the Fekete-Szegö theorem really different concepts of capacity are needed, and therefore generally justified and useful.

Altogether, the author has not only written a very deep-going research monograph, but also - at the same time - an excellent textbook on classical and generalized capacity theory. All the proofs are carefully and detailedly carried out, and the ubiquitous motivating discussions and examples make this very advanced topic a real pleasure to the reader. A list of indications and conjectures concerning the possibility of strengthening, vaster applying, and extending (to higher dimensions) capacity theory serves as a good hint to current research.

Reviewer: W.Kleinert

##### MSC:

14H25 | Arithmetic ground fields for curves |

30F15 | Harmonic functions on Riemann surfaces |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14G25 | Global ground fields in algebraic geometry |

14G20 | Local ground fields in algebraic geometry |

14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |

30-02 | Research exposition (monographs, survey articles) pertaining to functions of a complex variable |