This comprehensive account of the Gross-Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerton-Dyer conjecture and Diophantine equations.

The book begins with a conceptual formulation of the Gross-Zagier formula in terms of incoherent quaternion algebras and incoherent automorphic representations with rational coefficients attached naturally to abelian varieties parametrized by Shimura curves. This is followed by a complete proof of its coherent analogue: the Waldspurger formula, which relates the periods of integrals and the special values of L-series by means of Weil representations. The Gross-Zagier formula is then reformulated in terms of incoherent Weil representations and Kudla's generating series. Using Arakelov theory and the modularity of Kudla's generating series, the proof of the Gross-Zagier formula is reduced to local formulas.

"The Gross-Zagier Formula on Shimura Curves" will be of great use to students wishing to enter this area and to those already working in it.

The second volume of this work contains Parts 2 and 3 of the "Handbook of Coding Theory". Part 2, "Connections", is devoted to connections between coding theory and other branches of mathematics and computer science. Part 3, "Applications", deals with a variety of applications for coding.

1 Klassifikation der einfachen Hyperftachen-Singularitaten . . . . . . . . . . . . . . 2 1. 1 Abbildungskeime, Rechtsaquivalenz, Einfachheit . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. 2 Endlich bestimmte FUnktionskeime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 1. 3 Klassifikation der einfachen Singularitaten in C ---- ------------. -. . . --. - 11 1. 4 Beweis des verallgemeinerten Morse-Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1. 5 Klassifikation der einfachen Singularitaten in C" . . . . . . . . . . . . . . . . . . . . . . . . 19 3 2 Die einfachen Flachensingularitaten in C als Quotientensingularitaten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2. 1 Die endlichen Untergruppen von SL(2, C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2. 2 Quotientensingularitaten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 2. 3 C jG, wo G eine endliche Untergruppe von SL(2, C) ist . . . . . . . . . . . . . . . . . 27 2. 4 Die Rationalitat der Quotientensingularitaten . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3 Die Aufloesung der einfachen zweidimensionalen Hyperftachensingularitaten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3. 1 Das Aufloesen von Kurvensingularitaten . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2 3. 2 Das Aufloesen von C jG, wo G eine endliche Untergruppe von SL(2, C) ist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4 Elementare lokale Eigenschaften von Singularitaten . . . . . . . . . . . . . . . . . . 49 4. 1 Der Umgebungsrand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4. 2 Gute Reprasentanten von Abbildungskeimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4. 3 Monodromie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4. 4 Die Monodromie einer quadratischen Singularitat (lokaler Fall) . . . . . . . . . . 65 5 Die U ntersuchung von Milnorfasern . . . . . . . . . . . . . . . . . -. . . -. . --. - -. . . . . . . 73 5. 1 Milnorfasem von ebenen Kurvensingularitaten . . . . . . -. . -. . . -. . -. . . . . . . . . 73 5. 2 Milnorfasem von Hyperfia. chensingularitaten . . . . . . . . . . . . . . -. . -. . . . . . . . . . 81 6 Die Beec r hnung d er M o n oro d mie . . . . . . . . . . . . . . . . . . . -. . . . . . -. . - . . . . . . . . . 87 6. 1 Die Morsifikation . . . ., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -. . . . . . . 87 6. 2 Die Monodromie der ebenen Kurvensingularitaten in Cl. . . . . . . . . . 88 6. 3 Dynkin-Dia. gramm und Monodromiegruppe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6. 4 Die Monodromie beim Addieren von FUnktionskeimen . . . . . . . . . . . . .

Written and revised by D. B. A. Epstein.

The description for this book, Topics in Transcendental Algebraic Geometry. (AM-106), will be forthcoming.

A fractal drum is a bounded open subset of R. m with a fractal boundary. A difficult problem is to describe the relationship between the shape (geo- metry) of the drum and its sound (its spectrum). In this book, we restrict ourselves to the one-dimensional case of fractal strings, and their higher dimensional analogues, fractal sprays. We develop a theory of complex di- mensions of a fractal string, and we study how these complex dimensions relate the geometry with the spectrum of the fractal string. We refer the reader to [Berrl-2, Lapl-4, LapPol-3, LapMal-2, HeLapl-2] and the ref- erences therein for further physical and mathematical motivations of this work. (Also see, in particular, Sections 7. 1, 10. 3 and 10. 4, along with Ap- pendix B. ) In Chapter 1, we introduce the basic object of our research, fractal strings (see [Lapl-3, LapPol-3, LapMal-2, HeLapl-2]). A 'standard fractal string' is a bounded open subset of the real line. Such a set is a disjoint union of open intervals, the lengths of which form a sequence which we assume to be infinite. Important information about the geometry of . c is contained in its geometric zeta function (c(8) = L lj. j=l 2 Introduction We assume throughout that this function has a suitable meromorphic ex- tension. The central notion of this book, the complex dimensions of a fractal string . c, is defined as the poles of the meromorphic extension of (c.

Schon vor fUnf J ahren hatte ich mit meinem inzwischen verstorbenen Freunde FABIO CONFORTO (1909-1954) vereinbart, eine deutsche Be- arbeitung seiner Vorlesungen uber ABELsche Funktionen *), die er im Studienjahr 1940/41 in Rom gehalten hat, herauszugeben. Da aber eine grundliche Oberarbeitung des aus dem Jahre 1942 stammenden Textes notwendig erschien, die CONFORTO selbst besorgen wollte**), wurde die Verwirklichung dieses Planes zunachst noch hinausgeschoben. Aber bald nach dem AbschluB des Vertrages mit dem Verleger wurde CONFORTO von einer unerbittlichen Krankheit befallen, die ihn innerhalb J ahresfrist zwang, die Feder fUr immer aus der Hand zu legen, noch beY r er mit diesem Werk hatte beginnen k6nnen. So blieb mir dessen Gestaltung und Vollendung als Vermachtnis zuruck, das ich im Sinne und Geiste meines verstorbenen Freundes bestens durchzufUhren be- strebt war, urn so mehr, als ich damit hoffen durfte, eine wirkliche Lucke auszufUllen und den heute Lebenden eines der reizvollsten Gebiete der klassischen Mathematik naherzubringen, das seit dem Aussterben der alten Mathematikergeneration, zu der auch noch mein verehrter Lehrer W. WIRTINGER geh6rt hatte, beinahe vergessen worden ist. Dabei war ich glucklich, in A. ANDREOTTI und M. ROSATI zwei aus- gezeichnete Mitarbeiter zu finden, die als ehemalige Schuler CONFORTOs sehr gut mit seinen Absichten und Planen vertraut waren und so ent- scheidend zuJ? Gelingen des Werkes beigetragen haben. Aus den letzten Jahren lag auch noch eine vervielfaltigte Vorlesung***) CONFORTOs vor, deren erstes Kapitel in das vorliegende Buch hineinverarbeitet werden konnte.

A central concern of number theory is the study of local-to-global principles, which describe the behavior of a global field K in terms of the behavior of various completions of K. This book looks at a specific example of a local-to-global principle: Weil's conjecture on the Tamagawa number of a semisimple algebraic group G over K. In the case where K is the function field of an algebraic curve X, this conjecture counts the number of G-bundles on X (global information) in terms of the reduction of G at the points of X (local information). The goal of this book is to give a conceptual proof of Weil's conjecture, based on the geometry of the moduli stack of G-bundles. Inspired by ideas from algebraic topology, it introduces a theory of factorization homology in the setting -adic sheaves. Using this theory, Dennis Gaitsgory and Jacob Lurie articulate a different local-to-global principle: a product formula that expresses the cohomology of the moduli stack of G-bundles (a global object) as a tensor product of local factors. Using a version of the Grothendieck-Lefschetz trace formula, Gaitsgory and Lurie show that this product formula implies Weil's conjecture. The proof of the product formula will appear in a sequel volume.

Homogeneous spaces of linear algebraic groups lie at the crossroads of algebraic geometry, theory of algebraic groups, classical projective and enumerative geometry, harmonic analysis, and representation theory. By standard reasons of algebraic geometry, in order to solve various problems on a homogeneous space, it is natural and helpful to compactify it while keeping track of the group action, i.e., to consider equivariant completions or, more generally, open embeddings of a given homogeneous space. Such equivariant embeddings are the subject of this book. We focus on the classification of equivariant embeddings in terms of certain data of "combinatorial" nature (the Luna-Vust theory) and description of various geometric and representation-theoretic properties of these varieties based on these data. The class of spherical varieties, intensively studied during the last three decades, is of special interest in the scope of this book. Spherical varieties include many classical examples, such as Grassmannians, flag varieties, and varieties of quadrics, as well as well-known toric varieties. We have attempted to cover most of the important issues, including the recent substantial progress obtained in and around the theory of spherical varieties.

For every mathematician, ring theory and K-theory are intimately connected: al- braic K-theory is largely the K-theory of rings. At ?rst sight, polytopes, by their very nature, must appear alien to surveyors of this heartland of algebra. But in the presence of a discrete structure, polytopes de?ne a?ne monoids, and, in their turn, a?ne monoids give rise to monoid algebras. Teir spectra are the building blocks of toric varieties, an area that has developed rapidly in the last four decades. From a purely systematic viewpoint, "monoids" should therefore replace "po- topes" in the title of the book. However, such a change would conceal the geometric ?avor that we have tried to preserve through all chapters. Before delving into a description of the contents we would like to mention three general features of the book: (?) the exhibiting of interactions of convex geometry, ring theory, and K-theory is not the only goal; we present some of the central results in each of these ?elds; (?) the exposition is of constructive (i. e., algorithmic) nature at many places throughout the text-there is no doubt that one of the driving forces behind the current popularity of combinatorial geometry is the quest for visualization and computation; (? ) despite the large amount of information from various ?elds, we have strived to keep the polytopal perspective as the major organizational principle.

I. Bucur: L'anneau de Chow d'une variete algebrique.- E. Eckmann: Cohomologie et classes caracteristiques.- C. Teleman: Sur le caractere de Chern d'un fibre vectoriel complexe differentiable.- E. Thomas: Characteristic classes and differentiable manifolds.- A. Van de Ven: Chern classes and complex manifolds."

Hilbert Functions play major roles in Algebraic Geometry and Commutative Algebra, and are becoming increasingly important also in Computational Algebra. They capture many useful numerical characters associated to a projective variety or to a filtered module over a local ring. Starting from the pioneering work of D.G. Northcott and J. Sally, we aim to gather together in one place many new developments of this theory by using a unifying approach which gives self-contained and easier proofs. The extension of the theory to the case of general filtrations on a module, and its application to the study of certain graded algebras which are not associated to a filtration are two of the main features of the monograph. The material is intended for graduate students and researchers who are interested in Commutative Algebra, in particular in the theory of the Hilbert Functions and related topics.

The aim of this book is to study various geometric properties and algebraic invariants of smooth projective varieties with infinite fundamental groups. This approach allows for much interplay between methods of algebraic geometry, complex analysis, the theory of harmonic maps, and topology. Making systematic use of Shafarevich maps, a concept previously introduced by the author, this work isolates those varieties where the fundamental group influences global properties of the canonical class. The book is primarily geared toward researchers and graduate students in algebraic geometry who are interested in the structure and classification theory of algebraic varieties. There are, however, presentations of many other applications involving other topics as well--such as Abelian varieties, theta functions, and automorphic forms on bounded domains. The methods are drawn from diverse sources, including Atiyah's L2 -index theorem, Gromov's theory of Poincare series, and recent generalizations of Kodaira's vanishing theorem. Originally published in 1995. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

This volume offers a systematic treatment of certain basic parts of algebraic geometry, presented from the analytic and algebraic points of view. The notes focus on comparison theorems between the algebraic, analytic, and continuous categories. Contents include: 1.1 sheaf theory, ringed spaces; 1.2 local structure of analytic and algebraic sets; 1.3 Pn 2.1 sheaves of modules; 2.2 vector bundles; 2.3 sheaf cohomology and computations on Pn; 3.1 maximum principle and Schwarz lemma on analytic spaces; 3.2 Siegel's theorem; 3.3 Chow's theorem; 4.1 GAGA; 5.1 line bundles, divisors, and maps to Pn; 5.2 Grassmanians and vector bundles; 5.3 Chern classes and curvature; 5.4 analytic cocycles; 6.1 K-theory and Bott periodicity; 6.2 K-theory as a generalized cohomology theory; 7.1 the Chern character and obstruction theory; 7.2 the Atiyah-Hirzebruch spectral sequence; 7.3 K-theory on algebraic varieties; 8.1 Stein manifold theory; 8.2 holomorphic vector bundles on polydisks; 9.1 concluding remarks; bibliography. Originally published in 1974. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

The first application of modern algebraic techniques to a comprehensive selection of classical geometric problems. Written with spirit and originality, this is a valuable book for anyone interested in the subject from other than the purely algebraic point of view. Originally published in 1953. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

In spite of some recent applications of ultraproducts in algebra, they remain largely unknown to commutative algebraists, in part because they do not preserve basic properties such as Noetherianity. This work wants to make a strong case against these prejudices. More precisely, it studies ultraproducts of Noetherian local rings from a purely algebraic perspective, as well as how they can be used to transfer results between the positive and zero characteristics, to derive uniform bounds, to define tight closure in characteristic zero, and to prove asymptotic versions of homological conjectures in mixed characteristic. Some of these results are obtained using variants called chromatic products, which are often even Noetherian. This book, neither assuming nor using any logical formalism, is intended for algebraists and geometers, in the hope of popularizing ultraproducts and their applications in algebra.

Many of the developments of modern algebraic geometry and topology stem from the ideas of S. Lefschetz. These are featured in this volume of contemporary research papers contributed by mathematical colleagues to celebrate his seventieth birthday. Originally published in 1957. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Cuts and metrics are well-known objects that arise - independently, but with many deep and fascinating connections - in diverse fields: in graph theory, combinatorial optimization, geometry of numbers, combinatorial matrix theory, statistical physics, VLSI design etc.

This book presents a wealth of results, from different mathematical disciplines, in a unified comprehensive manner, and establishes new and old links, which cannot be found elsewhere. It provides a unique and invaluable source for researchers and graduate students.

From the Reviews:

"This book is definitely a milestone in the literature of integer programming and combinatorial optimization. It draws from the Interdisciplinarity of these fields ...]. With knowledge about the relevant terms, one can enjoy special subsections without being entirely familiar with the rest of the chapter. This makes it not only an interesting research book but even a dictionary. ...] The longer one works with it, the more beautiful it becomes." Optima 56, 1997.

This volume contains two related, though independently written, mo- graphs. In Notes on Derived Functors and Grothendieck Duality the ?rst three chapters treat the basics of derived categories and functors, and of the rich formalism, over ringed spaces, of the derived functors, for unbounded com- ? plexes, ofthesheaffunctors?, Hom, f andf wheref isaringed-spacemap. ? Included are some enhancements, for concentrated (i.e., quasi-compact and quasi-separated) schemes, of classical results such as the projection and K] unneth isomorphisms. The fourth chapter presents the abstract foun- tions of Grothendieck Duality-existence and tor-independent base change for the right adjoint of the derived functor Rf when f is a quasi-proper ? map of concentrated schemes, the twisted inverse image pseudofunctor for separated ?nite-type maps of noetherian schemes, re?nements for maps of ?nite tor-dimension, and a brief discussion of dualizing complexes. In Equivariant Twisted Inverses the theory is extended to the context of diagrams of schemes, and in particular, to schemes with a group-scheme action. An equivariant version of the twisted inverse-image pseudofunctor is de?ned, and equivariant versions of some of its important properties are proved, including Grothendieck duality for proper morphisms, and ?at base change. Also, equivariant dualizing complexes are dealt with. As an appli- tion, ageneralizedversionofWatanabe'stheoremontheGorensteinproperty of rings of invariants is proved. More detailed overviews are given in the respective Introductions."

In dem Lehrbuch wird eine mathematisch orientierte Einfuhrung in die algorithmische Geometrie gegeben. Im ersten Teil werden "klassische" Probleme und Techniken behandelt, die sich auf polyedrische (= linear begrenzte) Objekte beziehen. Hierzu gehoeren beispielsweise Algorithmen zur Berechnung konvexer Hullen und die Konstruktion von Voronoi-Diagrammen. Im zweiten Teil werden grundlegende Methoden der algorithmischen algebraischen Geometrie entwickelt und anhand von Anwendungen aus Computergrafik, Kurvenrekonstruktion und Robotik illustriert. Das Buch eignet sich fur ein fortgeschrittenes Modul in den derzeit neu konzipierten Bachelor-Studiengangen in Mathematik und Informatik.

Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number p can be computed in time bounded by a fixed power of the logarithm of p. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program.

The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields.

The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations.

This textbook is directed towards students who are familiar with matrices and their use in solving systems of linear equations. The emphasis is on the algebra supporting the ideas that make linear algebra so important, both in theoretical and practical applications. The narrative is written to bring along students who may be new to the level of abstraction essential to a working understanding of linear algebra. The determinant is used throughout, placed in some historical perspective, and defined several different ways, including in the context of exterior algebras. The text details proof of the existence of a basis for an arbitrary vector space and addresses vector spaces over arbitrary fields. It develops LU-factorization, Jordan canonical form, and real and complex inner product spaces. It includes examples of inner product spaces of continuous complex functions on a real interval, as well as the background material that students may need in order to follow those discussions. Special classes of matrices make an entrance early in the text and subsequently appear throughout. The last chapter of the book introduces the classical groups.

Cet ouvrage est consacre a l'arithmetique des surfaces fibrees en courbes de genre 1 au-dessus de la droite projective, et a l'arithmetique des intersections de deux quadriques dans l'espace projectif. Swinnerton-Dyer introduisit en 1993 une technique permettant d'etudier les points rationnels des pinceaux de courbes de genre 1. La premiere moitie de l'ouvrage reprend et developpe cette technique ainsi que ses generalisations ulterieures. La seconde moitie, qui repose sur la premiere, porte sur les surfaces de del Pezzo de degre 4 et sur les intersections de deux quadriques de dimension superieure; les resultats annonces dans C. R. Math. Acad. Sci. Paris 342 (2006), no. 4, 223--227] y sont demontres."

This book uses finite field theory as a hook to introduce the reader to a range of ideas from algebra and number theory. It constructs all finite fields from scratch and shows that they are unique up to isomorphism. As a payoff, several combinatorial applications of finite fields are given: Sidon sets and perfect difference sets, de Bruijn sequences and a magic trick of Persi Diaconis, and the polynomial time algorithm for primality testing due to Agrawal, Kayal and Saxena. The book forms the basis for a one term intensive course with students meeting weekly for multiple lectures and a discussion session. Readers can expect to develop familiarity with ideas in algebra (groups, rings and fields), and elementary number theory, which would help with later classes where these are developed in greater detail. And they will enjoy seeing the AKS primality test application tying together the many disparate topics from the book. The pre-requisites for reading this book are minimal: familiarity with proof writing, some linear algebra, and one variable calculus is assumed. This book is aimed at incoming undergraduate students with a strong interest in mathematics or computer science.