Stephen S. Kudla, Michael Rapoport, and Tonghai Yang

Modular Forms and Special Cycles on Shimura Curves

Paper | 2006 | ISBN: 0-691-12551-1
Cloth | 2006 | ISBN: 0-691-12550-3
384 pp. | 6 x 9 | 1 line illus. 3 tables.
Annals of Mathematics Studies, vol.161.

Modular Forms and Special Cycles on Shimura Curves is a thorough study of the generating functions constructed from special cycles, both divisors and zero-cycles, on the arithmetic surface "M" attached to a Shimura curve "M" over the field of rational numbers. These generating functions are shown to be the q-expansions of modular forms and Siegel modular forms of genus two respectively, valued in the Gillet-Soule arithmetic Chow groups of "M". The two types of generating functions are related via an arithmetic inner product formula. In addition, an analogue of the classical Siegel-Weil formula identifies the generating function for zero-cycles as the central derivative of a Siegel Eisenstein series. As an application, an arithmetic analogue of the Shimura-Waldspurger correspondence is constructed, carrying holomorphic cusp forms of weight 3/2 to classes in the Mordell-Weil group of "M". In certain cases, the nonvanishing of this correspondence is related to the central derivative of the standard L-function for a modular form of weight 2. These results depend on a novel mixture of modular forms and arithmetic geometry and should provide a paradigm for further investigations. The proofs involve a wide range of techniques, including arithmetic intersection theory, the arithmetic adjunction formula, representation densities of quadratic forms, deformation theory of p-divisible groups, p-adic uniformization, the Weil representation, the local and global theta correspondence, and the doubling integral representation of L-functions.

Stephen S. Kudla is at the University of Maryland. Michael Rapoport is at the Mathematisches Institut der Universitat, Bonn, Germany. Tonghai Yang is at the University of Wisconsin, Madison.

Table of Contents:

Acknowledgments ix
Chapter 1. Introduction 1
Bibliography 21
Chapter 2. Arithmetic intersection theory on stacks 27
Chapter 3. Cycles on Shimura curves 45
Chapter 4. An arithmetic theta function 71
Chapter 5. The central derivative of a genus two Eisenstein series 105
Chapter 6. The generating function for 0-cycles 167
Chapter 6 Appendix. The case p = 2, p | D (B) 181
Chapter 7. An inner product formula 205
Chapter 8. On the doubling integral 265
Chapter 9. Central derivatives of L-functions 351
Index 371

Paul J. Nahin

Dr. Euler's Fabulous Formula:
Cures Many Mathematical Ills

Cloth | 2006 | ISBN: 0-691-11822-1
404 pp. | 6 x 9 | 2 halftones.77 line illus.

In the mid-eighteenth century, Swiss-born mathematician Leonhard Euler developed a formula so innovative and complex that it continues to inspire research, discussion, and even the occasional limerick. Dr. Euler's Fabulous Formula shares the fascinating story of this groundbreaking formula--long regarded as the gold standard for mathematical beauty--and shows why it still lies at the heart of complex number theory.

This book is the sequel to Paul Nahin's An Imaginary Tale: The Story of I [the square root of -1], which chronicled the events leading up to the discovery of one of mathematics' most elusive numbers, the square root of minus one. Unlike the earlier book, which devoted a significant amount of space to the historical development of complex numbers, Dr. Euler begins with discussions of many sophisticated applications of complex numbers in pure and applied mathematics, and to electronic technology. The topics covered span a huge range, from a never-before-told tale of an encounter between the famous mathematician G. H. Hardy and the physicist Arthur Schuster, to a discussion of the theoretical basis for single-sideband AM radio, to the design of chase-and-escape problems.

The book is accessible to any reader with the equivalent of the first two years of college mathematics (calculus and differential equations), and it promises to inspire new applications for years to come. Or as Nahin writes in the book's preface: To mathematicians ten thousand years hence, "Euler's formula will still be beautiful and stunning and untarnished by time."

Paul J. Nahin is Professor Emeritus of Electrical Engineering at the University of New Hampshire. He is the author of Duelling Idiots and Other Probability Puzzlers, When Least Is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible, and An Imaginary Tale: The Story of I [the square root of -1] (all Princeton).

Endorsements:

"If you ever wondered about the beauties and powers of mathematics, this book is a treasure trove. Paul Nahin uses Euler's formula as the magic key to unlock a wealth of surprising consequences, ranging from number theory to electronics, presented clearly, carefully, and with verve."--Peter Pesic, St. John's College

"The range and variety of topics covered here is impressive. I found many little gems that I have never seen before in books of this type. Moreover, the writing is lively and enthusiastic and the book is highly readable."--Des Higham, University of Strathclyde, Glasgow

Seydel, Rudiger U.

Tools for Computational Finance, 3rd ed.

Series: Universitext
2006, XIX, 304 p. 75 illus., Softcover
ISBN: 3-540-27923-7

About this textbook

This book is very easy to read and one can gain a quick snapshot of computational issues arising in financial mathematics. Researchers or students of the mathematical sciences with an interest in finance will find this book a very helpful and gentle guide to the world of financial engineering. SIAM review (46, 2004).

The third edition is thoroughly revised and significantly extended. The largest addition is a new section on analytic methods with main focus on interpolation approach and quadratic approximation. New sections and subsections are among others devoted to risk-neutrality, early-exercise curves, multidimensional Black-Scholes models, the integral representation of options and the derivation of the Black-Scholes equation.

New figures, more exercises, more background material make this guide to the world of financial engineering a real must-to-have for everyone working in FE.

Table of contents

Modelling Tools for Financial Options.- Generating Random Numbers with Specified Distributions.- Simulation with Stochastic Differential Equations.- Standard Methods for Standard Options.- Finite-Element Methods.- Pricing of Exotic Options.- Appendices

Markus Stroppel (University of Stuttgart, Germany):

Locally Compact Groups

EMS Textbooks in Mathematics
ISBN 3-03719-016-7
February 2006, 312 pages, hardcover, 16.5 cm x 23.5 cm.

Locally compact groups play an important role in many areas of mathematics as well as in physics. The class of locally compact groups admits a strong structure theory, which allows to reduce many problems to groups constructed in various ways from the additive group of real numbers, the classical linear groups and from finite groups. The book gives a systematic and detailed introduction to the highlights of that theory.

In the beginning, a review of fundamental tools from topology and the elementary theory of topological groups and transformation groups is presented. Completions, Haar integral, applications to linear representations culminating in the Peter-Weyl Theorem are treated. Pontryagin duality for locally compact Abelian groups forms a central topic of the book. Applications are given, including results about the structure of locally compact Abelian groups, and a structure theory for locally compact rings leading to the classification of locally compact fields. Topological semigroups are discussed in a separate chapter, with special attention to their relations to groups. The last chapter reviews results related to Hilbert's Fifth Problem, with the focus on structural results for non-Abelian connected locally compact groups that can be derived using approximation by Lie groups.

The book is self-contained and is addressed to advanced undergraduate or graduate students in mathematics or physics. It can be used for one-semester courses on topological groups, on locally compact Abelian groups, or on topological algebra. Suggestions on course design are given in the preface. Each chapter is accompanied by a set of exercises that have been tested in classes.

Table of contents