Periods And Special Functions In Transcendence

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Paula Tretkoff

• Publisher: WSPC (EUROPE)
• Year: 2017
• eBook ISBN:
  9781786342966
• Topic: Scienze biologiche
• Subtopic: Scienza generale

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This book gives an introduction to some central results in transcendental number theory with application to periods and special values of modular and hypergeometric functions. It also includes related results on Calabi–Yau manifolds. Most of the material is based on the author's own research and appears for the first time in book form. It is presented with minimal of technical language and no background in number theory is needed. In addition, except the last chapter, all chapters include exercises suitable for graduate students. It is a nice book for graduate students and researchers interested in transcendence.

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Chapter 1

Group Varieties and Transcendence

A transcendental number τ is a complex number with P(τ) ≠ 0 for every non-zero polynomial P with coefficients in the field ℚ of rational numbers. In other words, the transcendental numbers are the complex numbers that are not algebraic. In particular, they are irrational; that is they do not lie in ℚ. Although this concept dates from antiquity, it was not until 1844 that Joseph Liouville constructed some specific numbers whose transcendence could be proven rigorously [Liouville (1844)], [Liouville (1851)]. The first number treated by Liouville is the Liouville number, also called the Liouville constant, given by ∑_k≥1 10^−k!. Liouville showed that this number was too well approximated by rational numbers to be an algebraic number. In 1874, Georg Cantor proved that the algebraic numbers are countable, and the real numbers are uncountable, thereby showing there are uncountably many transcendental numbers.

In antiquity, mathematicians asked whether or not π is a transcendental number. They understood, for example, that the transcendence of π implies that you cannot construct, using straightedge and compass, a square with the same area as the unit circle (“you cannot square the circle”). In 1737, Euler showed that e is irrational and published the result seven years later [Euler (1744)]. In 1761, Lambert showed that π is irrational, and rekindled interest in the transcendence of e and π [Lambert (1761)]. At least since this time, mathematicians had wondered about the transcendence of values at non-zero algebraic arguments of the exponential function e^z, which has period 2πi. This problem is the first occurrence of the theme of this book: 2πi (π times the algebraic number 2i) is a period of the classical special function e^z, in that e^z+2πi = e^z, whereas e^α, α ∈

, is a special value of this period...

Table of contents

• Cover Page
• Title
• Copyright
• Contents
• Preface
• Acknowledgments
• 1. Group Varieties and Transcendence
• 2. Transcendence Results for Exponential and Elliptic Functions
• 3. Modular Functions and Criteria for Complex Multiplication
• 4. Periods of 1-forms on Complex Curves and Abelian Varieties
• 5. Transcendence of Special Values of Hypergeometric Functions
• 6. Transcendence Criterion for Complex Multiplication on K3 Surfaces
• 7. Hodge Structures of Higher Level
• Bibliography
• Index