Many Variations of Mahler Measures: A Lasting Symphony

by François Brunault, Wadim Zudilin

Overview

The Mahler measure is a fascinating notion and an exciting topic in contemporary mathematics, interconnecting with subjects as diverse as number theory, analysis, arithmetic geometry, special functions and random walks. This friendly and concise introduction to the Mahler measure is a valuable resource for both graduate courses and self-study. It provides the reader with the necessary background material, before presenting the recent achievements and the remaining challenges in the field. The first part introduces the univariate Mahler measure and addresses Lehmer's question, and then discusses techniques of reducing multivariate measures to hypergeometric functions. The second part touches on the novelties of the subject, especially the relation with elliptic curves, modular forms and special values of L-functions. Finally, the Appendix presents the modern definition of motivic cohomology and regulator maps, as well as Deligne–Beilinson cohomology. The text includes many exercises to test comprehension and challenge readers of all abilities.

Product Details

ISBN-13:	9781108794459
Publisher:	Cambridge University Press
Publication date:	05/14/2020
Series:	Australian Mathematical Society Lecture Series , #28
Pages:	180
Product dimensions:	5.94(w) x 8.94(h) x 0.39(d)

About the Author

François Brunault is Associate Professor at École Normale Supérieure, Lyon in France, and is a member of the Mathematical Society of France. He is an arithmetic geometer with interest in elliptic curves, modular forms and L-functions, both from a theoretical and explicit point of view.

Wadim Zudilin is Professor of Pure Mathematics at Radboud University Nijmegen, known for his results that make use of special functions in number theory, in particular, about the irrationality for the values of Riemann's zeta function at positive integers. He co-authored the book Neverending Fractions: An Introduction to Continued Fractions (Cambridge, 2014).

Table of Contents

1. Some basics; 2. Lehmer's problem; 3. Multivariate setting; 4. The dilogarithm; 5. Differential equations for families of Mahler measures; 6. Random walk; 7. The regulator map for $K_2$ of curves; 8. Deninger's method for multivariate polynomials; 9. The Rogers–Zudilin method; 10. Modular regulators; Appendix. Motivic cohomology and regulator maps; References; Author Index; Subject index.

Editorial Reviews

‘… the book will serve as a great introduction to the subject of Mahler’s measure, in some of its manifold variations, with a special focus on its links with special values of L-functions. It is particularly suited for a student or research seminar, as well as for individual work, because of its concise nature, which emphasizes the most important points of the theory, while not leaving out crucial details when needed.’ Riccardo Pengo, zbMATH

From the Publisher