A Panorama of Number Theory <I>or</I> The View from Baker's Garden

doi:10.1017/CBO9780511542961

Alan Baker's 60th birthday in August 1999 offered an ideal opportunity to organize a conference at ETH Zurich with the goal of presenting the state of the art in number theory and geometry. Many of the leaders in the subject were brought together to present an account of research in the last century as well as speculations for possible further research. The papers in this volume cover a broad spectrum of number theory including geometric, algebrao-geometric and analytic aspects. This volume will appeal to number theorists, algebraic geometers, and geometers with a number-theoretic background. However, it will also be valuable for mathematicians (in particular research students) who would like to be informed of the state of number theory at the start of the 21st century and in possible developments for the future.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-vi
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Contributors
pp vii-x
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Introduction
pp xi-xvi
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1 - One Century of Logarithmic Forms
pp 1-10
- By G. Wüstholz
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2 - Report on p-adic Logarithmic Forms
pp 11-25
- By Kunrui Yu
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3 - Recent Progress on Linear Forms in Elliptic Logarithms
pp 26-37
- By Sinnou David, Noriko Hirata-Kohno
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4 - Solving Diophantine Equations by Baker's Theory
pp 38-72
- By Kálmán Győry
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5 - Baker's Method and Modular Curves
pp 73-88
- By Yuri F. Bilu
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6 - Application of the André–Oort Conjecture to some Questions in Transcendence
pp 89-106
- By Paula B. Cohen, Gisbert Wüstholz
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7 - Regular Dessins, Endomorphisms of Jacobians, and Transcendence
pp 107-120
- By Jürgen Wolfart
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8 - Maass Cusp Forms with Integer Coefficients
pp 121-127
- By Peter Sarnak
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9 - Modular Forms, Elliptic Curves and the ABC-Conjecture
pp 128-147
- By Dorian Goldfeld
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10 - On the Algebraic Independence of Numbers
pp 148-167
- By Yu. V. Nesterenko
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11 - Ideal Lattices
pp 168-184
- By Eva Bayer-Fluckiger
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12 - Integral Points and Mordell–Weil Lattices
pp 185-193
- By Tetsuji Shioda
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13 - Forty Years of Effective Results in Diophantine Theory
pp 194-213
- By Enrico Bombieri
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14 - Points on Subvarieties of Tori
pp 214-230
- By Jan-Hendrik Evertse
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15 - A New Application of Diophantine Approximations
pp 231-246
- By G. Faltings
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16 - Search Bounds for Diophantine Equations
pp 247-259
- By D. W. Masser
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17 - Regular Systems, Ubiquity and Diophantine Approximation
pp 260-279
- By V. V. Beresnevich, V. I. Bernik, M. M. Dodson
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18 - Diophantine Approximation, Lattices and Flows on Homogeneous Spaces
pp 280-310
- By Gregory Margulis
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19 - On Linear Ternary Equations with Prime Variables – Baker's Constant and Vinogradov's Bound
pp 311-324
- By Ming-Chit Liu, Tianze Wang
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20 - Powers in Arithmetic Progression
pp 325-336
- By T. N. Shorey
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21 - On the Greatest Common Divisor of Two Univariate Polynomials, I
pp 337-352
- By A. Schinzel
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22 - Heilbronn's Exponential Sum and Transcendence Theory
pp 353-356
- By D. R. Heath-Brown
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A Panorama of Number Theory or The View from Baker's Garden

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Contributors
pp vii-x

Introduction
pp xi-xvi

1 - One Century of Logarithmic Forms
pp 1-10

2 - Report on p-adic Logarithmic Forms
pp 11-25

3 - Recent Progress on Linear Forms in Elliptic Logarithms
pp 26-37

4 - Solving Diophantine Equations by Baker's Theory
pp 38-72

5 - Baker's Method and Modular Curves
pp 73-88

6 - Application of the André–Oort Conjecture to some Questions in Transcendence
pp 89-106

7 - Regular Dessins, Endomorphisms of Jacobians, and Transcendence
pp 107-120

8 - Maass Cusp Forms with Integer Coefficients
pp 121-127

9 - Modular Forms, Elliptic Curves and the ABC-Conjecture
pp 128-147

10 - On the Algebraic Independence of Numbers
pp 148-167

11 - Ideal Lattices
pp 168-184

12 - Integral Points and Mordell–Weil Lattices
pp 185-193

13 - Forty Years of Effective Results in Diophantine Theory
pp 194-213

14 - Points on Subvarieties of Tori
pp 214-230

15 - A New Application of Diophantine Approximations
pp 231-246

16 - Search Bounds for Diophantine Equations
pp 247-259

17 - Regular Systems, Ubiquity and Diophantine Approximation
pp 260-279

18 - Diophantine Approximation, Lattices and Flows on Homogeneous Spaces
pp 280-310

19 - On Linear Ternary Equations with Prime Variables – Baker's Constant and Vinogradov's Bound
pp 311-324

20 - Powers in Arithmetic Progression
pp 325-336

21 - On the Greatest Common Divisor of Two Univariate Polynomials, I
pp 337-352

22 - Heilbronn's Exponential Sum and Transcendence Theory
pp 353-356

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A Panorama of Number Theory or The View from Baker's Garden

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