## A Panorama of Number Theory Or The View from Baker's GardenAlan 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 are interested in being informed in the state of number theory at the start of the 21st century and in possible developments for the future. |

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### Contents

One Century of Logarithmic Forms G Wiistholz | 1 |

Report on padic Logarithmic Forms Kunrui Yu | 11 |

Recent Progress on Linear Forms in Elliptic Logarithms Sinnou David Noriko HirataKohno | 26 |

Sinnou David Universite P et M Curie Paris VI Institut Mathematique | 36 |

Solving Diophantine Equations by Bakers Theory Kdlmdn Gyory | 38 |

Bakers Method and Modular Curves Yuri F Bilu | 73 |

Application of the AndreOort Conjecture to some Questions in Transcendence Paula B Cohen Gisbert Wiistholz | 89 |

G Wiistholz ETH Zurich ETH Zentrum DMATH HG G 66 3 Raemistrasse | 101 |

Ideal Lattices Eva BayerFluckiger | 168 |

Integral Points and MordellWeil Lattices Tetsuji Shioda | 185 |

Forty Years of Effective Results in Diophantine Theory Enrico Bombieri | 194 |

Points on Subvarieties of Tori JanHendrik Evertse | 214 |

A New Application of Diophantine Approximations G Faltings | 231 |

Search Bounds for Diophantine Equations D W Masser | 247 |

Regular Systems Ubiquity and Diophantine Approximation V V Beresnevich V I Bernik MM Dodson | 260 |

Diophantine Approximation Lattices and Flows on Homogeneous Spaces Gregory Margulis | 280 |

Regular Dessins Endomorphisms of Jacobians and Transcendence Jurgen Wolfart | 107 |

Maass Cusp Forms with Integer Coefficients Peter Sarnak | 121 |

Modular Forms Elliptic Curves and the ABCConjecture Dorian Goldfeld | 128 |

JanHendrik Evertse Universiteit Leiden Mathematisch Instituut Postbus | 147 |

On the Algebraic Independence of Numbers Yu V Nesterenko | 148 |

On Linear Ternary Equations with Prime Variables Bakers Constant | 311 |

Powers in Arithmetic Progression T N Shorey | 325 |

On the Greatest Common Divisor of Two Univariate Polynomials I | 337 |

Heilbronns Exponential Sum and Transcendence Theory | 353 |

### Common terms and phrases

ABC-conjecture abelian varieties Acta Arith algebraic curve algebraic number field algebraic numbers algebraically independent applied arithmetic automorphism group Baker Baker's theory Bernik Bilu coefficients complex multiplication conjecture constant cusps defined over Q denote Diophantine approximation diophantine equations divisor effective elliptic curve Evertse Faltings finite form equations forms in logarithms function genus Gyory Hausdorff dimension height hence homogeneous ideal lattice implies inequality integral points irreducible isomorphic Jacobian Lemma linear forms lower bound Margulis Masser matrix method modular Mordell-Weil non-zero norm number field Number Theory obtained p-adic polynomial positive integers prime problem Proc proof of Theorem Proposition proved quadratic forms quotient rational result Riemann surfaces satisfying Shimura Shimura varieties Shorey Siegel solutions Sprindzuk subgroup subset subspace subvariety Tate-Shafarevich group Thue equations Tijdeman tion transcendence Transcendence Theory triangle group unit equations upper bound Wiistholz Wolfart zero

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Page vii - Tata Institute of Fundamental Research, Homi Bhabha Road. Mumbai 400 005, India Max-Planck Institut fur Mikrostrukturphysik.