A panorama of number theory, or, The view from Baker's garden

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 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. Half-title......Page 3 Title......Page 5 Copyright......Page 6 Contents......Page 7 Contributors......Page 9 Introduction......Page 13 1 Introduction......Page 19 2 Hilbert’s seventh problem......Page 20 3 Elliptic theory......Page 21 4 Group varieties......Page 22 5 The quantitative theory......Page 23 6 The abc-conjecture......Page 25 Bibliography......Page 27 1 Historical introduction......Page 29 2 A successful strategy......Page 32 3 New developments......Page 35 4 The application to the abc-conjecture......Page 38 References......Page 40 1 Introduction......Page 44 2 New result......Page 48 3 Key estimate for the refinement......Page 49 References......Page 53 Introduction......Page 56 1 General effective finiteness theorems......Page 58 1.1 Thue equations and Thue–Mahler equations......Page 59 1.2 Elliptic, hyperelliptic and superelliptic equations......Page 62 1.3 Unit equations and S-unit equations......Page 64 1.4 Discriminant form and index form equations......Page 67 1.5 Decomposable form equations of general type......Page 71 2 Explicit determination of the solutions......Page 73 2.1 Thue equations......Page 76 2.2 Unit equations and S-unit equations......Page 77 2.3 Discriminant form and index form equations......Page 78 2.4 Decomposable form equations of general type......Page 81 2.5 Elliptic equations......Page 82 References......Page 85 1 Introduction......Page 91 2 Heights......Page 92 3 Baker’s method......Page 93 Baker’s theorem......Page 94 Functional units......Page 96 Coverings......Page 98 Preliminaries......Page 99 Effective Siegel theorem for modular curves......Page 100 Proof of Theorem 10......Page 102 References......Page 103 Abstract......Page 107 1 Some problems on hypergeometric functions......Page 108 2 Fuchsian triangle groups......Page 114 3 Solving Problem 4 solves Problem 1......Page 118 4 The André–Oort Conjecture for the Siegel moduli space......Page 119 References......Page 122 1 Triangle groups and dessins......Page 125 2 Integration on regular dessins......Page 128 3 Shimura families and the Jacobi locus......Page 131 References......Page 136 8 Maass Cusp Forms with Integer Coefficients......Page 139 Remarks......Page 142 Bibliography......Page 144 1 The ABC-Conjecture......Page 146 2 Applications of the ABC-Conjecture......Page 148 3 Elliptic curves over Q (Global Minimal Models)......Page 151 4 Conjectures which are equivalent to ABC......Page 152 6 Modular symbols......Page 160 References......Page 162 1 E-functions......Page 166 2 Mahler functions......Page 168 3 Theorems about functions with addition properties......Page 170 4 Modular forms......Page 172 5 Hypergeometric functions......Page 176 6 Values of theta functions......Page 179 References......Page 184 1 Definitions, notation and basic facts......Page 186 2 Integral ideal lattices......Page 189 Quadratic fields......Page 191 Cyclotomic fields of prime power conductor......Page 193 Examples over cyclotomic fields......Page 194 Knot theory......Page 195 Symmetric, skew-symmetric and orthogonal matrices with a given characteristic polynomial......Page 196 4 Real ideal lattices......Page 197 5 Arakelov invariants......Page 198 References......Page 201 1 Introduction......Page 203 2 Main results......Page 204 3 Examples......Page 206 4 Many integral points......Page 208 References......Page 210 1 Thue, Siegel and Mahler......Page 212 2 Baker......Page 214 3 Applications of Baker’s theory......Page 218 4 The Padé method......Page 221 5 An alternative effective method......Page 222 6 The future: the abc-conjecture......Page 224 References......Page 228 1 Introduction......Page 232 2 Proof of a weaker version of Corollary 1......Page 238 Absolute values and heights......Page 239 Points of small height......Page 241 The Subspace Theorem......Page 242 Proof of Theorem 7......Page 245 References......Page 246 1 Known results......Page 249 2 Geometry of projections......Page 252 3 Dimensions and expectation values......Page 258 4 Intersection numbers......Page 260 5 Nondegeneracy......Page 262 References......Page 264 16 Search Bounds for Diophantine Equations......Page 265 References......Page 275 1 Introduction......Page 278 2 Regular systems......Page 280 3 Ubiquity......Page 283 A general lower bound......Page 285 4 Khintchine-type theorems on manifolds......Page 287 5 Hausdorff dimension on manifolds......Page 292 Simultaneous Diophantine approximation on manifolds......Page 293 References......Page 294 1 Diophantine approximation on manifolds......Page 298 2 Values of quadratic forms and of products of linear forms at integral points......Page 307 3 A quantitative version of the Oppenheim conjecture......Page 312 4 Counting lattice points on homogeneous varieties......Page 319 5 Translates of submanifolds and unipotent flows on homogeneous spaces......Page 322 References......Page 325 Introduction and Baker’s Problem......Page 329 Some Extensions of Baker’s Problem......Page 331 Relations with Linnik’s Theorem......Page 332 Outline of the Proof of Theorem 5......Page 334 2 Vinogradov’s Bound......Page 338 References......Page 340 1 Cubes and higher powers......Page 343 2 Squares......Page 351 References......Page 352 21 On the Greatest Common Divisor of Two Univariate Polynomials, I......Page 355 References......Page 370 22 Heilbronn’s Exponential Sum and Transcendence Theory......Page 371 References......Page 374 One Century Of Logarithmic Forms / G. Wüstholz -- Report On P-adic Logarithmic Forms / Kunrui Yu -- Recent Progress On Linear Forms In Elliptic Logarithms / Sinnou David & Noriko Hirata-kohno -- Solving Diophantine Equations By Baker's Theory / Kálmán Győry -- Baker's Method And Modular Curves / Yuri F. Bilu -- Application Of The Andrè-oort Conjecture To Some Questions In Transcendence / Paula B. Cohen & Gilbert Wüstholz -- Regular Dessins, Endomorphisms Of Jacobians, And Transcendence / Jürgen Wolfart -- Maass Cusp Forms With Integer Coefficients / Peter Sarnak -- Modular Forms, Elliptic Curves And The Abc-conjecture / Dorian Goldfeld -- On The Algebraic Independence Of Numbers / Yu. V. Nesterenko -- Ideal Lattices / Eva Bayer-fluckiger -- Integral Points And Mordell-weil Lattices / Tetsuji Shioda -- Forty Years Of Effective Results In Diophantine Theory / Enrico Bombieri -- Points On Subvarieties Of Tori / Jan-hendrik Evertse -- A New Application Of Diophantine Approximations / G. Faltings -- Search Bounds For Diophantine Equations / D.w. Masser -- Regular Systems, Ubiquity And Diophantine Approximation / V.v. Beresnevich, V.i. Bernik & M.m. Dodson -- Diophantine Approximation, Lattices And Flows On Homogeneous Spaces / Gregory Margulis -- On Linear Ternary Equations With Prime Variables / Ming-chit Liu & Tianze Wang -- Powers In Arithmetic Progression / T.n. Shorey -- On The Greatest Common Divisor Of Two Univariate Polynomials / A. Schinzel -- Heilbronn's Exponential Sum And Transcendence Theory / D.r. Heath-brown. Edited By Gisbert Wüstholz. Contains Papers Presented At A Conference Held In Zürich In 1999 To Mark The 60th Birthday Of Alan Baker. Includes Bibliographical References. 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.