Cover -- Half-title -- Title -- Copyright -- Contents -- Contributors -- Introduction -- 1 One Century of Logarithmic Forms -- 1 Introduction -- 2 Hilbert's seventh problem -- 3 Elliptic theory -- 4 Group varieties -- 5 The quantitative theory -- 6 The abc-conjecture -- Bibliography -- 2 Report on p-adic Logarithmic Forms -- 1 Historical introduction -- 2 A successful strategy -- 3 New developments -- 4 The application to the abc-conjecture -- References -- 3 Recent Progress on Linear Forms in Elliptic Logarithms -- 1 Introduction -- 2 New result -- 3 Key estimate for the refinement -- References -- 4 Solving Diophantine Equations by Baker's Theory -- Abstract -- Introduction -- 1 General effective finiteness theorems -- 1.1 Thue equations and Thue-Mahler equations -- 1.2 Elliptic, hyperelliptic and superelliptic equations -- 1.3 Unit equations and S-unit equations -- 1.4 Discriminant form and index form equations -- 1.5 Decomposable form equations of general type -- 2 Explicit determination of the solutions -- 2.1 Thue equations -- 2.2 Unit equations and S-unit equations -- 2.3 Discriminant form and index form equations -- 2.4 Decomposable form equations of general type -- 2.5 Elliptic equations -- References -- 5 Baker's Method and Modular Curves -- Abstract -- 1 Introduction -- 2 Heights -- 3 Baker's method -- Baker's theorem -- Functional units -- Coverings -- 4 Modular curves -- Preliminaries -- Effective Siegel theorem for modular curves -- Proof of Theorem 10 -- Consequences for elliptic curves -- References -- 6 Application of the André-Oort Conjecture to some Questions in Transcendence -- Abstract -- 1 Some problems on hypergeometric functions -- 2 Fuchsian triangle groups -- 3 Solving Problem 4 solves Problem 1 -- 4 The André-Oort Conjecture for the Siegel moduli space -- References.

7 Regular Dessins, Endomorphisms of Jacobians, and Transcendence -- 1 Triangle groups and dessins -- 2 Integration on regular dessins -- 3 Shimura families and the Jacobi locus -- References -- 8 Maass Cusp Forms with Integer Coefficients -- Remarks -- Bibliography -- 9 Modular Forms, Elliptic Curves and the ABC-Conjecture -- 1 The ABC-Conjecture -- 2 Applications of the ABC-Conjecture -- 3 Elliptic curves over Q (Global Minimal Models) -- 4 Conjectures which are equivalent to ABC -- 5 Large Tate-Shafarevich groups -- 6 Modular symbols -- References -- 10 On the Algebraic Independence of Numbers -- Abstract -- 1 E-functions -- 2 Mahler functions -- 3 Theorems about functions with addition properties -- 4 Modular forms -- 5 Hypergeometric functions -- 6 Values of theta functions -- References -- 11 Ideal Lattices -- Introduction -- 1 Definitions, notation and basic facts -- 2 Integral ideal lattices -- Quadratic fields -- Cyclotomic fields of prime power conductor -- Examples over cyclotomic fields -- 3 Generalization and examples -- Knot theory -- Symmetric, skew-symmetric and orthogonal matrices with a given characteristic polynomial -- 4 Real ideal lattices -- 5 Arakelov invariants -- References -- 12 Integral Points and Mordell-Weil Lattices -- Abstract -- 1 Introduction -- 2 Main results -- 3 Examples -- 4 Many integral points -- References -- 13 Forty Years of Effective Results in Diophantine Theory -- 1 Thue, Siegel and Mahler -- 2 Baker -- 3 Applications of Baker's theory -- 4 The Padé method -- 5 An alternative effective method -- 6 The future: the abc-conjecture -- References -- 14 Points on Subvarieties of Tori -- 1 Introduction -- 2 Proof of a weaker version of Corollary 1 -- Absolute values and heights -- Points of small height -- The Subspace Theorem -- Proof of Theorem 7 -- References.

15 A New Application of Diophantine Approximations -- 1 Known results -- 2 Geometry of projections -- 3 Dimensions and expectation values -- 4 Intersection numbers -- 5 Nondegeneracy -- References -- 16 Search Bounds for Diophantine Equations -- References -- 17 Regular Systems, Ubiquity and Diophantine Approximation -- 1 Introduction -- 2 Regular systems -- 3 Ubiquity -- A general lower bound -- 4 Khintchine-type theorems on manifolds -- 5 Hausdorff dimension on manifolds -- Simultaneous Diophantine approximation on manifolds -- References -- 18 Diophantine Approximation, Lattices and Flows on Homogeneous Spaces -- Introduction -- 1 Diophantine approximation on manifolds -- 2 Values of quadratic forms and of products of linear forms at integral points -- 3 A quantitative version of the Oppenheim conjecture -- 4 Counting lattice points on homogeneous varieties -- 5 Translates of submanifolds and unipotent flows on homogeneous spaces -- References -- 19 On Linear Ternary Equations with Prime Variables - Baker's Constant and Vinogradov's Bound -- 1 Baker's Constant -- Introduction and Baker's Problem -- Some Extensions of Baker's Problem -- Relations with Linnik's Theorem -- Some Recent Numerical Results on Baker's Constant -- Outline of the Proof of Theorem 5 -- 2 Vinogradov's Bound -- Acknowledgement -- References -- 20 Powers in Arithmetic Progression -- 1 Cubes and higher powers -- 2 Squares -- References -- 21 On the Greatest Common Divisor of Two Univariate Polynomials, I -- References -- 22 Heilbronn's Exponential Sum and Transcendence Theory -- References.