Linear forms in logarithms and their applications

From elementary height and approximation bounds to Baker's theorem, effective logarithmic estimates and their Diophantine applications.

This edition contains Lessons 1–9, the available sections of Lesson 10, and Lessons 11–22, with proofs, worked examples, original figures, exercises and solutions. It covers qualitative and effective Baker estimates, complex logarithmic forms, p-adic analysis, determinant bounds and Yu-type estimates, followed by applications to powers of two and three, irrationality measures, S-unit gaps, 3x+1 cycles, the many-to-two logarithm reduction, regulator bounds, the effective unit equation, Thue equations, effective irrationality bounds, binomial families, S-unit and primitive Thue–Mahler equations, superelliptic and hyperelliptic equations, affine curves of genus one, Mordell bounds, the Ramanujan–Nagell equation, primitive divisors, effective Catalan finiteness, fixed-variable Pillai equations, quadratic forms and class numbers, products of Dirichlet L-functions and their explicit limit formula, the complete class-number-one argument and the genus-character reduction for class number two, the Stewart–Tijdeman construction for abc triples and effective bounds in terms of their radicals, the Stewart–Yu one-third bound and its least-greatest-prime refinement, and complete equation resolutions by continued fractions and certified lattice reduction. The concluding lesson treats algebraic groups and the analytic subgroup theorem, abelian logarithms and periods, geometric relations between 1-periods, explicit elliptic logarithm bounds, integral points, and precisely stated open problems. The full programme has 22 lessons.

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