Introduction to G-Functions. (AM-133), Volume 133 (eBook)

Bernard Dwork, Giovanni Gerotto, Francis J. Sullivan (Autoren)

eBook Download: PDF

2016
352 Seiten
Princeton University Press (Verlag)
978-1-4008-8254-0 (ISBN)

Infos

Bernard Dwork is Professor of Mathematics at Princeton University. Giovanni Gerotto and Francis J. Sullivan are Associate Professors of Mathematics at the University of Padova.

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Written for advanced undergraduate and first-year graduate students, this book aims to introduce students to a serious level of p-adic analysis with important implications for number theory. The main object is the study of G-series, that is, power series y=aij=0 Ajxj with coefficients in an algebraic number field K. These series satisfy a linear differential equation Ly=0 with LIK(x) [d/dx] and have non-zero radii of convergence for each imbedding of K into the complex numbers. They have the further property that the common denominators of the first s coefficients go to infinity geometrically with the index s. After presenting a review of valuation theory and elementary p-adic analysis together with an application to the congruence zeta function, this book offers a detailed study of the p-adic properties of formal power series solutions of linear differential equations. In particular, the p-adic radii of convergence and the p-adic growth of coefficients are studied. Recent work of Christol, Bombieri, Andre, and Dwork is treated and augmented. The book concludes with Chudnovsky's theorem: the analytic continuation of a G -series is again a G -series. This book will be indispensable for those wishing to study the work of Bombieri and Andre on global relations and for the study of the arithmetic properties of solutions of ordinary differential equations.

Produkt-Details

• Erscheint lt. Verlag: 2.3.2016
• Reihe/Serie: Annals of Mathematics Studies
• Reihe/Serie: Annals of Mathematics Studies
• Zusatzinfo: 22 figs.
• Verlagsort: Princeton
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Schlagworte: Adjoint • Algebraically closed field • algebraic closure • Algebraic Method • algebraic number field • algebraic number theory • algebraic variety • analytic continuation • analytic function • Argument principle • arithmetic • automorphism • Bearing (navigation) • Binomial series • Calculation • Cardinality • Cartesian coordinate system • Cauchy Sequence • Cauchy's theorem (geometry) • coefficient • cohomology • Commutative Ring • complete intersection • Complex Analysis • conjecture • Density theorem • differential equation • Dimension (vector space) • direct sum • discrete valuation • Eigenvalues and Eigenvectors • Elliptic Curve • Equation • equivalence class • estimation • existential quantification • exponential function • Exterior algebra • Field of fractions • finite field • Formal power series • Fuchs' theorem • Galois extension • Galois group • general linear group • Generic point • Geometry • G-module • hypergeometric function • Identity matrix • Inequality (mathematics) • Intercept method • Irreducible element • irreducible polynomial • Laurent Series • Limit of a sequence • linear differential equation • Lowest common denominator • Mathematical Induction • Meromorphic Function • modular arithmetic • Module (mathematics) • Monodromy • Monotonic Function • Multiplicative group • Natural number • Newton Polygon • Number Theory • P-adic number • Parameter • Permutation • Polygon • polynomial • Projective line • Q.E.D. • Quadratic residue • radius of convergence • rational function • Rational number • residue field • Riemann hypothesis • Ring of integers • Root of unity • Separable polynomial • Sequence • Siegel's lemma • Special case • square root • Subring • Subset • Summation • Theorem • Topology of uniform convergence • transpose • Triangle Inequality • Unipotent • Valuation ring • Weil conjecture • Wronskian • Y-intercept
• ISBN-10: 1-4008-8254-0 / 1400882540
• ISBN-13: 978-1-4008-8254-0 / 9781400882540