LEADER 06959nam 22017175 450 001 9910154744303321 005 20230808192407.0 010 $a1-4008-8254-0 024 7 $a10.1515/9781400882540 035 $a(CKB)3710000000631352 035 $a(MiAaPQ)EBC4738740 035 $a(DE-B1597)468035 035 $a(OCoLC)979968812 035 $a(DE-B1597)9781400882540 035 $a(EXLCZ)993710000000631352 100 $a20190708d2016 fg 101 0 $aeng 135 $aurcnu|||||||| 181 $2rdacontent 182 $2rdamedia 183 $2rdacarrier 200 13$aAn Introduction to G-Functions. (AM-133), Volume 133 /$fBernard Dwork, Francis J. Sullivan, Giovanni Gerotto 210 1$aPrinceton, NJ : $cPrinceton University Press, $d[2016] 210 4$d©1994 215 $a1 online resource (349 pages) $cillustrations 225 0 $aAnnals of Mathematics Studies ;$v316 311 $a0-691-03681-0 311 $a0-691-03675-6 320 $aIncludes bibliographical references and index. 327 $tFrontmatter -- $tCONTENTS -- $tPREFACE / $rDwork, B. -- $tINTRODUCTION -- $tLIST OF SYMBOLS -- $tCHAPTER I. VALUED FIELDS -- $tCHAPTER II. ZETA FUNCTIONS -- $tCHAPTER III. DIFFERENTIAL EQUATIONS -- $tCHAPTER IV. EFFECTIVE BOUNDS. ORDINARY DISKS -- $tCHAPTER V. EFFECTIVE BOUNDS. SINGULAR DISKS -- $tCHAPTER VI. TRANSFER THEOREMS INTO DISKS WITH ONE SINGULARITY -- $tCHAPTER VII. DIFFERENTIAL EQUATIONS OF ARITHMETIC TYPE -- $tCHAPTER VIII. G-SERIES. THE THEOREM OF CHUDNOVSKY -- $tAPPENDIX I. CONVERGENCE POLYGON FOR DIFFERENTIAL EQUATIONS -- $tAPPENDIX II. ARCHIMEDEAN ESTIMATES -- $tAPPENDIX III. CAUCHY'S THEOREM -- $tBIBLIOGRAPHY -- $tINDEX 330 $aWritten for advanced undergraduate and first-year graduate students, this book aims to introduce students to a serious level of p-adic analysis with important implications for number theory. The main object is the study of G-series, that is, power series y=aij=0 Ajxj with coefficients in an algebraic number field K. These series satisfy a linear differential equation Ly=0 with LIK(x) [d/dx] and have non-zero radii of convergence for each imbedding of K into the complex numbers. They have the further property that the common denominators of the first s coefficients go to infinity geometrically with the index s. After presenting a review of valuation theory and elementary p-adic analysis together with an application to the congruence zeta function, this book offers a detailed study of the p-adic properties of formal power series solutions of linear differential equations. In particular, the p-adic radii of convergence and the p-adic growth of coefficients are studied. Recent work of Christol, Bombieri, André, and Dwork is treated and augmented. The book concludes with Chudnovsky's theorem: the analytic continuation of a G -series is again a G -series. This book will be indispensable for those wishing to study the work of Bombieri and André on global relations and for the study of the arithmetic properties of solutions of ordinary differential equations. 410 0$aAnnals of mathematics studies ;$vno. 133. 606 $aH-functions 606 $ap-adic analysis 610 $aAdjoint. 610 $aAlgebraic Method. 610 $aAlgebraic closure. 610 $aAlgebraic number field. 610 $aAlgebraic number theory. 610 $aAlgebraic variety. 610 $aAlgebraically closed field. 610 $aAnalytic continuation. 610 $aAnalytic function. 610 $aArgument principle. 610 $aArithmetic. 610 $aAutomorphism. 610 $aBearing (navigation). 610 $aBinomial series. 610 $aCalculation. 610 $aCardinality. 610 $aCartesian coordinate system. 610 $aCauchy sequence. 610 $aCauchy's theorem (geometry). 610 $aCoefficient. 610 $aCohomology. 610 $aCommutative ring. 610 $aComplete intersection. 610 $aComplex analysis. 610 $aConjecture. 610 $aDensity theorem. 610 $aDifferential equation. 610 $aDimension (vector space). 610 $aDirect sum. 610 $aDiscrete valuation. 610 $aEigenvalues and eigenvectors. 610 $aElliptic curve. 610 $aEquation. 610 $aEquivalence class. 610 $aEstimation. 610 $aExistential quantification. 610 $aExponential function. 610 $aExterior algebra. 610 $aField of fractions. 610 $aFinite field. 610 $aFormal power series. 610 $aFuchs' theorem. 610 $aG-module. 610 $aGalois extension. 610 $aGalois group. 610 $aGeneral linear group. 610 $aGeneric point. 610 $aGeometry. 610 $aHypergeometric function. 610 $aIdentity matrix. 610 $aInequality (mathematics). 610 $aIntercept method. 610 $aIrreducible element. 610 $aIrreducible polynomial. 610 $aLaurent series. 610 $aLimit of a sequence. 610 $aLinear differential equation. 610 $aLowest common denominator. 610 $aMathematical induction. 610 $aMeromorphic function. 610 $aModular arithmetic. 610 $aModule (mathematics). 610 $aMonodromy. 610 $aMonotonic function. 610 $aMultiplicative group. 610 $aNatural number. 610 $aNewton polygon. 610 $aNumber theory. 610 $aP-adic number. 610 $aParameter. 610 $aPermutation. 610 $aPolygon. 610 $aPolynomial. 610 $aProjective line. 610 $aQ.E.D. 610 $aQuadratic residue. 610 $aRadius of convergence. 610 $aRational function. 610 $aRational number. 610 $aResidue field. 610 $aRiemann hypothesis. 610 $aRing of integers. 610 $aRoot of unity. 610 $aSeparable polynomial. 610 $aSequence. 610 $aSiegel's lemma. 610 $aSpecial case. 610 $aSquare root. 610 $aSubring. 610 $aSubset. 610 $aSummation. 610 $aTheorem. 610 $aTopology of uniform convergence. 610 $aTranspose. 610 $aTriangle inequality. 610 $aUnipotent. 610 $aValuation ring. 610 $aWeil conjecture. 610 $aWronskian. 610 $aY-intercept. 615 0$aH-functions. 615 0$ap-adic analysis. 676 $a515/.55 700 $aDwork$b Bernard, $053745 702 $aGerotto$b Giovanni, 702 $aSullivan$b Francis J., 801 0$bDE-B1597 801 1$bDE-B1597 906 $aBOOK 912 $a9910154744303321 996 $aAn Introduction to G-Functions. (AM-133), Volume 133$91892113 997 $aUNINA