cam a22 4500 551415398 CHVBK 20201017121206.0 m d cr |n |||||||| 181214s2011 nju sb 001 0 eng d 2010053185 978-0-691-14201-2 (hardback) 978-0-691-14202-9 (pbk.) (SERSOL)ssj0000483479 (NEBIS)009946567 (WaSeSS)ssj0000483479 DLC DLC DLC WaSeSS QA247 .C638 2011eb 512/.32 22 MAT001000 MAT012010 bisacsh Computational aspects of modular forms and Galois representations [Elektronische Daten] how one can compute in polynomial time the value of Ramanujan's tau at a prime edited by Bas Edixhoven and Jean-Marc Couveignes Princeton Princeton University Press c2011 1 online resource (xi, 425 p.) Annals of mathematics studies 176 Includes bibliographical references (p. [403]-421) and index. Lizenzbedingungen können den Zugang einschränken. License restrictions may limit access. "Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number P can be computed in time bounded by a fixed power of the logarithm of P. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program.The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields.The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations"-- "This book represents a major step forward from explicit class field theory, and it could be described as the start of the 'explicit Langlands program'"-- MATHEMATICS / Advanced bisacsh MATHEMATICS / Geometry / Algebraic bisacsh Galois modules (Algebra) Class field theory ALGORITHMISCHE KOMPLEXITÄT (MATHEMATIK) ger (ETHUDK)000012496 ethudk KLASSENKÖRPERTHEORIE (ZAHLENTHEORIE) ger (ETHUDK)000012767 ethudk MODULARE FORMEN (ZAHLENTHEORIE) ger (ETHUDK)000012550 ethudk NORMALERWEITERUNGEN + GALOISERWEITERUNGEN (KÖRPERTHEORIE) ger (ETHUDK)000012750 ethudk u MODULARE FORMEN (ZAHLENTHEORIE) ger 511.38*1 nebis E1 u NORMALERWEITERUNGEN + GALOISERWEITERUNGEN (KÖRPERTHEORIE) ger 512.623.27 nebis E1 u KLASSENKÖRPERTHEORIE (ZAHLENTHEORIE) ger 511.236 nebis E1 u ALGORITHMISCHE KOMPLEXITÄT (MATHEMATIK) ger 510.52 nebis E1 u FORMES MODULAIRES (THÉORIE DES NOMBRES) fre 511.38*1 nebis E1 u MODULAR FORMS (NUMBER THEORY) eng 511.38*1 nebis E1 u EXTENSIONS NORMALES + EXTENSIONS GALOISIENNES (THÉORIE DES CORPS) fre 512.623.27 nebis E1 u NORMAL EXTENSIONS + GALOIS EXTENSIONS (FIELD THEORY) eng 512.623.27 nebis E1 u THÉORIE DES CORPS DE CLASSES (THÉORIE DES NOMBRES) fre 511.236 nebis E1 u CLASS FIELD THEORY (NUMBER THEORY) eng 511.236 nebis E1 u ALGORITHMIC COMPLEXITY (MATHEMATICS) eng 510.52 nebis E1 u COMPLEXITÉ ALGORITHMIQUE (MATHÉMATIQUES) fre 510.52 nebis E1 Edixhoven B. Bas 1962- Couveignes Jean-Marc https://www.degruyter.com/openurl?genre=book&isbn=9781400839001 Uni Basel: Volltext https://ebookcentral.proquest.com/lib/unibern/detail.action?docID=670341 Uni Bern: Volltext https://www.degruyter.com/openurl?genre=book&isbn=9781400839001 Uni Bern: Volltext BK020053 XK020053 XK020000 De Gruyter PDA STM eBooks E-Books von 360MarcUpdates Ebook Central All Subscribed Titles PDA Package: All eBook Content, incl. OWV/AV PDA5EBK EText nebis ED E01-Myilibrary-201309 nebis ER E01-Safari201409 nebis ER noalma ids I 122 E01-20140325 Safari E01-20120101 NEBIS E01 E01 EL IDSBB A145 A145 145VT IDSBB B405 B405 405VT NELB4052004 IDSBB B405 B405 405VT NELB4052004 IDSBB 490 0- Annals of mathematics studies 176 IDSBB 700 1- Edixhoven B. Bas 1962- IDSBB 700 1- Couveignes Jean-Marc IDSBB 856 40 https://www.degruyter.com/openurl?genre=book&isbn=9781400839001 Uni Basel: Volltext IDSBB 856 40 https://ebookcentral.proquest.com/lib/unibern/detail.action?docID=670341 Uni Bern: Volltext IDSBB 856 40 https://www.degruyter.com/openurl?genre=book&isbn=9781400839001 Uni Bern: Volltext NEBIS 490 -- Annals of mathematics studies no. 176 NEBIS 700 1- Couveignes Jean-Marc NEBIS 700 1- Edixhoven Bas NEBIS 856 -- http://sfx.ethz.ch/sfx_locater?sid=ALEPH:EBI01&genre=book&isbn=9780691142029 Online via SFX Volltext SWISSBIB 484291815