Computational aspects of modular forms and Galois representations [[electronic resource] ] : how one can compute in polynomial time the value of Ramanujan's tau at a prime / / edited by Jean-Marc Couveignes and Bas Edixhoven

• Record Nr.: UNINA9910789850303321
• Pubbl/distr/stampa: Princeton, N.J., : Princeton University Press, c2011
• ISBN: 1-283-05180-X; 9786613051806; 1-4008-3900-9
• Edizione: [Course Book]
• Descrizione fisica: 1 online resource (438 p.)
• Collana: Annals of mathematics studies ; ; 176
• Classificazione: MAT001000MAT012010
• Altri autori (Persone): EdixhovenB <1962-> (Bas); CouveignesJean-Marc
• Disciplina: 512/.32
• Soggetti: Galois modules (Algebra); Class field theory
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Front matter -- Contents -- Preface -- Acknowledgments -- Author information -- Dependencies between the chapters -- Chapter 1. Introduction, main results, context / Edixhoven, Bas -- Chapter 2. Modular curves, modular forms, lattices, Galois representations / Edixhoven, Bas -- Chapter 3. First description of the algorithms / Couveignes, Jean-Marc / Edixhoven, Bas -- Chapter 4. Short introduction to heights and Arakelov theory / Edixhoven, Bas / de Jong, Robin -- Chapter 5. Computing complex zeros of polynomials and power series / Couveignes, Jean-Marc -- Chapter 6. Computations with modular forms and Galois representations / Bosman, Johan -- Chapter 7. Polynomials for projective representations of level one forms / Bosman, Johan -- Chapter 8. Description of X1(5l) / Edixhoven, Bas -- Chapter 9. Applying Arakelov theory / Edixhoven, Bas / de Jong, Robin -- Chapter 10. An upper bound for Green functions on Riemann surfaces / Merkl, Franz -- Chapter 11. Bounds for Arakelov invariants of modular curves / Edixhoven, B. / de Jong, R. -- Chapter 12. Approximating Vf over the complex numbers / Couveignes, Jean-Marc ; -- Chapter 13. Computing Vf modulo p / Couveignes, Jean-Marc -- Chapter 14. Computing the residual Galois representations / Edixhoven, Bas -- Chapter 15. Computing coefficients of modular forms / Edixhoven, Bas -- Epilogue -- Bibliography -- Index
• Sommario/riassunto: "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'"--

Nécropoles protohistoriques de la région de Castres (Tarn) : Le Causse, Gourjade, Le Martinet

• Record Nr.: UNINA9910744511903321
• Pubbl/distr/stampa: Paris, : Éditions de la Maison des sciences de l'homme, 2003
• ISBN: 2-7351-2642-0
• Descrizione fisica: 1 online resource (276 p.)
• Collana: Documents d'archéologie française
• Soggetti: Archaeology
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Sommario/riassunto: Between 1987 and 1995, rescue excavations were carried out on three Late Bronze Age / Early Iron Age cremation burial cemeteries in the Castres area of the Tarn region (Gourjade and Le Martinet near Castres, Le Causse at Labruguière). These vast funerary complexes, containing more than 1,200 burials, were considered to form an indivisible whole and as such were listed as sites of national importance in 1995. Study of archaeological finds from the sites, comprised of almost 5,300 vessels and some 1,300 metal objects, has led to developing a simple common typology similar to those defined for cemeteries in the western lower Languedoc region. Typological characteristics are examined in detail and presented in conjunction with distribution maps. A detailed study of funerary architecture and deposit distribution, using the same approach as the spatial analysis, is presented in the following chapter. These observations provide the basis for suggestions concerning the chronological sequencing of the burials. The resulting seriation conclusively demonstrates the presence of five groups, thus shedding light on the evolution of material culture and funerary assemblages in the Castres cemeteries over more than three centuries (from the 9th to the first half of the 6th c BCE) and indicating that significant changes took place during the Bronze Age / Iron Age transition period. The study of topographical evolution points to logical patterns of development which were concentric at Causse and Goujade and linear at Martinet. Anthropological analysis, which is ; based on a remarkable sample comprised of more than 1,200 bone assemblages, unique in France for this period, has provided information on selection within the cemeteries and on changes in deposits and their constituents. A chapter is also given over to animal remains which are both numerous and varied, particularly at Causse. The study demonstrates changes in both the quantity and the qua lit y of meat offerings over time and provides a pertinent view of this practice. The paleosociological analysis which follows adopts a classical approach in which ail parameters are taken into consideration. The results indicate that deposits are relatively uniform and point to a degree of social levelling within the communities. The final chapter considers the cemeteries cultural identity and their regional connections. Study of available documentation over a wide area, extending from Provence to Aquitaine, demonstrates strong links with Mediterranean groups, primarily those from the western lower Languedoc area. The studies and syntheses are based upon abundant documentary evidence which is presented in its entirety in the second and third volumes.