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Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (Annals of Mathematics Studies)
Lingua: Inglese
Editore: Princeton University Press, 2011
ISBN 10: 0691142025 ISBN 13: 9780691142029
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Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau At a Prime
Lingua: Inglese
Editore: Princeton University Press, Princeton, 2011
ISBN 10: 0691142025 ISBN 13: 9780691142029
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Paperback. Condizione: Near Fine. A nice, solid copy. ; Annals Of Mathematics Studies; Vol. 176; 6.5 X 1 X 9.5 inches; 425 pages.
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Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176) (Annals of Mathematics Studies (176))
Lingua: Inglese
Editore: Princeton University Press, 2011
ISBN 10: 0691142025 ISBN 13: 9780691142029
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Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau At a Prime
Lingua: Inglese
Editore: Princeton University Press, Princeton, 2011
ISBN 10: 0691142025 ISBN 13: 9780691142029
Da: Daedalus Books, Portland, OR, U.S.A.
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Paperback. Condizione: Near Fine. A nice, solid copy. ; Annals of Mathematics Studies; Vol. 176; 6.5 X 1 X 9.5 inches; 425 pages.
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Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176) (Annals of Mathematics Studies (176))
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Computational Aspects of Modular Forms and Galois Representations : How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime
Lingua: Inglese
Editore: Princeton University Press, 2011
ISBN 10: 0691142025 ISBN 13: 9780691142029
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Computational Aspects of Modular Forms and Galois Representations : How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime
Lingua: Inglese
Editore: Princeton University Press, 2011
ISBN 10: 0691142025 ISBN 13: 9780691142029
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Introduction to Applications of Modular Forms : Computational Aspects
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Computational Aspects of Modular Forms and Galois Representations
Lingua: Inglese
Editore: Princeton University Press, 2011
ISBN 10: 0691142025 ISBN 13: 9780691142029
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Introduction to Applications of Modular Forms: Computational Aspects (Synthesis Lectures on Mathematics & Statistics)
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Introduction to Applications of Modular Forms Computational Aspects
Lingua: Inglese
Editore: Springer International Publishing AG, 2023
ISBN 10: 3031326288 ISBN 13: 9783031326288
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Introduction to Applications of Modular Forms: Computational Aspects (Synthesis Lectures on Mathematics & Statistics)
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Introduction to Applications of Modular Forms : Computational Aspects
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Computational Aspects of Modular forms and Galois Representations
Lingua: Inglese
Editore: Princeton University Press, 2011
ISBN 10: 0691142025 ISBN 13: 9780691142029
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Computational Aspects of Modular Forms and Galois Representations : How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime
Lingua: Inglese
Editore: Princeton University Press, 2011
ISBN 10: 0691142025 ISBN 13: 9780691142029
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Introduction to Applications of Modular Forms : Computational Aspects
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Introduction to Applications of Modular Forms: Computational Aspects (Synthesis Lectures on Mathematics & Statistics)
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Computational Aspects of Modular forms and Galois Representations
Lingua: Inglese
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Introduction to Applications of Modular Forms: Computational Aspects (Synthesis Lectures on Mathematics & Statistics)
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Introduction to Applications of Modular Forms: Computational Aspects (Synthesis Lectures on Mathematics & Statistics)
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Computational Aspects of Modular Forms and Galois Representations
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Introduction to Applications of Modular Forms : Computational Aspects
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Introduction to Applications of Modular Forms: Computational Aspects (Synthesis Lectures on Mathematics & Statistics)
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Computational Aspects of Modular Forms and Galois Representations : How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime
Lingua: Inglese
Editore: Princeton University Press, 2011
ISBN 10: 0691142025 ISBN 13: 9780691142029
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Computational Aspects of Modular Forms and Galois Representations
Lingua: Inglese
Editore: Princeton University Press, US, 2011
ISBN 10: 0691142025 ISBN 13: 9780691142029
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Aggiungi al carrelloPaperback. Condizione: New. 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.
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Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)
Lingua: Inglese
Editore: Princeton University Press, 2011
ISBN 10: 0691142025 ISBN 13: 9780691142029
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Introduction to Applications of Modular Forms: Computational Aspects
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Computational Aspects of Modular Forms and Galois Representations
Lingua: Inglese
Editore: Princeton University Press, US, 2011
ISBN 10: 0691142025 ISBN 13: 9780691142029
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Aggiungi al carrelloPaperback. Condizione: New. 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.
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Introduction to Applications of Modular Forms : Computational Aspects
Lingua: Inglese
Editore: Springer International Publishing, 2024
ISBN 10: 3031326318 ISBN 13: 9783031326318
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Aggiungi al carrelloTaschenbuch. Condizione: Neu. Druck auf Anfrage Neuware - Printed after ordering - This book is a self-contained treatment for those who study or work on the computational aspects of classical modular forms. The author describes the theory of modular forms and its applications in number theoretic problems such as representations by quadratic forms and the determination of asymptotic formulas for Fourier coefficients of different types of special functions. A detailed account of recent applications of modular forms in number theory with a focus on using computer algorithms is provided. Computer algorithms are included for each presented application to help readers put the theory in context and make new conjectures.
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Aggiungi al carrelloTaschenbuch. Condizione: Neu. Introduction to Applications of Modular Forms | Computational Aspects | Zafer Selcuk Aygin | Taschenbuch | x | Englisch | 2024 | Springer | EAN 9783031326318 | Verantwortliche Person für die EU: Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg, juergen[dot]hartmann[at]springer[dot]com | Anbieter: preigu.
