0
Your cart

Your cart is empty

Books > Science & Mathematics > Mathematics > Geometry > Algebraic geometry

Buy Now

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) (Paperback) Loot Price: R1,914
Discovery Miles 19 140
Computational Aspects of Modular Forms and Galois Representations - How One Can Compute in Polynomial Time the Value of...

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) (Paperback)

Bas Edixhoven, Jean-Marc Couveignes; Contributions by Robin De Jong, Franz Merkl, Johan Bosman

Series: Annals of Mathematics Studies

 (sign in to rate)
Loot Price R1,914 Discovery Miles 19 140 | Repayment Terms: R179 pm x 12*

Bookmark and Share

Expected to ship within 12 - 17 working days

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.

General

Imprint: Princeton University Press
Country of origin: United States
Series: Annals of Mathematics Studies
Release date: June 2011
First published: June 2011
Editors: Bas Edixhoven • Jean-Marc Couveignes
Contributors: Robin De Jong • Franz Merkl • Johan Bosman
Dimensions: 235 x 152 x 22mm (L x W x T)
Format: Paperback
Pages: 440
ISBN-13: 978-0-691-14202-9
Categories: Books > Science & Mathematics > Mathematics > Geometry > Algebraic geometry
Promotions
LSN: 0-691-14202-5
Barcode: 9780691142029

Is the information for this product incomplete, wrong or inappropriate? Let us know about it.

Does this product have an incorrect or missing image? Send us a new image.

Is this product missing categories? Add more categories.

Review This Product

No reviews yet - be the first to create one!

You might also like..

Sasakian Geometry
Charles Boyer, the late Krzysztof Galicki Hardcover R5,349 Discovery Miles 53 490
Michael Atiyah Collected Works - Volume…
Michael Atiyah Hardcover R9,656 Discovery Miles 96 560
Cyclic Modules and the Structure of…
S.K. Jain, Ashish K. Srivastava, … Hardcover R5,490 Discovery Miles 54 900
Riemann Surfaces
Simon Donaldson Hardcover R4,199 Discovery Miles 41 990
Sets, Groups, and Mappings - An…
Andrew D. Hwang Hardcover R2,330 Discovery Miles 23 300
Deformation Theory of Discontinuous…
Ali Baklouti Hardcover R5,537 Discovery Miles 55 370
The Classification of the Finite Simple…
Inna Capdeboscq, Daniel Gorenstein, … Paperback R2,558 Discovery Miles 25 580
One-cocycles And Knot Invariants
Thomas Fiedler Hardcover R3,326 Discovery Miles 33 260
Algebras, Lattices, Varieties - Volume…
Ralph S Freese, Ralph N. McKenzie, … Paperback R3,219 Discovery Miles 32 190
Infinity Operads And Monoidal Categories…
Donald Yau Hardcover R4,094 Discovery Miles 40 940
Geometric Sculptures - an Exploration of…
Rocky Lau Hardcover R1,776 Discovery Miles 17 760
Associative Algebraic Geometry
Arvid Siqveland Hardcover R3,822 Discovery Miles 38 220

See more

Partners