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) - Image 1

£70.00

Princeton University Press 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)

Name: 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)
Brand: Princeton University Press
Price: 70.0 GBP
Availability: InStock

Algebra

Price History & Forecast

Last 31 days • 31 data points (No recent data available)

Historical

Generating forecast...

Price Distribution

Price distribution over 31 days • 1 price levels

Days at Price

Price Analysis

Most common price: £70 (31 days, 100.0%)

Price range: £70 - £70

Price levels: 1 different prices over 31 days

Description

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.

Product Specifications

Brand: Princeton University Press
Format: Paperback
ASIN: 0691142025
Category: Books > Subjects > Science, Nature & Maths > Mathematics > Algebra
Domain: Amazon UK
Release Date: 19 June 2011
Listed Since: 02 January 2009

Barcode

No barcode data available

Princeton University Press 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)

Price History & Forecast

Price Distribution

Price Analysis

Description

Product Specifications

Barcode

Similar Products You Might Like

Automorphic Representations Of Low Rank Groups

Moduli Stacks of Étale (ϕ, Γ)-Modules and the Existence of Crystalline Lifts: (AMS-215) (Annals of Mathematics Studies, 215)

Modular Forms and Special Cycles on Shimura Curves. (AM-161) (Annals of Mathematics Studies)

Number Theory (P): A Study in Analytic Number Theory and Computational Complexity (Canadian Mathematical Society Series of Monographs and Advan)

Theta Constants, Riemann Surfaces and the Modular Group: An Introduction with Applications to Uniformization Theorems, Partition Identities and ... ... Theory (Graduate Studies in Mathematics)

The Geometry and Cohomology of Some Simple Shimura Varieties.: 151 (Annals of Mathematics Studies, 151)

Derived Langlands: Monomial Resolutions of Admissible Representations: 15 (Series on Number Theory and Its Applications)

GEOMETRIC MODULAR FORMS AND ELLIPTIC CURVES (2ND EDITION)

Supersingular p-adic L-functions, Maass-Shimura Operators and Waldspurger Formulas: (AMS-212) (Annals of Mathematics Studies, 212)

Arithmetic and Geometry: Ten Years in Alpbach - Princeton Press

Commensurabilities among Lattices in PU (1, n). (AM-132) (Annals of Mathematics Studies, 132)

Cohomology of Drinfeld Modular Varieties, Part 2, Automorphic Forms, Trace Formulas and Langlands Correspondence: 56 (Cambridge Studies in Advanced Mathematics, Series Number 56)

Supersingular p-adic L-functions, Maass-Shimura Operators and Waldspurger Formulas: (AMS-212) (Annals of Mathematics Studies, 212)

Rigid Character Groups, Lubin-Tate Theory, and $( arphi ,Gamma )$-Modules (Memoirs of the American Mathematical Society)

Arithmetic Compactifications of PEL-Type Shimura Varieties (London Mathematical Society Monographs)

The Absolute Galois Group of a Semi-Local Field (Springer Monographs in Mathematics)

Number Theory: Algebraic Numbers and Functions (Graduate Studies in Mathematics)

Advanced Analytic Number Theory: L-functions (Mathematical Surveys and Monographs)

Princeton University Press - The Gross-Zagier Formula on Shimura Curves

Modular Forms: A Classical Approach (Graduate Studies in Mathematics)

Alegebraic Curves and Cryptography (Fields Institute Communications) (Fields Institute Monographs)

The Gross-Zagier Formula on Shimura Curves (Annals of Mathematics Studies): (AMS-184)

Elementary Modular Iwasawa Theory: 16 (Series On Number Theory And Its Applications)

Effective Methods in Algebraic Geometry: 94 (Progress in Mathematics)