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

It has never been this cheap. We have no record of a lower price.

£70 today · cheaper than every other day in the last 5 months

NEW HERE?

Amazon shows you one price. We show you all of them.

Tosheroon watches Amazon prices so you don't have to. Every product on Amazon has a price history — we make it visible. Set the price you'd actually pay, and we'll email you the second it gets there. No app, no account, one email.

WHAT'S ON THIS PAGE

↓ Price chart
when this has been cheap or pricey

↓ Forecast
where the price is heading next

↓ Statistics
all-time high & low, recent range

↑ Price alert
name your number, we'll email you

Price History & Forecast

Grey patches = out of stock. Cheaper = lower on the chart. Hover for exact prices.

Last 60 days • 60 data points (No recent data available)

Historical

Generating forecast...

Price Distribution

Price distribution over 60 days • 1 price levels

Days at Price

Price Analysis

Most common price: £70 (60 days, 100.0%)

Price range: £70 - £70

Price levels: 1 different prices over 60 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)

It has never been this cheap. We have no record of a lower price.

Price History & Forecast

Price Distribution

Price Analysis

Description

Product Specifications

Barcode

Similar Products You Might Like

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

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

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

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

Arithmetic Fundamental Groups and Noncommutative Algebra Book

GEOMETRIC MODULAR FORMS AND ELLIPTIC CURVES (2ND EDITION)

Galois Groups over ?: Proceedings of a Workshop Held March 23–27, 1987: 16 (Mathematical Sciences Research Institute Publications, 16)

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

Inverse Galois Theory (Springer Monographs in Mathematics)

A Classical Introduction to Galois Theory

Galois Theory (Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts)

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

Elementary Galois Theory: A Constructive Approach: 3 (Mathematics Study Resources, 3)

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

Ottawa Lectures on Admissible Representations of Reductive P-adic Groups (Fields Institute Monographs)

Cyclotomic Fields I and II: 121 (Graduate Texts in Mathematics, 121)

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

Automorphic Representations Of Low Rank Groups

Lectures in Abstract Algebra: III. Theory of Fields and Galois Theory: 32 (Graduate Texts in Mathematics, 32)

Local Fields: 67 (Graduate Texts in Mathematics, 67)

Traces of Hecke Operators (Mathematical Surveys & Monographs) (Mathematical Surveys and Monographs)

L-Functions and Galois Representations: 320 (London Mathematical Society Lecture Note Series, Series Number 320)

Inverse Galois Theory (Springer Monographs in Mathematics)

Algebraic Number Fields (Graduate Studies in Mathematics)