Andrew Wiles

Bio: Andrew Wiles is an academic researcher from University of Oxford. The author has contributed to research in topics: Modular form & Iwasawa theory. The author has an hindex of 17, co-authored 23 publications receiving 5049 citations. Previous affiliations of Andrew Wiles include Princeton University & Harvard University.

Modular elliptic curves and Fermat’s Last Theorem

Abstract: When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n > 2 such that an + bn = cn. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.

Ring-Theoretic Properties of Certain Hecke Algebras

Abstract: The purpose of this article is to provide a key ingredient of [W2] by establishing that certain minimal Hecke algebras considered there are complete intersections. As is recorded in [W2], a method going back to Mazur [M] allows one to show that these algebras are Gorenstein, but for the complete intersection property a new approach is required. The methods of this paper are related to those of Chapter 3 of [W2]. The methods of Section 3 of this paper are based on a previous approach of one of us (A.W.). We would like to thank Henri Darmon, Fred Diamond and Gerd Faltings for carefully reading the first version of this article. Gerd Faltings has also suggested a simplification of our argument as well as of the argument of Chapter 3 of [W2] and we would like to thank him for allowing us to reproduce these in the appendix to this paper. R. T. would like to thank A. W. for his invitation to collaborate and for sharing his many insights into the questions considered. R. T. would also like to thank Princeton University, Universite de Paris 7 and Harvard University for their hospitality during this collaboration. A. W. was supported by an NSF grant.

Class fields of abelian extensions of Q.

Abstract: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 0. Notation and preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 1. Iwasawa theory, p-adic L-functions, and Fitting ideals . . . . . . . . . . . . . . . . . . . . 191 2. Models and moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 3. A study of abelian varieties which are \"good\" quotients of Jz (N) . . . . . . . . . . . . . . 261 4. The cuspidal group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 5. The kernel of the Eisenstein ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Appendix: Fitting ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

The Iwasawa conjecture for totally real fields

Abstract: Let F be a totally real number field. Let p be a prime number and for any integer n let Fun denote the group of nth roots of unity. Let 41 be a p-adic valued Artin character for F and let F,, be the extension of F attached to 4, i.e., so that 4 is the character of a faithful representation of Gal(F,,/F). We will assume that F,, is also totally real. For a number field K let K., denote the cyclotomic Zp-extension of K. Following Greenberg we say that 4 is of type S if F., n Fc, = F and of type W if 4 is one-dimensional with F.,p c Fcc. Deligne and Ribet (in [DR], following Kubota and Leopoldt for the case F= Q) have proved the existence of a p-adic L-function associated to a one-dimensional Artin character 4 with F,, totally real. This function Lp(s, 4) is continuous for s e Zp{ 1}, and even at s = 1 if 4, is not trivial, and satisfies the following interpolation property:

The Arithmetic of Elliptic Curves

Abstract: Algebraic Varieties.- Algebraic Curves.- The Geometry of Elliptic Curves.- The Formal Group of Elliptic Curves.- Elliptic Curves over Finite Fields.- Elliptic Curves over C.- Elliptic Curves over Local Fields.- Elliptic Curves over Global Fields.- Integral Points on Elliptic Curves.-Computing the Mordell Weil Group.- Appendix A: Elliptic Curves in Characteristics.-Appendix B: Group Cohomology (H0 and H1).

Analytic Number Theory

Abstract: Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

A Course in Computational Algebraic Number Theory

Abstract: A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.

How to Solve It: Modern Heuristics

Abstract: I What Are the Ages of My Three Sons?.- 1 Why Are Some Problems Difficult to Solve?.- II How Important Is a Model?.- 2 Basic Concepts.- III What Are the Prices in 7-11?.- 3 Traditional Methods - Part 1.- IV What Are the Numbers?.- 4 Traditional Methods - Part 2.- V What's the Color of the Bear?.- 5 Escaping Local Optima.- VI How Good Is Your Intuition?.- 6 An Evolutionary Approach.- VII One of These Things Is Not Like the Others.- 7 Designing Evolutionary Algorithms.- VIII What Is the Shortest Way?.- 8 The Traveling Salesman Problem.- IX Who Owns the Zebra?.- 9 Constraint-Handling Techniques.- X Can You Tune to the Problem?.- 10 Tuning the Algorithm to the Problem.- XI Can You Mate in Two Moves?.- 11 Time-Varying Environments and Noise.- XII Day of the Week of January 1st.- 12 Neural Networks.- XIII What Was the Length of the Rope?.- 13 Fuzzy Systems.- XIV Everything Depends on Something Else.- 14 Coevolutionary Systems.- XV Who's Taller?.- 15 Multicriteria Decision-Making.- XVI Do You Like Simple Solutions?.- 16 Hybrid Systems.- 17 Summary.- Appendix A: Probability and Statistics.- A.1 Basic concepts of probability.- A.2 Random variables.- A.2.1 Discrete random variables.- A.2.2 Continuous random variables.- A.3 Descriptive statistics of random variables.- A.4 Limit theorems and inequalities.- A.5 Adding random variables.- A.6 Generating random numbers on a computer.- A.7 Estimation.- A.8 Statistical hypothesis testing.- A.9 Linear regression.- A.10 Summary.- Appendix B: Problems and Projects.- B.1 Trying some practical problems.- B.2 Reporting computational experiments with heuristic methods.- References.

Modular elliptic curves and Fermat’s Last Theorem

Abstract: When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n > 2 such that an + bn = cn. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.