Proof of Fermat's Last Theorem

Proof of Fermat's Last Theorem
Name	Proof of Fermat's Last Theorem
Caption	Sir Andrew Wiles, 1997
Date	1994 (completed proof)
Location	Princeton University; Cambridge University
Principal	Andrew Wiles
Collaborators	Richard Taylor
Field	Number theory

Proof of Fermat's Last Theorem

Fermat's Last Theorem asserts that no nonzero integer solutions exist for xn + yn = zn with integer n > 2, a conjecture famously stated by Pierre de Fermat. The eventual proof by Sir Andrew Wiles built on deep work by Gerhard Frey, Jean-Pierre Serre, Ken Ribet, Goro Shimura, and others, linking questions about Diophantine equations to the theory of elliptic curves, modular forms, and Galois representations.

Statement of the theorem

Fermat's Last Theorem states that for integer exponent n > 2, the equation xn + yn = zn has no nontrivial integer solutions. The statement originates with Pierre de Fermat and was communicated in marginalia connected to his correspondence with Basile,Claude-Gaspar Bachet de Méziriac and later circulated among mathematicians such as Marin Mersenne and Christian Huygens. The conjecture stimulated work by Leonhard Euler, Sophie Germain, Adrien-Marie Legendre, Peter Gustav Lejeune Dirichlet, Ernst Kummer, Richard Dedekind, and David Hilbert.

Historical attempts and partial results

Efforts toward Fermat's Last Theorem spanned centuries, with milestones by many figures. Leonhard Euler proved the case n = 3, while Sophie Germain developed techniques for primes p where 2p+1 is prime, influencing later work by Adrien Sophie Germain allies and critics such as Ernst Kummer, who introduced ideal theory addressing regular primes. Kummer proved many cases using concepts later formalized by Richard Dedekind and Leopold Kronecker. The 20th century saw contributions from Ernst Selmer, Harold Davenport, Hans Heilbronn, Kurt Heegner, Alan Baker, Louis Mordell, Goro Shimura, Yutaka Taniyama, and Yoshitaka Manin, culminating in conjectures by Taniyama and Shimura that linked elliptic curves and modular forms. Work by Jean-Pierre Serre and Ken Ribet connected the hypothetical Frey curve of Gerhard Frey to a failure of modularity, setting the stage for a proof that would make use of results from Nicholas Katz, Barry Mazur, and John Coates.

Overview of Wiles' strategy

Wiles' strategy reduced Fermat's Last Theorem to a statement about modularity: show that all semistable elliptic curves over the rationals are modular. This reduction relied on the proposed Taniyama–Shimura–Weil conjecture (commonly called the Modularity Theorem) and a link via the Frey curve proposed by Gerhard Frey and proved conditional by Ken Ribet that nontrivial Fermat solutions would produce a nonmodular elliptic curve. Wiles combined techniques from Iwasawa theory, deformation theory, and the study of Galois representations pioneered by Jean-Pierre Serre and Barry Mazur to establish modularity for a class of elliptic curves.

Key ingredients: modular forms, elliptic curves, and Galois representations

The proof invokes several advanced theories. Elliptic curves over Q provide algebraic geometry objects studied by André Weil and Goro Shimura, while modular forms—classical objects from the work of Srinivasa Ramanujan, Erich Hecke, and Atkin and Lehner—encode analytic properties tied to L-series. Galois representations arose from work by Évariste Galois and were developed in arithmetic contexts by Serre, Jean-Marc Fontaine, Barry Mazur, and Richard Taylor. Tools from algebraic geometry as advanced by Alexander Grothendieck, Jean-Pierre Serre (again), and Pierre Deligne were essential, alongside the theory of Hecke algebras studied by Haruzo Hida and Nicholas Katz, and deformation techniques elaborated by Mazur and Richard Taylor.

The Modularity Theorem and its role

The Modularity Theorem (formerly the Taniyama–Shimura–Weil conjecture) asserts that every elliptic curve over Q is modular, meaning its L-series matches that of a modular form. Early formulations by Yutaka Taniyama and Goro Shimura were refined through work by André Weil and Hecke theory, and the conjecture became central after Gerhard Frey suggested that a counterexample to Fermat would yield a nonmodular elliptic curve. Ken Ribet proved Frey's criterion linking a hypothetical Fermat solution to nonmodularity, so proving modularity for semistable elliptic curves over Q implied Fermat's Last Theorem. Wiles proved modularity for semistable curves, leveraging techniques from Iwasawa theory developed by John Coates and Andrew Wiles himself, and later work extended full modularity by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor.

Proof outline and main steps

Wiles' proof proceeds by establishing modularity lifting theorems for certain Galois representations attached to elliptic curves. He constructed deformation rings for residual representations following ideas of Barry Mazur and matched them to appropriate Hecke algebras, a technique echoing the Taylor–Wiles method developed with Richard Taylor. Key inputs included the study of local deformation conditions inspired by Jean-Marc Fontaine and William Messing, control theorems related to Iwasawa theory from Barry Mazur and Ken Ribet, and patching arguments that applied commutative algebra from David Eisenbud and homological methods influenced by Pierre Deligne. After an initial manuscript in 1993 containing a gap, Wiles, with Richard Taylor, repaired the argument in 1994 using an improved patching argument and the introduction of new auxiliary primes, completing the proof that semistable elliptic curves over Q are modular and thus resolving Fermat's Last Theorem.

Aftermath and subsequent developments

The proof transformed number theory, inspiring advances by many researchers. Subsequent work by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor completed the proof of the Modularity Theorem for all elliptic curves over Q, a result announced in the mid-2000s. The Taylor–Wiles method influenced progress on the Fontaine–Mazur conjecture, Serre's conjectures (settled in many cases by Chandrashekhar Khare and Jean-Pierre Wintenberger), and reciprocity laws in the Langlands program advanced by Robert Langlands. The proof also affected computational projects involving John Cremona's tables of elliptic curves, inspired public interest via biographies of Andrew Wiles and histories involving figures like Simon Singh, and led to awards for many contributors including the Abel Prize and knighthood for Andrew Wiles.

Category:Fermat's Last Theorem