Andrew Wiles

How to use this course

Fermat’s Last Theorem says that \(x^n+y^n=z^n\) has no solutions in positive integers when \(n\ge3\). It resisted proof for more than 350 years. Andrew Wiles proved it in 1994, with a final step completed jointly with his former student Richard Taylor. The proof did not attack the equation directly. It established a large case of a conjecture linking two very different kinds of object, elliptic curves and modular forms. That link is the real achievement, and it changed number theory.

Each technical lesson opens with a paragraph for the general reader, then states results precisely, sketches the ideas of proof and ends with exercises with folded solutions. Lessons 04 to 06 can be read with undergraduate background. Lessons 07 to 10 are a guided tour of the proof: full details fill hundreds of pages, and the references point to the best expositions. Wiles is living; this course covers his mathematics, career and public honours only.

Timeline

Cambridge and a library book, 1953–1971

Andrew John Wiles was born in Cambridge on 11 April 1953. His father, Maurice Wiles, was an Anglican priest and theologian who later held the Regius Chair of Divinity at Oxford.

Wiles attended King’s College School in Cambridge and then The Leys School. He has told the story of how his interest began many times. Aged about ten, on the way home from school, he went into the local public library and found Eric Temple Bell’s book The Last Problem. It described Fermat’s claim about \(x^n+y^n=z^n\). The statement was simple enough for a ten-year-old to understand, and the book said nobody had proved it in three centuries. He decided then that he would solve it.

As a teenager he tried the methods that Pierre de Fermat himself might have used, and read about the nineteenth-century attempts. A teacher at The Leys with a doctorate in number theory encouraged him. These attempts did not succeed, and as a university student he set the problem aside.

Oxford, Cambridge and John Coates, 1971–1977

Wiles read mathematics at Merton College, Oxford, from 1971 and graduated in 1974. He then went to Clare College, Cambridge, for doctoral work. In the summer of 1975 he began working with John Coates, an Australian number theorist who had just arrived in Cambridge.

Fermat then looked inaccessible to serious research, and Coates introduced Wiles to the arithmetic of elliptic curves instead. The training turned out to be exactly what the problem would later need. The two worked on curves with complex multiplication, using the tools of Iwasawa theory and explicit reciprocity laws.

The collaboration quickly produced a major result. In 1977 Coates and Wiles proved the first general theorem toward the conjecture of Bryan Birch and Peter Swinnerton-Dyer: for a class of elliptic curves with complex multiplication, if the L-function does not vanish at \(s=1\), then the curve has only finitely many rational points (Lesson 05). Wiles was elected a Junior Research Fellow at Clare in 1977. His thesis, Reciprocity laws and the conjecture of Birch and Swinnerton-Dyer, earned the Cambridge PhD in 1980.

Harvard, Princeton and Iwasawa theory, 1977–1986

From 1977 to 1980 Wiles combined the Clare fellowship with a Benjamin Peirce Assistant Professorship at Harvard. There he began working with Barry Mazur on modular curves and Iwasawa theory. After spending 1981 at the Sonderforschungsbereich in Bonn and the Institute for Advanced Study in Princeton, he was appointed professor at Princeton University in 1982. He remained there for nearly three decades.

In 1984 Mazur and Wiles published Class fields of abelian extensions of \(\mathbb Q\), a proof of the main conjecture of Iwasawa theory over the rationals (Lesson 06). It related the class groups of cyclotomic fields to \(p\)-adic L-functions, and its method, which studied the Galois representations cut out by congruences between modular forms, foreshadowed Wiles’s later work. He spent 1985–86 as a Guggenheim Fellow in Paris, at the IHÉS and the École Normale Supérieure.

Then, in the summer of 1986, he heard that Ken Ribet had proved a link between Fermat’s Last Theorem and the Taniyama–Shimura conjecture on modularity of elliptic curves (Lesson 08). To prove Fermat, it would now be enough to prove that every semistable elliptic curve over \(\mathbb Q\) is modular. That was a problem in exactly the area in which Wiles was expert. He decided to work on it.

Fermat’s problem before 1985

For the general reader. Around 1637 Pierre de Fermat wrote in the margin of his copy of Diophantus that a cube cannot be split into two cubes, nor any higher power into two like powers, and that he had a marvellous proof which the margin was too narrow to contain. The note was published by his son in 1670. Fermat did prove the case of fourth powers. For three centuries mathematicians proved more cases, and each method stopped short of the whole.

If \(m\mid n\), a solution for \(n\) gives one for \(m\): \((x^{n/m})^m+(y^{n/m})^m=(z^{n/m})^m\). Every \(n\ge3\) is divisible by 4 or by an odd prime. So it suffices to treat \(n=4\) and \(n=p\) an odd prime.

Descent for \(n=4\)

Sketch. Take a solution with \(z\) minimal; then \(\gcd(x,y)=1\) and \((x^2,y^2,z)\) is a primitive Pythagorean triple. With \(y\) even, \(x^2=m^2-n^2\), \(y^2=2mn\), \(z=m^2+n^2\), with \(\gcd(m,n)=1\) and \(m\) odd, \(n\) even. Then \((x,n,m)\) is again a primitive triple, so \(n=2rs\), \(m=r^2+s^2\). From \(y^2=2mn=4mrs\) with \(m,r,s\) pairwise coprime, each of \(m,r,s\) is a square: \(m=w^2\), \(r=u^2\), \(s=v^2\). Then \(u^4+v^4=w^2\) with \(w\le w^2=m<z\). This contradicts minimality. The method, building a smaller solution from a larger one, is Fermat’s infinite descent.

Odd primes

Leonhard Euler treated \(n=3\) (1770, with a gap later repaired). Peter Gustav Lejeune Dirichlet and Adrien-Marie Legendre settled \(n=5\) around 1825, and Gabriel Lamé \(n=7\) in 1839. Sophie Germain proved the “first case” (\(p\nmid xyz\)) for many primes. In 1847 Lamé proposed factoring \(x^p+y^p=\prod_{k}(x+\zeta^k y)\) in \(\mathbb Z[\zeta]\), \(\zeta=e^{2\pi i/p}\), assuming unique factorization. Ernst Kummer had already found that unique factorization fails, and invented ideal numbers to repair it.

The irregular primes, starting 37, 59, 67, 101, are infinite in number (K. L. Jensen, 1915). Kummer and his successors found further criteria for irregular primes, and by 1993 computers had checked all prime exponents below four million, but no method covered every prime.

Elliptic curves and the Coates–Wiles theorem

For the general reader. An elliptic curve is the set of solutions of a cubic equation like \(y^2=x^3+ax+b\). Its points can be added: draw a line through two points, find the third point where it meets the curve, and reflect. The rational points form a group, and the central question is how large it is. Wiles’s first major theorem, with John Coates, gave the first real evidence for the Birch and Swinnerton-Dyer conjecture, which predicts the answer.

Definitions

An elliptic curve over \(\mathbb Q\) can be written \(E: y^2=x^3+ax+b\) with \(a,b\in\mathbb Q\) and \(\Delta=-16(4a^3+27b^2)\ne0\), together with a point \(O\) at infinity. The chord-and-tangent rule makes \(E(\mathbb Q)\) an abelian group with identity \(O\).

The theorem is due to Louis Mordell and André Weil. Take a minimal integral model. For a prime \(p\) of good reduction, set \(a_p=p+1-\#E(\mathbb F_p)\). Helmut Hasse proved \(|a_p|\le2\sqrt p\). The conductor \(N\) is divisible exactly by the primes of bad reduction; \(E\) is semistable if \(N\) is squarefree, meaning that at each bad prime the reduction is a nodal cubic. The L-function is

convergent for \(\operatorname{Re}s>3/2\). Its continuation to \(s=1\) was known for curves with complex multiplication and, after Wiles, follows for all curves from modularity (Lesson 07).

Idea. Complex multiplication makes the Galois action on torsion points of \(E\) abelian over \(K\), so class field theory over \(K\) describes it. Coates and Wiles chose a prime \(p\) that splits in \(K\) and studied the tower of fields generated by \(p\)-power torsion points. Elliptic units in this tower, through an explicit reciprocity law, produce a \(p\)-adic quantity that is a non-zero multiple of \(L(E,1)\). If \(L(E,1)\ne0\), this forces a certain Selmer group to be finite, and so the rank to be 0. Benedict Gross, Don Zagier and Victor Kolyvagin later extended such results to all modular elliptic curves of analytic rank 0 or 1.

The main conjecture of Iwasawa theory

For the general reader. Kummer saw that whether Fermat’s method works for a prime \(p\) depends on a class group, which measures failure of unique factorization, and on Bernoulli numbers, which are values of the Riemann zeta function. Kenkichi Iwasawa conjectured a precise identity between the two, uniformly up a tower of fields. Mazur and Wiles proved it in 1984.

Setting

Fix an odd prime \(p\). Let \(A_n\) be the \(p\)-part of the class group of \(\mathbb Q(\zeta_{p^{n+1}})\), and \(X=\varprojlim A_n\). The Galois group \(\mathrm{Gal}(\mathbb Q(\zeta_{p^\infty})/\mathbb Q)\cong\mathbb Z_p^\times\cong\mu_{p-1}\times(1+p\mathbb Z_p)\) acts on \(X\). The \(\mu_{p-1}\) part splits \(X\) into eigenspaces \(X^{(i)}\) for the powers \(\omega^i\) of the Teichmüller character, and each is a finitely generated torsion module over the Iwasawa algebra \(\Lambda=\mathbb Z_p[[T]]\). By the structure theory of such modules, each has a characteristic ideal \(\mathrm{char}(X^{(i)})=(g_i(T))\).

On the analytic side, Tomio Kubota and Heinrich-Wolfgang Leopoldt constructed \(p\)-adic L-functions \(L_p(s,\omega^{j})\) interpolating values of Dirichlet L-functions, and Iwasawa showed they come from power series in \(\Lambda\).

A finite-level consequence refines a classical theorem of Jacques Herbrand and Ken Ribet.

Idea. Herbrand’s direction uses the theorem of Ludwig Stickelberger on annihilators of class groups. For the converse, Ribet used an Eisenstein series congruent to a cusp form modulo \(p\) when \(p\mid B_{p-i}\); the Galois representation of the cusp form, reduced mod \(p\), produces an unramified extension of \(\mathbb Q(\zeta_p)\) of the predicted type. Mazur and Wiles carried this out in the whole tower, studying the Jacobians of modular curves \(X_1(Np^n)\), to show \(g_i\) divides the \(p\)-adic L-function; the analytic class number formula then forces equality. Wiles extended the theorem to totally real fields in 1990. Seeing Galois representations through congruences of modular forms is the thread that leads to Lesson 09.

Modular forms and the modularity conjecture

For the general reader. Modular forms are functions on the upper half-plane with an enormous amount of symmetry. Their Fourier coefficients often encode arithmetic. The modularity conjecture, which grew from questions of Yutaka Taniyama in 1955 and was sharpened by Goro Shimura and André Weil, says that the point counts \(a_p\) of every elliptic curve over \(\mathbb Q\) are the Fourier coefficients of a modular form. Wiles proved it for semistable curves.

Cusp forms

Let \(\Gamma_0(N)=\{\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{SL}_2(\mathbb Z):N\mid c\}\). A cusp form of weight 2 and level \(N\) is a holomorphic \(f\) on \(\mathbb H=\{\operatorname{Im}\tau>0\}\) with \(f\big(\frac{a\tau+b}{c\tau+d}\big)=(c\tau+d)^2f(\tau)\) for all such matrices, vanishing at every cusp. Then \(f(\tau)=\sum_{n\ge1}a_nq^n\) with \(q=e^{2\pi i\tau}\). The space \(S_2(\Gamma_0(N))\) is finite-dimensional, of dimension equal to the genus of the modular curve \(X_0(N)\). Hecke operators \(T_p\) act on it, and a newform is a normalized simultaneous eigenform (\(a_1=1\)) not coming from lower level; for it, \(T_pf=a_pf\).

Consequences: \(L(E,s)\) continues to an entire function and satisfies a functional equation relating \(s\) and \(2-s\), so the Birch and Swinnerton-Dyer conjecture makes sense for every curve over \(\mathbb Q\).

The example of level 11

\(X_0(11)\) has genus 1, so \(S_2(\Gamma_0(11))\) is one-dimensional, spanned by

The curve \(E: y^2+y=x^3-x^2\) has conductor 11, and its point counts give \(a_2=-2\), \(a_3=-1\), \(a_5=1\), \(a_7=-2\), \(a_{13}=4\), \(a_{17}=-2\), matching the coefficients of \(f\).

Frey, Serre and Ribet: from Fermat to modularity

For the general reader. In 1984 Gerhard Frey suggested a startling idea. A solution to Fermat’s equation would produce an elliptic curve so strange that it could not be modular. Jean-Pierre Serre identified exactly what was needed to make this precise, and in 1986 Ken Ribet proved it. After Ribet, Fermat’s Last Theorem was a consequence of the modularity conjecture for semistable curves.

The Frey curve

Suppose \(a^p+b^p=c^p\) with \(p\ge5\) prime and \(a,b,c\) coprime non-zero integers. After permuting and changing signs we may assume \(a^p\equiv-1\pmod4\) and \(b\) even. Consider

Its discriminant is \(16\,(a^pb^pc^p)^2\). The curve was studied by Yves Hellegouarch in the 1970s; Frey saw its significance for modularity.

For an odd prime \(\ell\mid abc\), exactly two of the three roots \(0,a^p,-b^p\) collide mod \(\ell\), giving a node. At 2, since \(b\) is even and \(p\ge5\), \(32\mid b^p\), and with \(a^p\equiv-1\pmod 4\) a change of variables gives a minimal model with multiplicative reduction. For every odd prime \(\ell\) of bad reduction, \(v_\ell(\Delta_{\min})=2p\,v_\ell(abc)\) is divisible by \(p\). At 2 the exponent \(2p\,v_2(b)-8\) need not be.

Mod-\(p\) representations

The \(p\)-torsion points \(E[p]\cong(\mathbb Z/p)^2\) carry an action of \(G_{\mathbb Q}=\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)\), giving \(\bar\rho_{E,p}:G_{\mathbb Q}\to\mathrm{GL}_2(\mathbb F_p)\). For a semistable curve and \(\ell\ne p\), John Tate’s uniformization of curves with multiplicative reduction shows \(\bar\rho_{E,p}\) is unramified at \(\ell\) whenever \(p\mid v_\ell(\Delta_{\min})\). For the Frey curve this holds at every odd prime \(\ell\ne p\) dividing \(abc\). At \(p\) itself the representation is “finite flat” (it comes from a finite flat group scheme), by the same divisibility if \(p\mid abc\) and by good reduction otherwise. So the only possible bad prime left is 2: Serre’s recipe predicts that \(\bar\rho_{E,p}\) should come from a modular form of level 2.

The deduction

1. Suppose the Frey curve \(E\) is modular, attached to a newform of level \(N=\mathrm{rad}(abc)\).
2. \(\bar\rho_{E,p}\) is irreducible: otherwise \(E\) would have a rational subgroup of order \(p\). Since all 2-torsion points of \(E\) are rational, this is ruled out for \(p\ge5\) by Barry Mazur’s theorems on rational torsion and isogenies (1977–78).
3. Apply Ribet’s theorem repeatedly to remove every odd prime from \(N\), including \(p\) if it divides \(abc\). The result is a newform of weight 2 and level 2.
4. But \(S_2(\Gamma_0(2))=0\) (Exercise 7.1). Contradiction.

So: modularity of semistable elliptic curves implies Fermat’s Last Theorem for primes \(p\ge5\). The cases \(p=3\) and \(n=4\) were already known.

Galois deformations and R = T

For the general reader. To prove an elliptic curve is modular, Wiles worked with its symmetries under the Galois group, packaged as a representation. He started from the known case of its reduction modulo a prime. Then he showed that every way of lifting that reduction, subject to natural local conditions, comes from a modular form. The key statement is that two rings are equal: one parametrizes all the lifts, the other parametrizes the modular ones.

Galois representations

The \(p\)-adic Tate module \(T_pE=\varprojlim E[p^n]\cong\mathbb Z_p^2\) gives a continuous representation \(\rho_{E,p}:G_{\mathbb Q}\to\mathrm{GL}_2(\mathbb Z_p)\). For primes \(\ell\nmid Np\) it is unramified at \(\ell\) and

where \(\chi_p\) is the \(p\)-adic cyclotomic character. Martin Eichler and Goro Shimura attached similar representations \(\rho_{f,\lambda}\) to newforms. So \(E\) is modular if and only if \(\rho_{E,p}\cong\rho_{f,\lambda}\) for some newform \(f\), for one (equivalently every) prime \(p\).

Deformations and Hecke algebras

Fix \(\bar\rho:G_{\mathbb Q}\to\mathrm{GL}_2(\mathbb F)\), absolutely irreducible. Barry Mazur (1989) showed that lifts of \(\bar\rho\) to complete local rings, with prescribed local behaviour at a finite set \(\Sigma\) of primes, are classified by a universal deformation ring \(R_\Sigma\). The modular lifts are classified by a Hecke algebra \(\mathbb T_\Sigma\), generated by Hecke operators acting on a suitable space of modular forms. If \(\bar\rho\) is modular, there is a natural surjection \(R_\Sigma\twoheadrightarrow\mathbb T_\Sigma\).

The numerical criterion

Let \(\mathcal O\) be the ring of integers of a finite extension of \(\mathbb Q_p\). A modular form gives a homomorphism \(\pi:\mathbb T_\Sigma\to\mathcal O\), and so \(\pi_R:R_\Sigma\to\mathcal O\). Write \(\mathfrak p_R=\ker\pi_R\) and let \(\eta_{\mathbb T}\) be the congruence ideal, measuring congruences between \(f\) and other forms. One always has \(\#\mathcal O/\eta_{\mathbb T}\le\#\mathfrak p_{\mathbb T}/\mathfrak p_{\mathbb T}^2\le\#\mathfrak p_R/\mathfrak p_R^2\). Wiles proved a commutative-algebra criterion: if \(\#\mathfrak p_R/\mathfrak p_R^2\le\#\mathcal O/\eta_{\mathbb T}\), then \(R\to\mathbb T\) is an isomorphism of complete intersections. The left side is the size of a Selmer group, a Galois cohomology group bounding deformations; the right side relates to a special value of an L-function. So R = T becomes a case of a Birch and Swinnerton-Dyer-type formula for the adjoint representation.

Patching, the 3–5 switch, and after

For the general reader. The final proof needs three things: a starting point where the mod-\(p\) representation is known to be modular, a way to bound the Selmer group, and a trick to handle the curves where the starting point fails. The bound came from the Taylor–Wiles method, which adds carefully chosen auxiliary primes. The trick is the 3–5 switch.

Starting at 3

The starting point is a theorem of Robert Langlands and Jerrold Tunnell.

\(\mathrm{GL}_2(\mathbb F_3)\) has order 48 and is solvable. It embeds in \(\mathrm{GL}_2(\mathbb Z[\sqrt{-2}])\), so \(\bar\rho_{E,3}\) lifts to a complex representation, and Langlands–Tunnell together with a congruence argument shows that \(\bar\rho_{E,3}\) is modular. Thus if \(\bar\rho_{E,3}\) satisfies the irreducibility hypothesis, the theorem of Lesson 09 with \(p=3\) shows \(E\) is modular.

The 3–5 switch

If \(\bar\rho_{E,3}\) fails the hypothesis, Wiles showed that \(\bar\rho_{E,5}\) satisfies it, using the fact that the modular curve \(X_0(15)\) has only finitely many rational points, none giving a semistable curve of the relevant kind. Wiles found a second semistable curve \(E'\) with \(E'[5]\cong E[5]\) as Galois modules and with \(\bar\rho_{E',3}\) good. Such curves are rational points on a twist of the modular curve \(X(5)\), which has genus 0 and so has infinitely many, and Hilbert’s irreducibility theorem picks one with the right mod-3 behaviour. Then \(E'\) is modular by the \(p=3\) argument, so \(\bar\rho_{E,5}=\bar\rho_{E',5}\) is modular, and the theorem with \(p=5\) shows \(E\) is modular.

Bounding the Selmer group

Wiles’s 1993 argument bounded the Selmer group using an Euler system, extending work of Matthias Flach; that step failed. The repair, with Richard Taylor, works in the minimal case by adjoining sets \(Q\) of auxiliary primes \(q\equiv1\pmod{p^n}\), chosen so that the Selmer groups for \(\Sigma\cup Q\) are controlled. The rings \(R_Q\) and \(\mathbb T_Q\) become algebras over a power series ring, and passing to a limit over \(n\) (“patching”) shows that \(\mathbb T\) is a complete intersection and \(R=\mathbb T\). Wiles then reached the non-minimal cases by the numerical criterion, using level-raising congruences.

Fred Diamond (1996) relaxed the semistability hypotheses. In 2001 Christophe Breuil, Brian Conrad, Diamond and Taylor proved modularity of every elliptic curve over \(\mathbb Q\). Modularity lifting theorems of Taylor–Wiles type became the main tool of the Langlands programme, used in the proofs of the Sato–Tate conjecture and of Serre’s modularity conjecture by Chandrashekhar Khare and Jean-Pierre Wintenberger.

Seven years, a gap and a repair, 1986–1995

After Ribet’s result, Wiles set aside most other research and worked on modularity largely alone. He continued to teach and publish papers from earlier projects, so that colleagues would not guess what he was doing. He told almost no one. The first years went into finding the right starting point; he settled on the mod-3 representation and Langlands–Tunnell, and on Mazur’s deformation theory. The hardest part was bounding the Selmer group, for which he tried Iwasawa theory and then an Euler system of the kind Kolyvagin and Flach had introduced. In 1988–90 he spent two years in Oxford as a Royal Society Research Professor.

By early 1993 he believed the argument complete. He confided in his Princeton colleague Nicholas Katz, who audited it through a graduate course. In June 1993 Wiles gave three lectures at the Isaac Newton Institute in Cambridge, under a deliberately bland title on modular forms, elliptic curves and Galois representations. At the end of the third, on 23 June, he wrote Fermat’s Last Theorem on the board as a corollary. The news reached newspapers around the world.

The manuscript went to six referees. In the late summer Katz found a problem in the Euler system bound. In December 1993 Wiles acknowledged publicly that a gap remained. In January 1994 Richard Taylor, his former student, came to Princeton to work with him. On 19 September 1994, reconsidering why the earlier Iwasawa-theoretic approach had failed, Wiles saw that it could be combined with the Euler system idea to give exactly what was needed. In October he circulated two manuscripts, Modular elliptic curves and Fermat’s Last Theorem and, with Taylor, Ring-theoretic properties of certain Hecke algebras. Both appeared in the Annals of Mathematics in May 1995.

Honours, Oxford and legacy

The proof brought honours on a scale rare in mathematics. In 1995 and 1996 Wiles received the Rolf Schock Prize, the Prix Fermat, the Ostrowski Prize, the Wolf Prize (shared with Robert Langlands), the Royal Medal of the Royal Society and the NAS Award in Mathematics. In 1997 he received the Cole Prize and the Wolfskehl Prize, endowed in 1908 for a proof of Fermat’s theorem. Because he was over 40 at the 1998 International Congress of Mathematicians in Berlin, he was ineligible for a Fields Medal; the International Mathematical Union gave him a special silver plaque instead. He was knighted in 2000 and received the Shaw Prize in 2005.

At Princeton he became Eugene Higgins Professor in 1994. His doctoral students there and earlier at Harvard include Karl Rubin, Richard Taylor, Fred Diamond, Ehud de Shalit, Brian Conrad, Christopher Skinner, Vinayak Vatsal, Manjul Bhargava (Fields Medal 2014) and Ritabrata Munshi. Skinner and Wiles extended modularity lifting to residually reducible representations. Around 2010–11 Wiles returned to Oxford as a Royal Society Research Professor. The university’s new Mathematical Institute, opened in 2013, is the Andrew Wiles Building.

In 2016 he received the Abel Prize, presented in Oslo on 24 May, for his proof of Fermat’s Last Theorem through the modularity conjecture for semistable elliptic curves; the committee described the work as the start of a new era in number theory. The Royal Society awarded him the Copley Medal in 2017. In 2018 he became the first Regius Professor of Mathematics at Oxford, and in 2019 the London Mathematical Society awarded him its De Morgan Medal.

Legacy

• Fermat: the most famous open problem in number theory was solved, by methods that Fermat could not have imagined.
• Modularity: the bridge between elliptic curves and modular forms, a central case of the Langlands programme, is now a theorem for all elliptic curves over \(\mathbb Q\), with extensions to many number fields.
• Method: the identification \(R=\mathbb T\) and Taylor–Wiles patching are the standard tools for proving that Galois representations are automorphic.
• Iwasawa theory: the Mazur–Wiles and Wiles main conjectures and the Coates–Wiles theorem underpin much current work on the Birch and Swinnerton-Dyer conjecture.

Sources and further reading

Biography and honours

1. J. J. O’Connor and E. F. Robertson, Andrew Wiles, MacTutor History of Mathematics.
2. Andrew Wiles, Wikipedia.
3. The Abel Prize, 2016 laureate: Sir Andrew J. Wiles, with citation and press release.
4. University of Oxford, Sir Andrew Wiles.
5. S. Singh, Fermat’s Last Theorem, Fourth Estate, 1997.

Mathematics

1. J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39 (1977), 223–251.
2. B. Mazur and A. Wiles, Class fields of abelian extensions of \(\mathbb Q\), Invent. Math. 76 (1984), 179–330.
3. A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. 131 (1990), 493–540.
4. K. Ribet, On modular representations of \(\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)\) arising from modular forms, Invent. Math. 100 (1990), 431–476.
5. A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. of Math. 141 (1995), 443–551.
6. R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. 141 (1995), 553–572.
7. C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the modularity of elliptic curves over \(\mathbb Q\): wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), 843–939.
8. H. Darmon, F. Diamond and R. Taylor, Fermat’s Last Theorem, Current Developments in Mathematics, 1995.
9. G. Cornell, J. H. Silverman and G. Stevens (eds.), Modular Forms and Fermat’s Last Theorem, Springer, 1997.
10. F. Diamond and J. Shurman, A First Course in Modular Forms, Springer, 2005.
11. L. Washington, Introduction to Cyclotomic Fields, 2nd ed., Springer, 1997 (Iwasawa theory and the main conjecture).