# "Modular elliptic curves and Fermat's Last Theorem" - Sir Andrew Wiles

## "Modular elliptic curves and Fermat's Last Theorem" - Sir Andrew Wiles

Modular elliptic curves

and

Fermat’s Last Theorem

By ANDREW WILES

For Nada, Glare, Kate and Olivia

"Cubum autem in duos cubos, aut quadratoquadratum in

duos quadratoquadratos, et generaliter nullam in infinitum

ultra quadratum potestatem in duos ejusdem nominis fas est

dividere:

cujus rei demonstrationem mirabilem sane detexi.

Hanc

marginis

exiguitas

non caperet"

Introduction

An elliptic curve over Q is said to be modular if it has a finite covering by a

modular curve of the form Xc(S). Any such elliptic curve has the property

that its Hasse-Weil zeta function has an analytic continuation and satisfies a

functional equation of the standard type. If an elliptic curve over Q with a

given j-invariant is modular then it is easy to see that all elliptic curves with

the same j-invariant are modular (in which case we say that the j-invariant

is modular). A well-known conjecture which grew out of the work of Shimura

and Taniyama in the 1950’s and 1960’s asserts that every elliptic curve over Q

is modular. However, it only became widely known through its publication in a

paper of Weil in 1967 [We] (as an exercise for the interested reader!). in which,

moreover, IVeil gave conceptual evidence for the conjecture. Although it had

been numerically verified in many cases, prior to the results described in this

paper it had only been known that finitely many j-invariants u-ere modular.

111 1985 Frey made the remarkable observation that this conjecture should

imply Fermat’s Last Theorem. The precise mechanism relating the two was

formulated by Serre as the s-conjecture and this was then proved by Ribec in

the summer of 1936. Ribet‘s result only requires one to prove the conjecture

for semistable elliptic curves in order to deduce Fermat’s Last Theorem.

http://math.stanford.edu/~lekheng/flt/wiles.pdf

and

Fermat’s Last Theorem

By ANDREW WILES

For Nada, Glare, Kate and Olivia

"Cubum autem in duos cubos, aut quadratoquadratum in

duos quadratoquadratos, et generaliter nullam in infinitum

ultra quadratum potestatem in duos ejusdem nominis fas est

dividere:

cujus rei demonstrationem mirabilem sane detexi.

Hanc

marginis

exiguitas

non caperet"

Introduction

An elliptic curve over Q is said to be modular if it has a finite covering by a

modular curve of the form Xc(S). Any such elliptic curve has the property

that its Hasse-Weil zeta function has an analytic continuation and satisfies a

functional equation of the standard type. If an elliptic curve over Q with a

given j-invariant is modular then it is easy to see that all elliptic curves with

the same j-invariant are modular (in which case we say that the j-invariant

is modular). A well-known conjecture which grew out of the work of Shimura

and Taniyama in the 1950’s and 1960’s asserts that every elliptic curve over Q

is modular. However, it only became widely known through its publication in a

paper of Weil in 1967 [We] (as an exercise for the interested reader!). in which,

moreover, IVeil gave conceptual evidence for the conjecture. Although it had

been numerically verified in many cases, prior to the results described in this

paper it had only been known that finitely many j-invariants u-ere modular.

111 1985 Frey made the remarkable observation that this conjecture should

imply Fermat’s Last Theorem. The precise mechanism relating the two was

formulated by Serre as the s-conjecture and this was then proved by Ribec in

the summer of 1936. Ribet‘s result only requires one to prove the conjecture

for semistable elliptic curves in order to deduce Fermat’s Last Theorem.

http://math.stanford.edu/~lekheng/flt/wiles.pdf

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