Taniyama-Shimura theorem

Taniyama-Shimura theorem

For any

natural number

N

≥

1

, define the

modular group

Γ

0

⁢

(

N

)

to be the following subgroup of the group

SL

⁡

(

2

,

ℤ

)

of

integer

coefficient matrices of

determinant

1:

Γ

0

⁢

(

N

)

:=

{

(

a

b

c

d

)

∈

SL

⁡

(

2

,

ℤ

)

|

c

≡

0

(

mod

N

)

}

.

Let

ℍ

*

be the subset of the

Riemann sphere

consisting of all
points in the

upper half plane

(i.e.,

complex numbers

with strictly

positive

imaginary part

), together with the

rational numbers

and the
point at infinity. Then

Γ

0

⁢

(

N

)

acts on

ℍ

*

, with group
action given by the

operation

(

a

b

c

d

)

⋅

z

:=

a

⁢

z

+

b

c

⁢

z

+

d

.

Define

X

0

⁢

(

N

)

to be the

quotient

of

ℍ

*

by the action of

Γ

0

⁢

(

N

)

. The quotient space

X

0

⁢

(

N

)

inherits a quotient
topology and

holomorphic structure

from

ℂ

making it into a compact
Riemann surface. (Note:

ℍ

*

itself is not a

Riemann surface

; only
the quotient

X

0

⁢

(

N

)

is.) By a general

theorem

in complex

algebraic
geometry

, every compact Riemann surface admits a unique realization as
a complex

nonsingular

projective curve

; in particular,

X

0

⁢

(

N

)

has
such a realization, which by abuse of notation we will also denote

X

0

⁢

(

N

)

. This curve is defined over

ℚ

, although the proof of this
fact is beyond the scope of this entry

1

1

Explicitly, the curve

X

0

⁢

(

N

)

is the unique nonsingular projective curve which has

function
field

equal to

ℂ

⁢

(

j

⁢

(

z

)

,

j

⁢

(

N

⁢

z

)

)

, where

j

denotes the elliptic
modular

j

–

function

. The curve

X

0

⁢

(

N

)

is essentially the

algebraic
curve

defined by the polynomial equation

Φ

N

⁢

(

X

,

Y

)

=

0

where

Φ

N

is the modular

polynomial

, with the caveat that this
procedure yields

singularities

which must be resolved manually. The
fact that

Φ

N

has integer coefficients provides one proof that

X

0

⁢

(

N

)

is defined over

ℚ

.

.

Taniyama-Shimura Theorem

(weak form):

For any

elliptic curve

E

defined
over

ℚ

, there exists a

positive integer

N

and a

surjective

algebraic morphism

ϕ

:

X

0

⁢

(

N

)

→

E

defined over

ℚ

.

This theorem was first conjectured (in a much more precise, but

equivalent

formulation) by Taniyama, Shimura, and Weil in
the 1970’s. It attracted considerable interest in the 1980’s when
Frey

[

2

]

proposed that the

Taniyama-Shimura conjecture

implies
Fermat’s Last Theorem. In 1995, Andrew Wiles

[

3

]

proved a
special case of the Taniyama-Shimura theorem which was strong enough
to yield a proof of Fermat’s Last Theorem. The full Taniyama-Shimura
theorem was finally proved in 1997 by a team of a half-dozen
mathematicians who, building on Wiles’s work, incrementally chipped
away at the remaining cases until the full result was proved. As of this writing, the proof of the full theorem can still be found on

http://abel.math.harvard.edu/ rtaylor/

Richard Taylor’s preprints page.

References

1

Breuil, Christophe; Conrad, Brian;

Diamond

, Fred; Taylor, Richard;

On the modularity of elliptic curves over

Q

: wild 3-adic exercises

. J. Amer. Math. Soc.

14

(2001), no. 4, 843–939

2

Frey, G.

Links between stable elliptic curves and
certain

Diophantine equations

.

Ann. Univ. Sarav.

1

(1986), 1–40.

3

Wiles, A.

Modular elliptic curves and Fermat’s Last
Theorem.

Annals of Math.

141

(1995), 443–551.

Title

Taniyama-Shimura theorem

Canonical name

TaniyamaShimuraTheorem

Date of creation

2013-03-22 12:16:27

Last modified on

2013-03-22 12:16:27

Owner

djao (24)

Last modified by

djao (24)

Numerical id

14

Author

djao (24)

Entry type

Theorem

Classification

msc 11F06

Classification

msc 14H52

Synonym

Taniyama-Shimura-Weil conjecture

Synonym

Taniyama-Weil conjecture

Synonym

Taniyama-Shimura conjecture

Synonym

Taniyama-Shimura-Weil theorem

Related topic

FermatsLastTheorem

Related topic

ModularForms

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