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Additional
modularity theorem
English Wikipedia - The Free Encyclopedia
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Modularity theorem
In
mathematics
, the
modularity theorem
(formerly called the
Taniyama–Shimura–Weil conjecture
and several related names) states that
elliptic curves
over the field of
rational numbers
are related to
modular forms
.
Andrew Wiles
proved the modularity theorem for
semistable elliptic curves
, which was enough to imply
Fermat's last theorem
. Later,
Christophe Breuil
,
Brian Conrad
,
Fred Diamond
, and
Richard Taylor
extended Wiles' techniques to prove the full modularity theorem in 2001. The modularity theorem is a special case of more general conjectures due to
Robert Langlands
. The
Langlands program
seeks to attach an
automorphic form
or
automorphic representation
(a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a
number field
. Most cases of these extended conjectures have not yet been proved. However, proved that elliptic curves defined over real quadratic fields are modular.
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