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The '''Taniyama-Shimura theorem''' establishes an important connection between [[elliptic curve]]s, which are objects from [[algebraic geometry]], and [[modular form]]s, which are certain periodic [[holomorphic function]]s investigated in [[number theory]].
If ''p'' is a [[prime number]] and ''E'' is an elliptic curve over '''Q''', we can reduce the equation defining ''E'' [[modular arithmetic|modulo]] ''p''; for all but finitely many values of ''p'' we will get an elliptic curve over the [[finite field]] '''F'''<sub>''p''</sub>, with ''n''<sub>''p''</sub> elements , say. One then considers the sequence ''a''<sub>''p''</sub> = ''n''<sub>''p''</sub> - ''p'', which is an important invariant of the elliptic curve ''E''. Every modular form also gives rise to a sequence of numbers, by [[Fourier transform]]. An elliptic curve whose sequence agrees with that from a modular form is called '''modular'''. The Taniyma-Shimura theorem states:
:"All elliptic curves over '''Q''' are modular."
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== References ==
* Henri Darmon: ''[http://www.ams.org/notices/199911/comm-darmon.pdf A Proof of the Full Shimura-Taniyama-Weil Conjecture Is Announced]'', Notices of the American Mathematical Society, Vol. 46 (1999), No. 11. Contains a gentle introduction to the theorem and an outline of the proof.
* Brian Conrad, Fred Diamond, Richard Taylor: ''Modularity of certain potentially Barsotti-Tate Galois representations'', Journal of the American Mathematical Society 12 (1999), pp.
== Some notation ==
'''Q''' denotes the field of [[rational number]]s.<BR>
'''F'''<sub>''p''</sub> is also called a [[Evariste Galois|Galois]] [[field]].
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