Modularity theorem: Difference between revisions

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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 [[holomorphic function]]s investigated in [[number theory]]. Every modular form gives rise to an elliptic curve over the [[field]] '''Q''' of [[rational number]]s; roughly speaking, the Taniyma-Shimura theorem states that ''every'' elliptic curve over '''Q''' arises from a modular form:

Informally, the '''Taniyama-Shimura theorem''' states:

:"allAll [[elliptic curve]]scurves areover [[modular'''Q''' are form|modular]]."

This [[theorem]] was first [[conjecture]]d by [[Yutaka Taniyama]] in September [[1955]]. With [[Goro Shimura]] he improved its rigor until [[1957]]. Taniyama died in [[1958]]. In the [[1960s]] it became associated with the [[Langlands program]] of unifying conjectures in mathematics, and was a key component thereof. The conjecture was picked up and promoted by [[Andre Weil|André Weil]] in the [[1970s|1970's]], and Weil's name was associated with it in some quarters. Despite the interest, some considered it beyond proving.

It attracted considerable interest in the [[1980s|1980's]] when [[Gerhard Frey]] proposedproved that the '''Taniyama-Shimura conjecture''' (as it was then called) implies [[Fermat's last theorem]]. He did this by showing that any counterexample to Fermat's last theorem would give rise to a non-modular elliptic curve. In [[1995]], [[Andrew Wiles]] and [[Richard Taylor]] proved a special case of the '''Taniyama-Shimura theorem''' (the case of [[semistable elliptic curve]]s) which was strong enough to yield a proof of '''Fermat's Last Theorem'''.

The full '''Taniyama-Shimura theorem''' was finally proved in [[19971999]] by aBreuil, teamConrad, ofDiamond, aand half-dozen mathematiciansTaylor who, building on '''Wiles''''s work, incrementally chipped away at the remaining cases until the full result was proved.

Several theorems in number theory similar to Fermat's last theorem follow from the Taniyama-Shimura theorem. For example: no cube can be written as a sum of two [[relatively prime]] ''n''-th powers, ''n'' ≥ 3. (The case ''n'' = 3 was already known by [[Euler]].)

In March [[1996]] Wiles shared the [[Wolf Prize]] with [[Robert Langlands]]. Although neither of them had originated nor finished the proof of the full theorem that had enabled their achievements, they were recognized as having had the decisive influences that led to its finally being proven.

== 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. 521–567. Contains the proof.