Editing Algebraic number theory (section)

===Modern theory===
Around 1955, Japanese mathematicians [[Goro Shimura]] and [[Yutaka Taniyama]] observed a possible link between two apparently completely distinct, branches of mathematics, [[elliptic curve]]s and [[modular form]]s. The resulting [[modularity theorem]] (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is [[modular elliptic curve|modular]], meaning that it can be associated with a unique [[modular form]].

It was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist [[André Weil]] found evidence supporting it, yet no proof; as a result the "astounding"<ref name="Singh">{{citation |title=[[Fermat's Last Theorem (book)|Fermat's Last Theorem]] |author-link=Simon Singh |first=Simon |last=Singh |year=1997 |publisher=Fourth Estate |isbn=1-85702-521-0}}</ref> conjecture was often known as the Taniyama–Shimura-Weil conjecture. It became a part of the [[Langlands program]], a list of important conjectures needing proof or disproof.

From 1993 to 1994, [[Andrew Wiles]] provided a proof of the [[modularity theorem]] for [[semistable elliptic curve]]s, which, together with [[Ribet's theorem]], provided a proof for Fermat's Last Theorem. Almost every mathematician at the time had previously considered both Fermat's Last Theorem and the Modularity Theorem either impossible or virtually impossible to prove, even given the most cutting-edge developments. Wiles first announced his proof in June 1993<ref name=nyt>{{cite news|last=Kolata|first=Gina|title=At Last, Shout of 'Eureka!' In Age-Old Math Mystery|url=https://www.nytimes.com/1993/06/24/us/at-last-shout-of-eureka-in-age-old-math-mystery.html|access-date=21 January 2013|newspaper=The New York Times|date=24 June 1993}}</ref> in a version that was soon recognized as having a serious gap at a key point. The proof was corrected by Wiles, partly in collaboration with [[Richard Taylor (mathematician)|Richard Taylor]], and the final, widely accepted version was released in September 1994, and formally published in 1995. The proof uses many techniques from [[algebraic geometry]] and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the [[category (mathematics)|category]] of [[scheme (mathematics)|schemes]] and [[Iwasawa theory]], and other 20th-century techniques not available to Fermat.