Editing Modularity theorem (section)

==History==
{{see also|Taniyama's problems}}
{{further|Fermat's Last Theorem|Wiles's proof of Fermat's Last Theorem}}
[[Yutaka Taniyama]]{{sfn|Taniyama|1956}}<!--{{harvs|txt|authorlink=Yutaka Taniyama|last=Taniyama|first=Yutaka|year=1956}}--> stated a preliminary (slightly incorrect) version of the conjecture  at the 1955 international symposium on [[algebraic number theory]] in [[Tokyo]] and [[Nikkō, Tochigi|Nikkō]] as the twelfth of his [[Taniyama's problems|set of 36 unsolved problems]].  [[Goro Shimura]] and Taniyama worked on improving its rigor until 1957. [[André Weil]]{{sfn|Weil|1967}}<!--{{harvs|txt|authorlink=André Weil|last=Weil|first=André|year= 1967}}--> rediscovered the conjecture, and showed in 1967 that it would follow from the (conjectured) functional equations for some twisted {{mvar|L}}-series of the elliptic curve; this was the first serious evidence that the conjecture might be true. Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form. The Taniyama–Shimura–Weil conjecture became a part of the [[Langlands program]].<ref name="Harris Virtues of Priority">{{cite arXiv | last=Harris | first=Michael | title=Virtues of Priority | year=2020 | eprint=2003.08242 | page=| class=math.HO }}</ref><ref>{{cite journal
| last      = Lang
| first     = Serge
| date       = November 1995
| title      = Some History of the Shimura-Taniyama Conjecture
| journal    = Notices of the American Mathematical Society
| volume     = 42
| issue      = 11
| pages      = 1301–1307
| url        = https://www.ams.org/notices/199511/forum.pdf
| access-date = 2022-11-08
}}</ref>

The conjecture attracted considerable interest when [[Gerhard Frey]]{{sfn|Frey|1986}}<!--{{harvs|txt|authorlink=Gerhard Frey|last=Frey|first=Gerhard|year=1986}}--> suggested in 1986 that it implies [[Fermat's Last Theorem]]. He did this by attempting to show that any counterexample to Fermat's Last Theorem would imply the existence of at least one non-modular elliptic curve. This argument was completed in 1987 when Jean-Pierre Serre{{sfn|Serre|1987}}<!--{{harvs|txt|authorlink=Jean-Pierre Serre|last=Serre|first=Jean-Pierre|year=1987}}--> identified a missing link (now known as the [[epsilon conjecture]] or Ribet's theorem) in Frey's original work, followed two years later by Ken Ribet's completion of a proof of the epsilon conjecture.{{sfn|Ribet|1990}}<!--{{harvs|txt|authorlink=Ken Ribet|last=Ribet|first=Ken|year=1990}}-->

Even after gaining serious attention, the Taniyama–Shimura–Weil conjecture was seen by contemporary mathematicians as extraordinarily difficult to prove or perhaps even inaccessible to prove<!--{{harv|Singh|1997|pp=203–205, 223, 226}}-->.{{sfn|Singh|1997|pp=203–205, 223, 226}} For example, Wiles's Ph.D. supervisor [[John H. Coates|John Coates]] states that it seemed "impossible to actually prove", and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".

In 1995, Andrew Wiles, with some help from [[Richard Taylor (mathematician)|Richard Taylor]], proved the Taniyama–Shimura–Weil conjecture for all [[semistable elliptic curve]]s. Wiles used this to prove Fermat's Last Theorem,{{sfnm|Wiles|1995a|Wiles|1995b}}<!--{{harvs|txt|authorlink=Andrew Wiles|last=Wiles|year=1995}}--> and the full Taniyama–Shimura–Weil conjecture was finally proved by Diamond,{{sfn|Diamond|1996}}<!--{{harvtxt|Diamond|1996}}--> Conrad, Diamond & Taylor; and Breuil, Conrad, Diamond & Taylor; building on Wiles's work, they incrementally chipped away at the remaining cases until the full result was proved in 1999.{{sfn|Conrad|Diamond|Taylor|1999}}<!--{{harvtxt|Conrad|Diamond|Taylor|1999}}-->{{sfn|Breuil|Conrad|Diamond|Taylor|2001}}<!--{{harvtxt|Breuil|Conrad|Diamond|Taylor|2001}}--> Once fully proven, the conjecture became known as the modularity theorem.

Several theorems in number theory similar to Fermat's Last Theorem follow from the modularity theorem. For example: no cube can be written as a sum of two [[coprime]] {{mvar|n}}th powers, {{math|''n'' ≥ 3}}.{{efn|The case {{math|''n'' {{=}} 3}} was already known by [[Euler]].}}