Modularity theorem: Difference between revisions

Revision as of 17:33, 6 August 2021

Revision as of 16:48, 1 March 2022

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==History==

[[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]]. [[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 <math>L</math>-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]].

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{{sfn|Ribet|1990}}<!--{{harvs|txt|authorlink=Ken Ribet|last=Ribet|first=Ken|year=1990}}-->'s completion of a proof of the epsilon conjecture.

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 proof<!--{{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, which he used 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, 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}}-->

{{further|Fermat's Last Theorem|Wiles's proof of Fermat's Last Theorem}}