Modularity theorem: Difference between revisions

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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 proveproof<!--{{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.