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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
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.
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For prime numbers {{mvar|l}} not equal to 37, one can verify the property about the coefficients. Thus, for {{math|''l'' {{=}} 3}}, there are 6 solutions of the equation modulo 3: {{math|(0, 0)}}, {{math|(0, 1)}}, {{math|(1, 0)}}, {{math|(1, 1)}}, {{math|(2, 0)}}, {{math|(2, 1)}}; thus {{math|1=''a''(3) = 3 − 6 = −3}}.
The conjecture, going back to the 1950s, was completely proven by 1999 using the ideas of [[Andrew Wiles]], who proved it in 1994 for a large family of elliptic curves.<ref>A synthetic presentation (in French) of the main ideas can be found in [http://www.numdam.org/item/SB_1994-1995__37__319_0/ this] [[Nicolas Bourbaki|Bourbaki]] article of [[Jean-Pierre Serre]]. For more details see {{Harvard citations |last=Hellegouarch |year=2001 |nb=yes}}</ref>
There are several formulations of the conjecture. Showing that they are equivalent was a main challenge of number theory in the second half of the 20th century. The modularity of an elliptic curve {{mvar|E}} of conductor {{mvar|N}} can be expressed also by saying that there is a non-constant [[rational map]] defined over {{math|ℚ}}, from the modular curve {{math|''X''<sub>0</sub>(''N'')}} to {{mvar|E}}. In particular, the points of {{mvar|E}} can be parametrized by [[modular function]]s.
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