Modularity theorem: Difference between revisions

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TheIn '''modularity[[number theorem''' (formerly calledtheory]], the '''Taniyama–Shimura conjecture''', '''Taniyama–Shimura–Weil conjecture''' or '''modularity conjecture for elliptic curvestheorem''') states that [[elliptic curve]]s over the field of [[rational number]]s are related to [[modular form]]s in a particular way. [[Andrew Wiles]] and [[Richard Taylor (mathematician)|Richard Taylor]] proved the modularity theorem for [[semistable elliptic curve]]s, which was enough to imply [[Fermat's Last Theorem]] (FLT). Later, a series of papers by Wiles's former students [[Brian Conrad]], [[Fred Diamond]] and Richard Taylor, culminating in a joint paper with [[Christophe Breuil]], extended Wiles's techniques to prove the full modularity theorem in 2001. Before that, the statement was known as the '''Taniyama–Shimura conjecture''', '''Taniyama–Shimura–Weil conjecture''', or the '''modularity conjecture for elliptic curves'''.

==Statement==

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The conjecture attracted considerable interest when [[Gerhard Frey (mathematician)|Gerhard Frey]]{{sfn|Frey|1986}}<!--{{harvs|txt|authorlink=Gerhard Frey (mathematician)|last=Frey|first=Gerhard|year=1986}}--> suggested in 1986 that it implies [[Fermat's Last Theorem|FLT]]. He did this by attempting to show that any counterexample to Fermat's Last TheoremFLT would imply the existence of at least one non-modular elliptic curve. This argument was completedmoved closer to its goal 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".

With Ribet’s proof of the epsilon conjecture, Andrew Wiles saw an opportunity: Fermat’s Last Theorem was a respectable research project because it was now a corollary of the TSW conjecture. He had expertise in [[Iwasawa theory]]; maybe there was a path from Iwasawa theory to Taniyama–Shimura–Weil.

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.

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 TheoremFLT,{{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]].}}

Several theorems in number theory similar to Fermat's Last TheoremFLT 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]].}}

In 2025, modularity was extended to over 10% of [[abelian surfaces]].<ref>{{Cite web |last=Howlett |first=Joseph |date=2025-06-02 |title=The Core of Fermat’s Last Theorem Just Got Superpowered |url=https://www.quantamagazine.org/the-core-of-fermats-last-theorem-just-got-superpowered-20250602/ |access-date=2025-08-06 |website=Quanta Magazine |language=en}}</ref>

==Generalizations==

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Another formulation depends on the comparison of [[Galois representation]]s attached on the one hand to elliptic curves, and on the other hand to modular forms. The latter formulation has been used in the proof of the conjecture. Dealing with the level of the forms (and the connection to the conductor of the curve) is particularly delicate.

The most spectacular application of the conjecture is the proof of [[Fermat's Last Theorem|FLT]] (FLT). Suppose that for a prime {{math|''p'' ≥ 5}}, the Fermat equation

:<math>a^p + b^p = c^p</math>

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:<math>\Delta = \frac{1}{256}(abc)^{2p}</math>

cannot be modular.{{sfn|Ribet|1990}} Thus, the proof of the Taniyama–Shimura–Weil conjecture for this family of elliptic curves (called Hellegouarch–Frey curves) implies FLT. The proof of the link between these two statements, based on an idea of [[Gerhard Frey (mathematician)|Gerhard Frey]] (1985), is difficult and technical. It was established by [[Kenneth Ribet]] in 1987.<ref>See the survey of {{cite journal |first=K. |last=Ribet |title=From the Taniyama–Shimura conjecture to Fermat's Last Theorem |journal=Annales de la Faculté des Sciences de Toulouse |volume=11 |year=1990b |pages=116–139 |doi= 10.5802/afst.698|url=http://www.numdam.org/item?id=AFST_1990_5_11_1_116_0 |doi-access=free }}</ref>

==Notes==