Modularity theorem: Difference between revisions

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==Statement==

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The [[theorem]] states that any [[elliptic curve]] over <math>\mathbf{Q}</math> can be obtained via a [[rational map]] with [[integer]] [[coefficient]]s from the [[classical modular curve]] <math>X_0(N)</math> for some integer <math>N</math>; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level <math>N</math>. If <math>N</math> is the smallest integer for which such a parametrization can be found (which by the modularity theorem itself is now known to be a number called the ''[[conductor of an elliptic curve|conductor]]''), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level ''<math>N''</math>, a normalized [[newform]] with integer ''<math>q''</math>-expansion, followed if need be by an [[Elliptic curve#Isogeny|isogeny]].

===Related statements===

The modularity theorem implies a closely related analytic statement:

to an elliptic curve ''E'' over <math>\mathbf{Q}</math> we may attach a corresponding [[L-series of an elliptic curve|L-series]]. The ''<math>L''</math>-series is a [[Dirichlet series]], commonly written

:<math>L(E, s) = \sum_{n=1}^\infty \frac{a_n}{n^s}.</math>

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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 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 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 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.

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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]] ''<math>n''</math>-th powers, ''<math>n''&nbsp;≥&nbsp; \geq 3</math>. (The case ''<math>n''&nbsp; =&nbsp; 3</math> was already known by [[Euler]].)

==Generalizations==