Editing Modular form (section)

==History==
{{Unreferenced section|date=October 2019}}
The theory of modular forms was developed in four periods:

* In connection with the theory of [[elliptic function]]s, in the early nineteenth century
* By [[Felix Klein]] and others towards the end of the nineteenth century as the automorphic form concept became understood (for one variable)
* By [[Erich Hecke]] from about 1925
* In the 1960s, as the needs of number theory and the formulation of the [[modularity theorem]] in particular made it clear that modular forms are deeply implicated.

Taniyama and Shimura identified a 1-to-1 matching between certain modular forms and elliptic curves. [[Robert Langlands]] built on this idea in the construction of his expansive [[Langlands program]], which has become one of the most far-reaching and consequential research programs in math.

In 1994 [[Andrew Wiles]] used modular forms to prove [[Fermat’s Last Theorem]]. In 2001 all elliptic curves were proven to be  modular over the rational numbers. In 2013 elliptic curves were proven to be modular over real [[quadratic fields]]. In 2023 elliptic curves were proven to be modular over about half of imaginary quadratic fields, including fields formed by combining the [[rational numbers]] with the [[square root]] of integers down to −5.<ref name=":0" />