Editing Ribet's theorem (section)

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{{short description|Result concerning properties of Galois representations associated with modular forms}}
'''Ribet's theorem''' (earlier called the '''epsilon conjecture''' or '''ε-conjecture''') is part of [[number theory]]. It concerns properties of [[Galois representation]]s associated with [[modular form]]s. It was proposed by [[Jean-Pierre Serre]] and [[mathematical proof|proven]] by [[Ken Ribet]]. The proof was a significant step towards the proof of [[Fermat's Last Theorem]] (FLT). As shown by Serre and Ribet, the [[Taniyama–Shimura conjecture]] (whose status was unresolved at the time) and the epsilon conjecture together imply that FLT is true.

In mathematical terms, Ribet's theorem shows that if the Galois representation associated with an [[elliptic curve]] has certain properties, then that curve cannot be modular (in the sense that there cannot exist a modular form that gives rise to the same representation).<ref>{{cite web | url=http://cgd.best.vwh.net/home/flt/flt08.htm | archive-url=https://web.archive.org/web/20081210102243/http://cgd.best.vwh.net/home/flt/flt08.htm | url-status=dead | archive-date=2008-12-10 | title=The Proof of Fermat's Last Theorem| date=2008-12-10}}</ref>