Serre conjecture (number theory)

In mathematics, Serre's modularity conjecture, introduced by Jean-Pierre Serre (Serre 1975, 1987), states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form, and a stronger version of his conjecture specifies the weight and level of the modular form. It was proved by Chandrashekhar Khare in the level 1 case, in 2005 and later in 2008 a proof of the full conjecture was worked out jointly by Khare and Jean-Pierre Wintenberger.

The conjecture concerns the absolute Galois group ${\displaystyle G_{\mathbb {Q} }}$ of the rational number field ${\displaystyle \mathbb {Q} }$.

Let ${\displaystyle \rho }$ be an absolutely irreducible, continuous, two-dimensional representation of ${\displaystyle G_{\mathbb {Q} }}$ over a finite field ${\displaystyle F=\mathbb {F} _{\ell ^{r}}}$.

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