Editing Ribet's theorem (section)

== Statement ==

Let {{math|''f''}} be a weight 2 [[Atkin–Lehner theory|newform]] on {{math|Γ<sub>0</sub>(''qN'')}} &ndash; i.e. of level {{math|''qN''}} where {{math|''q''}} does not divide {{math|''N''}} &ndash; with absolutely irreducible 2-dimensional mod {{math|''p''}} Galois representation {{math|''ρ<sub>f,p</sub>''}} unramified at {{math|''q''}} if {{math|''q'' ≠ ''p''}} and finite flat at {{math|''q'' {{=}} ''p''}}. Then there exists a weight 2 newform {{math|''g''}} of level {{math|''N''}} such that
:<math> \rho_{f,p} \simeq \rho_{g,p}. </math>

In particular, if {{math|''E''}} is an [[elliptic curve]] over <math>\mathbb{Q}</math> with [[conductor of an elliptic curve|conductor]] {{math|''qN''}}, then the [[modularity theorem]] guarantees that there exists a weight 2 newform {{math|''f''}} of level {{math|''qN''}} such that the 2-dimensional mod {{math|''p''}} Galois representation {{math|''ρ<sub>f, p</sub>''}} of {{math|''f''}} is isomorphic to the 2-dimensional mod {{math|''p''}} Galois representation {{math|''ρ<sub>E, p</sub>''}} of {{math|''E''}}. To apply Ribet's Theorem to {{math|''ρ''<sub>''E'', ''p''</sub>}}, it suffices to check the irreducibility and ramification of {{math|''ρ<sub>E, p</sub>''}}. Using the theory of the [[Tate curve]], one can prove that {{math|''ρ<sub>E, p</sub>''}} is unramified at {{math|''q'' ≠ ''p''}} and finite flat at {{math|''q'' {{=}} ''p''}} if {{math|''p''}} divides the power to which {{math|''q''}} appears in the minimal discriminant {{math|Δ<sub>''E''</sub>}}. Then Ribet's theorem implies that there exists a weight 2 newform {{math|''g''}} of level {{math|''N''}} such that {{math|''ρ''<sub>''g'', ''p''</sub> ≈ ''ρ''<sub>''E'', ''p''</sub>}}.