The global deformation theory of residually reducible Galois representations with fixed auxiliary conditions is studied. We show that ρ̄ : Gal(ℚ̄/ℚ) →GL2(𝔽̄p) lifts to a Hida line for which the weights range over a congruence class modulo-p2. The advantage of the purely Galois theoretic approach is that it allows us to construct p-adic families of Galois representations lifting the actual representation ρ̄, and not just the semisimplification.

中文翻译：

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Hida 族的可约伽罗瓦表示的变形

研究了具有固定辅助条件的剩余可约伽罗瓦表示的全局变形理论。我们表明ρ̄ ： 加尔(ℚ̄/ℚ) →总帐2(𝔽̄p)提升到 Hida 线，其权重范围为同余类模p2. 纯伽罗瓦理论方法的优点是它允许我们构造p- 伽罗瓦表示的 adic 家族提升实际表示ρ̄，而不仅仅是半简化。