On deformation rings of residually reducible Galois representations and R = T theorems
On deformation rings of residually reducible Galois representations and R = T theorems
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关于剩余可约伽罗瓦表示的变形环和R = T定理
DOI：
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发表时间：
2011
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通讯作者：
Krzysztof Klosin
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作者：
Tobias Berger;Krzysztof Klosin
本文介绍了在剩余可约情形下R = T定理的一种新的证明方法。研究了n维模p Galois表示ρ0的晶泛变形环R（及其约化理想I），ρ 0的半单化是两个互不同构的绝对不可约分量ρ1和ρ2的直和.在与ρ1和ρ2有关的塞尔默群的某些假设下，我们证明了R/I是循环的且常常是有限的.利用Bellaïche和Chenevier的思想和结果（但有些不同的假设），我们证明了I是基本自对偶表示的主要部分，并推导出关于R的结构的陈述。使用一个新的交换代数标准，我们表明，在Hecke侧的足够的信息得到一个R = T-定理。然后，我们应用该技术的模块化问题的2维表示在虚二次域和4维表示在Q。
We introduce a new method of proof for R = T theorems in the residually reducible case. We study the crystalline universal deformation ring R (and its ideal of reducibility I) of a mod p Galois representation ρ0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non-isomorphic constituents ρ1 and ρ2. Under some assumptions on Selmer groups associated with ρ1 and ρ2 we show that R/I is cyclic and often finite. Using ideas and results of (but somewhat different assumptions from) Bellaïche and Chenevier we prove that I is principal for essentially self-dual representations and deduce statements about the structure of R. Using a new commutative algebra criterion we show that given enough information on the Hecke side one gets an R = T-theorem. We then apply the technique to modularity problems for 2-dimensional representations over an imaginary quadratic field and a 4-dimensional representation over Q.