Algebraic cycles, modular forms and Euler systems
Algebraic cycles, modular forms and Euler systems
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代数环、模形式和欧拉系统
DOI：
10.1515/crll.2002.011
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
T. Weston
中科院分区：
文献类型：
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作者：
T. Weston
固定一个无平方因子的整数N，设f是Γ0（N）的权为2的新形式;我们假设f没有复数乘法。在[14]和[15]中表明，对于密度为1的素数集合l，与f相关联的模l伽罗瓦表示的朴素变形理论是畅通的（在这个意义上，通用变形环是Witt向量上的幂级数环）。在[31]中，这些方法被修改以获得泰勒-怀尔斯研究的变形问题的结果。本文将Flach和Mazur的结果推广到权为κ ≥ 2的新形式f的情形。我们现在更精确地陈述我们的结果。固定l > max{5，κ+1}，设f如上，H为相应的l-adic表示：H是某个Hecke代数A上秩为2的自由模，A本身是有限平坦局部Gorenstein Zl-代数。设T是H的迹零自同态模的Tate twist EndAH（1）.利用Flach的技巧，我们构造了H（Q，T）中具有严格控制分歧的上同调类{c}的集合.在适当的附加假设下，将Kolyvagin的方法应用于这些类，得到T的Cartier对偶的塞尔默群Hf（Q，T）的某个零化子η ∈ A.这个塞尔默群与微分ΩR <$RA对偶，其中R是H的剩余表示的泛极小分歧变形环。在η是单位的情况下，这意味着R和A都同构于A的剩余域上的维特向量环。在一般情况下，遵循Mazur，我们证明了我们的构造产生了从A到塞尔默群H f（Q，T/ηT）的导子;它遵循了自然满射R A诱导同构ΩR <$RA <$= ΩA的形式论证。虽然不是最强的可能结果，但它确实提供了关于环R结构的大量信息。(It任何这样的映射R A都有可能是同构的，尽管据我所知这个问题还没有解决。）我们还证明了同构ΩR <$RA <$= ΩA的特征是ΩA <$= ΩR <$RA <$= HomZl（Hf（Q，T），Ql/Zl）
Fix a squarefree integer N and let f be a newform of weight 2 for Γ0(N); we assume that f does not have complex multiplication. It was shown in [14] and [15] that for a set of primes l of density 1 the naive deformation theory of the mod l Galois representation associated to f is unobstructed (in the sense that the universal deformation ring is a power series ring over the Witt vectors). In [31] these methods were modified to obtain results on the deformation problems studied by Taylor-Wiles. In this paper we extend the results of Flach and Mazur to the case of newforms f of weight κ ≥ 2 for Γ1(N). We now state our results more precisely. Fix l > max{5, κ+1}, let f be as above and let H be the associated l-adic representation: H is a free module of rank 2 over a certain Hecke algebra A, which itself is a finite, flat, local, Gorenstein Zl-algebra. Let T be the Tate twist EndAH(1) of the module of trace zero endomorphisms of H. Using techniques of Flach we construct a collection of cohomology classes {c} in H(Q, T ) with tightly controlled ramification. With some mild additional hypotheses, applying the methods of Kolyvagin to these classes yields a certain annihilator η ∈ A of the Selmer group H f (Q, T ∗) of the Cartier dual of T . This Selmer group is dual to the differentials ΩR⊗RA, where R is the universal minimally ramified deformation ring of the residual representation of H. In the case that η is a unit this then implies that both R and A are isomorphic to the ring of Witt vectors over the residue field of A. In the general case, following Mazur we show that our construction yields a derivation from A to the Selmer group H f (Q, T/ηT ); it follows by a formal argument that the natural surjection R A induces an isomorphism ΩR⊗RA ∼= ΩA. Although not the strongest possible result, this does provide a great deal of information on the structure of the ring R. (It is possible that any such map R A must be an isomorphism, although as far as I know this question remains open.) We also show that the isomorphism ΩR⊗RA ∼= ΩA is characterized by the fact that ΩA ∼= ΩR⊗RA ∼= HomZl ( H f (Q, T ),Ql/Zl )
模形式和 p-adic Hodge 理论
DOI：
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发表时间：
1997
期刊：
Invent Math 129
影响因子：
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作者：
T. Saito
通讯作者：
T. Saito