MODULARITY OF RESIDUAL GALOIS EXTENSIONS AND THE EISENSTEIN IDEAL
MODULARITY OF RESIDUAL GALOIS EXTENSIONS AND THE EISENSTEIN IDEAL
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DOI：
10.1090/tran/7851
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发表时间：
2019-12-01
影响因子：
1.3
通讯作者：
Klosin, Krzysztof
中科院分区：
文献类型：
--
作者：
Berger, Tobias;Klosin, Krzysztof
对于全真实的域F，F-p的有限扩张F，以及从{p竖线p}的有限位置Sigma超集未分歧的伽罗瓦特征符chi：G（F）-> F-x，考虑Bloch-Kato塞尔默群H：= H-Sigma（1）（F，chi（-1））.作者以前证明了在H中产生被某个rho（f）模化的线（在Sigma外也是非分歧的）的棒上的（非半单的，可约的）剩余表示（rho）的同构类的个数d满足d >= n：= dim（F）H。这是在同余模的阶大于或等于可分塞尔默群的阶的假设下证明的。我们在这里证明，如果另外相关的局部爱森斯坦理想J是非主的，则d > n。当F = Q时，我们证明了同余模和塞尔默群的期望界.我们还制定了一个全等条件，暗示J的非主性，可以在实践中检查，让我们提供的例子，其中d > n。
For a totally real field F, a finite extension F of F-p, and a Galois character chi : G(F) -> F-x unramified away from a finite set of places Sigma superset of {p vertical bar p}, consider the Bloch-Kato Selmer group H := H-Sigma(1)(F, chi(-1)). The authors previously proved that the number d of isomorphism classes of (nonsemisimple, reducible) residual representations (rho) over bar giving rise to lines in H which are modular by some rho(f) (also unramified outside Sigma) satisfies d >= n := dim(F) H. This was proved under the assumption that the order of a congruence module is greater than or equal to that of a divisible Selmer group. We show here that if in addition the relevant local Eisenstein ideal J is nonprincipal, then d > n. When F = Q we prove the desired bounds on the congruence module and the Selmer group. We also formulate a congruence condition implying the nonprincipality of J that can be checked in practice, allowing us to furnish examples where d > n.