Let be a prime number. We consider the Iwasawa -invariants associated to modular Bloch–Kato Selmer groups, considered over the cyclotomic -extension of . Let be a -ordinary cuspidal newform of weight 2 and trivial nebentype. We assume that the -invariant of vanishes, and that the image of the residual representation associated to is suitably large. We show that for any number greater than or equal to the -invariant of , there are infinitely many newforms that are -congruent to , with -invariant equal to . We also prove quantitative results regarding the levels of such modular forms with prescribed -invariant.

中文翻译：

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用规定的 Iwasawa λ 不变量构造伽罗瓦表示

让是素数。我们考虑岩泽-与模块化 Bloch-Kato Selmer 群相关的不变量，在分圆法上考虑-的扩展。让成为一个-重量为2的普通尖头新形式和普通neben类型。我们假设- 不变的消失，并且与关联的剩余表示的图像适当大。我们证明对于任何数字大于或等于- 不变的，有无穷多个新形式那是- 符合， 和- 不变等于。我们还证明了有关此类模块化形式水平的定量结果-不变的。