论文标题
在具有可还原和非$ p $变化的残留galois表示形式的模块化形式的Iwasawa上
On Iwasawa invariants of modular forms with reducible and non-$p$-distinguished residual Galois representations
论文作者
论文摘要
在本文中，我们研究了$ p $ -adic $ l $ - functions和（严格的）Selmer组，超过$ \ Mathbb {q} _ {\ infty} $，环环$ \ Mathbb {z} _p {z} _p $ extensive $ \ mathbb {q} $ p $ p $ - $ p $ - 稳定的重量一个Eisenstein系列，假设$ p $ p $ - 非凡的通用cuspidal Hecke代数的某个Eisenstein组件是Gorenstein。作为一个应用程序，我们计算了普通模块化的重量$ k \ geq 2 $的iwasawa不变性，具有与$ f $相同的剩余galois表示，在我们的环境中，它是可还原且非$ p $ distanceed的。将其与kato \ cite [定理〜17.4.2] {kato04}的结果结合起来，我们证明了这些形式的伊瓦沙（Iwasawa）主要猜想。另外，我们提供了满足Gorenstein假设的数值示例。 分析柜台的关键点是，在戈伦斯坦假设下，我们能够在格林伯格（Greenberg） - 瓦特斯（Greenberg） - 瓦特斯（Greenberg），$ p $ - ad的$ l $ l $ l $ - $ p $ - aadig weigations $ p $ - aadig weightem-aadig weightem-f $ f $ f $ f $ f $ f $ f $作为一维iwasawa algebra中的一个元素，通过使用Mazur-l- $ kitag $ $ kitaga，$ kitaga $ p $ - 通过本地明确的互惠法明确计算它们。在代数计数器部分上，我们通过\ cite {bdp}研究的$ f $的galois表示，计算$ f $ $ f $ $ f $ $ f $的（严格）selmer组。
In the present paper, we study the $p$-adic $L$-functions and the (strict) Selmer groups over $\mathbb{Q}_{\infty}$, the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, of the $p$-adic weight one cusp forms $f$, obtained via the $p$-stabilization of weight one Eisenstein series, under the assumption that a certain Eisenstein component of the $p$-ordinary universal cuspidal Hecke algebra is Gorenstein. As an application, we compute the Iwasawa invariants of ordinary modular forms of weight $k\geq 2$ with the same residual Galois representations as the one of $f$, which in our setting, is reducible and non-$p$-distinguished. Combining this with a result of Kato \cite[Theorem~17.4.2]{kato04}, we prove the Iwasawa main conjecture for these forms. Also, we give numerical examples that satisfy the Gorenstein hypothesis. The crucial point on the analytic counter part is that under the Gorenstein hypothesis, we are able to define, following Greenberg--Vatsal, the $p$-adic $L$-functions of $p$-adic weight one forms $f$ as an element in the one-dimensional Iwasawa algebra by using Mazur--Kitagawa two-variable $p$-adic $L$-function and then, to compute them explicitly via local explicit reciprocity law. On the algebraic counter part, we compute the (strict) Selmer groups of $f$ over $\mathbb{Q}_{\infty}$ via the knowledge of the Galois representations of $f$ studied in \cite{BDP}.
