论文标题
兰金·塞尔伯格（Rankin-Selberg）在爱森斯坦Prime上的兰金·塞尔伯格（Rankin-Selberg）卷积理论
Iwasawa theory for Rankin-Selberg convolution at an Eisenstein prime
论文作者
论文摘要
让$ p $是一个奇怪的素数，$ f $是$ p $ - 非凡的重量$ k $，$ h $是标准化的cuspidal $ p $ p $ p $ p $ p $ hecke eigke eigen eigenform $ l <k $。在本文中，我们研究了$ p $ -Adic $ l $ - 功能和$ p^{\ infty} $ -Selmer组的Rankin-Selberg产品的$ f $和$ h $，假设$ p $是$ h $ $ h $的eisenstein prime，即$ h $。我们表明，$ p $ -adic $ l $ function和$ f，h $的rankin-selberg产品的$ p^\ infty $ -Selmer组的特征理想在Iwasawa代数中生成了相同的理想模式$ p $。作为我们结果的应用，我们明确描述了上述一致性所具有的一些示例。
Let $p$ be an odd prime, $ f$ be a $ p $-ordinary newform of weight $ k $ and $ h $ be a normalized cuspidal $ p $-ordinary Hecke eigenform of weight $ l < k$. In this article, we study the $p$-adic $ L $-function and $ p^{\infty} $-Selmer group of the Rankin-Selberg product of $f$ and $h$ under the assumption that $ p $ is an Eisenstein prime for $ h $ i.e. the residual Galois representation of $ h $ at $ p $ is reducible. We show that the $ p $-adic $ L $-function and the characteristic ideal of the $p^\infty$-Selmer group of the Rankin-Selberg product of $f, h$ generate the same ideal modulo $ p $ in the Iwasawa algebra i.e. the Rankin-Selberg Iwasawa main conjecture for $f \otimes h$ holds mod $p$. As an application to our results, we explicitly describe a few examples where the above congruence holds.
