论文标题
在Rankin-Selberg设置中的Mazur-Tate类型猜想的类似物上
On analogues of Mazur-Tate type conjectures in the Rankin-Selberg setting
论文作者
论文摘要
我们研究了$ \ mathbb {q} $ selmer组的$ \ mathbb {q} $的cylotomic $ \ mathbb {z} _p $ - extension的拟合理想的合适理想。受到Mazur和Tate定义的模块化形式的theta元素的启发，我们在``桦木和Swinnerton-Dyer型的精制猜想''中，为Rankin-Selberg汇总$ F $和$ G $使用Loeeffler-- szerbes的几何$ p $ $ $ $ l $ - $ l $ lunctions和$ g $ for $ f $ f $ f $ f $ f $ lunctions $ f $ l $ lunctions y-functions for $ f $ f $ f $ f $ f $及其。 在某些技术假设下，我们概括了Kim-kurihara在椭圆曲线上的最新作品，以证明结果非常接近Mazur的\ emph {弱的主要猜想}，而兰金氏症则是selberg的卷积。特别强调$ f $对应于椭圆曲线$ e $和$ g $的情况下，二维奇数不可约的artin代表$ρ$带有分裂字段$ f $。作为一个应用程序，我们给出了$ e $ $ e $的$ρ$ - 异型组件的上限，与$ \ mathbb {z} _p $ extension $ f $ f $的有限层相比，我们的theta emlements消失的顺序。
We study the Fitting ideals over the finite layers of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of Selmer groups attached to the Rankin--Selberg convolution of two modular forms $f$ and $g$. Inspired by the Theta elements for modular forms defined by Mazur and Tate in ``Refined conjectures of the Birch and Swinnerton-Dyer type'', we define new Theta elements for Rankin--Selberg convolutions of $f$ and $g$ using Loeffler--Zerbes' geometric $p$-adic $L$-functions attached to $f$ and $g$. Under certain technical hypotheses, we generalize a recent work of Kim--Kurihara on elliptic curves to prove a result very close to the \emph{weak main conjecture} of Mazur and Tate for Rankin--Selberg convolutions. Special emphasis is given to the case where $f$ corresponds to an elliptic curve $E$ and $g$ to a two dimensional odd irreducible Artin representation $ρ$ with splitting field $F$. As an application, we give an upper bound of the dimension of the $ρ$-isotypic component of the Mordell-Weil group of $E$ over the finite layers of the cyclotomic $\mathbb{Z}_p$-extension of $F$ in terms of the order of vanishing of our Theta elements.
