论文标题
在模块化形式的家族中的Shafarevich-Tate组和分析等级上
On Shafarevich-Tate groups and analytic ranks in families of modular forms, I. Hida families
论文作者
论文摘要
让$ f $成为重量$ 2 $的新形式，无方面的水平和琐碎的角色，让$ a_f $是$ f $附加的abelian品种，对于$ f $ for $ f $ for $ f $ f $ \ boldsymbol f^{（p）} $ be $ p $ hida hida hida fiakan $ f $ f $ f $ f $ f $。我们证明，对于上述所有几乎有限的数量数量$ p $，如果$ a_f $是椭圆曲线，以至于$ a_f（\ mathbb q）$具有排名$ 1 $，而$ p $ - shafarevich-tate的$ a_f $ a_f $ a_f $ a_f $ yathbb q $ y的$ \ quys $到$ 2 $ modulo $ 2（P-1）$，而微不足道的角色具有有限的（$ P $ - 主要）Shafarevich-Tate Group和$ 1 $维度的相关$ P $ -P $ -ADIC-ADICétaleAbel-Jacobi Map的图像。同样的结果也可以在等级$ 0 $案例中获得。作为第二个贡献，没有限制$ a_f $的尺寸，但假设Heegner Cycles之间的某些高度配对的非降级率，我们表明，如果$ f $具有分析等级$ 1 $，那么所有这些，但对于所有$ p $，$ \ boldsymbol f^（p $ hosembol f^（p y） $ 2（P-1）$和微不足道的角色具有分析排名$ 1 $。该结果提供了一些排名$ 1 $的证据，重量大于$ 2 $，这是格林伯格的猜想，预测HIDA家族中均匀的权重模块化形式的分析等级应与功能方程式所允许的范围一样小，并且最有限的例外。
Let $f$ be a newform of weight $2$, square-free level and trivial character, let $A_f$ be the abelian variety attached to $f$ and for every good ordinary prime $p$ for $f$ let $\boldsymbol f^{(p)}$ be the $p$-adic Hida family through $f$. We prove that, for all but finitely many primes $p$ as above, if $A_f$ is an elliptic curve such that $A_f(\mathbb Q)$ has rank $1$ and the $p$-primary part of the Shafarevich-Tate group of $A_f$ over $\mathbb Q$ is finite then all specializations of $\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have finite ($p$-primary) Shafarevich-Tate group and $1$-dimensional image of the relevant $p$-adic étale Abel-Jacobi map. Analogous results are obtained also in the rank $0$ case. As a second contribution, with no restriction on the dimension of $A_f$ but assuming the non-degeneracy of certain height pairings à la Gillet-Soulé between Heegner cycles, we show that if $f$ has analytic rank $1$ then, for all but finitely many $p$, all specializations of $\boldsymbol f^{(p)}$ of weight congruent to $2$ modulo $2(p-1)$ and trivial character have analytic rank $1$. This result provides some evidence in rank $1$ and weight larger than $2$ for a conjecture of Greenberg predicting that the analytic ranks of even weight modular forms in a Hida family should be as small as allowed by the functional equation, with at most finitely many exceptions.
