论文标题
Iwasawa二次希尔伯特模块化形式的理论
Iwasawa theory for quadratic Hilbert modular forms
论文作者
论文摘要
我们研究了P-Cyclotomic Tower上的二次希尔伯特模块化形式的硫泽主要猜想。在Siegel模块化品种的共同体中使用Euler系统，我们证明了在某些技术假设下的Iwasawa Main猜想的“ KATO划分”。通过将该结果与WAN引起的相反的分裂性进行比较，我们在环形ZP扩展上获得了完整的主要猜想。结果，我们证明了二次希尔伯特模块形式的kato猜想的新案例，以及分析等级0中椭圆形曲线中椭圆形曲线中的近似birch-swinnerton-dyer猜测，而真实的Quadratic曲线在由Dirichlet字符扭曲的真实Quadratic领域上。 作为此处开发的理论的“副产品”，我们还为兰金（Rankberg）的硫岛理论提出了新的结果 - 塞尔伯格（Selberg）模块化形式的卷积，放松的假设$ p $ distrinction或$ p $ - 以前的作品中假定。这为椭圆形曲线的Equivariant BSD猜想提供了新的案例，该曲线超过了$ \ mathbf {q} $，由2维奇数Artin表示扭曲，从而使Tate-Shafarevich Group的$ P $ - 零件有限，但对于所有人来说，几乎有限的许多普通的普通素数。
We study the Iwasawa main conjecture for quadratic Hilbert modular forms over the p-cyclotomic tower. Using an Euler system in the cohomology of Siegel modular varieties, we prove the "Kato divisibility" of the Iwasawa main conjecture under certain technical hypotheses. By comparing this result with the opposite divisibility due to Wan, we obtain the full Main Conjecture over the cyclotomic Zp-extension. As a consequence, we prove new cases of the Bloch--Kato conjecture for quadratic Hilbert modular forms, and of the equivariant Birch--Swinnerton-Dyer conjecture in analytic rank 0 for elliptic curves over real quadratic fields twisted by Dirichlet characters. As a "by-product" of the theory developed here, we also present new results on Iwasawa theory for Rankin--Selberg convolutions of modular forms, relaxing hypotheses of $p$-distinction or $p$-regularity assumed in previous works. This gives new cases of the equivariant BSD conjecture for elliptic curves over $\mathbf{Q}$ twisted by 2-dimensional odd Artin representations, giving finiteness of the $p$-part of the Tate--Shafarevich group for all but finitely many ordinary primes.
