Diagonal cycles and Euler systems II: the Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin L-functions
Diagonal cycles and Euler systems II: the Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin L-functions
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对角循环和欧拉系统 II：Hasse-Weil-Artin L 函数的 Birch 和 Swinnerton-Dyer 猜想
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
V. Rotger
中科院分区：
文献类型：
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作者：
H. Darmon;V. Rotger
本文建立了在解析秩为0的$ mathbb{Q}$上的椭圆曲线的Birch猜想和Swinnerton-Dyer猜想的新情况，这些椭圆曲线是在维数不超过$ 4的某些自对偶Artin表示切割的域上观察的。当相关的$ L$-函数在其中心点消失（偶阶$ g 2$）时，构造了相应Selmer群中的两个正则类，并证明了它们是线性无关的，假设Garrett-Hida $ p$-adic $ L$-函数在其经典插值范围之外的点上不消失。这两个结果的关键工具是研究由模曲线的三重积塔中的Gross-Kudla-Schoen对角环产生的全局伽罗维上同类的某些$ p$进族。
This article establishes new cases of the Birch and Swinnerton-Dyer conjecture in analytic rank 0, for elliptic curves over $ mathbb{Q}$ viewed over the fields cut out by certain self-dual Artin representations of dimension at most $ 4$. When the associated $ L$-function vanishes (to even order $ ge 2$) at its central point, two canonical classes in the corresponding Selmer group are constructed and shown to be linearly independent assuming the non-vanishing of a Garrett-Hida $ p$-adic $ L$-function at a point lying outside its range of classical interpolation. The key tool for both results is the study of certain $ p$-adic families of global Galois cohomology classes arising from Gross-Kudla-Schoen diagonal cycles in a tower of triple products of modular curves.
一般簇的有限多项式上同调
DOI：
10.1007/s40316-015-0041-7
发表时间：
2016
期刊：
Annales mathématiques du Québec
影响因子：
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作者：
Besser A
通讯作者：
Besser A