Theta 循环和贝林松-布洛赫-卡托猜想

Pub Date : 2024-05-06 DOI:10.1016/j.jnt.2024.04.001
Daniel Disegni
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引用次数: 0

摘要

我们在某些具有共轭交错对称性的伽罗瓦表示的塞尔默群中引入了 "典型 "类。它们是单元志村变中特殊循环的图像,并且是唯一定义的标量。这一构造是对刘玉良的构造的细微改进,它基于库德拉特殊循环的 Theta 序列的猜想模块性。对于二维表示,Theta 循环是(希格纳点的塞尔玛图像)。一般而言,它们在秩 1 中与贝林森-布洛赫-加藤猜想有类似的紧密联系,我们收集了这方面的现有证据。
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Theta cycles and the Beilinson–Bloch–Kato conjectures
We introduce ‘canonical’ classes in the Selmer groups of certain Galois representations with a conjugate-symplectic symmetry. They are images of special cycles in unitary Shimura varieties, and defined uniquely up to a scalar. The construction is a slight refinement of one of Y. Liu, based on the conjectural modularity of Kudla's theta series of special cycles. For 2-dimensional representations, Theta cycles are (the Selmer images of) Heegner points. In general, they conjecturally exhibit an analogous strong relation with the Beilinson–Bloch–Kato conjectures in rank 1, for which we gather the available evidence.
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