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arXiv · 2303.17817

Theta cycles and the Beilinson--Bloch--Kato conjectures

Daniel Disegni
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Abstract

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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Daniel Disegni. 2023-03-31. Theta cycles and the Beilinson--Bloch--Kato conjectures. https://doi.org/10.1016/j.jnt.2024.04.001

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