Theta cycles and the Beilinson--Bloch--Kato conjectures
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.
Keywords
Cite
@article{arxiv.2303.17817,
title = {Theta cycles and the Beilinson--Bloch--Kato conjectures},
author = {Daniel Disegni},
journal= {arXiv preprint arXiv:2303.17817},
year = {2026}
}
Comments
24 pages, final version, to appear in Journal of Number Theory (special issue: Proceedings of the Second JNT Biennial Conference 2022)