Theta cycles and the Beilinson–Bloch–Kato conjectures

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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.

Original languageAmerican English
JournalJournal of Number Theory
DOIs
StateAccepted/In press - 1 Jan 2024

Keywords

  • Beilinson–Bloch–Kato conjectures
  • Galois representations
  • Kudla program
  • Unitary Shimura varieties

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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