Theta cycles

Abstract: For any elliptic curve E over Q, an explicit construction yields a point P in E(Q) that is canonical, in the following sense: (*) P is non-torsion <=> the group E(Q) and all p^\infty-Selmer groups of E have rank 1. I will discuss a partial generalization of this picture to higher-rank motives M enjoying a `conjugate-symplectic’ symmetry; examples arise from symmetric products of elliptic curves. The construction of the “canonical algebraic cycle on M", based on works of Kudla and Y. Liu, uses theta series valued in Chow groups of Shimura varieties, and it relies on two very different modularity conjectures. Assuming those, I will present a version of the " => " part of (*), whose proof uses recent advances in the theory of Euler systems.

algebraic geometrynumber theory

Audience: researchers in the topic

UCSB Seminar on Geometry and Arithmetic