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Explore a mathematical strategy for proving the Tate conjecture on algebraic cycles for quaternionic Shimura varieties in this advanced lecture. Delve into the key components of this approach, including twisted adjoint L-value formulas relative to quaternion algebras D/F over totally real fields F and their scalar extensions B to totally real quadratic extensions E/F. Examine how theta base-change lifts of Hilbert modular forms to multiplicative groups relate to period integrals over Shimura subvarieties, and discover how these period integrals connect to adjoint L-values twisted by quadratic characters evaluated at 1. Learn why the non-vanishing property of adjoint L-values at 1, due to their abscissa of convergence, enables Shimura subvarieties Sh(D) to generate non-trivial Tate cycles in degree 2r cohomology groups, where r represents the dimension of Sh(D). This presentation was delivered as part of the Workshop on "Eisenstein Series, Spaces of Automorphic Forms, and Applications" at the Erwin Schrödinger International Institute for Mathematics and Physics.
