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Explore a mathematical strategy for proving the Tate conjecture on algebraic cycles for quaternionic Shimura varieties in this 56-minute lecture. Discover how 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 form the foundation of this approach. Learn about the theta base-change lift f of Hilbert modular forms to multiplicative groups of B, and understand how their period integrals over Shimura subvarieties Sh(D) inside Sh(B) connect to adjoint L-values twisted by quadratic characters of E/F at 1. Examine why the non-vanishing property of adjoint L-values at 1 (due to their abscissa of convergence) enables Sh(D) to generate non-trivial Tate cycles in the degree 2r cohomology group of Sh(D), where r represents the dimension of Sh(D). Gain insights into advanced topics in algebraic geometry, automorphic forms, and L-functions through this specialized mathematical presentation delivered at the Erwin Schrödinger International Institute for Mathematics and Physics.
