Number/Representation Theory

The study of special values and derivatives of automorphic L-functions reveals deep connections between arithmetic geometry and harmonic analysis. A central theme is the Arithmetic Fundamental Lemma (AFL), which predicts precise identities between orbital integrals and intersection numbers of cycles. While methods based on perverse sheaves have achieved remarkable results in the function field case, they often obscure the underlying local geometry.

In this talk I will present recent progress on the higher linear AFL through the framework of the Relative Trace Formula (RTF). This approach provides explicit structural links between analytic orbital integrals and local intersection theory, enabling direct local proofs beyond global sheaf-theoretic methods. I will also outline several new directions: extending the AFL by adding conductors, and exploring their applications to global conjectures on derivatives of L-functions, including variants of Gross–Zagier type formulas.