# Non-reductive geometry, arithmetic fundamental lemmas and relative trace formulas: (twisted) Asai motives

We care about arithmetic invariants of polynomial equations / motives e.g. conductors or L-functions, which (conjecturally) are often automorphic and related to cycles on Shimura varieties. In this talk, I will focus on L-functions of Asai motives (e.g. Rankin-Selberg motives for GL_n x GL_n) with mild ramifications in the context of twisted Gan-Gross-Prasad (TGGP) conjecture. I will compare global relative trace distributions and prove new cases of TGGP conjecture (joint with W. Lu and D. Wang). Moreover, I will define an arithmetic relative trace distribution using Shimura varieties with local-global decompositions and prove a key twisted arithmetic fundamental lemma (TAFL) towards the arithmetic TGGP conjecture. I will emphasize the use of non-reductive geometry and mysterious new cycles of “half integral dimension”, from the mirabolic cycles on Rapoport-Zink spaces (and real circles on unitary Shimura varieties). Time permitting, I will formulate a conjectural higher AFL for higher L-derivatives in char p>0 (in progress with Z. Yun).