In 1983, Ribet presented the level-raising theorem for modular forms in his ICM report. One approach to prove this is to show the surjectivity of the Abel–Jacobi map for modular curves. Recently, many high-dimensional generalizations along this approach have been established, including the works of Liu–Tian (2020), Rong Zhou (2023) on quaternionic Shimura varieties, and LTXZZ (2022), LTX (2024) on U(2r − 1, 1) Shimura varieties.
In this talk, I will introduce the ongoing work with Hao Fu to show an arithmetic level- raising result for the special fiber of U(2r, 1) Shimura variety at an inert prime. Inspired by Rong’s work, we exhibit elements in the higher Chow group of the supersingular locus and use this to prove the surjectivity of the Abel–Jacobi map. A key ingredient of the proof is to show a form of Ihara’s lemma.
