Arithmetic Geometry and Automorphic L-Functions

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The research project concerns one of the basic questions in mathematics: solving algebraic equations. Information about solutions is encoded in various mathematical objects, including arithmetic varieties and automorphic L-functions. This research aims to deepen the understanding of these mathematical objects, especially in higher dimension, which requires developing new tools and interactions in diverse areas and appealing to new perspectives that may shed new light on longstanding open questions. The project also aims to advance the techniques for understanding the arithmetic of elliptic curves, particularly the Birch and Swinnerton-Dyer conjecture, one of the seven Millennium Prize Problems of the Clay Mathematics Institute.The research consists of several projects relating arithmetic geometry with automorphic L-functions, centered on the common theme of the generalization and applications of the Gross–Zagier formula. The PI will investigate the Kudla–Rapoport conjecture for ramified unitary groups and orthogonal groups. The PI plans to extend the arithmetic inner product formula and apply it to the Beilinson–Bloch and Bloch–Kato conjectures of symmetric power motives of elliptic curves. The PI will also investigate a new arithmetic relative trace formula approach towards a Gross–Zagier type formula for orthogonal Shimura varieties.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.