Arakelov geometry, Shimura varieties, and L-functions of motives

As the title suggests, the main theme of this proposal is to study L-functions of motives, via studying various aspects of Shimura varieties and the method of Arakelov geometry. The L-functions of motives play an important role in the current research of number theory, automorphic representations and arithmetic geometry, as they encode crucial information from all these aspects. The Langlands Program predicts that motives, like rational elliptic curves, are linked with automorphic forms – objects from harmonic analysis and representation theory of Lie groups – through L-functions. More precisely, in this proposal, the PI will describe five projects which are respectively about: (1) The Bloch–Kato conjecture for motives appearing in the Gan–Gross–Prasad conjecture; (2) Modularity of Hecke actions on special cycles on Shimura varieties; (3) Potential theory on non-archimedean spaces and its connection to Arakelov geometry; (4) Higher derivative version of the Rallis inner product formula over function fields; (5) Nearby cycles for Artin stacks over general bases. Intellectual merits. The expected outcomes of the proposal will contain the following aspects. There will be new crucial evidence for the Bloch–Kato conjecture for motives of higher ranks, which generalizes famous results about rational elliptic curves in the context of the Birch and Swinnerton-Dyer conjecture. The modularity property for the Hecke action is a key property that will be needed in the attempt of Gross–Zagier type formulae, which are formulae computing first central derivative of L-functions for motives of higher ranks. One of the most important applications of the development of the non-archimedean potential theory will be in the computation of height of algebraic cycles and thus also related to the central derivative of L-functions. Over function fields, we have more flexibility in geometry such that a formula for higher central derivatives of L-functions would be possible – besides the one recently proved by Z. Yun and W. Zhang, we expect to obtain a new formula in the framework of theta lifting. The last outcome is not directly attached to the main stream of the proposal, however still applicable in geometric methods of automorphic forms which themselves are related to L-functions. Broader impacts. The research results of the PI will be disseminated through publication in standard research journals and lectures in workshops and conferences. The PI will continue to advise graduate students and be mentor of postdoctoral researchers. Projects described in this proposal can be used to form thesis problems for graduate students, to stimulate collaborations with postdocs, as well as to generate research projects suitable for undergraduate students. The PI plans to organize workshops or conferences related to the areas described in the proposal, which will provide students, postdocs, and other mathematicians in the area substantial opportunities for instruction, discussion and collaboration. The PI will continue to give lectures in the form of colloquiums and public lectures to introduce recent progress in related fields to general mathematicians and to promote mathematics to general publics.