Arithmetic Geometry, Modularity, and L-Functions of Motives

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Number theory has a long history in mathematics, beginning with the ancient Greeks. One of the most fundamental and important problems in number theory is to solve Diophantine equations, that is, to find integer solutions to polynomial equations with integer coefficients. For example, the famous Fermat's Last Theorem concerns one class of Diophantine equations. In modern mathematical language, Diophantine equations can be encoded into a geometric notion, called algebraic varieties. Then the information of the solution will give rise to an arithmetic invariant of the corresponding algebraic variety. However, these algebraic varieties carry other important invariants related to the arithmetic one via the so-called L-function. It is often possible to transfer information from different invariants and achieve better understanding of all of them. This project aims to deepen understanding of fundamental relationships among such mathematical constructs.The main theme of this project is to study L-functions of motives, which are a systematic generalization of algebraic varieties, 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 in 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 through L-functions. This project investigates five topics: (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) a higher derivative version of the Rallis inner product formula over function fields; and (5) nearby cycles for Artin stacks over general bases.