This proposal is to support the work of the Wei Zhang who propose to work on several projects on arithmetic algebraic geometry, in particular, automorphic forms and Shimura varieties. In particular to prove the refined Gross-Prasad conjecture for unitary groups in three variables using the relative trace formulae of Jacquet-Rallis secondly to prove a Gross-Zagier type formula for unitary Shimura varieties using the approach of the arithmetic relative trace formulae initiated by W.Zhang recently,and thirdly to study their applications to the Beilinson-Bloch conjecture and Bloch-Kato conjecture on the relation between Chow groups, Selmer groups and L-values.
This proposal will provide support for a recent PhD, now a PostDoc at Harvard to pursue research in Number Theory and Representation Theory. This research concerns a special type of function known as an L-function which encodes information about geometry and arithmetic.
As the fist outcome of this project, the PI developed new tools to attack several questions in the theory of the relative trace formula. The relative trace formula is a tool iniatiated by H. Jacquet in 1980s to study period integrals of automorphic forms, a special type of functions on groups with many nice properties. These functions are central in the so-called "Langlands program" in current mathematics. The study of period integral has great applicatons in number theoretical questions. The PI developed effective methods to prove some existence theorem in matching test functions in the theory of the relative trace formula, when the relevant groups are all reductive. This type of existence theorem is complementary to the celebated "fundamental lemma"-type theorem. The PI is then able to prove new theorems relating the Gan--Gross--Prasad type period integral to some Rankin--Selberg type L-functions. The second outcome of this project is that the PI extended the relative trace formula due to Jacquet and Rallis to an arithmetic setting, where the period integral on groups is replaced by intersection numbers of algebraic cyles on Shimura varieties. Shimura varieties, defined over number fields, are important geometric objects directly releated to automorphic forms. Algebraic cycls on them are expected to relate to the derivative of relevant L-functions, following the example of the celebarted Gross--Zagier formula first proved in 1980s for the modular curve (a special case of Shimura curves, namely one-dimensional Shimura varieties). The PI has formulated an approach to prove the high dimensional generalization of the Gross-Zagier formula. An ingredient is the so-called "arithmetic fundamental lemma", an identity between two counting functions of different flavors: one counting analytic objects (reltiave orbital integrals) and the other counting arithmetic-geometric objects (arithmetic intersection numbers). The PI is able to prove new nontrivial case of this "arithmetic fundamental lemma". The third outcome is a general Gross-Zagier formula for Shimura curves, from the joint work with X. Yuan and S. Zhang. This formula is funfamental for Diophantine application to elliptic curves and its generalization to abelian varieties parameterized by Shimura curves.
