## Project Details

### Description

The study of L-functions or zeta functions has a long history in mathematical study, dating back to Gauss, Dirichlet, Hecke and Artin, for example. In the recent fifty years, major progress has been achieved after Langlands introduce his deep philosophy between Number Theory and Automorphic Forms. Among these, the relations between special values of L-functions and periods, which is the major focus of the PI’s proposal, is one of the most concerned problems. The periods we consider have two meanings: period integrals and invariants in arithmetic geometry, stated as follows.
(1) Classical periods: to understand the relation between central L-values of automorphic representations for unitary groups and the classical period integrals of Bessel or Fourier–Jacobi types. This is the content of the recently proposed global Gan–Gross–Prasad conjecture for unitary groups. The approach we will use is the relative trace formula, through which the PI plans particularly to study a series of identities between orbital integrals, known as relative fundamental lemmas and smooth matching.
(2) Arithmetic periods: to understand the relation between central L-derivatives of automorphic
representations and arithmetic invariants of Shimura varieties: the heights of cycles
which we would like to refer as arithmetic periods. We will study such relation via arithmetic
theta lifting. The PI plans to work in the next unknown case where Shimura 3-folds show up,
as well as other problems such as the calculation of several arithmetic intersection numbers,
and the modularity of compactified generating series.
Intellectual Merit. The proposal to investigate the fundamental relation between certain important L-values (resp. L-derivatives) and period integrals (resp. heights) has potential application to many questions in number theory and representation theory. It provides formulae to compute such L-values (resp. L-derivatives) and help to derive their properties, such as the positivity of central values (resp. derivatives) and subconvexity bound, as evidences of the Generalized Riemann Hypothesis. Conversely, one can use L-functions to study periods or cycles, which are further related to the geometry and arithmetics of the corresponding spaces or varieties.
According to the philosophy of Langlands, each geometric object should have an automorphic representation attached. The proposed projects take a step forward, by setting up a bridge connecting deeper information on both sides.
Broader Impacts. The research of the PI will be disseminated through respected journals and through conferences, seminars and lectures. One special case in the PI’s proposal, the global Gan–Gross–Prasad conjecture in the equal rank case, has inspired a research project of a current graduate student.
The combinatorial part of this proposal could generate several undergraduate projects such as: (a) combinatorics appearing in the calculation of local intersection numbers in the second part of the proposal; (b) computation of some integrals on groups over local fields, for example, explicit formulas for Whittaker–Shintani functions in some cases.

Status | Finished |
---|---|

Effective start/end date | 10/13/15 → 8/31/16 |

### Funding

- National Science Foundation (DMS-1602149)

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