Arithmetic inner product formula for unitary groups

Title:

Arithmetic inner product formula for unitary groups

Author(s):

Liu, Yifeng

Thesis Advisor(s):

Zhang, ShouWu

Date:

2012

Type:

Dissertations

Department:

Mathematics

Permanent URL:

http://hdl.handle.net/10022/AC:P:13049

Notes:

Ph.D., Columbia University.

Abstract:

We study central derivatives of Lfunctions of cuspidal automorphic representations for unitary groups of even variables defined over a totally real number field, and their relation with the canonical height of special cycles on Shimura varieties attached to unitary groups of the same size. We formulate a precise conjecture about an arithmetic analogue of the classical Rallis' inner product formula, which we call arithmetic inner product formula, and confirm it for unitary groups of two variables. In particular, we calculate the NéronTate height of special points on Shimura curves attached to certain unitary groups of two variables. For an irreducible cuspidal automorphic representation of a quasisplit unitary group, we can associate it an εfactor, which is either 1 or 1, via the dichotomy phenomenon of local theta liftings. If such factor is 1, the central Lvalue of the representation always vanishes and the Rallis' inner product formula is not interesting. Therefore, we are motivated to consider its central derivative, and propose the arithmetic inner product formula. In the course of such formulation, we prove a modularity theorem of the generating series on the level of Chow groups. We also show the cohomological triviality of the arithmetic theta lifting, which is a necessary step to consider the canonical height. As evidence, we also prove an arithmetic local SiegelWeil formula at archimedean places for unitary groups of arbitrary sizes, which contributes as a part of the local comparison of the conjectural arithmetic inner product formula.

Subject(s):

Mathematics
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