2012 Theses Doctoral

Arithmetic inner product formula for unitary groups

Liu, Yifeng

We study central derivatives of L-functions 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 Néron-Tate height of special points on Shimura curves attached to certain unitary groups of two variables.

For an irreducible cuspidal automorphic representation of a quasi-split unitary group, we can associate it an ε-factor, which is either 1 or -1, via the dichotomy phenomenon of local theta liftings. If such factor is -1, the central L-value 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 Siegel-Weil 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.