We study special cycles on integral models of Shimura varieties
associated with unitary similitude groups of signature (n.1, 1).We construct
an arithmetic theta lift from harmonic Maass forms of weight 2 . n to the
arithmetic Chow group of the integral model of a unitary Shimura variety,
by associating to a harmonic Maass form f a linear combination of Kudla–
Rapoport divisors, equipped with the Green function given by the regularized
theta lift of f . Our main result is an equality of two complex numbers: (1)
the height pairing of the arithmetic theta lift of f against a CM cycle, and (2)
the central derivative of the convolution L-function of a weight n cusp form
(depending on f ) and the theta function of a positive definite hermitian lattice
of rank n − 1. When specialized to the case n = 2, this result can be viewed
as a variant of the Gross–Zagier formula for Shimura curves associated to
unitary groups of signature (1, 1). The proof relies on, among other things, a
new method for computing improper arithmetic intersections.
We formulate and prove a log-algebraicity theorem for arbitrary rank Drinfeld modules dened over the polynomial ring Fq[\theta]. This generalizes results of Anderson for the
rank one case. As an application we show that certain special values of Goss L-functions are linear forms in Drinfeld logarithms and are transcendental.
In this paper we derived a nice general formula for the local integrals of triple product formula whenever one of the representations has sufficiently higher level than the other two. As an application we generalized Venkatesh and Woodbury’s work on the subconvexity bound of triple product L-function in level aspect, allowing joint ramifications, higher ramifications, general unitary central characters and general special values of local epsilon factors.
We define the Heegner--Drinfeld cycle on the moduli stack of Drinfeld Shtukas of rank two with r-modifications for an even integer r. We prove an identity between (1) The r-th central derivative of the quadratic base change L-function associated to an everywhere unramified cuspidal automorphic representation π of PGL2; (2) The self-intersection number of the π-isotypic component of the Heegner--Drinfeld cycle. This identity can be viewed as a function-field analog of the Waldspurger and Gross--Zagier formula for higher derivatives of L-functions.