Green forms and the arithmetic Siegel–Weil formula

Abstract

We construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in all codimensions, and, for compact Shimura varieties of type O(p, 2) and U(p, 1), we show that the resulting local archimedean height pairings are related to special values of derivatives of Siegel Eisentein series. A conjecture put forward by Kudla relates these derivatives to arithmetic intersections of special cycles, and our results settle the part of his conjecture involving local archimedean heights.

Kudla program for unitary Shimura varieties

“…本节回顾 U(n − 1, 1) 所对应的酉志村簇以及上面的算术闭链. 除了 Green 函数 [1,32] 以及 Green 流动形 (current) 的定义 (参见文献 [12]), 我们将主要参考文献 [13]. 然后将讨论 Kudla 纲领在模性以 及算术 Siegel-Weil 公式方面的进展.…”

“…Liu [11] 证明了以下局部算术 Siegel-Weil 公式以及在 ∞ 处的算术 Siegel-Weil 公式. Garcia 和 Sankaran [12] 给出了一个更加一般的证明 (包括了退化项). 定理 3.3 (∞ 处的局部算术 Siegel-Weil 公式 [11] ) 记号同上, 则…”

On the arithmetic Siegel--Weil formula for GSpin Shimura varieties

“…The arithmetic Siegel-Weil formula was established by Kudla, Rapoport and Yang ([KRY99, Kud97b, KR00b, KRY06]) for n = 1, 2. The archimedean part of the formula for all n was also known by Garcia-Sankaran [GS19] and Bruinier-Yang [BY21]. However, for the nonarchimedean part for higher n, the only known cases were when n = 3 due to Gross-Keating [GK93] (cf.…”

The construction of Green currents and singular theta lifts for unitary groups

“…The principle of this construction and its properties were already outlined in [15] for O(p, q) × SL 2 (R) for the form ψ mentioned above and was also implicit in [6] for the Hermitian case O(p, 2). Garcia and Sankaran [17] also follow these lines but use superconnections to solve (1.1). We have not checked the details but it seems likely that for n = 1 their form ν is equal to our form ψ. Garcia and Sankaran then succeed to construct Green forms for n > 1 using a similar integral as above.…”

“…In Kudla's original work [21], Liu [27], and Bruinier and Yang [11] star products are used to construct Green forms for cycles of higher codimension for O(p, 2) and U(p, 1). In recent groundbreaking work, Garcia and Sankaran [17] employed Quillen's theory of superconnections to construct Green forms in O(p, 2) and U(p, q) in any codimension.…”