On two arithmetic theta lifts
On two arithmetic theta lifts
复制标题
关于两个算术 theta 提升
DOI：
10.1112/s0010437x18007327
复制
发表时间：
2016
影响因子：
1.8
通讯作者：
Siddarth Sankaran
中科院分区：
文献类型：
--
作者：
Stephan Ehlen;Siddarth Sankaran
我们的目的是阐明Kudla的格林函数和Bruinier的格林函数之间的关系，这些格林函数附属于正交和酉型Shimura簇上的特殊循环，在Kudla程序的背景下，它们在这些循环的算术几何中起着关键作用。特别地，我们证明了取两族格林函数之差得到的生成级数是一个非全纯模形式，并且具有平凡(尖)全纯投影。在此过程中，我们使用正则化的theta提升，构造了Weil表示中所值的中等增长形式的Maaú降低算子的一部分，这可能是独立感兴趣的，因为它特别应用于模拟模形式。我们还考虑了算术几何在$U(n，1)$Shimura簇的积分模型中的应用。每个Green函数族都产生一个形式算术theta函数，取值于算术Chow群中，它被猜想为模；我们的主要结果是两个算术theta函数之差的模性。最后，我们将特殊圈的算术高度与Kudla猜想所预言的Eisenstein级数的特殊导数联系起来，并描述了Bruinier，Howard和Yang关于CM点的算术交的一个定理的改进。
Our aim is to clarify the relationship between Kudla’s and Bruinier’s Green functions attached to special cycles on Shimura varieties of orthogonal and unitary type, which play a key role in the arithmetic geometry of these cycles in the context of Kudla’s program. In particular, we show that the generating series obtained by taking the differences of the two families of Green functions is a non-holomorphic modular form and has trivial (cuspidal) holomorphic projection. Along the way, we construct a section of the Maaß lowering operator for moderate growth forms valued in the Weil representation using a regularized theta lift, which may be of independent interest, as it in particular has applications to mock modular forms. We also consider arithmetic-geometric applications to integral models of $U(n,1)$ Shimura varieties. Each family of Green functions gives rise to a formal arithmetic theta function, valued in an arithmetic Chow group, that is conjectured to be modular; our main result is the modularity of the difference of the two arithmetic theta functions. Finally, we relate the arithmetic heights of the special cycles to special derivatives of Eisenstein series, as predicted by Kudla’s conjecture, and describe a refinement of a theorem of Bruinier, Howard and Yang on arithmetic intersections against CM points.
Borcherds 产品的算术
影响因子：
1.1
作者：
Howard, Benjamin;Madapusi Pera, Keerthi
通讯作者：
Madapusi Pera, Keerthi
影响因子：
0.9
作者：
J. B. Gil;Jürg Kramer;U. Kuehn
通讯作者：
J. B. Gil;Jürg Kramer;U. Kuehn