Intersection numbers of cycles on locally symmetric spaces and fourier coefficients of holomorphic modular forms in several complex variables
Intersection numbers of cycles on locally symmetric spaces and fourier coefficients of holomorphic modular forms in several complex variables
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局部对称空间上的循环交集数与多复变量全纯模形式的傅里叶系数
DOI：
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发表时间：
1990
期刊：
影响因子：
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通讯作者：
J. Millson
中科院分区：
文献类型：
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作者：
S. Kudla;J. Millson
利用theta对应，我们构造了与O（p，q）（resp. U（p，q））的度q（resp. Hodge typenq，nq）到权（p +q）/2和亏格n（resp.权p +q和亏格n的全纯埃尔米特模形式）。值得注意的是，具有紧支撑的上同调包含Borel [3]的尖点调和形式。我们可以用η在某些全测地圈上的周期来表示η提升的傅立叶系数--推广了Shintani对Shimura猜想的解[21]。然后，我们选择η作为（有限）圈的Poincaré对偶，并获得类似于Hirzebruch-Zagier [8]的公式集合。在我们以前的工作中，我们构造了上述提升，但我们无法证明它取值于全纯形式。此外，我们无法计算提升类的不定傅立叶系数。根据柯彻定理，我们现在可以得出这样的结论：所有这些系数都是零。
Using the theta correspondence we construct liftings from the cohomology with compact supports of locally symmetric spaces associated to O(p, q) (resp. U(p, q)) of degreenq (resp. Hodge typenq, nq) to the space of classical holomorphic Siegel modular forms of weight (p +q)/2 and genusn (resp. holomorphic hermitian modular forms of weightp +q and genusn). It is important to note that the cohomology with compact supports contains the cuspidal harmonic forms by Borel [3]. We can express the Fourier coefficients of the lift of η in terms of periods of η over certain totally geodesic cycles—generalizing Shintani’s solution [21] of a conjecture of Shimura. We then choose η to be the Poincaré dual of a (finite) cycle and obtain a collection of formulas analogous to those of Hirzebruch-Zagier [8]. In our previous work we constructed the above lifting but we were unable to prove that it took values in theholomorphic forms. Moreover, we were unable to compute the indefinite Fourier coefficients of a lifted class. By Koecher’s Theorem we may now conclude that all such coefficients are zero.