论文标题
关于特殊周期的替代
On the subring of special cycles
论文作者
论文摘要
对于完全d> 1的完全真实的f，以及签名的二次空间v（m，2）^{d_+} x（m+2,0）^{d-d_+}与相关的shimura品种sh（v），我们考虑由加权特殊周期的类别产生的同谱。我们假设D _+<d。我们通过与之配对的限制的限制来采用该环的商SC（V）。我们表明，SC（V）中类的内部产物由Hilbert-Siegel Eisenstein系列M的傅立叶系数确定为较小的Siegel空间的产物，而SC（V）类中的类产品由较小Siegel空间的Triple产品的傅立叶系数确定。结果，我们表明，对于二次空间V和V'，在所有有限的地方都是同构的，但是除了必要的条件外，对d _+（v）和d _+（v'）没有任何限制，除了它们具有相同的奇偶校验的必要条件外，特殊的周期环（V）和SC（v）和SC（V'）是相同的。这是Siegel-Weil公式和匹配原则的结果。最后，我们给出了与F上的尺寸m+2的完全确定的二次空间V_+相关的环SC（V_+）的组合构造，并表明比较同构延伸至这种情况。
For a totally real field F of degree d>1 and a quadratic space V of signature (m,2)^{d_+} x (m+2,0)^{d-d_+} with associated Shimura variety Sh(V), we consider the subring of cohomology generated by the classes of weighted special cycles. We assume that d_+<d. We take the quotient SC(V) of this ring by the radical of the restriction of the intersection pairing to it. We show that the inner products of classes in SC(V) are determined by Fourier coefficients of pullbacks of Hilbert-Siegel Eisenstein series of genus m to products of smaller Siegel spaces and that the products of classes in SC(V) are determined by Fourier coefficients of pullbacks to triple products of smaller Siegel spaces. As a consequence, we show that, for quadratic spaces V and V' over F that are isomorphic at all finite places, but with no restriction on d_+(V) and d_+(V') other than the necessary condition that they have the same parity, the special cycles rings SC(V) and SC(V') are isometrically isomorphic. This is a consequence of the Siegel-Weil formula and the matching principle. Finally, we give a combinatorial construction of a ring SC(V_+) associated to a totally positive definite quadratic space V_+ of dimension m+2 over F and show that the comparison isomorphism extends to this case.
