摘要翻译：
高阶Green函数是在同余子群作用下在上半平面上双不变的两个变量的实值函数，在对角线上具有对数奇异性，但我们用方程$delta f=0$代替了通常的方程$delta f=k(1-k)f$。这里$k$是一个正整数。这些函数的性质与权值$2k$的模形式的空间有关。在没有尖点形式的权值$2k$的情况下，推测复乘点处的格林函数值是代数数对数的代数倍数。我们证明了如果在椭圆曲线族的幂上构造某些高阶Chow群的元素族，这个猜想在任何特殊情况下都可以证明。这些家庭必须满足某些性质。复杂乘法的不同点需要一个不同的高周群元素家族。我们给出了这类族的一个例子，从而证明了当群为$PSL_2(\mathbf Z)$,$k=2$且其中一个参数为$\sqrt{-1}$时的猜想。
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英文标题：
《Higher Green's functions for modular forms》
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作者：
Anton Mellit
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最新提交年份：
2008
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分类信息：

一级分类：Mathematics        数学
二级分类：Number Theory        数论
分类描述：Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数，丢番图方程，解析数论，代数数论，算术几何，伽罗瓦理论
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  Higher Green functions are real-valued functions of two variables on the upper half plane which are bi-invariant under the action of a congruence subgroup, have logarithmic singularity along the diagonal, but instead of the usual equation $\Delta f=0$ we have equation $\Delta f = k(1-k) f$. Here $k$ is a positive integer. Properties of these functions are related to the space of modular forms of weight $2k$. In the case when there are no cusp forms of weight $2k$ it was conjectured that the values of the Green function at points of complex multiplication are algebraic multiples of logarithms of algebraic numbers. We show that this conjecture can be proved in any particular case if one constructs a family of elements of certain higher Chow groups on the power of a family of elliptic curves. These families have to satisfy certain properties. A different family of elements of Higher Chow groups is needed for a different point of complex multiplication. We give an example of such families, thereby proving the conjecture for the case when the group is $PSL_2(\mathbf Z)$, $k=2$ and one of the arguments is $\sqrt{-1}$.
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PDF链接：
https://arxiv.org/pdf/0804.3184