论文标题
标量价值深度两个尖端形式的Eichler-shimura积分
Scalar-valued depth two Eichler-Shimura Integrals of Cusp Forms
论文作者
论文摘要
给定尖式形式的$ f $和$ g $的整体权重$ k \ geq 2 $，深度为两个全态迭代的eichler-shimura积分$ i_ {f，g} $由$ {\int_τ^{i \ infty} f（i \ infty} $ i_g $是$ g $的eichler积分，$ x，y $是正式变量。我们提供了一个显式矢量值模块化表单，其顶部组件由$ i_ {f，g} $给出。 We show that this vector-valued modular form gives rise to a scalar-valued iterated Eichler integral of depth two, denoted by $\mathcal{E}_{f,g}$, that can be seen as a higher-depth generalization of the scalar-valued Eichler integral $\mathcal{E}_f$ of depth one.顺便说一句，我们的论点提供了对最初由于Paşol-popa最初由多项式满足的正交关系的替代解释。我们表明，$ \ Mathcal {e} _ {f，g} $可以用矢量价值的Eisenstein系列的乘积总和表示，具有经典模块化形式，并具有适当的歧视模块化模块化表格$δ$的乘法。这允许有效地计算$ \ Mathcal {e} _ {f，g} $。
Given cusp forms $f$ and $g$ of integral weight $k \geq 2$, the depth two holomorphic iterated Eichler-Shimura integral $I_{f,g}$ is defined by ${\int_τ^{i\infty}f(z)(X-z)^{k-2}I_g(z;Y)\mathrm{d}z}$, where $I_g$ is the Eichler integral of $g$ and $X,Y$ are formal variables. We provide an explicit vector-valued modular form whose top components are given by $I_{f,g}$. We show that this vector-valued modular form gives rise to a scalar-valued iterated Eichler integral of depth two, denoted by $\mathcal{E}_{f,g}$, that can be seen as a higher-depth generalization of the scalar-valued Eichler integral $\mathcal{E}_f$ of depth one. As an aside, our argument provides an alternative explanation of an orthogonality relation satisfied by period polynomials originally due to Paşol-Popa. We show that $\mathcal{E}_{f,g}$ can be expressed in terms of sums of products of components of vector-valued Eisenstein series with classical modular forms after multiplication with a suitable power of the discriminant modular form $Δ$. This allows for effective computation of $\mathcal{E}_{f,g}$.
