论文标题
极性谐波maaß形式和全态投影
Polar harmonic Maaß forms and holomorphic projection
论文作者
论文摘要
最近，梅尔滕斯（Mertens），奥诺（Ono）和第三作者研究了爱森斯坦系列的模拟模块化类似物。它们的系数由小的除数函数给出，并且具有经典的Shimura theta函数给出的阴影。在这里，我们构造一类小除数函数$σ^{\ text {sm}} _ {2，χ} $，并证明这些生成了极性谐波（弱）Maaß形式的重量$ \ frac $ \ frac {3} {2} {2} $。为此，我们从本质上计算了混合谐波maß形式的全态投影，但没有假设这种形式的结构。取而代之的是，我们对傅立叶系数施加了翻译不变性和合适的生长条件。专门针对某些字符选择，我们获得了$σ^{\ text {sm}} _ {2，\ \ text {id}} $和hurwitz class编号之间的身份。此外，当$ p $是一个奇怪的素数时，我们证明了$ p $ ad的一致性。如果$χ$是非平地的，我们将$σ^{\ text {sm}} _ {2，χ} $的生成函数作为上cluss sums的线性组合及其前两个归一化衍生物的线性组合。最后，我们提供了与索引$ -1 $和False Theta功能的Meromorphic Jacobi形式的建筑联系。
Recently, Mertens, Ono, and the third author studied mock modular analogues of Eisenstein series. Their coefficients are given by small divisor functions, and have shadows given by classical Shimura theta functions. Here, we construct a class of small divisor functions $σ^{\text{sm}}_{2,χ}$ and prove that these generate the holomorphic part of polar harmonic (weak) Maaß forms of weight $\frac{3}{2}$. To this end, we essentially compute the holomorphic projection of mixed harmonic Maaß forms in terms of Jacobi polynomials, but without assuming the structure of such forms. Instead, we impose translation invariance and suitable growth conditions on the Fourier coefficients. Specializing to a certain choice of characters, we obtain an identitiy between $σ^{\text{sm}}_{2,\ \text{Id}}$ and Hurwitz class numbers, and ask for more such identities. Moreover, we prove $p$-adic congruences of our small divisor functions when $p$ is an odd prime. If $χ$ is non-trivial we rewrite the generating function of $σ^{\text{sm}}_{2,χ}$ as a linear combination of Appell-Lerch sums and their first two normalized derivatives. Lastly, we offer a connection of our construction to meromorphic Jacobi forms of index $-1$ and false theta functions.
