AI 中文总结
该研究引入向量值模块复兴级数，将模块复兴范式扩展至向量场景，推测相关q级数向量为向量值量子模形式，还将q-Pochhammer符号向量、特定权的向量值模形式Eichler积分纳入该框架并推导其结构。
AI 中文摘要
基于已有成果[1]，我们引入了向量值模块复兴级数，其分量在Borel平面中呈现单个无限奇点塔、平凡次级复兴级数，且Stokes常数由L-函数向量的系数线性组合给出。我们将模块复兴范式扩展至该场景，强调Stokes常数的作用、关联q级数向量与Dirichlet级数的相互作用，以及由此产生的向量值模块复兴级数标准对之间的对称性。此外，我们推测部分具有模块复兴渐近性的q级数向量为向量值量子模形式，可通过中值重求和重构。最后，我们表明此前在[2]中研究的q-Pochhammer符号向量，以及权为1/2和3/2的向量值模形式的Eichler积分，可在向量值模块复兴框架内研究；前者相当于标量模块复兴的便捷重新打包，后者涉及模群的非平凡表示，故体现了向量值框架的必要性。我们一般地建立了其模块复兴结构，并对一元theta级数详细推导，其Eichler积分是量子拓扑中的伪theta函数。
英文摘要
Building on prior results [1], we introduce vector-valued modular resurgent series, whose components exhibit a single infinite tower of singularities in the Borel plane, trivial secondary resurgent series, and Stokes constants given by linear combinations of the coefficients of a vector of $L$-functions. We extend the paradigm of modular resurgence to this setting, emphasizing the role of the Stokes constants and the interplay between the associated vectors of $q$-series and Dirichlet series, and describing the resulting symmetry relating canonical pairs of vector-valued modular resurgent series. Moreover, we conjecture that certain vectors of $q$-series with modular resurgent asymptotics are vector-valued quantum modular forms and can be reconstructed via median resummation. Finally, we show that vectors of $q$-Pochhammer symbols, previously considered in [2], and Eichler integrals of vector-valued modular forms of weight $1/2$ and $3/2$ can be studied within the framework of vector-valued modular resurgence. While the first case amounts to a convenient repackaging of scalar modular resurgence, the second involves a non-trivial representation of the modular group and therefore illustrates the necessity of the vector-valued framework; we establish its modular resurgent structure in general, and work it out in full detail for the unary theta series, whose Eichler integrals are the false theta functions of quantum topology.
Comments52 pages
