Smoot Nicolas Allen
Johannes Kepler University Linz, Research Institute for Symbolic Computation, Linz, Austria.
Ramanujan J. 2023;62(1):1-45. doi: 10.1007/s11139-023-00747-9. Epub 2023 Jun 28.
In 2018, Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third-order mock theta function . Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 -module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1. We show that Wang and Yang's family of congruences, which is associated with a genus 0 modular curve, can be proved using a single-variable approach, via a ring of modular functions isomorphic to a localization of . To our knowledge, this is the first time that such an algebraic structure has been applied to the theory of partition congruences. Our induction is more complicated, and relies on sequences of functions which exhibit a somewhat irregular 5-adic convergence. However, the proof ultimately rests upon the direct verification of only 10 initial relations, and is similar to the classical methods of Ramanujan and Watson.
2018年，王柳泉和杨一帆证明了与三阶模拟theta函数相对应的最小部分函数存在无穷多个同余式。他们的证明采用归纳法形式，需要20个初始关系，并利用了一个与自由秩2 -模同构的模函数空间。这种证明策略最初是由保勒和拉杜开发的，用于研究与亏格为1的模曲线相关的同余式族。我们表明，王和杨的同余式族与亏格为0的模曲线相关，可以通过一个与局部化同构的模函数环，采用单变量方法来证明。据我们所知，这是这种代数结构首次应用于分拆同余式理论。我们的归纳更复杂，并且依赖于表现出某种不规则5进收敛的函数序列。然而，证明最终仅依赖于对10个初始关系的直接验证，并且与拉马努金和沃森的经典方法类似。
