发表机构
Universität Wien; Chongqing Normal University（维也纳大学; 重庆师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明了拉马努金θ函数φ(-q)与ψ(q)相关无穷级数的八类模3、5任意次幂的同余关系，明确二者同余族的同构性并预测额外同构的存在。
AI 中文摘要
近期，第一作者与第三作者开启了对拉马努金θ函数φ(-q)与ψ(q)之间算术关系的研究。本研究中，我们证明了与这两个θ函数相关的无穷级数在模3和5的任意次幂下的八类内部同余关系。我们还表明，φ(-q)与ψ(q)的同余族之间存在同构，这类同构可通过Garvan、Sellers及第二作者此前研究的同余重数来实现。我们认为这类等价关系至少在同余子群Γ₀(6)和Γ₀(10)上是独有的。最后，我们展示了函数域扩张的简单操作如何帮助预测是否会出现与我们的同余相关的额外同构。
英文摘要
Recently, the first and third authors initiated a study of arithmetical relationships between Ramanujan's theta functions $φ(-q)$ and $ψ(q)$. In this work, we prove eight families of internal congruences modulo arbitrary powers of $3$ and $5$ for infinite series related to the two theta functions. We also show that there exist isomorphisms between the congruence families for $φ(-q)$ and $ψ(q)$, which can be realized by the study of congruence multiplicities previously studied by Garvan, Sellers, and the second author. We believe that such equivalences are exclusive, at least on the congruence subgroups $Γ_0(6)$ and $Γ_0(10)$. In the end, we show how a simple manipulation of function field extensions allows us to predict whether additional isomorphisms to our congruences occur.
