arXiv 2112.094812021-12math.NT

分拆函数的Atkin同余式

Congruences like Atkin's for the partition function

Scott Ahlgren, Patrick B. Allen, Shiang Tang

备注：: Minor revisions to improve exposition

摘要：

设$p(n)$为普通的分拆函数。在20世纪60年代，阿特金发现了一些形式为$p( Q^3 \ell n+β)\equiv0\pmod\ell$的同余式的例子，其中$\ell$和$Q$是素数，而$5\leq \ell\leq 31$是素数;这些都属于两个自然家族，由$1-24β\pmod\ell$的平方类区分开来。近几十年来，人们做了大量的工作来理解$p(Q^m\ell n+β)\equiv 0\pmod\ell$形式的同余。现在知道，当$m\geq 4$时，有许多这样的同余，当$m=1, 2$时，这样的同余是稀少的（如果它们存在的话），而对于$m=0$，这样的同余只存在于当$\ell=5, 7, 11$时。对于像阿特金的同余（当$m=3$），更多的例子已经发现了$5\leq \ell\leq 31$，但很少有人知道。在这里，我们使用的理论模块伽罗瓦表示证明，每一个总理$\ell\geq 5$，有无限多的同余像阿特金的第一个自然家庭，他发现，并至少为$17/24$的总理$\ell$有无限多的同余在第二个家庭。

英文摘要：

Let $p(n)$ be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form $p( Q^3 \ell n+β)\equiv0\pmod\ell$ where $\ell$ and $Q$ are prime and $5\leq \ell\leq 31$; these lie in two natural families distinguished by the square class of $1-24β\pmod\ell$. In recent decades much work has been done to understand congruences of the form $p(Q^m\ell n+β)\equiv 0\pmod\ell$. It is now known that there are many such congruences when $m\geq 4$, that such congruences are scarce (if they exist at all) when $m=1, 2$, and that for $m=0$ such congruences exist only when $\ell=5, 7, 11$. For congruences like Atkin's (when $m=3$), more examples have been found for $5\leq \ell\leq 31$ but little else seems to be known. Here we use the theory of modular Galois representations to prove that for every prime $\ell\geq 5$, there are infinitely many congruences like Atkin's in the first natural family which he discovered and that for at least $17/24$ of the primes $\ell$ there are infinitely many congruences in the second family.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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