论文标题
稀缺的分区功能的一致性
Scarcity of congruences for the partition function
论文作者
论文摘要
普通分区功能的算术属性$ p（n）$一直是过去一个世纪的密集研究主题。 Ramanujan证明了$ p（\ ell n+β）\ equiv 0 \ pmod \ ell $ for Primes $ \ ell = 5、7、11 $的线性一致性，并且众所周知，此形式没有其他形式。另一方面，对于每一个prime $ \ ell \ geq 5 $，都有无限的示例$ p（\ ell q^m n+β）\ equiv 0 \ pmod \ ell $，其中$ q \ geq 5 $是prime和$ m \ geq 3 $。当$ m = 1 $或$ m = 2 $时，这留下了关于这种一致性的问题（在这些情况下尚无示例）。从精确的意义上说，这样的一致性（如果存在）极为稀缺。我们的方法涉及对与分区函数相关的完整模块组中半积分重量的模块化形式的仔细研究。在许多其他工具中，我们使用RADU的工作，描述了模块化曲线$ x（\ ell Q）$，GALOIS表示和算术大筛子的cusps的这种模块化形式的扩展。
The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+β)\equiv 0\pmod\ell$ for the primes $\ell=5, 7, 11$, and it is known that there are no others of this form. On the other hand, for every prime $\ell\geq 5$ there are infinitely many examples of congruences of the form $p(\ell Q^m n+β)\equiv 0\pmod\ell$ where $Q\geq 5$ is prime and $m\geq 3$. This leaves open the question of the existence of such congruences when $m=1$ or $m=2$ (no examples in these cases are known). We prove in a precise sense that such congruences, if they exist, are exceedingly scarce. Our methods involve a careful study of modular forms of half integral weight on the full modular group which are related to the partition function. Among many other tools, we use work of Radu which describes expansions of such modular forms along square classes at cusps of the modular curve $X(\ell Q)$, Galois representations and the arithmetic large sieve.
