论文标题
Ramanujan型一致性I
Relations among Ramanujan-Type Congruences I
论文作者
论文摘要
我们证明，Ramanujan型的一致性远离水平的整体权重模块化形式，一致性素数等同于特定的Hecke eigenvalues。特别是，我们表明Ramanujan型的一致性是由浅Hecke代数的作用保留的。更普遍地，我们显示出弱的全态模块化形式的整体权重，这表明Ramanujan型的一致性自然是在两个方形级的联合中发生的变化，而不是在分区函数上出现的单个方形级别的相反。我们还排除了无方时期的可能性，最近研究了在分区功能的情况下的稀缺性。我们补充了与几个存在陈述的最大Ramanujan型一致性的障碍。我们的结果基于一个框架，该框架通过模块化表示理论利用模块化曲线的积分模型的经典结果，并适用于所有弱塑形模块化形式的一致性。 Steinberg表示的所有最大Ramanujan型的一致性都具有不可或缺的权重。通过示例计算，我们辨别框架的范围。
We prove that Ramanujan-type congruences for integral weight modular forms away from the level and the congruence prime are equivalent to specific congruences for Hecke eigenvalues. In particular, we show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra. More generally, we show for weakly holomorphic modular forms of integral weight, that Ramanujan-type congruences naturally occur for shifts in the union of two square-classes as opposed to single square-classes that appear in the literature on the partition function. We also rule out the possibility of square-free periods, whose scarcity in the case of the partition function was investigated recently. We complement our obstructions on maximal Ramanujan-type congruences with several existence statements. Our results are based on a framework that leverages classical results on integral models of modular curves via modular representation theory, and applies to congruences of all weakly holomorphic modular forms. Steinberg representations govern all maximal Ramanujan-type congruences for integral weights. We discern the scope of our framework in the case of half-integral weights through example calculations.
