定正交模形式：计算、偏移与发现
Definite orthogonal modular forms: Computations, Excursions and Discoveries
摘要：
我们考虑空间的模形式附加到明确的正交群的低偶数秩和非平凡的水平，配备Hecke运营商定义的Kneser邻居。在回顾了这些空间的计算算法后，我们研究了使用θ系列和拉利斯定理的内窥镜检查。一路上，我们展示了许多例子，并提出了一些建议。作为第一个应用程序，我们表示计数的Kneser邻居的系数的经典或Siegel模形式，补充工作的Chenevier-Lannes。作为第二个应用，我们证明了Ramanujan和Kurokawa-Mizumoto型的Eisenstein同余的新例子。
英文摘要：
We consider spaces of modular forms attached to definite orthogonal groups of low even rank and nontrivial level, equipped with Hecke operators defined by Kneser neighbours. After reviewing algorithms to compute with these spaces, we investigate endoscopy using theta series and a theorem of Rallis. Along the way, we exhibit many examples and pose several conjectures. As a first application, we express counts of Kneser neighbours in terms of coefficients of classical or Siegel modular forms, complementing work of Chenevier-Lannes. As a second application, we prove new instances of Eisenstein congruences of Ramanujan and Kurokawa-Mizumoto type.
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