Assaf Eran, Fretwell Dan, Ingalls Colin, Logan Adam, Secord Spencer, Voight John
Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755 USA.
School of Mathematics, Fry Building, University of Bristol, Woodland Road, Bristol, BS8 1UG UK.
Res Number Theory. 2022;8(4):70. doi: 10.1007/s40993-022-00373-2. Epub 2022 Sep 16.
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.
我们考虑与低偶秩且非平凡层数的定正交群相关联的模形式空间，这些空间配备了由克内泽邻域定义的赫克算子。在回顾了用于这些空间计算的算法之后，我们利用西塔级数和拉利斯定理研究内窥镜。在此过程中，我们展示了许多例子并提出了几个猜想。作为第一个应用，我们用经典或西格尔模形式的系数来表示克内泽邻域的计数，补充了谢内维耶 - 兰内斯的工作。作为第二个应用，我们证明了拉马努金和黑川 - 水本类型的艾森斯坦同余的新实例。
