Definite orthogonal modular forms: computations, excursions, and discoveries Article Swipe

PDF

Related Concepts

Modular form Ramanujan's sum Eisenstein series Mathematics Congruence relation Pure mathematics Modular design Algebra over a field Rank (graph theory) Computation Hecke operator Type (biology) Combinatorics Computer science Algorithm Operating system Biology Ecology

Eran Assaf , Dan Fretwell , Colin Ingalls , Adam Logan , Spencer Secord , John Voight ·

YOU? · · 2022 · Open Access · · DOI: https://doi.org/10.1007/s40993-022-00373-2 · OA: W4226262664

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.