TY - JOUR
T1 - Definite orthogonal modular forms
T2 - Computations, Excursions and Discoveries
AU - Assaf, Eran
AU - Fretwell, Dan
AU - Ingalls, Colin
AU - Logan, Adam
AU - Secord, Spencer
AU - Voight, John
N1 - Article is included in the Springer Collection: Fifteenth Algorithmic Number Theory Symposium (ANTS XV)
ISSN: 2522-0160 (Print) 2363-9555 (Online)
PY - 2022/9/16
Y1 - 2022/9/16
N2 - 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.
AB - 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.
KW - math.NT
U2 - 10.48550/arXiv.2203.06405
DO - 10.48550/arXiv.2203.06405
M3 - Article
C2 - 36283022
VL - 8
JO - Research In Number Theory
JF - Research In Number Theory
SN - 2363-9555
IS - 4
ER -