Eisenstein series for orthogonal groups and Kneser neighbours
Abstract
We study Eisenstein series for orthogonal groups of even unimodular lattices of signature (n,2) which arise as lifts of elliptic modular forms for SL2(ℤ) on 1-dimensional cusps. When the weight is small they have to be defined by analytic continuation. In general, they will be smooth but non-holomorphic after regularization. We show exactly in which cases we obtain holomorphic modular forms by computing the intertwining factors appearing in Langlands' constant term formula in terms of standard parameters and Rankin-Selberg L-values. This is related to Kneser's neighbour theory of positive-definite even lattices, which also governs the eigenvalues of these Eisenstein series. In particular, we describe the space of singular weight modular forms and construct the function Φ12, the denominator function of the Fake Monster Algebra, as an Eisenstein series. We apply these results to show that the corresponding orthogonal modular varieties are of general type.
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BibTeX
@misc{muller2026eisenstein,
title = {{Eisenstein series for orthogonal groups and Kneser neighbours}},
author = {Manuel K. -H. Müller and Nils R. Scheithauer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26697},
eprint = {2610.26697},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26697},
}
RIS
TY - UNPB
TI - Eisenstein series for orthogonal groups and Kneser neighbours
AU - Manuel K. -H. Müller
AU - Nils R. Scheithauer
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26697
UR - https://arxiv.org/abs/2610.26697
AB - We study Eisenstein series for orthogonal groups of even unimodular lattices of signature (n,2) which arise as lifts of elliptic modular forms for SL_2(ℤ) on 1-dimensional cusps. When the weight is small they have to be defined by analytic continuation. In general, they will be smooth but non-holomorphic after regularization. We show exactly in which cases we obtain holomorphic modular forms by computing the intertwining factors appearing in Langlands' constant term formula in terms of standard parameters and Rankin-Selberg L-values. This is related to Kneser's neighbour theory of positive-definite even lattices, which also governs the eigenvalues of these Eisenstein series. In particular, we describe the space of singular weight modular forms and construct the function Φ_12, the denominator function of the Fake Monster Algebra, as an Eisenstein series. We apply these results to show that the corresponding orthogonal modular varieties are of general type.
ER -