# Automorphic forms for some even unimodular lattices

Automorphic forms for some even unimodular lattices We look at genera of even unimodular lattices of rank 12 over the ring of integers of Q(5)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathbb {Q}}}(\sqrt{5})$$\end{document} and of rank 8 over the ring of integers of Q(3)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathbb {Q}}}(\sqrt{3})$$\end{document}, using Kneser neighbours to diagonalise spaces of scalar-valued algebraic modular forms. We conjecture most of the global Arthur parameters, and prove several of them using theta series, in the manner of Ikeda and Yamana. We find instances of congruences for non-parallel weight Hilbert modular forms. Turning to the genus of Hermitian lattices of rank 12 over the Eisenstein integers, even and unimodular over Z\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathbb {Z}}}$$\end{document}, we prove a conjecture of Hentschel, Krieg and Nebe, identifying a certain linear combination of theta series as an Hermitian Ikeda lift, and we prove that another is an Hermitian Miyawaki lift. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

# Automorphic forms for some even unimodular lattices

, Volume 91 (1) – Apr 1, 2021
39 pages

/lp/springer-journals/automorphic-forms-for-some-even-unimodular-lattices-AQtsOFmkWA
Publisher
Springer Journals
Copyright © The Author(s), under exclusive licence to Mathematisches Seminar der Universität Hamburg 2021
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/s12188-021-00231-5
Publisher site
See Article on Publisher Site

### Abstract

We look at genera of even unimodular lattices of rank 12 over the ring of integers of Q(5)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathbb {Q}}}(\sqrt{5})$$\end{document} and of rank 8 over the ring of integers of Q(3)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathbb {Q}}}(\sqrt{3})$$\end{document}, using Kneser neighbours to diagonalise spaces of scalar-valued algebraic modular forms. We conjecture most of the global Arthur parameters, and prove several of them using theta series, in the manner of Ikeda and Yamana. We find instances of congruences for non-parallel weight Hilbert modular forms. Turning to the genus of Hermitian lattices of rank 12 over the Eisenstein integers, even and unimodular over Z\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${{\mathbb {Z}}}$$\end{document}, we prove a conjecture of Hentschel, Krieg and Nebe, identifying a certain linear combination of theta series as an Hermitian Ikeda lift, and we prove that another is an Hermitian Miyawaki lift.

### Journal

Abhandlungen aus dem Mathematischen Seminar der Universität HamburgSpringer Journals

Published: Apr 1, 2021

Keywords: Algebraic modular forms; Even unimodular lattices; Theta series; Hilbert modular forms; Hermitian modular forms; 11F41; 11F27; 11F33; 11E12; 11E39