Automorphic forms for some even unimodular lattices

Neil Dummigan, Dan Fretwell

Research output: Contribution to journalArticlepeer-review

Abstract

We look at genera of even unimodular lattices of rank 12 over the ring of integers of Q(5) and of rank 8 over the ring of integers of Q(3), 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, 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.

Original languageEnglish
Pages (from-to)29-67
Number of pages39
JournalAbhandlungen aus dem Mathematischen Seminar der Universitat Hamburg
Volume91
Issue number1
Early online date20 Feb 2021
DOIs
Publication statusPublished - Apr 2021
Externally publishedYes

Keywords

  • Algebraic modular forms
  • Even unimodular lattices
  • Hermitian modular forms
  • Hilbert modular forms
  • Theta series

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