Automorphic forms for some even unimodular lattices. (English) Zbl 1483.11089
B. Venkov and the reviewer [J. Reine Angew. Math. 531, 49–60 (2001; Zbl 0997.11039)] developed elementary linear algebra methods to compute Hecke eigenforms as formal linear combinations of isometry classes of lattices in the genus of 24-dimensional even unimodular lattices. With the help of an inner product and a multiplication on this space, they could determine most of the degrees of the corresponding Siegel modular forms. The remaining ones have been computed in the fundamental book [G. Chenevier and J. Lannes, Automorphic forms and even unimodular lattices. Kneser neighbors of Niemeier lattices. Translated by R. Erné. Cham: Springer (2019; Zbl 1430.11001)] using more sophisticated methods. The present paper continues in this spirit investigating Hilbert modular forms that arise as linear combinations of theta series of even unimodular lattices of rank \(\leq 12\) respectively \(\leq 8\) over the three real quadratic fields of discriminant \(5\) respectively \(8\) and \(12\).
The authors also investigate degree questions for the two genera of Hermitian Eisenstein lattices of rank 12 that are either even unimodular of dimension \(24\) over \({\mathbb Z}\) or Hermitian unimodular over the Eisenstein integers. For all these genera the authors explicitly computed Kneser neighbouring operators for certain small prime ideals using Kirschmer’s Magma library for lattices over number rings. These are self-adjoint operators whose eigenvalues are often rational, though some quadratic irrationalities arise, apparently unrelated to the ground field. Computing some coefficients of the Siegel theta series of these eigenvectors gives a hint on their degrees. The authors use clever arguments to prove almost all of these degrees and conjecture the corresponding global Arthur parameters.
The authors also investigate degree questions for the two genera of Hermitian Eisenstein lattices of rank 12 that are either even unimodular of dimension \(24\) over \({\mathbb Z}\) or Hermitian unimodular over the Eisenstein integers. For all these genera the authors explicitly computed Kneser neighbouring operators for certain small prime ideals using Kirschmer’s Magma library for lattices over number rings. These are self-adjoint operators whose eigenvalues are often rational, though some quadratic irrationalities arise, apparently unrelated to the ground field. Computing some coefficients of the Siegel theta series of these eigenvectors gives a hint on their degrees. The authors use clever arguments to prove almost all of these degrees and conjecture the corresponding global Arthur parameters.
Reviewer: Gabriele Nebe (Aachen)
MSC:
| 11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
| 11F27 | Theta series; Weil representation; theta correspondences |
| 11F33 | Congruences for modular and \(p\)-adic modular forms |
| 11E12 | Quadratic forms over global rings and fields |
| 11E39 | Bilinear and Hermitian forms |
Keywords:
algebraic modular forms; even unimodular lattices; theta series; Hilbert modular forms; Hermitian modular formsReferences:
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