Tsukuba Journal of Mathematics

A numerical study of Siegel theta series of various degrees for the 48-dimensional even unimodular extremal lattices

Michio Ozeki

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Abstract

Salvati Manni showed that the difference of the Siegel theta series of degree 4 associated with the two even unimodular 48-dimensional extremal lattices is a constant multiple of the cube J3 of the Schottky modular form J, which is a Siegel cusp form of degree 4 and weight 8. His result implies that the Siegel theta series of degree up to 3 is unique. But apparently his method does not supply us the process to compute the Fourier coefficients of these series.

In the present paper we show that the Fourier coefficients of the Siegel theta series associated with the even unimodular 48-dimensional extremal lattices of degrees 2 and 3 can be computed explicitly, and the Fourier coefficients of the Siegel theta series of degree 4 for those lattices are computed almost explicitly.

Article information

Source
Tsukuba J. Math., Volume 40, Number 2 (2016), 139-186.

Dates
Received: 24 June 2016
Revised: 13 October 2016
First available in Project Euclid: 13 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.tkbjm/1492104601

Digital Object Identifier
doi:10.21099/tkbjm/1492104601

Mathematical Reviews number (MathSciNet)
MR3635383

Zentralblatt MATH identifier
06710502

Subjects
Primary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Secondary: 11E20: General ternary and quaternary quadratic forms; forms of more than two variables 11T71: Algebraic coding theory; cryptography

Keywords
Theta series with spherical functions Siegel theta series 48-dimensional extremal lattices

Citation

Ozeki, Michio. A numerical study of Siegel theta series of various degrees for the 48-dimensional even unimodular extremal lattices. Tsukuba J. Math. 40 (2016), no. 2, 139--186. doi:10.21099/tkbjm/1492104601. https://projecteuclid.org/euclid.tkbjm/1492104601


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