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Moscow Mathematical Journal, 2006, Volume 6, Number 2, Pages 225–264
DOI: https://doi.org/10.17323/1609-4514-2006-6-2-225-264
(Mi mmj245)
 

This article is cited in 15 scientific papers (total in 15 papers)

Spherical designs attached to extremal lattices and the modulo $p$ property of Fourier coefficients of extremal modular forms

E. Bannai, M. Koike, M. Shinohara, M. Tagami

Graduate school of Mathematics, Kyushu University
Full-text PDF Citations (15)
References:
Abstract: A theorem of Venkov says that each nontrivial shell of an extremal even unimodular lattice in $\mathbb R^n$ with $24\mid n$ is a spherical 11-design. It is a difficult open question whether there exists any 12-design among them. In the first part of this paper, we consider the following problem: When do all shells of an even unimodular lattice become 12-designs? We show that this does not happen in many cases, though there are also many cases yet to be answered. In the second part of this paper, we study the modulo p property of the Fourier coefficients of the extremal modular forms $f=\sum_{i\ge 0}a_iq^i$ (where $q=e^{2\pi i\tau}$) of weight $k$ with $k$ even. We are interested in determining, for each pair consisting of $k$ and a prime $p$, which of the following three (exclusive) cases holds: (1) $p\mid a_i$ for all $i\ge 1$ (2) $p\mid a_i$ for all $i\ge 1$ with $p\nmid i$, and there exists at least one $j\ge 1$ with $p\nmid a_j$ (3) there exists at least one $j\ge 1$ with $p\nmid j$ such that $p\nmid a_j$. We first prove that case (1) holds if and only if $(p-1)\mid k$. Then we obtain several conditions which guarantee that case (2) holds. Finally, we propose a conjecture that may characterize situations in which case (2) holds.
Key words and phrases: Spherical design, extremal lattice, extremal modular form, Assmus–Mattson theorem, Lehmer conjecture.
Received: February 18, 2005
Bibliographic databases:
MSC: Primary 05Exx; Secondary 05B05, 11E12, 11F11, 11F30, 11F33
Language: English
Citation: E. Bannai, M. Koike, M. Shinohara, M. Tagami, “Spherical designs attached to extremal lattices and the modulo $p$ property of Fourier coefficients of extremal modular forms”, Mosc. Math. J., 6:2 (2006), 225–264
Citation in format AMSBIB
\Bibitem{BanKoiShi06}
\by E.~Bannai, M.~Koike, M.~Shinohara, M.~Tagami
\paper Spherical designs attached to extremal lattices and the modulo $p$ property of Fourier coefficients of extremal modular forms
\jour Mosc. Math.~J.
\yr 2006
\vol 6
\issue 2
\pages 225--264
\mathnet{http://mi.mathnet.ru/mmj245}
\crossref{https://doi.org/10.17323/1609-4514-2006-6-2-225-264}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2270613}
\zmath{https://zbmath.org/?q=an:1121.11046}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000208595800001}
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  • This publication is cited in the following 15 articles:
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