Height of flat tori
HTML articles powered by AMS MathViewer
- by Patrick Chiu PDF
- Proc. Amer. Math. Soc. 125 (1997), 723-730 Request permission
Abstract:
Relations between the height and the determinant of the Laplacian on the space of $n$-dimensional flat tori and the classical formulas of Kronecker and Epstein are established. Extrema of the height are shown to exist, and results for a global minimum for 2-d tori and a local minimum for 3-d tori are given, along with more general conjectures of Sarnak and Rankin.References
- J. W. S. Cassels, On a problem of Rankin about the Epstein zeta-function, Proc. Glasgow Math. Assoc. 4 (1959), 73β80 (1959). MR 117193, DOI 10.1017/S2040618500033906
- P. Chiu, Covering with Hecke points, J. Number Theory 53 (1995), 25β44.
- J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 290, Springer-Verlag, New York, 1988. With contributions by E. Bannai, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov. MR 920369, DOI 10.1007/978-1-4757-2016-7
- Veikko Ennola, A lemma about the Epstein zeta-function, Proc. Glasgow Math. Assoc. 6 (1964), 198β201 (1964). MR 168536, DOI 10.1017/S2040618500035024
- Isaac Efrat, On a $\textrm {GL}(3)$ analog of $|\eta (z)|$, J. Number Theory 40 (1992), no.Β 2, 174β186. MR 1149736, DOI 10.1016/0022-314X(92)90038-Q
- Veikko Ennola, A lemma about the Epstein zeta-function, Proc. Glasgow Math. Assoc. 6 (1964), 198β201 (1964). MR 168536, DOI 10.1017/S2040618500035024
- Veikko Ennola, On a problem about the Epstein zeta-function, Proc. Cambridge Philos. Soc. 60 (1964), 855β875. MR 168535, DOI 10.1017/S0305004100038330
- B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), no.Β 1, 148β211. MR 960228, DOI 10.1016/0022-1236(88)90070-5
- Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82β96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
- S. S. Ryshkov, On the question of final $\zeta$-optimality of lattices providing the closest lattice packing of $n$-dimensional spheres, Sibirsk. Mat. Zh. 14 (1973), 1065β1075; English transl., Siberian Math. J. 14 (1973), 743β750.
- Peter Sarnak, Determinants of Laplacians; heights and finiteness, Analysis, et cetera, Academic Press, Boston, MA, 1990, pp.Β 601β622. MR 1039364
- Audrey Terras, Harmonic analysis on symmetric spaces and applications. I, Springer-Verlag, New York, 1985. MR 791406, DOI 10.1007/978-1-4612-5128-6
- Audrey A. Terras, Bessel series expansions of the Epstein zeta function and the functional equation, Trans. Amer. Math. Soc. 183 (1973), 477β486. MR 323735, DOI 10.1090/S0002-9947-1973-0323735-6
Additional Information
- Patrick Chiu
- Affiliation: P.O. Box 7486, Palo Alto, California 94309
- Received by editor(s): October 15, 1995
- Communicated by: Dennis A. Hejhal
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 723-730
- MSC (1991): Primary 11M36; Secondary 11F20, 11E45, 11H50, 11H55
- DOI: https://doi.org/10.1090/S0002-9939-97-03872-0
- MathSciNet review: 1396970