Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Height of flat tori


Author: Patrick Chiu
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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [Ca] J. W. S. Cassels, On a problem of Rankin about the Epstein zeta function, Proc. Glasgow Math. Assoc. 4 (1959), 73-80; 6 (1963), 116. MR 22:7975; MR 27:4803
  • [Ch] P. Chiu, Covering with Hecke points, J. Number Theory 53 (1995), 25-44. CMP 95:16
  • [CS] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups, Springer-Verlag, 1988. MR 89a:11067
  • [Di] P. H. Diananda, Notes on two lemmas concerning the Epstein zeta-function, Proc. Glasgow Math. Assoc. 6 (1964), 202-204. MR 29:5798b
  • [Ef] I. Efrat, On a $GL(3)$ analog of $|\eta (z)|$, J. Number Theory 40 (1992), 174-186. MR 93a:11040
  • [En1] V. Ennola, A lemma about the Epstein zeta function, Proc. Glasgow Math. Assoc. 6 (1964), 198-201. MR 29:5798a
  • [En2] -, On a problem about the Epstein zeta function, Proc. Cambridge Philos. Soc. 60 (1964), 855-875. MR 29:5797
  • [OPS] B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Analysis 80 (1988), 148-211. MR 90d:58159
  • [Ra] R. A. Rankin, A minimum problem for the Epstein zeta function, Proc. Glasgow Math. Assoc. 1 (1953), 149-158. MR 15:507c
  • [Ry] 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), 1065-1075; English transl., Siberian Math. J. 14 (1973), 743-750.
  • [Sa] P. Sarnak, Determinants of Laplacians; height and finiteness, Analysis et cetera, ed. P. Rabinowitz and E. Zehnder, Academic Press, 1990, pp. 601-622. MR 91d:58260
  • [Te1] A. Terras, Harmonic Analysis on Symmetric Spaces and Applications, Vol. I, Springer-Verlag, 1985. MR 87f:22010
  • [Te2] -, Bessel series expansions of the Epstein zeta function and the functional equation, Trans. Amer. Math. Soc. 183 (1973), 477-486. MR 48:2091

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 11M36, 11F20, 11E45, 11H50, 11H55

Retrieve articles in all journals with MSC (1991): 11M36, 11F20, 11E45, 11H50, 11H55


Additional Information

Patrick Chiu
Affiliation: P.O. Box 7486, Palo Alto, California 94309

DOI: https://doi.org/10.1090/S0002-9939-97-03872-0
Received by editor(s): October 15, 1995
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society