A note on the Hermite–Rankin constant

Kazuomi Sawatani; Takao Watanabe; Kenji Okuda[1]

  • [1] Department of Mathematics Graduate School of Science Osaka University Toyonaka 1-1, Osaka, Japan

Journal de Théorie des Nombres de Bordeaux (2010)

  • Volume: 22, Issue: 1, page 209-217
  • ISSN: 1246-7405

Abstract

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We generalize Poor and Yuen’s inequality to the Hermite–Rankin constant γ n , k and the Bergé–Martinet constant γ n , k ′ . Moreover, we determine explicit values of some low- dimensional Hermite–Rankin and Bergé–Martinet constants by applying Rankin’s inequality and some inequalities proven by Bergé and Martinet to explicit values of γ 5 ′ , γ 7 ′ , γ 4 , 2 and γ n ( n ≦ 8 ).

How to cite

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Sawatani, Kazuomi, Watanabe, Takao, and Okuda, Kenji. "A note on the Hermite–Rankin constant." Journal de Théorie des Nombres de Bordeaux 22.1 (2010): 209-217. <http://eudml.org/doc/116397>.

@article{Sawatani2010,
abstract = {We generalize Poor and Yuen’s inequality to the Hermite–Rankin constant $\gamma _\{n,k\}$ and the Bergé–Martinet constant $\gamma ^\prime_\{n,k\}$. Moreover, we determine explicit values of some low- dimensional Hermite–Rankin and Bergé–Martinet constants by applying Rankin’s inequality and some inequalities proven by Bergé and Martinet to explicit values of $\gamma ^\prime_5, \gamma ^\prime_7$, $\gamma _\{4,2\}$ and $\gamma _n$ ($n \leqq 8$).},
affiliation = {Department of Mathematics Graduate School of Science Osaka University Toyonaka 1-1, Osaka, Japan},
author = {Sawatani, Kazuomi, Watanabe, Takao, Okuda, Kenji},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Hermite-Rankin constant; generalized Bergé-Martinet constants},
language = {eng},
number = {1},
pages = {209-217},
publisher = {Université Bordeaux 1},
title = {A note on the Hermite–Rankin constant},
url = {http://eudml.org/doc/116397},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Sawatani, Kazuomi
AU - Watanabe, Takao
AU - Okuda, Kenji
TI - A note on the Hermite–Rankin constant
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 1
SP - 209
EP - 217
AB - We generalize Poor and Yuen’s inequality to the Hermite–Rankin constant $\gamma _{n,k}$ and the Bergé–Martinet constant $\gamma ^\prime_{n,k}$. Moreover, we determine explicit values of some low- dimensional Hermite–Rankin and Bergé–Martinet constants by applying Rankin’s inequality and some inequalities proven by Bergé and Martinet to explicit values of $\gamma ^\prime_5, \gamma ^\prime_7$, $\gamma _{4,2}$ and $\gamma _n$ ($n \leqq 8$).
LA - eng
KW - Hermite-Rankin constant; generalized Bergé-Martinet constants
UR - http://eudml.org/doc/116397
ER -

References

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  1. E. S. Barnes and M. J. Cohn, On the inner product of positive quadratic forms. J. London Math. Soc. (2) 12 (1975), 32–36. Zbl0312.10013MR387196
  2. A.-M. Bergé and J. Martinet, Sur un problème de dualité lié aux sphères en géométrie des nombres. J. Number Theory 32 (1989), 14–42. Zbl0677.10022MR1002112
  3. R. Coulangeon, Réseaux k -extrêmes. Proc. London Math. Soc. 73 (1996), 555–574. Zbl0861.11040MR1407461
  4. J. Martinet, Perfect Lattices in Euclidean Spaces. Springer-Verlag, 2003. Zbl1017.11031MR1957723
  5. B. Mayer, Constantes d’Hermite et théorie de Voronoï. Thése, Université Bordeaux 1, 2008. 
  6. C. Poor and D. S. Yuen, Linear dependence among Siegel modular forms. Math. Ann. 318 (2000), 205–234. Zbl0972.11035MR1795560
  7. C. Poor and D. S. Yuen, The extreme core. Abh. Math. Sem. Univ. Hamburg 75 (2005), 1–25. Zbl1082.11026MR2187578
  8. C. Poor and D. S. Yuen, The Bergé–Martinet constant and slopes of Siegel cusp forms. Bull. London Math. Soc. 38 (2006), 913–924. Zbl1104.11039MR2285245
  9. R. A. Rankin, On positive definite quadratic forms. J. London Math. Soc. 28 (1953), 309–314. Zbl0050.27401MR55380

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