no. 1
A note on the Hermite–Rankin constant
Sawatani, Kazuomi ; Watanabe, Takao ; Okuda, Kenji1
1 Department of Mathematics Graduate School of Science Osaka University Toyonaka 1-1, Osaka, Japan
Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 209-217.

Nous généralisons l’inégalité de Poor et Yuen au cas des constantes γ n,k d’Hermite-Rankin et γ n,k ′ de Bergé–Martinet. En outre, nous donnons les valeurs exactes de certaines constantes d’Hermite-Rankin et de Bergé–Martinet de petite dimension en appliquant certaines inégalités démontrées par Bergé et Martinet aux valeurs explicites de γ 5 ′ , γ 7 ′ , γ 4,2 et γ n ,n≤8.

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).

DOI : 10.5802/jtnb.712
Sawatani, Kazuomi  ; Watanabe, Takao  ; Okuda, Kenji 1

1 Department of Mathematics Graduate School of Science Osaka University Toyonaka 1-1, Osaka, Japan 
@article{JTNB_2010__22_1_209_0,
     author = {Sawatani, Kazuomi and Watanabe, Takao and Okuda, Kenji},
     title = {A note on the {Hermite{\textendash}Rankin} constant},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {209--217},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {1},
     year = {2010},
     doi = {10.5802/jtnb.712},
     zbl = {1257.11068},
     mrnumber = {2675881},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.712/}
}
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
SP  - 209
EP  - 217
VL  - 22
IS  - 1
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.712/
DO  - 10.5802/jtnb.712
LA  - en
ID  - JTNB_2010__22_1_209_0
ER  - 
%0 Journal Article
%A Sawatani, Kazuomi
%A Watanabe, Takao
%A Okuda, Kenji
%T A note on the Hermite–Rankin constant
%J Journal de théorie des nombres de Bordeaux
%D 2010
%P 209-217
%V 22
%N 1
%I Université Bordeaux 1
%U http://www.numdam.org/articles/10.5802/jtnb.712/
%R 10.5802/jtnb.712
%G en
%F JTNB_2010__22_1_209_0
Sawatani, Kazuomi; Watanabe, Takao; Okuda, Kenji. A note on the Hermite–Rankin constant. Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 209-217. doi : 10.5802/jtnb.712. http://www.numdam.org/articles/10.5802/jtnb.712/

[BC] E. S. Barnes and M. J. Cohn, On the inner product of positive quadratic forms. J. London Math. Soc. (2) 12 (1975), 32–36. | MR | Zbl

[BM] 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. | MR | Zbl

[C] R. Coulangeon, Réseaux k-extrêmes. Proc. London Math. Soc. 73 (1996), 555–574. | MR | Zbl

[M] J. Martinet, Perfect Lattices in Euclidean Spaces. Springer-Verlag, 2003. | MR | Zbl

[Ma] B. Mayer, Constantes d’Hermite et théorie de Voronoï. Thése, Université Bordeaux 1, 2008.

[PY] C. Poor and D. S. Yuen, Linear dependence among Siegel modular forms. Math. Ann. 318 (2000), 205–234. | MR | Zbl

[PY2] C. Poor and D. S. Yuen, The extreme core. Abh. Math. Sem. Univ. Hamburg 75 (2005), 1–25. | MR | Zbl

[PY3] C. Poor and D. S. Yuen, The Bergé–Martinet constant and slopes of Siegel cusp forms. Bull. London Math. Soc. 38 (2006), 913–924. | MR | Zbl

[R] R. A. Rankin, On positive definite quadratic forms. J. London Math. Soc. 28 (1953), 309–314. | MR | Zbl

Cité par Sources :