Linear dependence of quasi-periods over the rationals

K. Senthil Kumar

doi:10.5802/crmath.171

Théorie des nombres

Linear dependence of quasi-periods over the rationals

K. Senthil Kumar ¹

¹ National Institute of Science Education and Research, HBNI, P.O.Jatni, Khurda, Odisha-752 050, India

Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 409-414.

Résumé

In this note we shall show that a lattice $ℤ ω_{1} + ℤ ω_{2}$ in $ℂ$ has $ℚ$ -linearly dependent quasi-periods if and only if $ω_{2} / ω_{1}$ is equivalent to a zero of the Eisenstein series $E_{2}$ under the action of ${SL}_{2} (ℤ)$ on the upper half plane of $ℂ$ .

Reçu le : 2020-03-22
Révisé le : 2020-08-06
Accepté le : 2020-12-16
Publié le : 2021-05-27

MR Zbl

DOI : 10.5802/crmath.171

Classification : 11J72, 11J89

Affiliations des auteurs :

K. Senthil Kumar ¹

¹ National Institute of Science Education and Research, HBNI, P.O.Jatni, Khurda, Odisha-752 050, India

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2021__359_4_409_0,
     author = {K. Senthil Kumar},
     title = {Linear dependence of quasi-periods over the rationals},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {409--414},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {4},
     year = {2021},
     doi = {10.5802/crmath.171},
     mrnumber = {4264323},
     zbl = {07362161},
     language = {en},
}

TY  - JOUR
AU  - K. Senthil Kumar
TI  - Linear dependence of quasi-periods over the rationals
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 409
EP  - 414
VL  - 359
IS  - 4
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.171
LA  - en
ID  - CRMATH_2021__359_4_409_0
ER  -

%0 Journal Article
%A K. Senthil Kumar
%T Linear dependence of quasi-periods over the rationals
%J Comptes Rendus. Mathématique
%D 2021
%P 409-414
%V 359
%N 4
%I Académie des sciences, Paris
%R 10.5802/crmath.171
%G en
%F CRMATH_2021__359_4_409_0

K. Senthil Kumar. Linear dependence of quasi-periods over the rationals. Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 409-414. doi : 10.5802/crmath.171. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.171/

Bibliographie
Cité par

[1] Alan Baker On the quasi-periods of the Weierstrass $ζ$ -function, Nachr. Akad. Wiss. Gött., II. Math.-Phys. Kl., Volume 1969 (1970), pp. 145-157 | Zbl

[2] W. Dale Brownawell; Kenneth K. Kubota The algebraic independence of Weierstrass functions and some related numbers, Acta Arith., Volume 33 (1977), pp. 111-149 | DOI | MR | Zbl

[3] John Coates Linear forms in the periods of the exponential and elliptic functions, Invent. Math., Volume 12 (1971), pp. 290-299 | DOI | MR | Zbl

[4] John Coates The transcendence of linear forms in $ω_{1}, ω_{2}, η_{1}, η_{2}, 2 π i,$ , Am. J. Math., Volume 93 (1971), pp. 385-397 | DOI | MR | Zbl

[5] Maurice H. Heins On the pseudo-periods of the Weierstrass zeta-functions, SIAM J. Numer. Anal., Volume 3 (1966), pp. 266-268 | DOI | Zbl

[6] Maurice H. Heins On the pseudo-periods of the Weierstrass zeta functions. II., Nagoya Math. J., Volume 30 (1967), pp. 113-119 | DOI | MR | Zbl

[7] S. Lang Elliptic functions, Graduate Texts in Mathematics, 112, Springer, 1987 | MR | Zbl

[8] David Masser Elliptic functions and transcendence, Lecture Notes in Mathematics, 437, Springer, 1975 | MR | Zbl

[9] Theodor Schneider Einführung in die transzendenten Zahlen, Grundlehren der Mathematischen Wissenschaften, 81, Springer, 1957 | MR | Zbl

[10] Carl L. Siegel Über die Perioden elliptischer Funktionen, J. Reine Angew. Math., Volume 167 (1932), pp. 62-69 | Zbl

Cité par Sources :

Commentaires - Politique