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WEIERSTRASS ZETA FUNCTIONS AND p-ADIC LINEAR RELATIONS

Part of: Arithmetic algebraic geometry Diophantine approximation, transcendental number theory Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  11 March 2024

DUC HIEP PHAM Open the ORCID record for DUC HIEP PHAM [Opens in a new window]
DUC HIEP PHAM*
Affiliation:
University of Education, Vietnam National University, Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam
*
e-mail: phamduchiep@vnu.edu.vn
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Abstract

We discuss the p-adic Weierstrass zeta functions associated with elliptic curves defined over the field of algebraic numbers and linear relations for their values in the p-adic domain. These results are extensions of the p-adic analogues of results given by Wüstholz in the complex domain [see A. Baker and G. Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs, 9 (Cambridge University Press, Cambridge, 2007), Theorem 6.3] and also generalise a result of Bertrand to higher dimensions [‘Sous-groupes à un paramètre p-adique de variétés de groupe’, Invent. Math. 40(2) (1977), 171–193].

Keywords

elliptic curveWeierstrass zeta functionp-adic transcendence

MSC classification

Primary: 11J89: Transcendence theory of elliptic and abelian functions
Secondary: 11G07: Elliptic curves over local fields 11S99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 10
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This research has been done under the research project QG.23.48 ‘Some selected topics in number theory’ of Vietnam National University, Hanoi.

References

Baker, A. and Wüstholz, G., Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs, 9 (Cambridge University Press, Cambridge, 2007).Google Scholar
Bertrand, D., ‘Sous-groupes à un paramètre $p$ -adique de variétés de groupe’, Invent. Math. 40(2) (1977), 171–193.CrossRefGoogle Scholar
Bourbaki, N., Elements of Mathematics. Lie groups and Lie algebras. Part I: Chapters 1–3, Actualities scientifiques et industrielles (Herman, Paris, 1975); English translantion.Google Scholar
Caveny, D. and Tubbs, R., ‘Well-approximated points on linear extensions of elliptic curves’, Proc. Amer. Math. Soc. 138 (2010), 2745–2754.CrossRefGoogle Scholar
Chandrasekharan, K., Elliptic Functions, Grundlehren der mathematischen Wissenschaften, 281 (Springer-Verlag, Berlin, 1985).CrossRefGoogle Scholar
Coates, J., ‘The transcendence of linear forms in ${\omega}_1,{\omega}_2,{\eta}_1,{\eta}_2,2\pi i$ ’, Amer. J. Math. 93 (1971), 385–397.CrossRefGoogle Scholar
Faltings, G. and Wüstholz, G., ‘Einbettungen kommutativer algebraischer Gruppen und einige ihrer Eigenschaften’, J. reine angew. Math. 354 (1984), 175–205.Google Scholar
Fuchs, C. and Pham, D. H., ‘The $p$ -adic analytic subgroup theorem revisited’, p-Adic Numbers Ultrametric Anal. Appl. 7 (2015), 143–156.CrossRefGoogle Scholar
Lutz, E., ‘Sur l’équation ${Y}^2=A{X}^3- AX-B$ dans les corps $p$ -adiques’, J. reine angew. Math. 177 (1937), 238–247.CrossRefGoogle Scholar
Matev, T., ‘The $p$ -adic analytic subgroup theorem and applications’, Preprint, 2010, arXiv:1010.3156v1.Google Scholar
Robert, A. M., A Course in $p$ -adic Analysis, Graduate Texts in Mathematics, 198 (Springer-Verlag, New York, 2000).Google Scholar
Schneider, T., ‘Transzendenzuntersuchungen periodischer Funktionen: I Transzendenz von Potenzen; II Transzendenzeigenschaften elliptischer Funktionen’, J. reine angew. Math. 172 (1934), 65–74.Google Scholar
Serre, J. P., ‘Quelques propriétés des groupes algébriques commutatifs’, Astérisque 69–70 (1979), 191–202.Google Scholar
Weil, A., ‘Sur les fonctions elliptiques $\mathfrak{p}$ -adiques’, C. R. Hebdomadaires Séances L’Acad. Sci. 203(1) (1936), 22–24.Google Scholar
Wüstholz, G., ‘Algebraische Punkte auf Analytischen Untergruppen algebraischer Gruppen’, Ann. of Math. (2) 129 (1989), 501–517.CrossRefGoogle Scholar