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ON THE NUMBER OF ALGEBRAIC POINTS ON THE GRAPH OF THE WEIERSTRASS SIGMA FUNCTIONS

Part of: Diophantine approximation, transcendental number theory Other special functions

Published online by Cambridge University Press:  13 January 2023

GOREKH PRASAD SENA Open the ORCID record for GOREKH PRASAD SENA [Opens in a new window]  and
K. SENTHIL KUMAR Open the ORCID record for K. SENTHIL KUMAR [Opens in a new window]
GOREKH PRASAD SENA*
Affiliation:
National Institute of Science Education and Research, Bhubaneswar, An OCC of Homi Bhabha National Institute, Khurda 752050, Odisha, India
K. SENTHIL KUMAR
Affiliation:
National Institute of Science Education and Research, Bhubaneswar, An OCC of Homi Bhabha National Institute, Khurda 752050, Odisha, India e-mail: senthil@niser.ac.in
*
e-mail: gorekh.sena@niser.ac.in
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Abstract

Let $\Omega =\mathbb {Z}\omega _1+\mathbb {Z}\omega _2$ be a lattice in $\mathbb {C}$ with invariants $g_2,g_3$ and $\sigma _{\Omega }(z)$ the associated Weierstrass $\sigma $-function. Let $\eta _1$ and $\eta _2$ be the quasi-periods associated to $\omega _1$ and $\omega _2$, respectively. Assuming $\eta _2/\eta _1$ is a nonzero real number, we give an upper bound for the number of algebraic points on the graph of $\sigma _{\Omega }(z)$ of bounded degrees and bounded absolute Weil heights in some unbounded region of $\mathbb {C}$ in the following three cases: (i) $\omega _1$ and $\omega _2$ algebraic; (ii) $g_2$ and $g_3$ algebraic; (iii) the algebraic points are far from the lattice points.

Keywords

Weierstrass σ-functionalgebraic pointsquasi-periodsWeil heightMahler measure

MSC classification

Primary: 11J89: Transcendence theory of elliptic and abelian functions
Secondary: 11J82: Measures of irrationality and of transcendence 33E05: Elliptic functions and integrals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 108 , Issue 2 , October 2023 , pp. 205 - 216
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This research is partially supported by the MATRICS grant MTR/2021/000476 and the second author is thankful to the SERB, India.

References

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