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

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. Assumi...

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Bibliographic Details
Published in:Bulletin of the Australian Mathematical Society 2023-10, Vol.108 (2), p.205-216
Main Authors: SENA, GOREKH PRASAD, KUMAR, K. SENTHIL
Format: Article
Language:English
Subjects:
Algebra
Real numbers
Upper bounds
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Summary: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.
ISSN:0004-9727
1755-1633
DOI:10.1017/S0004972722001575