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ON THE NUMBER OF ALGEBRAIC POINTS ON THE GRAPH OF THE WEIERSTRASS SIGMA FUNCTIONS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 108 / Issue 2 / October 2023
- Published online by Cambridge University Press:
- 13 January 2023, pp. 205-216
- Print publication:
- October 2023
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- Article
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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.
