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Preprint Number 1887

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1887. Gareth Boxall and Taboka Chalebgwa and Gareth Jones
On algebraic values of Weierstrass σ-functions

Submission date: 24 November 2020


Suppose that Ω is a lattice in the complex plane and let σ be the corresponding Weierstrass σ-function. Assume that the point τ associated to Ω in the standard fundamental domain has imaginary part at most 1.9. Assuming that Ω has algebraic invariants g_2, g_3 we show that a bound of the form cd^m (log H)^n holds for the number of algebraic points of height at most H and degree at most d lying on the graph of σ. To prove this we apply results by Masser and Besson. What is perhaps surprising is that we are able to establish such a bound for the whole graph, rather than some restriction. We prove a similar result when, instead of g_2, g_3, the lattice points are algebraic. For this we naturally exclude those (z,σ(z)) for which z in Ω.

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Full text arXiv 2011.11980: pdf, ps.

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