On algebraic values of Weierstrass -functions

  • Gareth Boxall

    Stellenbosch University, Matieland, South Africa
  • Taboka Chalebgwa

    McMaster University, Ontario, Canada; Stellenbosch University, Matieland, South Africa
  • Gareth A. Jones

    University of Manchester, UK
On algebraic values of Weierstrass $\sigma$-functions cover

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Abstract

Suppose that is a lattice in the complex plane and let be the corresponding Weierstrass -function. Assume that the point associated with in the standard fundamental domain has imaginary part at most 1.9. Assuming that has algebraic invariants we show that a bound of the form holds for the number of algebraic points of height at most and degree at most lying on the graph of . 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 , the lattice points are algebraic. For this we naturally exclude those for which .

Cite this article

Gareth Boxall, Taboka Chalebgwa, Gareth A. Jones, On algebraic values of Weierstrass -functions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 32 (2021), no. 4, pp. 819–833

DOI 10.4171/RLM/960