On algebraic values of Weierstrass $\sigma$-functions

Abstract

Suppose that $\Omega$ is a lattice in the complex plane and let $\sigma$ be the corresponding Weierstrass $\sigma$-function. Assume that the point $\tau$ associated with $\Omega$ in the standard fundamental domain has imaginary part at most 1.9. Assuming that $\Omega$ has algebraic invariants $g_2,g_3$ we show that a bound of the form $c{d}^m(\log H{)}^n$ holds for the number of algebraic points of height at most $H$ and degree at most $d$ lying on the graph of $\sigma$. 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 $(z,\sigma (z))$ for which $z\in \Omega$.