On algebraic values of Weierstrass $\sigma$-functions

TL;DR

This paper establishes bounds on the number of algebraic points of bounded height and degree on the graph of the Weierstrass sigma-function, assuming algebraic invariants or algebraic lattice points, using results by Masser and Besson.

Contribution

It provides the first known bounds for the entire graph of the sigma-function under algebraic conditions, extending previous results to a broader setting.

Findings

Bound of the form c d^m (log H)^n for algebraic points on sigma

Applicable to points with algebraic invariants g2,g3

Similar results for algebraic lattice points, excluding certain z

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 to $\Omega$ 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 $z\in\Omega$.