On algebraic values of Weierstrass $\sigma$-functions

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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 .

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