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 .

On the Number of Algebraic Points on the Graph of the Weierstrass Sigma Functions

“…In the same paper, Masser suggested some possible extensions of his method to other classes of functions. There have been several results already published based on his suggestions of which a recent result by Boxall et al [2] is closely related to our work. To state the main results of [2], we need to introduce some notation.…”

“…So z is an integer multiple of z * . Thus, if we show that n 2 ≤ c 18 d 5 (log d) 2 (log H)(1 + d log H) 3 , whenever nz * ∈ S for some n ∈ Z, we are done. Indeed, just now we have seen that if z ∈ S, then z = nz * for some n ∈ N. Accordingly, we assume nz * ∈ S for some n ∈ Z.…”

“…In 2011, Masser [6] proved that for any positive integer d, there exists an effective constant C > 0 such that for all H > e e , there are at most C(log H/log log H) 2 algebraic numbers α ∈ (2, 3), with ζ(α) also algebraic, such that both α, ζ(α) have degrees at most d and multiplicative heights at most H, where ζ(z) is the Riemann zeta function. Recall that for an algebraic number α of degree d, its multiplicative height is defined by H(α) = (M(α)) 1/d , where M(α) is its Mahler measure.…”

“…For a pair α, β of algebraic numbers, we put H(α, β) = max{H(α), H( β)}. In [2], the authors proved two results.…”

“…If we take T = c 26 d 10 (log d)2 log H for a sufficiently large c 26 > 0, then (1.2) is satisfied. Thus, by Proposition 1.5, we deduce that there exists a nonzero polynomial P ∈ Z[X, Y] of total degree at most T such that P(z, σ Ω (z)) = 0 for all z ∈ Z 2 .Finally taking R = C 11 d 9 (log d)2 log H and T = c 26 d 10 (log d) 2 log H in Proposition 1.4, we deduce that there are at most c 27 d 20 (log d)5 (log H) 2 log log H zeros of P(z, σ Ω (z)) in the region |z| ≤ R. Hence, the number of elements in the set Z 2 is at most c 27 d 20 (log d)5 (log H) 2 log log H. Sincec 24 d 5 (log d) 2 (log H)(1 + d log H) 3 ≤ c 27 d 20 (log d) 5 (log H) 2 log log H, from Lemma 4.3, there are at most 2c 27 d 20 (log d) 5 (log H) 2 log log H [11]The Weierstrass sigma functions 215algebraic numbers z such that z ∈ A ρ , [Q(z, σ Ω (z)) : Q] ≤ d and H(z, σ Ω (z)) ≤ H.This completes the proof of the theorem. Proof of Theorem 1.Throughout this section, let δ, r denote the constants from the statements of Theorem 1.3 and Proposition 2.1.…”