ON THE NUMBER OF ALGEBRAIC POINTS ON THE GRAPH OF THE WEIERSTRASS SIGMA FUNCTIONS

Abstract

Let $\Omega =\mathbb {Z}\omega _1+\mathbb {Z}\omega _2$ be a lattice in $\mathbb {C}$ with invariants $g_2,g_3$ and $\sigma _{\Omega }(z)$ the associated Weierstrass $\sigma $-function. Let $\eta _1$ and $\eta _2$ be the quasi-periods associated to $\omega _1$ and $\omega _2$, respectively. Assuming $\eta _2/\eta _1$ is a nonzero real number, we give an upper bound for the number of algebraic points on the graph of $\sigma _{\Omega }(z)$ of bounded degrees and bounded absolute Weil heights in some unbounded region of $\mathbb {C}$ in the following three cases: (i) $\omega _1$ and $\omega _2$ algebraic; (ii) $g_2$ and $g_3$ algebraic; (iii) the algebraic points are far from the lattice points.

1 Introduction

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 $\alpha \in (2,3)$ , with $\zeta (\alpha )$ also algebraic, such that both $\alpha ,\zeta (\alpha )$ have degrees at most d and multiplicative heights at most $H,$ where $\zeta (z)$ is the Riemann zeta function. Recall that for an algebraic number $\alpha $ of degree $d,$ its multiplicative height is defined by $H(\alpha )=(M(\alpha ))^{1/d},$ where $M(\alpha )$ is its Mahler measure. 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.

Let $\Omega $ be a lattice in $\mathbb {C}.$ Throughout our discussion, we fix a $\mathbb {Z}$ -basis $\{\omega _1,\omega _2\}$ of $\Omega $ such that $\tau =\omega _2/\omega _1$ lies in the upper half plane $\mathbb {H}$ of $\mathbb {C}$ with $|\tau |\geq 1$ and the real part of $\tau $ lies in the interval $\big[{-}\tfrac 12,\tfrac 12\big].$ Such a basis always exists. Let $\eta _1$ and $\eta _2$ be the quasi-periods associated to $\omega _1$ and $\omega _2$ , respectively. For a pair $\alpha ,\beta $ of algebraic numbers, we put $H(\alpha ,\beta )=\max \{H(\alpha ),H(\,\beta )\}.$ In [2], the authors proved two results. The first [2, Theorem 1.1] is for lattices $\Omega $ for which $\omega _1,\omega _2$ are algebraic: if $\mbox {Im}(\tau )\leq 1.9$ and $\omega _1$ and $\omega _2$ are both algebraic, where $\mbox {Im}(z)$ is the imaginary part of $z,$ then there exists a constant $C_1=C_1(\Omega )>0$ such that for all $d\geq e$ and $H\geq e^e,$ there are at most

(1.1) $$ \begin{align} C_1d^6(\log d)(\log H)^2\log\log H \end{align} $$

algebraic numbers z such that $[\mathbb {Q}(z,\sigma _{\Omega }(z)):\mathbb {Q}]\leq d,H(z,\sigma _{\Omega }(z))\leq H$ and $z\notin \Omega .$ Their second result [2, Theorem 1.2] deals with the case in which the invariants

$$ \begin{align*}g_2=60\sum_{\omega\in\Omega'}^{}\omega^{-4},\quad g_3=140\sum_{\omega\in\Omega'}\omega^{-6}\end{align*} $$

are both algebraic ( $\omega _1,\omega _2$ need not be algebraic) and here also $\mbox {Im}(\tau )\leq 1.9$ , where $\Omega '=\Omega \setminus \{0\}$ . In this case, instead of the bound (1.1), they give the bound

$$ \begin{align*} C_2d^{20} (\log d)^5 (\log H )^2 (\log \log H) \end{align*} $$

for the number of algebraic numbers z such that $[\mathbb {Q}(z,\sigma _{\Omega }(z)):\mathbb {Q}]\leq d,H(z,\sigma _{\Omega }(z))\,{\leq}\, H.$ Here, $C_2$ is a constant depending only on $\Omega .$ The importance of the results in [2] is that they count algebraic points of bounded degrees and heights on the entire graph of $\sigma _{\Omega }(z)$ . Earlier, Besson [1] also proved similar results for the number of algebraic points on the graph of the Weierstrass $\sigma $ -function, but his results are restricted to bounded domains. One of the main ingredients in [2] is the lower bound of $\sigma _{\Omega }(z)$ in terms of the exponential function $e^z$ and the values of $\sigma _{\Omega }(z)$ on the fundamental domain P enclosed by the parallelogram with vertices $\tfrac 12(\pm \omega _1\pm \omega _2).$ In this paper, we extend the main results of [2] to a general $\tau \in \mathbb {H}$ under the assumption that $\rho =\eta _2/\eta _1$ is a nonzero real number. With this assumption, we are only able to count the algebraic points of $\sigma _{\Omega }(z)$ in an unbounded subset $\mathcal {A}_{\rho }$ of $\mathbb {C}$ defined as follows. First, put

$$ \begin{align*}\Omega^+=\{m\omega_1+n\omega_2:mn\geq 0\}\quad \mbox{and}\quad \Omega^-=\{m\omega_1+n\omega_2:mn\leq 0\},\end{align*} $$

$$ \begin{align*}\mathcal{A}^+=\{z\in\mathbb{C}: \mbox{there exists}~z_0\in P~\mbox{such that}~z-z_0\in\Omega^+\},\end{align*} $$

and

$$ \begin{align*}\mathcal{A}^-=\{z\in\mathbb{C}: \mbox{there exists}~z_0\in P~\mbox{such that}~z-z_0\in\Omega^-\}.\end{align*} $$

Finally, define

$$ \begin{align*} \mathcal{A}_{\rho}= \begin{cases} \mathcal{A}^+ & \text{if } \rho>0,\\ \mathcal{A}^- & \text{if } \rho<0. \end{cases} \end{align*} $$

Our first result is an analogue of [2, Theorem 1.1].

Theorem 1.1. Let $\Omega =\mathbb {Z}\omega _1+\mathbb {Z}\omega _2$ be a lattice in $\mathbb {C}$ such that $\omega _1$ and $\omega _2$ are both algebraic. Assume that $\rho = \eta _2/\eta _1$ is a nonzero real number. Then there exists a constant $C_3=C_3(\Omega )>0$ such that for all $d\geq e$ and $H\geq e^e,$ there are at most

$$ \begin{align*}C_3d^6(\log d)(\log H)^2\log\log H\end{align*} $$

algebraic numbers z such that $z\in \mathcal {A}_{\rho }$ , $[\mathbb {Q}(z,\sigma _{\Omega }(z)):\mathbb {Q}]\leq d~\mbox {and}~H(z,\sigma _{\Omega }(z))\leq H.$

Our second result is for $g_2,g_3$ algebraic, analogous to [2, Theorem 1.2].

Theorem 1.2. Let $\Omega =\mathbb {Z}\omega _1+\mathbb {Z}\omega _2$ be a lattice in $\mathbb {C}$ such that $g_2$ and $g_3$ are both algebraic. Assume that $\rho = \eta _2/\eta _1$ is a nonzero real number. Then there exists a constant $C_4=C_4(\Omega )>0$ such that for all $d\geq e$ and $H\geq e^e,$ there are at most

$$ \begin{align*}C_4d^{20}(\log d)^5(\log H)^2\log\log H\end{align*} $$

algebraic numbers z such that $z\in \mathcal {A}_{\rho }$ , $[\mathbb {Q}(z,\sigma _{\Omega }(z)):\mathbb {Q}]\leq d~\mbox {and}~H(z,\sigma _{\Omega }(z))\leq H.$

We also prove the following more general result with no assumptions on the quantities $\omega _1,\omega _2,g_2$ and $g_3.$ In this case, we are only able to count the algebraic points of $\sigma _{\Omega }(z)$ which are not close to the lattice points.

Theorem 1.3. Let $\Omega =\mathbb {Z}\omega _1+\mathbb {Z}\omega _2$ be a lattice in $\mathbb {C}.$ Assume that $\rho = \eta _2/\eta _1$ is a nonzero real number. Let $0<\delta <\min \{1,|\omega _1+\omega _2|/2,|\omega _1-\omega _2|/2\}.$ Then there exists a constant $C_5=C_5(\delta ,\Omega )>0$ such that for all $d\geq e$ and $H\geq e^e,$ there are at most

$$ \begin{align*}C_5d^4(\log d)(\log H)^2\log\log H\end{align*} $$

algebraic numbers z such that $z\in \mathcal {A}_{\rho },[\mathbb {Q}(z,\sigma _{\Omega }(z)):\mathbb {Q}]\leq d,~H(z,\sigma _{\Omega }(z))\leq H$ , $\mbox {dist}(z,\Omega )\geq \delta ,$ where

$$ \begin{align*}\mbox{dist}(z,\Omega)=\min_{w\in\Omega}|z-w|.\end{align*} $$

Like [2], in all our results, we count algebraic points of $\sigma _{\Omega }(z)$ on some unbounded regions of $\mathbb {C}.$ Since $\mathbb {R}$ is unbounded, by a result of Heins [4, page 114], we deduce that there are uncountably many lattices $\Omega $ with $|\tau |>1.9$ and $\eta _2/\eta _1\in \mathbb {R}\setminus \{0\}.$ However, despite Theorem 1.2, we do not know a single example of a lattice $\Omega $ with algebraic invariants $g_2,g_3$ for which there exists a nonzero algebraic number $\alpha $ such that $\sigma _{\Omega }(\alpha )$ is algebraic. It is, in fact, expected that there is no such $\alpha .$ However, the number of algebraic numbers $\alpha $ with degrees at most d and multiplicative heights at most H is at most $4^d H^d,$ which means that such $\alpha $ are very rare.

We end this section with three results which are needed for the proof of our theorems.

Proposition 1.4 [1, Théorème 1.2]

Let $T\geq 1$ be an integer and $R\geq 2$ be a real number. Consider any nonzero polynomial $P(X,Y)\in \mathbb {C}[X,Y]$ of degree at most T in each variable. Then there exists an effective constant $C_6>0$ such that the function $P(z,\sigma _{\Omega }(z))$ has at most

$$ \begin{align*}C_6T\Big(R+\sqrt{T}\Big)^2\log(R+T)\end{align*} $$

zeros in the disk $|z|\leq R.$

Proposition 1.5 [6, Proposition 2]

Fix integers $d\geq 1,T\geq \sqrt {8d}$ and positive real numbers $A,Z,M$ and H with $H\geq 1.$ Let $f_1,f_2$ be two analytic functions on a neighbourhood of the disk $|z|\leq 2Z.$ Suppose that $|f_1(z)|\leq M,|f_2(z)|\leq M$ for all $|z|\leq 2Z.$ Let $\mathcal {Z}\subseteq \mathbb {C}$ be finite and such that for all $z,z'\in \mathcal{Z}{:}$

1. (1) $|z|\leq Z;$

2. (2) $|z-z'|\leq 1/A;$

3. (3) $[\mathbb {Q}(f_1(z),f_2(z)):\mathbb {Q}]\leq d;$

4. (4) $H(f_1(z),f_2(z))\leq H.$

If

(1.2) $$ \begin{align} (AZ)^T>(4T)^{96d^2/T}(M+1)^{16d}H^{48d^2}, \end{align} $$

then there exists a nonzero polynomial $P\in \mathbb {Z}[X,Y]$ of total degree at most T such that $P(f_1(z),f_2(z))=0 \ \mbox {for all}~z\in \mathcal {Z}.$

Proposition 1.6 [5, Lemma 7.1]

For any Weierstrass $\sigma $ -function $\sigma _{\Omega }(z),$ there exists a constant $C_7=C_7(\Omega )$ such that for any $R\geq 1,$

$$ \begin{align*}|\sigma_{\Omega}(z)|\leq C_7^{R^2} \quad\mbox{for all}~|z|\leq R.\end{align*} $$

Our paper is organised as follows. In Section 2, we prove an analogue of [2, Proposition 2.1] for $z\in \mathcal {A}_{\rho }.$ Then we prove Theorem 1.1 in Section 3, Theorem 1.2 in Section 4 and Theorem 1.3 in Section 5.

2 Lower bound

Recall that P is the fundamental domain of the lattice $\Omega =\mathbb {Z}\omega _1+\mathbb {Z}\omega _2$ enclosed by the parallelogram with vertices $\tfrac 12(\pm \omega _1\pm \omega _2).$

Proposition 2.1. Let $\Omega =\mathbb {Z}\omega _1+\mathbb {Z}\omega _2$ be a lattice in $\mathbb {C}$ with $\rho = \eta _2/\eta _1$ a nonzero real number. Then there exist positive constants r and C depending only on $\Omega $ such that for all $z\in \mathcal {A}_{\rho }$ with $|z|\geq r$ , there exists $z_0\in P$ with

$$ \begin{align*} |\sigma_{\Omega}(z)|\geq |\sigma_{\Omega}(z_0)|e^{C|z|^2}. \end{align*} $$

Proof. As in the proof of [2, Proposition 2.1], we may assume that $\Omega =\mathbb {Z}+\mathbb {Z}\tau .$ Let $z\in \mathcal {A}_{\rho }$ and $z_0\in P$ be such that $z=z_0+m+n\tau $ for some integers m and $n.$ Then

$$ \begin{align*} \sigma_{\Omega}(z_0+m+n\tau)=(-1)^{m+n+mn}\sigma_{\Omega}(z_0)e^{(m\eta_1+n\eta_2)(z_0+m/2+(n/2)\tau)} \end{align*} $$

(see [7, page 255]). Hence,

$$ \begin{align*}|\sigma_{\Omega}(z_0+m+n\tau)|=|\sigma_{\Omega}(z_0)|e^{R(m,n,z_0)},\end{align*} $$

where $R(m,n,z_0)=\operatorname {Re}[(m\eta _1+n\eta _2)(z_0+m/2+(n/2)\tau )].$ Note that

$$ \begin{align*}R(m,n,z_0)=\operatorname{Re}\bigg(\frac{\eta_1}{2}\bigg)m^2+\operatorname{Re}\bigg(\frac{\eta_1\tau+\eta_2}{2}\bigg)mn+\operatorname{Re}\bigg(\frac{\eta_2\tau}{2}\bigg)n^2+\operatorname{Re}(\eta_1z_0)m+\operatorname{Re}(\eta_2z_0)n.\end{align*} $$

Further, from Legendre’s relation $\eta _1\tau -\eta _2=2\pi i,$ we obtain

$$ \begin{align*}\operatorname{Re}(\eta_1\tau)=\operatorname{Re}(\eta_2) \quad\mbox{and}\quad \operatorname{Re}\bigg(\frac{\eta_1\tau+\eta_2}{2}\bigg)=\operatorname{Re}(\eta_2).\end{align*} $$

Moreover,

$$ \begin{align*}\operatorname{Re}(\eta_2\tau)=\operatorname{Re}(\eta_1\rho\tau)=\rho\operatorname{Re}(\eta_1\tau)=\rho\operatorname{Re}(\eta_2)=\rho^2\operatorname{Re}(\eta_1).\end{align*} $$

Therefore,

$$ \begin{align*} \operatorname{Re}\bigg(\frac{\eta_1}2\bigg)m^2+\operatorname{Re}\bigg(\frac{\eta_1\tau+\eta_2}{2}\bigg)mn+\operatorname{Re}\bigg(\frac{\eta_2\tau}{2}\bigg)n^2&=\operatorname{Re}(\eta_1)\bigg(\frac{m^2} 2+\rho mn+\frac{\rho^2}2 n^2\bigg)\\&=\frac {1} 2\operatorname{Re}(\eta_1)( n\rho + m )^2. \end{align*} $$

Hence,

$$ \begin{align*} R(m,n,z_0)&=\frac {1} 2\operatorname{Re}(\eta_1)( n\rho + m )^2+\operatorname{Re}(\eta_1z_0)(n\rho+m)\\&=(n\rho+ m )\bigg[\frac{\operatorname{Re}(\eta_1)}{2} (n\rho+ m )+\operatorname{Re}(\eta_1z_0)\bigg]. \end{align*} $$

(Recall that by Dirichlet’s theorem, there are infinitely many pairs of integers $(m,n)$ such that either $ \rho + m/n=0$ or $| \rho + m/n|< 1 /{n^2}.$ For this reason, we need to restrict the values of $m,n.$ ) Also, since $\eta _2/\eta _1=\tau -2\pi i/\eta _1$ and $\eta _2/\eta _1$ is real, we have $\mbox {Im}(\tau -2\pi i/\eta _1)=0.$ So,

$$ \begin{align*}\mbox{Im}(\tau)=2\pi\operatorname{Re}\bigg(\frac{1}{\eta_1}\bigg)=2\pi \frac{\operatorname{Re}(\eta_1)}{\operatorname{Re}(\eta_1)^2+\mbox{Im}(\eta_1)^2}.\end{align*} $$

Since $\mbox {Im}(\tau )>0,$ we have $\operatorname {Re}(\eta _1)>0.$

Case 1: $\rho>0.$ Suppose $ m>0,n>0$ . Then there exists a positive constant $r=r(\Omega )$ such that whenever $|z|>r,$ we have

$$ \begin{align*}R(m,n,z_0)\geq c_1(n\rho+m)^2\geq c_2\max(|m|,|n|)^2\end{align*} $$

for some positive constants $c_1,c_2$ depending only on $\Omega .$ However,

$$ \begin{align*}|z_0+m+n\tau|\leq c_3\max(|m|,|n|)\end{align*} $$

for some constant $c_3=c_3(\Omega )>0.$ Hence,

(2.1) $$ \begin{align} |\sigma_{\Omega}(z_0+m+n\tau)|\geq |\sigma_{\Omega}(z_0)|e^{c_4|z_0+m+n\tau|^2} \end{align} $$

for some constant $c_4=c_4(\Omega )>0.$ If $m<0, n<0$ , then consider the point $-z_0-m-n\tau .$ Clearly, $-z_0\in P.$ Therefore, from (2.1), we obtain

$$ \begin{align*}|\sigma_{\Omega}(-z_0-m-n\tau)|\geq |\sigma_{\Omega}(-z_0)|e^{c_4|-z_0-m-n\tau|^2}=|\sigma_{\Omega}(-z_0)|e^{c_4|z_0+m+n\tau|^2}.\end{align*} $$

However, since $\sigma _{\Omega }(z)$ is an odd function,

$$ \begin{align*}|\sigma_{\Omega}(-z_0-m-n\tau)|=|\sigma_{\Omega}(z_0+m+n\tau)|\quad \text{and}\quad |\sigma_{\Omega}(-z_0)=|\sigma_{\Omega}(z_0)|,\end{align*} $$

and the required result follows.

Case 2: $\rho <0.$ The proof of this case is similar to Case 1 and therefore we omit it.

3 Proof of Theorem 1.1

Recall that $\Omega =\mathbb {Z}\omega _1+\mathbb {Z}\omega _2$ is a lattice in $\mathbb {C}$ and $\rho = \eta _2/\eta _1$ is a nonzero real number. Throughout this section, let r and C denote the constants from Proposition 2.1. In the following, $c_5,\ldots ,c_{17}$ denote positive constants depending only on $\Omega $ (and independent of d and H). Since

$$ \begin{align*}\lim_{z\rightarrow 0}\frac{\sigma_{\Omega}(z)}{z}=1,\end{align*} $$

there exists an $\varepsilon $ with $0<\varepsilon <1/2$ such that

(3.1) $$ \begin{align} |\!\log |\sigma_{\Omega}(z)|-\log |z||\leq 1 \end{align} $$

whenever $|z|<\varepsilon .$ We fix such an $\varepsilon .$

Lemma 3.1. Let $z\in \mathcal {A}_{\rho } $ and $z_0\in P$ be such that $z-z_0\in \Omega $ with $|z_0|\geq \varepsilon .$ Assume that $|z|\geq r$ and both z, $\sigma _{\Omega }(z)$ are algebraic with $[\mathbb {Q}(z,\sigma _{\Omega }(z)):\mathbb {Q}]\leq d$ and $H(z,\sigma _{\Omega }(z))\leq H$ for some $d\geq e$ and $H\geq e.$ Then $|z|\leq C_8\sqrt {d\log H}$ for some constant $C_8=C_8(\Omega )>0.$

Proof. Let $S=\{z\in P:|z|<\varepsilon \}.$ Note that $P\setminus S$ is compact. Since $\sigma _{\Omega }(z)$ is continuous and nonzero in $P\setminus S,$ for all $z\in P\setminus S,$ we have $|\sigma _{\Omega }(z)|\geq c_5.$ Since $|z_0|\geq \varepsilon ,$ we have $|\sigma _{\Omega }(z_0)|\geq c_5.$ Now from Proposition 2.1,

$$ \begin{align*}|\sigma_{\Omega}(z)|\geq |\sigma_{\Omega}(z_0)|e^{C|z|^2}. \end{align*} $$

However, since $[\mathbb {Q}(\sigma _{\Omega }(z)):\mathbb {Q}]\leq d$ and $H(\sigma _{\Omega }(z))\leq H,$ we have $|\sigma _{\Omega }(z)|\leq H^d.$ So

$$ \begin{align*}C|z|^2\leq \log |\sigma_{\Omega}(z)|-\log|\sigma_{\Omega}(z_0)|\leq d\log H-\log c_5\leq c_6d\log H,\end{align*} $$

and therefore,

$$ \begin{align*} |z|\leq c_7\sqrt{d\log H}. \end{align*} $$

This completes the proof of the lemma.

Lemma 3.2. Let $z\in \mathcal {A}_{\rho } $ and $z_0\in P$ be such that $z-z_0\in \Omega $ with $|z_0|< \varepsilon .$ Assume that $|z|\geq r$ and both z, $\sigma _{\Omega }(z)$ are algebraic with $[\mathbb {Q}(z,\sigma _{\Omega }(z)):\mathbb {Q}]\leq d$ and $H(z,\sigma _{\Omega }(z))\leq H$ for some $d\geq e$ and $H\geq e.$ For all $B>0$ and for all $N\geq \sqrt {d\log H},$ if $|z|\geq \sqrt {{(2+B)}/{C}}N,$ then $\log |z_0|\leq -BN^2.$

Proof. Let $z\in \mathcal {A}_{\rho }$ with $|z|\geq r.$ Let $z_0\in P$ be such that $z-z_0\in \Omega .$ From Proposition 2.1,

$$ \begin{align*}|\sigma_{\Omega}(z)|\geq |\sigma_{\Omega}(z_0)|e^{C|z|^2}.\end{align*} $$

Using $|\sigma _{\Omega }(z)|\leq H^d$ and $N\geq \sqrt {d\log H},$ we obtain

(3.2) $$ \begin{align} C|z|^2+\log |\sigma_{\Omega}(z_0)|\leq \log |\sigma_{\Omega}(z)|\leq d\log H\leq N^2. \end{align} $$

For any $B>0,$ put

$$ \begin{align*}A=\sqrt{\frac{2+B}{C}}.\end{align*} $$

If $|z|\geq AN,$ then from (3.2), we deduce that $CA^2N^2+\log |\sigma _{\Omega }(z_0)|\leq N^2.$ So, $\log |\sigma _{\Omega }(z_0)|\leq (1-CA^2)N^2.$ Since $|z_0|\leq \varepsilon ,$ applying (3.1), we obtain

$$ \begin{align*} \log |z_0|\leq \log |\sigma_{\Omega}(z_0)|+1 \leq (1-CA^2)N^2+1 \leq (2-CA^2)N^2 =-BN^2. \end{align*} $$

Thus, the result follows.

Lemma 3.3. Assume that $\omega _1$ and $\omega _2$ are both algebraic. Let $z\in \mathcal {A}_{\rho }$ be such that $|z|\geq r$ and both $z,\sigma _{\Omega }(z)$ are algebraic with $[\mathbb {Q}(z,\sigma _{\Omega }(z)):\mathbb {Q}]\leq d$ and ${H(z,\sigma _{\Omega }(z))\leq H}$ , where $d\geq e$ and $H\geq e.$ Then there exists a constant $C_9=C_9(\Omega )>0$ such that $|z|\leq C_9d\sqrt {\log H}.$

Proof. Suppose $z\in \mathcal {A}_{\rho }.$ Choose $z_0\in P$ such that $z-z_0\in \Omega .$ If $|z_0|\geq \varepsilon ,$ then by Lemma 3.1, $|z|\leq c_8\sqrt {d\log H}.$ So we assume that $|z_0|<\varepsilon .$ Let $\omega =z-z_0.$ By [2, Lemma 3.5], if $\omega =k\omega _1+l\omega _2$ , then

$$ \begin{align*}\max(|k|,|l|)\leq c_{9}|\omega|\leq c_{9}(|z|+|z_0|)\leq c_{9}(|z|+|\omega_1|+|\omega_2|)\leq c_{10}|z|.\end{align*} $$

However, since $H(z)\leq H$ and $[\mathbb {Q}(z):\mathbb {Q}]\leq d,$ we deduce that $|z|\leq H^d.$ So $H(k)=|k|\leq c_{10}|z|\leq c_{10}H^d$ and similarly $H(l)\leq c_{10}H^d.$ Now, using the inequality

$$ \begin{align*}H(z_0)\leq 2H(z)H(\omega)\leq 4H(z)H(k)H(\omega_1)H(l)H(\omega_2)\leq c_{11}H^{2d+1}\end{align*} $$

together with the bounds

$$ \begin{align*}[\mathbb{Q}(z_0):\mathbb{Q}]=[\mathbb{Q}(z-\omega):\mathbb{Q}]\leq [\mathbb{Q}(\omega_1,\omega_2):\mathbb{Q}]d\leq c_{12}d\end{align*} $$

and

$$ \begin{align*} M(z_0)=M(z_0^{-1})\geq 1/|z_0|, \end{align*} $$

we deduce that

$$ \begin{align*} \log |z_0|\geq \log(1/M(z_0))&= -[\mathbb{Q}(z_0):\mathbb{Q}]\log (H(z_0))\\&\geq -c_{12}d((2d+1)\log H+\log c_{11})\geq -c_{13}d^2\log H, \end{align*} $$

where $M(\alpha )$ is the Mahler measure of $\alpha .$ Applying Lemma 3.2 with $B=c_{13}$ and $N=d\sqrt {\log H},$ we deduce that $|z|\leq c_{14}d\sqrt {\log H},$ where $c_{14}=\sqrt {{(2+c_{13})}/{C}}.$ Taking $C_9=\max (c_8, c_{14}),$ we obtain the required result.

Proof of Theorem 1.1

Define

$$ \begin{align*} \mathcal{Z}_1=\{z\in\mathcal{A}_{\rho}:[\mathbb{Q}(z,\sigma_{\Omega}(z)):\mathbb{Q}]\leq d,~H(z,\sigma_{\Omega}(z))\leq H\}. \end{align*} $$

Put

$$ \begin{align*} Z=4C_9d\sqrt{\log H},\quad A=2/Z. \end{align*} $$

From Lemma 3.3, $|z|\leq C_9d\sqrt {\log H}\leq Z$ and $|z-z'|\leq 1/A$ for all $z,z'\in \mathcal {Z}_1.$ However, from Proposition 1.6, there exists a constant $c_{15}\geq 1$ such that for all $z\in \mathcal {A}_{\rho },$

$$ \begin{align*}|\sigma_{\Omega}(z)|\leq c_{15}^{|z|^2}.\end{align*} $$

Put $M=c_{15}^{Z^2}.$ Then, $|z|\leq M$ and $|\sigma _{\Omega }(z)|\leq M$ for all $|z|\leq 2Z.$ With these choices of $A,Z$ and $M,$ the conditions of Proposition 1.5 are satisfied. If we take $T=c_{16}d^3\log H$ for a sufficiently large $c_{16}>0,$ then (1.2) is satisfied. Hence by Proposition 1.5, there exists a nonzero polynomial $P\in \mathbb {Z}[X,Y]$ of total degree at most T such that $P(z,\sigma _{\Omega }(z))=0$ for all $z\in \mathcal {Z}_1.$ Finally taking $R=C_9d\sqrt {\log H}$ and $T=c_{16}d^3\log H$ in Proposition 1.4, we deduce that there are at most

$$ \begin{align*}c_{17}d^6(\log d)(\log H)^2\log \log H\end{align*} $$

zeros of $P(z,\sigma _{\Omega }(z))$ in the region $|z|\leq R.$ Hence, the number of elements in the set $\mathcal {Z}_1$ is at most $c_{17}d^6(\log d)(\log H)^2\log \log H.$ This completes the proof.

4 Proof of Theorem 1.2

Throughout this section, let $\Omega $ denote a lattice in $\mathbb {C}$ with algebraic invariants $g_2,g_3.$ In this section, $c_{18},\ldots ,c_{27}$ denote various constants which depend only on $\Omega .$ We first state the following transcendence measure for the nonzero elements of $\Omega $ , due to David and Hirata-Kohno.

Lemma 4.1 [3, Theorem 1.6]

Let $\Omega $ be a lattice in $\mathbb {C}.$ Let $d\geq 1$ and $H\geq 3$ be real numbers. Let $\alpha $ be an algebraic number with $[\mathbb {Q}(\alpha ):\mathbb {Q}]\leq d$ and $H(\alpha )\leq H.$ Then there exists a constant $C_{10}=C_{10}(\Omega )>0$ such that

$$ \begin{align*} \log |\alpha-\omega|\geq -C_{10}d^4(\log d)^2(\log H)|\omega|^2(1+\max\{0,\log |\omega|\})^3 \end{align*} $$

for all $\omega \in \Omega \setminus \{0\}.$

The following is an analogue of [2, Proposition 4.2].

Lemma 4.2. Assume that $\rho =\eta _2/\eta _1$ is a nonzero real number. Let $d\geq 1, H\geq 3$ be real numbers. There exist positive constants $C_{11},C_{12}$ depending only on $\Omega $ such that the following holds. If $z,z'\in \mathcal {A}_{\rho }$ with $[\mathbb {Q}(z,\sigma _{\Omega }(z)):\mathbb {Q}]\leq d, [\mathbb {Q}(z',\sigma _{\Omega }(z')):\mathbb {Q}]\leq d$ , $H(z,\sigma _{\Omega }(z))\leq H$ and $H(z',\sigma _{\Omega }(z'))\leq H,$ then

$$ \begin{align*} \min\{|z|,|z'|\}\leq C_{11}\sqrt{d^9(\log d)^2\log H} \end{align*} $$

or there exists $\omega ,\omega '\in \Omega $ such that

$$ \begin{align*}\max\{\log |z-\omega|,\log |z'-\omega'|\}\leq -C_{12}d^9(\log d)^2\log H\end{align*} $$

with $z'/z=\omega '/\omega \in \mathbb {Q}.$

Proof. The proof follows the same line of argument as in [2, Proposition 4.2], so we omit it here.

Lemma 4.3. Assume that $\rho =\eta _2/\eta _1$ is a nonzero real number and $g_2,g_3$ are both algebraic. Let $d\geq 1, H\geq 3$ be real numbers. Let r be from Proposition 2.1 and $C_{11}$ be from Lemma 4.2. Consider the set

$$ \begin{align*} S=\bigg\{z\in \mathcal{A}_{\rho}: &\, [\mathbb{Q}(z,\sigma_{\Omega}(z)): \mathbb{Q}]\leq d, H(z,\sigma_{\Omega}(z))\leq H,\\ &\text{and}\ |z|>\max\bigg\{r,C_{11}\sqrt{d^9(\log d)^2\log H}\bigg\}\bigg\}. \end{align*} $$

Then there exists a positive constant $C_{13}=C_{13}(\Omega )$ such that the number of elements of S is at most

$$ \begin{align*}C_{13}\sqrt{d^5(\log d)^2(\log H)(1+d\log H)^3}.\end{align*} $$

Proof. We follow the strategy given in [2]. Suppose $z,z'\in S.$ Then,

$$ \begin{align*}\min\{|z|,|z'|\}> C_{11}\sqrt{d^9(\log d)^2\log H}.\end{align*} $$

So by Lemma 4.2, there exist $\omega ,\omega '\in \Omega $ such that

$$ \begin{align*}\max\{\log |z-\omega|,\log |z'-\omega'|\}\leq -C_{12}d^9(\log d)^2\log H\end{align*} $$

with $z'/z=\omega '/\omega \in \mathbb {Q}.$ This implies $z, z'$ are not periods of $\Omega .$

Put $\omega '/\omega =q.$ Let $\omega ^*\in \Omega \setminus \{0\}$ be of minimum modulus on the line joining $0$ and  $\omega .$ Then $\omega =m\omega ^*$ for some nonzero integer $m.$ Let $z^*=z/m.$ Note that $z^*\in \mathcal {A}_{\rho }.$ Also, since $\omega '$ and $\omega $ lie on the same line, $\omega '=m_1\omega ^*$ for some $m_1\in \mathbb {Z}.$ So $qm\omega ^*=m_1\omega ^*.$ Hence, $qm=m_1.$ Now, $z'=qz=qmz^*=m_1z^*.$ So $z'$ is an integer multiple of $z^*.$ Thus, if we show that

$$ \begin{align*}n^2\leq c_{18}d^5(\log d)^2(\log H)(1+d\log H)^3,\end{align*} $$

whenever $nz^*\in S$ for some $n\in \mathbb {Z}$ , we are done. Indeed, just now we have seen that if $z'\in S$ , then $z'=nz^*$ for some $n\in \mathbb {N}.$ Accordingly, we assume $nz^*\in S$ for some $n\in \mathbb {Z}.$ Put $nz^*=z".$ Thus, $z"\in \mathcal {A}_{\rho }.$ Let $z_0\in P$ be such that $z"-z_0=\omega"\in \Omega .$ Since both $z"$ and z belong to $S,$ we have $z"/z=\omega "/\omega .$ Hence, $nz^*/mz^*=\omega "/m\omega ^*,$ or equivalently $\omega "=n\omega ^*.$ Since $z"\in \mathcal {A}_{\rho },$ from Proposition 2.1, we obtain

$$ \begin{align*}\log |\sigma_{\Omega}(z")|\geq \log |\sigma_{\Omega}(z_0)|+C|z"|^2.\end{align*} $$

Note that $z_0\neq 0.$ Therefore, $|\sigma _{\Omega }(z_0)/z_0|>e^{c_{18}}.$ Hence,

$$ \begin{align*} \log |\sigma_{\Omega}(z")| & \geq \log |z_0|+c_{19}+C|z"|^2\\ & = \log |z"-\omega"|+c_{19}+C|z"|^2\\ & \geq \log |nz^*-n\omega^*|+c_{19}+Cn^2|z^*|^2\\ & \geq \log |z^*-\omega^*|+c_{19}+Cn^2|z^*|^2. \end{align*} $$

Write $\omega =k\omega _1+l\omega _2$ with integers $k,l.$ As we have seen earlier in the proof of Lemma 3.3, $\max (|k|,|l|)\leq c_{20}|z|\leq c_{20}H^d.$ Further, since $\omega =m\omega ^*,$ we see that m divides both $k,l.$ We deduce that $|m|\leq c_{20}H^d.$ So,

$$ \begin{align*}\log H(z^*)=\log H(z/m)\leq c_{21}d\log H.\end{align*} $$

Since $z^*=z/m$ is algebraic, by Lemma 4.1, we deduce that

$$ \begin{align*} \log |z^*-\omega^*|\geq &-c_{22}d^4(\log d)^2d(\log H)|\omega^*|^2(1+\max\{0,\log |\omega^*|\})^3\\ \geq &-c_{23}d^5(\log d)^2(\log H)|z^*|^2(1+\max\{0,\log |z^*|\})^3. \end{align*} $$

So,

$$ \begin{align*} -c_{23}d^5(\log d)^2(\log H)|z^*|^2(1+\max\{0,\log |z^*|\})^3+c_{19}+Cn^2|z^*|^2 & \leq \log |\sigma_{\Omega}(z")| \\ & \leq d\log H. \end{align*} $$

In other words,

$$ \begin{align*}n^2\leq c_{24}d^5(\log d)^2(\log H)(1+d\log H)^3.\end{align*} $$

This completes the proof of the lemma.

Proof of Theorem 1.2

To prove Theorem 1.2, by Lemma 4.3, we only need to count the number of elements in the set

$$ \begin{align*} \mathcal{Z}_2= \bigg\{z\in \mathcal{A}_{\rho}: &\, [\mathbb{Q}(z,\sigma_{\Omega}(z)):\mathbb{Q}]\leq d,~H(z,\sigma_{\Omega}(z))\leq H, \\ & \text{and}~ |z|\leq C_{11}\sqrt{d^9(\log d)^2\log H}\bigg\}. \end{align*} $$

Put

$$ \begin{align*}Z=4C_{11}\sqrt{d^9(\log d)^2\log H},\quad A=2/Z. \end{align*} $$

Then $|z|\leq Z$ and $|z-z'|\leq 1/A$ for all $z,z'\in \mathcal {Z}_2.$ By Proposition 1.6, for all $z\in \mathcal {A}_{\rho },$

$$ \begin{align*}|\sigma_{\Omega}(z)|\leq c_{25}^{|z|^2}.\end{align*} $$

Put $M=c_{25}^{Z^2}.$ Then, $|z|\leq M$ and $|\sigma _{\Omega }(z)|\leq M$ for all $|z|\leq 2Z.$ With these choices of $A,Z$ and $M,$ the conditions of Proposition 1.5 are satisfied. 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 $P\in \mathbb {Z}[X,Y]$ of total degree at most T such that $P(z,\sigma _{\Omega }(z))=0~\mbox {for all}~z\in \mathcal {Z}_2.$

Finally taking $R\ {=}\ C_{11}\sqrt {d^9(\log d)^2\log H}$ and $T\ {=}\ c_{26}d^{10}(\log d)^2\log H$ in Proposition 1.4, we deduce that there are at most

$$ \begin{align*}c_{27}d^{20}(\log d)^5(\log H)^2\log \log H\end{align*} $$

zeros of $P(z,\sigma _{\Omega }(z))$ in the region $|z|\leq R.$ Hence, the number of elements in the set $\mathcal {Z}_2$ is at most $c_{27}d^{20}(\log d)^5(\log H)^2\log \log H.$ Since

$$ \begin{align*}\sqrt{c_{24}d^5(\log d)^2(\log H)(1+d\log H)^3}\leq c_{27}d^{20}(\log d)^5(\log H)^2\log \log H,\end{align*} $$

from Lemma 4.3, there are at most

$$ \begin{align*}2c_{27}d^{20}(\log d)^5(\log H)^2\log\log H\end{align*} $$

algebraic numbers z such that $z\in \mathcal {A}_{\rho }$ , $[\mathbb {Q}(z,\sigma _{\Omega }(z)):\mathbb {Q}]\leq d$ and $H(z,\sigma _{\Omega }(z))\leq H.$ This completes the proof of the theorem.

5 Proof of Theorem 1.3

Throughout this section, let $\delta ,r$ denote the constants from the statements of Theorem 1.3 and Proposition 2.1.

Lemma 5.1. Let $\Omega =\mathbb {Z}\omega _1+\mathbb {Z}\omega _2$ be a lattice in $\mathbb {C}$ with $\rho = \eta _2/\eta _1$ a nonzero real number. Let $z\in \mathcal {A}_{\rho } $ be such that $|z|\geq r$ and $\mbox {dist}(z,\Omega )\geq \delta .$ For any $d\geq e$ and $H\geq e,$ if both z and $\sigma _{\Omega }(z)$ are algebraic with $[\mathbb {Q}(z,\sigma _{\Omega }(z)):\mathbb {Q}]\leq d$ and $H(z,\sigma _{\Omega }(z))\leq H,$ then there exists a constant $C_{14}=C_{14}(\delta ,\Omega )$ such that $|z|\leq C_{14}\sqrt {d\log H}.$

Proof. Let $z\in \mathcal {A}_{\rho }$ and $z_0\in P$ be such that $z-z_0=m_1\omega _1+n_1\omega _2\in \Omega .$ Since $\mbox {dist}(z,\Omega )\geq \delta ,$ we have $|z_0|=|z-m_1\omega _1-n_1\omega _2|\geq \delta .$ Let $\varepsilon $ denote the constant from Lemma 3.1.

Case 1: $\delta \geq \varepsilon .$ Since $|z_0|\geq \delta ,$ we have $|z_0|\geq \varepsilon .$ So by Lemma 3.1, there exists a constant $c_{28}=c_{28}(\Omega )$ such that $|z|\leq c_{28}\sqrt {d\log H}.$ Hence, the lemma is proved.

Case 2: $\delta < \varepsilon .$ Suppose $\delta \leq \mbox {dist}(z,\Omega )<\varepsilon $ , so $\delta \leq |z_0|<\varepsilon .$ Hence,

$$ \begin{align*}|z_0|\geq e^{-\log1/\delta}\geq e^{-d\log H\log 1/\delta}=e^{-BN^2},\end{align*} $$

where $B=\log 1/\delta $ and $N=\sqrt {d\log H}.$ Since $|z_0|<\varepsilon ,$ applying Lemma 3.2, we obtain

$$ \begin{align*}|z|\leq\sqrt{\frac{2+\log 1/\delta}{C}}\sqrt{d\log H},\end{align*} $$

where C is as in Lemma 3.2.

Now suppose $\mbox {dist}(z,\Omega )\geq \varepsilon $ , so $|z_0|\geq \varepsilon .$ As in Case 1, $|z|\leq c_{28}\sqrt {d\log H}.$ Taking $C_{14}=\max\kern-1.5pt\Big(c_{28}, \sqrt {{(2+\log 1/\delta )}/{C}}\Big),$ we get the required result.

Proof of Theorem 1.3

Define

$$ \begin{align*}\mathcal{Z}_3=\{z\in\mathcal{A}_{\rho}:\mbox{dist}(z,\Omega)\geq \delta,[\mathbb{Q}(z,\sigma_{\Omega}(z)):\mathbb{Q}]\leq d~\text{and}~H(z,\sigma_{\Omega}(z))\leq H\}.\end{align*} $$

Put

$$ \begin{align*}Z=4C_{14}\sqrt{d\log H},\quad A=2/Z.\end{align*} $$

From Lemma 5.1, $|z|\leq C_{14}\sqrt {d\log H}\leq Z$ and $|z-z'|\leq 1/A$ for all $z,z'\in \mathcal {Z}_3.$ By Proposition 1.6,

$$ \begin{align*}|\sigma_{\Omega}(z)|\leq c_{29}^{|z|^2}.\end{align*} $$

Put $M=c_{29}^{Z^2}.$ Then, $|z|\leq M$ and $|\sigma _{\Omega }(z)|\leq M$ for all $|z|\leq 2Z.$ With these choices of $A,Z$ and $M,$ the conditions of Proposition 1.5 are satisfied. If we take $T=c_{30}d^2\log H$ for some sufficiently large constant $c_{30}=c_{30}(\delta ,\Omega ),$ then (1.2) is satisfied. Thus, by applying Proposition 1.5, there exists a nonzero polynomial $P\in \mathbb {Z}[X,Y]$ of total degree at most T such that $P(z,\sigma _{\Omega }(z))=0~\mbox {for all}~z\in \mathcal {Z}_3.$

Finally taking $R=C_{14}\sqrt {d\log H}$ and $T=c_{30}d^2\log H$ in Proposition 1.4, we deduce that there are at most

$$ \begin{align*}c_{31}d^4(\log d)(\log H)^2\log \log H\end{align*} $$

zeros of $P(z,\sigma _{\Omega }(z))$ in the region $|z|\leq R$ for some constant $c_{31}=c_{31}(\delta ,\Omega )>0.$ Hence, the number of elements in the set $\mathcal {Z}_3$ is at most $c_{31}d^4(\log d)(\log H)^2\log \log H.$ This completes the proof of the theorem.