THEOREM OF WARD ON SYMMETRIES OF ELLIPTIC NETS

Abstract

We present a new version of a generalisation to elliptic nets of a theorem of Ward [‘Memoir on elliptic divisibility sequences’, Amer. J. Math. 70 (1948), 31–74] on symmetry of elliptic divisibility sequences. Our results cover all that is known today.

1. Introduction

This paper concerns a generalisation of a theorem of Ward [7] on symmetry of elliptic sequences to the case of nondegenerate elliptic nets of rank $d \ (d \in {\mathbb N})$ associated to an elliptic curve E and points on E. In our opinion, it is the most comprehensive form that we can hope to achieve.

Symmetries of such elliptic nets written explicitly in a form similar to Ward’s theorem [7] are only known for the cases $d=1$ [6] and $d=2$ [4, 6]. To get the right shape for all d, an essential point of our demonstration consists of showing that appropriate quotients of two elliptic nets follow a geometric progression. This new approach allows us to obtain a simple proof of the generalisation of the symmetry theorem in Ward’s form. In this way, we unify all the results known to date: for $d=1$ , Ward [7, Theorem 8.1], Stange [4, Theorem 10.2.2] and [6, Theorem 4], and the author [2, Theorem 1]; for $d=2$ , [4, Lemma 10.2.5] and [6, Theorem 5]; and for $d>2$ , [4, Theorem 10.2.3] and Akbary et al. [1, Theorems 1.12 and 1.13].

Let E be an elliptic curve over a field ${\mathbb K}$ (see [3]). To simplify, we assume that the characteristic is different from $2$ and $3$ . Then

$$ \begin{align*}E({\mathbb K})=\{[X:Y:Z] \in {\mathbb P}^2({\mathbb K}) \mid {\cal F}(X,Y,Z)=0\} = \{(x,y) \in {\mathbb K}^2 \mid {\cal F}(x,y,1)=0\} \cup \{0_E\},\end{align*} $$

with ${\cal F}(X,Y,Z)=Y^2Z-(X^3+ aXZ^2+bZ^3)$ , $a,b \in {\mathbb K}$ such that $4a^3+27b^2 \neq 0$ and $0_E$ the unique point at infinity of the curve. The group structure of $E({\mathbb K})$ is defined by the chord and tangent method with the neutral element $0_E$ .

We introduce division polynomials $\psi _m(x,y), m \in {\mathbb Z}$ , of an elliptic curve E over the field ${\mathbb K}$ with an affine equation $y^2=x^3+ax+b$ (see [8]) by

$$ \begin{align*} \psi_0(x,y)=0,& \quad \psi_1(x,y)=1, \quad \psi_2(x,y)=2y \quad \psi_3(x,y)=3x^4+6 a x^2 + 12 b x -a^2,\\ &\psi_4(x,y)=4y(x^6+5 a x^4 + 20 b x^3 - 5 a^2 x^2 - 4 a b x -8 b^2-a^3), \end{align*} $$

and for n a natural integer, $\psi _{-n}=-\psi _n$ . Then, for all $(m,n)$ in ${\mathbb Z}^2$ ,

(1.1) $$ \begin{align} \psi_{m+n}\psi_{m-n}=\psi_{m+1}\psi_{m-1}\psi_n^2-\psi_{n+1}\psi_{n-1}\psi_m^2. \end{align} $$

This equality can be used for the product $\psi _{\imath } \psi _{\jmath }$ when the integers $\imath $ and $\jmath $ have the same parity. Any solution over an arbitrary integral domain of (1.1) is called an elliptic sequence. Also,

$$ \begin{align*}\psi_{2n+1}=\psi_{n+2}\psi_n^3-\psi_{n+1}^3\psi_{n-1} \quad\text{and}\quad \psi_{2n}\psi_{2}=\psi_n(\psi_{n+2}\psi_{n-1}^2-\psi_{n-2}\psi_{n+1}^2)\end{align*} $$

for n in ${\mathbb Z}$ . Note also Stephen Nelson’s form (see [4, page 22]): for all $(\alpha ,\beta ,\gamma ,\delta ) \in {\mathbb Z}^4$ ,

(1.2) $$ \begin{align} \psi_{\alpha+\beta}\psi_{\alpha-\beta}\psi_{\gamma+\delta}\psi_{\gamma-\delta}+ \psi_{\alpha+\gamma}\psi_{\alpha-\gamma}\psi_{\delta+\beta}\psi_{\delta-\beta}+ \psi_{\alpha+\delta}\psi_{\alpha-\delta}\psi_{\beta+\gamma}\psi_{\beta-\gamma}=0. \end{align} $$

Division polynomials have partial periodicity, called symmetry.

Theorem 1.1 [2]

Let $\mathbb {F}_q$ be a finite field, let $E/\mathbb {F}_q$ be an elliptic curve and let $P \in E(\bar {\mathbb {F}}_q)$ be a point of exact order $u\geq 2$ . Then there exists $\omega \in \bar {\mathbb {F}}_q$ , depending on P, such that the following hold.

1. (1) If $u \geq 3$ , then for all k and v in $\mathbb {Z}$ :

   • • if $u=2m$ , we have $\psi _{ku+v}(P)=(-\omega ^m)^{k^2}\omega ^{kv}\psi _v(P);$

   • • if $u=2m+1$ , we have $\psi _{ku+v}(P)=(-\omega ^{2m+1})^{k^2}(\omega ^2)^{kv}\psi _v(P).$

2. (2) If $u=2$ , then for all $k \in \mathbb {Z}$ ,

   $$ \begin{align*}\psi_{4k+1}(P)=(-1)^k\psi_3^{k(2k+1)}, \quad \psi_{4k+3}(P)=(-1)^k\psi_3^{(k+1)(2k+1)}.\end{align*} $$

Note that the proof works for any field ${\mathbb K}$ and that $\psi _u(P)=0$ . Furthermore, if ${u=2m}$ , then $\omega =({\psi _{m+1}}/{\psi _{m-1}})(P)$ ; otherwise $\omega =({\psi _{m+1}}/{\psi _m})(P)$ . This result will become a particular case of our generalisation and is already a precision of Ward’s symmetry theorem for the elliptic sequence $(\psi _n)$ .

Theorem 1.2 [7]

Let W be an integer elliptic sequence such that $W(1)=1$ and $W(2)\mid W(4)$ . Let p be an odd prime and suppose that $W(2)W(3) \not \equiv 0 \bmod p$ . Let u be the rank of apparition of W with respect to p (that is, $W(u) \equiv 0$ and $W(m) \not \equiv 0$ for any $m \mid u$ ). Then there exist integers ${\cal A}$ and ${\cal C}$ such that

(1.3) $$ \begin{align} W(ku+v)={\cal A}^{kv}{\cal C}^{k^2}W(v) \quad \text{for all } k,v \in {\mathbb N}. \end{align} $$

We usually call the smallest positive index of a vanishing term the rank of zero-apparition. If we consider the elliptic sequence $W=\psi (P)$ , the rank of zero-apparition is the order of P on E.

In [5], Stange generalised the concept of an elliptic sequence to a d-dimensional array, called an elliptic net. An elliptic net in this article is a map $W : {\mathbb Z}^d \rightarrow {\mathbb K}$ such that, for all $\mathbf {p},\mathbf {q},\mathbf {r},\mathbf {s}$ in ${\mathbb Z}^d$ ,

(1.4) $$ \begin{align} \kern-8pt W(\mathbf{p}+\mathbf{q}+\mathbf{s})W(\mathbf{p}-\mathbf{q})W(\mathbf{r}+\mathbf{s}) W(\mathbf{r}) &+ W(\mathbf{q}+\mathbf{r}+\mathbf{s})W(\mathbf{q}-\mathbf{r})W(\mathbf{p}+\mathbf{s})W(\mathbf{p}) \notag\\& + W(\mathbf{r}+\mathbf{p}+\mathbf{s})W(\mathbf{r}-\mathbf{p})W(\mathbf{q}+\mathbf{s})W(\mathbf{q})=0. \end{align} $$

We have $W(\mathbf {0})=0$ , where $\mathbf {0}$ is the additive identity element of ${\mathbb Z}^d$ , since $\text {char}({\mathbb K}) \neq 3$ . Stange proved that we can compute $W(\mathbf {v})$ for all $\mathbf {v}$ in ${\mathbb Z}^d$ from (1.4) and initial values $W(\mathbf {v})$ with $\mathbf {v}=\mathbf {e}_i$ , $\mathbf {v}=2\mathbf {e}_i$ , $\mathbf {v}=\mathbf {e}_i+\mathbf {e}_j$ and $\mathbf {v}=2\mathbf {e}_i+\mathbf {e}_j$ with $\{\mathbf {e}_1,\mathbf {e}_2,\ldots ,\mathbf {e}_d\}$ the standard basis of ${\mathbb Z}^d$ . For $\mathbf {s}=\mathbf {0}$ , we deduce that

(1.5) $$ \begin{align} W(\mathbf{p}+\mathbf{q})W(\mathbf{p}-\mathbf{q})W(\mathbf{r})^2=W(\mathbf{p}+\mathbf{r})W(\mathbf{p}-\mathbf{r})W(\mathbf{q})^2- W(\mathbf{q}+\mathbf{r})W(\mathbf{q}-\mathbf{r})W(\mathbf{p})^2. \end{align} $$

An elliptic net W is called degenerate if one of the terms $W(\mathbf {e}_i), W(2\mathbf {e}_i), W(\mathbf {e}_i \pm \mathbf {e}_j)$ (where $i \neq j$ ) is zero, and $W(3\mathbf {e}_1)$ is zero if $d=1$ . As shown in [5], we can define an elliptic net ${\cal W}=W_{E,\mathbf {P}}$ associated to the elliptic curve E and a d-tuple of fixed points $\mathbf {P}=(P_1,P_2,\ldots ,P_d)$ on $E^d$ with $P_i=(x_i,y_i) \neq 0_E$ for $1 \leq i \leq d$ and $P_i \pm P_j \neq 0_E$ for $i\neq j$ , using the recurrence relation (1.4) and initial values

$$ \begin{align*}{\cal W}(\mathbf{e}_i)=1, \quad {\cal W}(2\mathbf{e}_i)=2y_i, \quad {\cal W}(\mathbf{e}_i+\mathbf{e}_j)=1, \quad {\cal W}(2\mathbf{e}_i+\mathbf{e}_j)=2x_i+x_j-\bigg( \frac{y_j-y_i}{x_j-x_i} \bigg).\end{align*} $$

From [1, Example 2.4], $W(\mathbf {e}_i-\mathbf {e}_j)=W(\mathbf {e}_i+2\mathbf {e}_j)-W(2\mathbf {e}_i+\mathbf {e}_j)$ , so ${\cal W}(\mathbf {e}_i-\mathbf {e}_j)=x_j-x_i$ . The nondegenerate case therefore reduces to ${\cal W}(2 \mathbf {e}_i) \neq 0 \ (1 \leq i \leq d)$ with ${\cal W}(3 \mathbf {e}_1) \neq 0$ when $d=1$ .

From (1.5) with $\mathbf {r}=\mathbf {e}_r$ , we obtain (1.1) when $d=1$ (note that, in general, $W_1=1$ [7, Ch. VII]). Therefore, elliptic nets are effectively a generalisation of elliptic sequences.

Even though it is not essential for our purpose, we take the opportunity to show the converse, that is, that (1.1) implies (1.4) for $d=1$ , by giving the missing elementary proof reported in [4, Ch. 3, page 22].

Proposition 1.3. For all $(p,q,r,s) \in {\mathbb Z}^4$ ,

(1.6) $$ \begin{align} \psi_{p+q+s}\psi_{p-q}\psi_{r+s}\psi_r+\psi_{q+r+s}\psi_{q-r}\psi_{p+s}\psi_p+\psi_{r+p+s}\psi_{r-p}\psi_{q+s}\psi_q=0. \end{align} $$

Proof. For any $(\alpha ,\beta ) \in {\mathbb Z}^2$ , the integers $\alpha +\beta +1$ and $\alpha -\beta $ have different parities. Thus, we obtain $\psi _{\alpha +\beta +1}\psi _{\alpha -\beta }\psi _2\psi _1=\psi _{\beta +2}\psi _{\beta -1}\psi _{\alpha +1}\psi _\alpha -\psi _{\alpha +2}\psi _{\alpha -1}\psi _{\beta +1}\psi _\beta $ from the expressions for $\psi _{2k+1}\psi _1$ and $\psi _{2k'}\psi _2$ for the left-hand side and from (1.1) for the right-hand side, since the terms on each side of the subtraction can be coupled in pairs of products $\psi _{\imath }\psi _{\jmath }$ whose indexes have the same parity, which can be written in terms of k and $k'$ . Accordingly, we deduce a modified version of Stephen Nelson’s form: for all $(\alpha ,\beta ,\gamma ,\delta ) \in {\mathbb Z}^4,$

(1.7) $$ \begin{align} \psi_{\alpha+\beta+1}\psi_{\alpha-\beta}\psi_{\gamma+\delta+1}\psi_{\gamma-\delta}+ \psi_{\alpha+\gamma+1}\psi_{\alpha-\gamma}\psi_{\delta+\beta+1}\psi_{\delta-\beta}+ \psi_{\alpha+\delta+1}\psi_{\alpha-\delta}\psi_{\beta+\gamma+1}\psi_{\beta-\gamma}=0. \end{align} $$

The equality (1.6) follows by setting $r=\beta -\alpha , p=\gamma -\alpha , q=\delta -\alpha $ and, according to the parity, $s=2\alpha $ in (1.2) or $s=2\alpha +1$ in (1.7).

For the symmetries, for the case $d=1$ [4, Theorem 10.2.2], with ${\cal W}(u)=0\ (u \in {\mathbb Z})$ at a point P of E, we have, for all $k \in {\mathbb Z}$ ,

$$ \begin{align*}{\cal W}(ku+v)={\cal A}^{kv}{\cal C}^{k^2}{\cal W}(v) \quad \mathrm{with} \ {\cal A}=\frac{{\cal W}(u+2)}{{\cal W}(u+1){\cal W}(2)} \text{ and } {\cal C}=\frac{{\cal W}(u+1)}{{\cal A}}.\end{align*} $$

For the case $d=2$ [4, Lemma 10.2.5], with ${\cal W}(\mathbf {u})={\cal W}({u_1,u_2})=0\ (\mathbf {u}=(u_1,u_2) \in {\mathbb Z}^2)$ , $\mathbf {P}=(P_1,P_2) \in E^2$ and $\mathbf {v}=(v_1,v_2) \in {\mathbb Z}^2$ , we have, for all $k \in {\mathbb Z}$ ,

$$ \begin{align*} {\cal W}(k\mathbf{u}+\mathbf{v}) ={\cal A}_1^{kv_1}{\cal A}_2^{kv_2}{\cal C}^{k^2} & {\cal W}(\mathbf{v}) \quad \mathrm{with} \ {\cal A}_1=\frac{{\cal W}(u_1+2,u_2)}{{\cal W}(u_1+1,u_2){\cal W}(2,0)}, \\ &{\cal A}_2=\frac{{\cal W}(u_1,u_2+2)}{{\cal W}(u_1,u_2+1){\cal W}(0,2)}, \ {\cal C}=\frac{{\cal W}(u_1+1,u_2+1)}{{\cal A}_1{\cal A}_2{\cal W}(1,1)}. \end{align*} $$

There are some general results in the literature [4, Theorem 10.2.3] and [1, Theorem 1.13] for any natural integer d, presented as a generalisation of Ward’s theorem (1.3), which we give here in a succinct form to avoid overloading the presentation. For the version ([4], [1, Theorem 1.12]), which deals with nondegenerate elliptic nets associated with an elliptic curve and a d-tuple of points on it,

(1.8) $$ \begin{align} {\cal W}(\mathbf{u}+\mathbf{v})=\delta(\mathbf{u},\mathbf{v}) {\cal W}(\mathbf{v}) \quad \text{for all } \mathbf{v} \in {\mathbb Z}^d, \end{align} $$

where ${\cal W}(\mathbf {u})=0$ and $\delta $ is a quadratic function that is linear in the second factor. Stange’s version has a rather complicated proof [4, Theorem 10.2.3, page 62] and a simplified version of its proof with ‘general’ elliptic nets W can be found in [1, Theorem 1.13] with a factorised form of $\delta $ into linear and quadratic forms: that is,

(1.9) $$ \begin{align} W(\mathbf{u}+\mathbf{v})=\xi(\mathbf{u})\chi(\mathbf{u},\mathbf{v}) W(\mathbf{v}) \quad \text{for all } \mathbf{v} \in {\mathbb Z}^d. \end{align} $$

To obtain their results, Ward and Stange use complex analysis, which requires the nondegeneracy hypothesis. The authors in [1] use the recurrence (1.4), which allows them to remove the nondegeneracy condition and deal with elliptic nets that do not necessarily come from elliptic curves but with the property that $\Lambda =W^{-1}(0)$ is a subgroup of ${\mathbb Z}^d$ and $|{\mathbb Z}^d/\Lambda |\geq 4$ . The result (1.9) is presented as a generalisation of (1.3) by letting ${\cal A}=\chi (v,1)$ and ${\cal C}=\xi (u)$ (see [1] for more details).

The purpose of this article is to prove the following result that unifies [7, Theorem 9.2], [2, Theorem 1], [4, Theorem 10.2.3] and [1, Theorem 1.13].

Theorem 1.4. For a nondegenerate elliptic net ${\cal W}=W_{E,\mathbf{P}}$ associated to an elliptic curve E and a d-tuple of fixed points $\mathbf {P}=(P_1,P_2,\ldots ,P_d)$ on $E^d$ such that ${\cal W}(\mathbf {u})=0$ with $\mathbf {u} \in ({\mathbb Z}^*)^d \ (d \in {\mathbb N})$ , we have, for all $k \in {\mathbb Z}$ and $\mathbf {v}=(v_1,v_2,\ldots ,v_d) \in {\mathbb Z}^d$ ,

(1.10) $$ \begin{align} {\cal W}(k\mathbf{u}+\mathbf{v})={\cal C}^{k^2} \bigg(\prod_{r=1}^d {\cal A}_r^{v_r} \bigg)^k \times {\cal W}(\mathbf{v}) \end{align} $$

with

$$ \begin{align*} {\cal A}_r & =\frac{{\cal W}(\mathbf{u}+2\mathbf{e}_r)}{{\cal W}(\mathbf{u}+\mathbf{e}_r){\cal W}(2\mathbf{e}_r)} \quad\text{for all } r \in \{1,2,\ldots,d\}, \\[6pt] {\cal C} & =\left\{ \begin{array}{@{}ll} \dfrac{{\cal W}(\mathbf{u}+\mathbf{1})}{{\cal W}(\mathbf{1}) \times \prod_{r=1}^d {\cal A}_r} & \mathrm{if} \ \mathbf{u} \neq \mathbf{1}, \\\\[-5pt] -{\cal A}_s {\cal W}(\mathbf{u}-\mathbf{e}_s) \ \ (s \in \{1,2,\ldots,d\}) \ & \mathrm{if} \ \mathbf{u} = \pm \mathbf{1}. \end{array} \right. \end{align*} $$

We limit ourselves to elliptic nets of the form ${\cal W}$ . Indeed, Ward [7] showed that almost all elliptic divisibility sequences are of the form ${\cal W}=W_{E,P}=\psi _n(P)$ and Stange [6] reports that ‘nearly all elliptic nets arise in this way’, and are hence of the form ${\cal W}=W_{E,\mathbf {P}}$ . On the other hand, in [1], to ensure that $\Lambda $ is a group, the authors use the hypothesis that each elliptic sequence $W(ne_i) \ (n \in \{1,2,\ldots ,d\})$ has a unique rank of zero-apparition. In our context, this means that all points $P_i$ are of finite order on E, which seems to be very restrictive in a field of characteristic different from zero.

Note that, from [5, Corollary 5.2], we have the equivalence between ${\cal W}(\mathbf {u})=0$ and $\mathbf {u}.\mathbf {P}=0_E$ . The zeros of an elliptic net then appear as a sublattice of ${\mathbb Z}^d$ , called the lattice of zero-apparition [6, Definition 3].

2. Periodicity

2.1. Generalities

In this paragraph, we consider, for d in ${\mathbb N}_{\geq 2}$ and $\pmb {\ell }=(\ell _1,\ell _2,\ldots , \ell _d)$ in ${\mathbb Z}^d$ , a multi-index sequence denoted by $G_{\pmb {\ell }}=G_{\ell _1,\ell _2,\ldots , \ell _d}$ of elements in the field ${\mathbb K}$ . We say that the sequence $G_{\pmb {\ell }}$ is ${\mathbb Z}$ -geometric if, for all k fixed in $\{1,2,\ldots ,d\}$ and $\pmb {\ell }$ fixed in ${\mathbb Z}^d$ , the sequence $G_{\ell _1,\ell _2,\ldots ,\ell _{k-1},\ell ,\ell _{k+1},\ldots , \ell _d}={\cal G}_\ell $ is geometric. To be more explicit, for all k in $\{1,2,\ldots ,d\}$ we set ${\pmb {\ell }}_k=(\ell _1, \ell _2, \ldots ,\ell _{k-1},\ell _{k+1},\ldots ,\ell _d)$ in ${\mathbb Z}^{d-1}$ and define the ratios $q^{(k)}_{{\pmb {\ell }}_k}$ in ${\mathbb K}$ such that $G_{\pmb {\ell }+\mathbf {e}_k}= q^{(k)}_{{\pmb {\ell }}_k} G_{\pmb {\ell }}$ .

We prove the following lemma, which is useful for obtaining our final result.

Lemma 2.1. Consider a ${\mathbb Z}$ -geometric sequence $(G_{\pmb {\ell }})_{\pmb {\ell } \in {\mathbb Z}^d}$ of elements in the field ${\mathbb K}$ such that

$$ \begin{align*}\text{for all } u\neq v \in \{1,2,\ldots,d\}, \quad G_{\pmb{\ell}+\mathbf{e}_u+\mathbf{e}_v}G_{\pmb{\ell}}=G_{\pmb{\ell}+\mathbf{e}_u}G_{\pmb{\ell}+\mathbf{e}_v}.\end{align*} $$

Then, the sequence $G_{\pmb {\ell }}$ is geometric in each direction $\mathbf {e}_k$ for $k \in \{1,2,\ldots ,d\}$ , namely,

$$ \begin{align*} \text{for all } k \in \{1,2,\ldots,d\} \text{ there exists } q_k \in {\mathbb K}, \quad G_{\pmb{\ell}+\mathbf{e}_k}=q_kG_{\pmb{\ell}}. \end{align*} $$

Proof. We show this result by induction on the integer d.

In the case $d=2$ , for $i\neq j$ in $\{1,2\}$ , from $G_{\pmb {\ell }+\mathbf {e}_j}G_{\pmb {\ell }-\mathbf {e}_j}=G_{\pmb {\ell }}^2$ since $G_{\pmb {\ell }}$ is ${\mathbb Z}$ -geometric, we deduce that $q_{{\pmb {\ell }}_j+1}^{(i)}G_{\pmb {\ell }-\mathbf {e}_i+\mathbf {e}_j}q_{{\pmb {\ell }}_j-1}^{(i)}G_{\pmb {\ell }-\mathbf {e}_i-\mathbf {e}_j}=(q_{{\pmb {\ell }}_j}^{(i)}G_{\pmb {\ell }-\mathbf {e}_i})^2$ so $q_{{\pmb {\ell }}_j}^{(i)}$ is a geometric sequence whose ratio is denoted $r_j$ . So, we have $q_{{\pmb {\ell }}_j}^{(i)}=r_j^{\ell _j}q_0^{(i)}$ . Expressing $G_{1,1}$ in terms of $G_{0,0}$ gives $r_1=r_2$ and, from $G_{1,1}G_{0,0}=G_{1,0}G_{0,1}$ , we find that $r_1=r_2=1$ . Finally, we obtain $G_{\pmb {\ell }+\mathbf {e}_i}=q_{{\pmb {\ell }}_j}^{(i)}G_{\pmb {\ell }}=r_j^{\ell _j}q_0^{(i)}G_{\pmb {\ell }}=q_0^{(i)}G_{\pmb {\ell }}=q_iG_{\pmb {\ell }}$ with $q_0^{(i)}=q_i$ .

For the case $d>2$ , in the same way, we deduce, for k in $\{1,2, \ldots ,d\}$ , that $q_{{\pmb {\ell }}_k}^{(k)}$ is ${\mathbb Z}$ -geometric. On the other hand, for $u \neq v$ , $q_{{\pmb {\ell }}_k}^{(k)}$ satisfies $\ q_{{\pmb {\ell }}_k+\mathbf {e}_u+\mathbf {e}_v}^{(k)}q_{{\pmb {\ell }}_k}^{(k)}=q_{{\pmb {\ell }}_k+\mathbf {e}_u}^{(k)}q_{{\pmb {\ell }}_k+\mathbf {e}_v}^{(k)}$ . Therefore, by the inductive hypothesis,

$$ \begin{align*} \text{for all } k \in \{1,2,\ldots,d\} \text{ and } \text{for all } j \neq k, \text{ there exists } r_{k,j} \in {\mathbb{K}}, \quad q_{{\pmb{\ell}}_k+{\bar {\mathbf{e}}_j}}^{(k)}=r_{k,j}q_{{\pmb{\ell}}_k}^{(k)},\\[-13pt]\end{align*} $$

where ${\bar {\mathbf {e}}_j}$ is the projection of ${\mathbf {e}}_j$ over ${\mathrm {span}}_{\mathbb {Z}} ({\mathbf {e}}_1,\ldots ,{\mathbf {e}}_{k-1},{\mathbf {e}}_{k+1},\ldots ,{\mathbf {e}}_d)$ . It follows that $q_{{\pmb {\ell }}_k}^{(k)}=\prod _{1\le j\le d,{j\neq k}}r_{k,j}^{\ell _j} q_{{\mathbf {0}}_{d-1}}^{(k)}$ with $\mathbf {0}_{d-1}=(0,0,\ldots ,0)$ in ${\mathbb Z}^{d-1}$ and thus we have $G_{\pmb {\ell }+\mathbf {e}_k}=\prod _{1\le j\le d,{j\neq k}}r_{k,j}^{\ell _j} q_{\mathbf {0}_{d-1}}^{(k)} G_{\pmb {\ell }}.$ So, for $u\neq v$ in $\{1,2,\ldots ,d\}$ , we can write $G_{\mathbf {e}_u+\mathbf {e}_v}=r_{v,u}q_{\mathbf {0}_{d-1}}^{(v)} q_{\mathbf {0}_{d-1}}^{(u)} G_{\mathbf {0}}= G_{\mathbf {e}_v+\mathbf {e}_u}$ . Hence, $r_{u,v}=r_{v,u}$ . Finally, from $G_{\mathbf {e}_u+\mathbf {e}_v}G_{\mathbf {0}}=G_{\mathbf {e}_u}G_{\mathbf {e}_v}$ , we obtain $r_{u,v}=1$ and so, for all k in $\{1,2,\ldots ,d\}$ , we have $G_{\pmb {\ell }+\mathbf {e}_k}=q_{\mathbf {0}_{d-1}}^{(k)} G_{\pmb {\ell }} =q_k G_{\pmb {\ell }}$ .

2.2. Geometric sequence of quotient of elliptic nets

We consider a nondegenerate elliptic net ${\cal W}=W_{E,\mathbf {P}}$ associated to the elliptic curve E and the d-tuple of fixed points $\mathbf {P}=(P_1,P_2,\ldots ,P_d)$ on $E^d$ . We assume that there is $\mathbf {u}=(u_1,\ldots ,u_d)$ in ${\mathbb Z}^d$ with ${\cal W}(\mathbf {u})={\cal W}_{E,\mathbf {P}}=0$ . In other words, $\mathbf {u}.\mathbf {P}=u_1P_1+\cdots +u_dP_d=0_E$ [5, Corollary 5.2].

In equation (1.5), we set $\mathbf {r}=\mathbf {e}_r \ (r \in \{1,2,\ldots ,d\})$ , $\mathbf {p} = \mathbf {i}-\pmb {\ell }$ and $\mathbf {q} = \mathbf {j}+\pmb {\ell }$ with $ \pmb {\ell }, \mathbf {i}, \mathbf {j} \in {\mathbb Z}^d$ and we consider $\mathbf {i}+\mathbf {j}=\mathbf {u}$ . We obtain, for all r in $\{ 1,2,\ldots ,d \}$ ,

(2.1) $$ \begin{align} {\cal W}(\mathbf{i}-\pmb{\ell}+\mathbf{e}_r){\cal W}(\mathbf{i}-\pmb{\ell}-\mathbf{e}_r){\cal W}(\mathbf{j}+\pmb{\ell})^2- {\cal W}(\mathbf{j}+\pmb{\ell}+\mathbf{e}_r){\cal W}(\mathbf{j}+\pmb{\ell}-\mathbf{e}_r){\cal W}(\mathbf{ i}-\pmb{\ell})^2=0. \end{align} $$

This equation does not provide any information in certain cases, for example, for $\pmb {\ell }=\mathbf {i}\pm \mathbf {e}_r, \mathbf {i}$ . We now define

$$ \begin{align*}G_{\pmb{\ell}}=\frac{{\cal W}(\mathbf{j}+\pmb{\ell})}{{\cal W}(\mathbf{i}-\pmb{\ell})},\end{align*} $$

which depends on $\mathbf {i}$ and $\mathbf {j}$ but we will fix them later. Note also that $G_{\pmb {\ell }}$ is not defined for some $\pmb {\ell }$ , for example, for $\pmb {\ell }=\mathbf {i}, \pmb {\ell }=-\mathbf {j}$ . From (2.1),

(2.2) $$ \begin{align} \text{for all } r \in \{1,2,\ldots,d \}, \quad G_{\pmb{\ell}+\mathbf{e}_r} \times G_{\pmb{\ell}-\mathbf{e}_r}=G_{\pmb{\ell}}^2. \end{align} $$

Again, (2.2) does not make sense for some values of $\pmb {\ell }$ . We will come back later to all these problematic cases (see Section 2.3) and we provisionally assume that $G_{\pmb {\ell }}$ is well defined for all $\pmb {\ell }$ in ${\mathbb Z}^d$ .

So, the sequence $G_{\pmb {\ell }}$ is ${\mathbb Z}$ -geometric. Furthermore, from (1.4) with $\mathbf {p}=-\mathbf {e}_u$ , $\mathbf {q}=\mathbf {j}+\pmb {\ell }+\mathbf {e}_v$ , $\mathbf {r}=\mathbf {i}-\pmb {\ell }-\mathbf {e}_u$ and $\mathbf {s}=\mathbf {e}_u-\mathbf {e}_v$ , we obtain

$$ \begin{align*}\text{for all } u\neq v \in \{1,2,\ldots,d\}, \quad G_{\pmb{\ell}+\mathbf{e}_u+\mathbf{e}_v}G_{\pmb{\ell}}=G_{\pmb{\ell}+\mathbf{e}_u}G_{\pmb{\ell}+\mathbf{e}_v}.\end{align*} $$

From the previous section, with $q_r={G_{\mathbf {e}_r}}/{G_{\mathbf {0}}}$ , we deduce that

$$ \begin{align*}\text{for all } r \in \{1,\ldots,d\}, \text{ there exists } q_r \in {\mathbb K}, \quad G_{\pmb{\ell}+\mathbf{e}_r}=q_r G_{\pmb{\ell}}.\end{align*} $$

Finally,

(2.3) $$ \begin{align} \text{for all } \pmb{\ell}=(\ell_1,\ell_2,\ldots,\ell_d) \in {\mathbb Z}^d, \quad G_{\pmb{\ell}}=\prod_{r=1}^d q_r^{\ell_r} G_{\mathbf{0}}. \end{align} $$

However, this result omits the problematic cases mentioned, which does not guarantee the existence of $G_{\pmb {\ell }}$ for some $\pmb {\ell }$ in ${\mathbb Z}^d$ . Thus, we do not know whether we are keeping the same ratio through certain points of ${\mathbb Z}^d$ in a given direction. We deal with these questions in the following section.

Before doing so, we fix ${\mathbf {i}}$ and $\mathbf {j}$ with ${\mathbf {u}}={\mathbf {i}}+\mathbf {j}$ . For that, for all r in $\{1,2,\ldots ,d\}$ , if $u_r=2w_r \ (\overline {u_r} \equiv u_r \bmod 2 =0)$ , we set $i_r=w_r-1$ ; but if $u_r=2w_r+1 \ (\overline {u_r}=1)$ , we set $i_r=w_r$ and, in all cases, $j_r=w_r+1$ . Thus, if ${{\mathbf {i}}}=(i_1,i_2,\ldots ,i_d)$ and ${\mathbf {j}}=(j_1,j_2,\ldots ,j_d)$ , writing ${\bar {\mathbf {u}}} \equiv {\mathbf {u}} \bmod 2$ and $\mathbf {1}=(1,1,\ldots ,1)$ in ${\mathbb Z}^d$ , we have

$$ \begin{align*}{\mathbf{i}}=\frac{{\mathbf{u}}+{\bar {\mathbf{u}}}}{2}-\mathbf{1} \quad \mathrm{and} \quad \mathbf{j}=\frac{{\mathbf{u}}-{\bar {\mathbf{u}}}}{2}+\mathbf{1}.\end{align*} $$

It can be observed that $G_{\pmb {\ell }}'=G_{\pmb {\ell }}^{-1}$ with ${\pmb {\ell }}'={\bar {\mathbf {u}}}-2\times {\mathbf {1}}-{\pmb {\ell }}$ .

2.3. Problematic cases

First, if ${\mathbf {u}}={\mathbf {u}}_1+{\mathbf {u}}_2$ in ${\mathbb Z}^d$ with ${\cal W}_{{\mathbf {u}}}=0$ , then ${\cal W}_{{\mathbf {u}}_1}=0 \Leftrightarrow {\cal W}_{{\mathbf {u}}_2}=0$ . Thus, the quantities $G_{\pmb {\ell }}$ do not cancel, but are not defined at some points of ${\mathbb Z}^d$ . Moreover, the nondegeneracy hypothesis tells us that a problematic case can only occur on one of three (four if $d=1$ ) consecutive terms of the sequence $G_{\pmb {\ell }}$ in one direction. We will come back to the special cases of points of order two or three in Section 2.6. On the other hand, if $G_{\pmb {\ell }}$ and $G_{\pmb {\ell }}'$ are not defined, then $(\pmb {\ell }-{\pmb {\ell }}').\mathbf {P}=0_E$ . We deduce that, if $G_{\pmb {\ell }}$ is not defined, then this is not the case for the $G_{\pmb {\ell } +\delta \mathbf {e}_r}$ such that $\delta $ is in $\{\pm 1, \pm 2\}$ for r in $\{ 1,2,\ldots ,d\}$ or even for $G_{\pmb {\ell } \pm \mathbf {e}_r \pm \mathbf {e}_s} \ (r \neq s)$ .

We show that we keep the same ratio $q_r \ ( r \in \{1,2,\ldots ,d\})$ through a problematic case of index $\pmb {\ell }$ in the direction $\mathbf {e}_r$ . This means that ${\cal W}(\mathbf {j}+\pmb {\ell })= {\cal W}({\mathbf {i}}-\pmb {\ell })=0$ . We define the value of $G_{\pmb {\ell }}$ by the expression ${G_{\pmb {\ell }-\mathbf {e}_r}^2}/{G_{\pmb {\ell }-2 \mathbf {e}_r}}=q_rG_{\pmb {\ell }-\mathbf {e}_r}$ . Then, from the addition formula on an elliptic curve expressing $x((\mathbf {r}+\mathbf {s}).\mathbf {P})$ and $x((\mathbf {r}-\mathbf {s}).\mathbf {P})$ for $\mathbf {r} \neq \mathbf {s}$ in $({\mathbb Z}^d)^*$ such that $x(\mathbf {r}.\mathbf {P}) \neq x(\mathbf {s}.\mathbf {P})$ and [5, Lemma 4.2], we obtain ${\cal W}(2\mathbf {r}){\cal W}(2\mathbf {s})=4 y(\mathbf {r}.\mathbf {P})y(\mathbf {s}.\mathbf {P}){\cal W}(\mathbf {r})^4{\cal W}(\mathbf {s})^4$ . Hence, if $\mathbf {s}=\mathbf {e}_s$ for $s\neq r$ in $\{1,2,\ldots ,d\}$ with ${x(\mathbf {r}.\mathbf {P})\neq x(P_s)}$ , we deduce that

(2.4) $$ \begin{align} {\cal W}(2 \mathbf{r}) =2y(\mathbf{r}.\mathbf{P}) {\cal W}(\mathbf{r})^4, \end{align} $$

for r in $\{1,2,\ldots ,d\}$ . With $\mathbf {r}=\mathbf {j}+\pmb {\ell }-\mathbf {e}_r$ , so that $y(\mathbf {r}.\mathbf {P})=-y_r$ in (2.4), we obtain ${\cal W}(2(\mathbf {j}+\pmb {\ell }-\mathbf {e}_r))=-{\cal W}(2\mathbf {e}_r){\cal W}(\mathbf {j}+\pmb {\ell }-\mathbf {e}_r)^4$ . Combining this with (1.5) for $\mathbf {p}=\mathbf {j}+\pmb {\ell }$ , $\mathbf {q}=\mathbf {j}+\pmb {\ell }-2 \mathbf {e}_r$ and $\mathbf {r}=\mathbf {e}_r$ gives

(2.5) $$ \begin{align} {\cal W}(\mathbf{j}+\pmb{\ell}+\mathbf{e}_r) {\cal W}(\mathbf{j}+\pmb{\ell}-2\mathbf{e}_r)^2=-{\cal W}(2 \mathbf{e}_r)^2 {\cal W}(\mathbf{j}+\pmb{\ell}-\mathbf{e}_r)^3. \end{align} $$

In the same way, with $\mathbf {r}=\mathbf {i}-\pmb {\ell }+\mathbf {e}_r$ in (2.4) and $\mathbf {p}=\mathbf {i}-\pmb {\ell }$ , $\mathbf {q}=\mathbf {i}-\pmb {\ell }+2 \mathbf {e}_r$ and $\mathbf {r}=\mathbf {e}_r$ in (1.5), we obtain

(2.6) $$ \begin{align} {\cal W}(\mathbf{i}-\pmb{\ell}-\mathbf{e}_r) {\cal W}(\mathbf{i}-\pmb{\ell}+2\mathbf{e}_r)^2=-{\cal W}(2 \mathbf{e}_r)^2 {\cal W}(\mathbf{i}-\pmb{\ell}+\mathbf{e}_r)^3. \end{align} $$

From (2.5) and (2.6), we deduce that

$$ \begin{align*}&{\cal W}(\mathbf{j}+\pmb{\ell}+\mathbf{e}_r) {\cal W}(\mathbf{j}+\pmb{\ell}-2\mathbf{e}_r)^2 {\cal W}(\mathbf{i}-\pmb{\ell}+\mathbf{e}_r)^3\\ &\quad= {\cal W}(\mathbf{i}-\pmb{\ell}-\mathbf{e}_r) {\cal W}(\mathbf{i}-\pmb{\ell}+2\mathbf{e}_r)^2 {\cal W}(\mathbf{j}+\pmb{\ell}-\mathbf{e}_r)^3,\end{align*} $$

and, therefore, $G_{\pmb {\ell }+\mathbf {e}_r}={G_{\pmb {\ell }}^2}/{G_{\pmb {\ell }-\mathbf {e}_r}}=q_r G_{\pmb {\ell }}$ with the new definition of $G_{\pmb {\ell }}$ .

Next, for all $\lambda $ and $\mu $ in ${\mathbb Z}^*$ , we set $\mathbf {p}=\mathbf {i}-\pmb {\ell }+\lambda \mathbf {e}_r$ , $\mathbf {q}=\lambda \mathbf {e}_r + \mu \mathbf {e}_r$ , $\mathbf {r}=\mathbf {j}+\pmb {\ell }+\lambda \mathbf {e}_r$ and $\mathbf {s}=-2 \lambda \mathbf {e}_r$ with $r \in \{1,2,\ldots ,d\}$ in (1.4). We obtain $G_{\pmb {\ell }+\lambda \mathbf {e}_r}G_{\pmb {\ell }-\lambda \mathbf {e}_r}=G_{\pmb {\ell }+\mu \mathbf {e}_r}G_{\pmb {\ell }+\mu \mathbf {e}_r},$ and, therefore, $G_{\pmb {\ell }+2 \mathbf {e}_r}/G_{\pmb {\ell }+\mathbf {e}_r}=G_{\pmb {\ell }- \mathbf {e}_r}/G_{\pmb {\ell }-2\mathbf {e}_r}=q_r$ .

Finally, we show that the definition of $G_{\pmb {\ell }}$ in the direction $\mathbf {e}_r$ is consistent with that in another direction $\mathbf {e}_s$ , which we denote by ${\widetilde G}_{\pmb {\ell }}$ . For that, we set $\mathbf {p}=\mathbf {j}+\pmb {\ell }-\mathbf {e}_r -\mathbf {e}_s$ , $\mathbf {q}=\mathbf {i}- \pmb {\ell } + \mathbf {e}_r + \mathbf {e}_s$ and $\mathbf {r}=\mathbf {e}_r-\mathbf {e}_s$ in (1.5) to obtain $G_{\pmb {\ell }-\mathbf { e}_r-\mathbf {e}_s}^2=G_{\pmb {\ell }-2\mathbf {e}_s}G_{\pmb {\ell }-2\mathbf {e}_r}$ , and so $G_{\pmb {\ell }-\mathbf {e}_r}^2G_{\pmb {\ell }-2\mathbf {e}_s}=G_{\pmb {\ell }-\mathbf {e}_s}^2G_{\pmb {\ell }-2\mathbf {e}_r}$ , that is, $G_{\pmb {\ell }}={\widetilde G}_{\pmb {\ell }}$ . So, for a problematic index $\pmb {\ell }$ , we can set $G_{\pmb {\ell }}=q_rG_{\pmb {\ell }-\mathbf {e}_r}$ to ensure that $G_{\pmb {\ell }}$ is geometric in each direction.

Example 2.2. For the curve $y^2=x^3+2x-4$ over ${\mathbb F}_{73}$ and the points $P_1=(36,71)$ , $P_2=(51,53)$ , $P_3=(7,34)$ , we have $U=(3,5,7)$ and $(q_1,q_2,q_3)=(22,71,58)$ . The values $G_{\mathbf {i}}$ and $G_{\mathbf {-j}}$ are not defined. We set $G_{\mathbf {i}}=q_rG_{\mathbf {i}-\mathbf {e}_r}=47$ and $G_{-\mathbf {j}}=q_rG_{-\mathbf {j}-\mathbf {e}_r}=14$ . The values of $G_{\mathbf {i}+k\mathbf {e}_r} \ (k \in \{-3;3\})$ are, for $r=1,2,3$ successively,

$$ \begin{align*}\{61,28,32,\mathbf{47},12,45,45 \}, \quad \{58,30,13,\mathbf{47},52,42,62\}, \quad \{23,20,65,\mathbf{47},25,63,4\},\end{align*} $$

and for $G_{-\mathbf {j}+k\mathbf {e}_r}$ ,

$$ \begin{align*}\{57,13,67,\mathbf{14},16,60,6\}, \quad \{53,40,66,\mathbf{14},45,56,34\}, \quad \{55,51,38,\mathbf{14},9,11,54\}.\end{align*} $$

We can give a harmonious formulation of the ratios $q_r$ in terms of G and, therefore, of ${\cal W}$ , if the quantities involved are well defined. Indeed, from (2.2) for $\pmb {\ell }=\mathbf {e}_r-\mathbf {1}$ , we obtain $G_{2\mathbf {e}_r-\mathbf {1}}G_{-\mathbf {1}}=G_{\mathbf {e}_r-\mathbf {1}}^2$ for all r in $\{1,2,\ldots ,d\}$ . With $G_{2\mathbf {e}_r-\mathbf {1}}=q_r G_{\mathbf {e}_r-\mathbf {1}}$ and ${G_{-\mathbf {1}}=G_{\bar {{\mathbf {u}}}-\mathbf {1}}^{-1},}$ we deduce that

(2.7) $$ \begin{align} \text{for all } r \in \{ 1,2,\ldots,d\}, \quad q_r=G_{{\bar {\mathbf{u}}}-\mathbf{1}}\times G_{\mathbf{e}_r-\mathbf{1}}= \frac{{\cal W}(\frac{{\mathbf{u}}+{\bar {\mathbf{u}}}}{2})}{{\cal W}(\frac{{\mathbf{u}}-{\bar {\mathbf{u}}}}{2})} \times \frac{{\cal W}(\frac{{\mathbf{u}}-{\bar {\mathbf{u}}}}{2}+\mathbf{e}_r)}{{\cal W}(\frac{{\mathbf{u}}+{\bar {\mathbf{u}}}}{2}-\mathbf{e}_r)}. \end{align} $$

Example 2.3. For the curve $y^2=x^3+x+1$ over ${\mathbb F}_{11}$ , we consider the points of order seven, that is, $P_1=(6,5)$ and $P_2=(3,3)$ . We have $3P_1+P_2=0_E=2P_1+3P_2=5P_1+4P_2$ , so ${\mathbf {u}}=(5,4)=(3,1)+(2,3)={\mathbf {u}}_{1}+{\mathbf {u}}_{2}$ . In this case, $G_{(-1,0)}$ and $G_{(0,-2)}$ are not defined since ${\cal W}_{2,3}={\cal W}_{3,1}=0$ and so $q_2$ is not defined. We define $G_{(0,-2)}=q_1G_{(-1,-2)}=4*5=9$ and $G_{(-1,0)}=G_{(0,0)}/q_1=9/4=5=9^{-1}$ . We also set $q_2=G_{(0,-1)}G_{(-1,0)}=2*5=10$ . Note that, at the end of the article, we show that $q_r({\mathbf {u}})=q_r({\mathbf {u}}_1)*q_r({\mathbf {u}}_2)\ (r \in \{1,2\})$ . Indeed, $q({\mathbf {u}})=(4,10)$ , $q({\mathbf {u}}_1)=(6,6)$ and $q({\mathbf {u}}_2)=(8,9)$ .

If we now consider ${\mathbf {u}}=2(3,1)=(6,2)$ , then $G_{-\mathbf 1}$ is not defined, nor are the quantities $q_1$ and $q_2$ . We have $q_1=G_{(1,0)}/G_{(0,0)}=3$ , $q_2=G_{(0,1)}/G_{(0,0)}=3$ and $G_{(-1,-1)}=G_{(0,0)}/(q_1q_2)=-1$ . Once again, we see that $q_r(2{\mathbf {u}})=q_r({\mathbf {u}})^2$ . Indeed, $q((6,2))=(3,3); q((3,1))=(6,6)$ .

For the case ${\mathbf {u}}=\mathbf {1}$ , the quantities $G_{-\mathbf {1}}$ , $G_{\mathbf {0}}$ , and thus the ratios $q_k$ , are not defined. But, we can set

$$ \begin{align*}\text{for all } k \in \{1,2,\ldots,n\}, \quad q_k \stackrel{k'\neq k}{=}\frac{G_{\mathbf{e}_{k'}}}{G_{\mathbf{e}_{k'}-\mathbf{e}_k}},\end{align*} $$

and $G_{-\mathbf {1}}=G_{-\mathbf {1}+\mathbf {e}_k}/q_k$ , $G_{\mathbf {0}}=q_kG_{-\mathbf {e}_k}$ .

For the curve $y^2=x^3+17x-53$ over ${\mathbb F}_{229}$ , we consider the points $P_1=(217,63)$ , $P_2=(153,59)$ , $P_3=(42,211)$ , $P_4=(40,222)$ and $P_5=(13,126)$ . We have ${\mathbf {u}}=\mathbf {1}$ . We can write $q_1=G_{\mathbf {e}_2}/G_{\mathbf {e}_2-\mathbf {e}_1}=211$ and so $q_2=55, q_3= 221, q_4=13, q_5=227$ and $G_{-\mathbf {1}}=G_{\mathbf {e}_1-\mathbf {1}}/q_1=181$ .

So we can have cases where the definition $q_r=G_{{\bar {\mathbf {u}}}-\mathbf {1}}\times G_{\mathbf {e}_r-\mathbf {1}}$ is problematic. However, we can always find $\pmb {\ell }$ in ${\mathbb Z}^d$ so that the ratio $q_r={G_{\pmb {\ell }+\mathbf {e}_r}}/{G_{\pmb {\ell }}}$ is well defined. Nevertheless, the expression (2.7) needs some ${\cal W}$ whose indexes are in the neighbourhood of ${\mathbf {u}}/2$ , which is the best that we can do for the computation of $G_{\pmb {\ell }}$ whose indexes are symmetric with respect to ${\mathbf {u}}/2$ .

2.4. Proof of Theorem 1.4

First, we set $\pmb {\ell }=\mathbf {i}+\mathbf {v}$ for $\mathbf {v}$ in ${\mathbb Z}^d \backslash \Gamma $ , giving

$$ \begin{align*}G_{\pmb{\ell}}=G_{\mathbf{i}+\mathbf{v}}=\frac{{\cal W}(\mathbf{i}+\mathbf{j}+\mathbf{v})}{{\cal W}(-\mathbf{v})}=\frac{{\cal W}({\mathbf{u}}+\mathbf{v})}{{\cal W}(-\mathbf{v})}= -\frac{{\cal W}({\mathbf{u}}+\mathbf{v})}{{\cal W}(\mathbf{v})}.\end{align*} $$

Therefore, from (2.3), we obtain, in the cases where $G_{-\mathbf {1}}$ is well defined,

$$ \begin{align*}{\cal W}({\mathbf{u}}+\mathbf{v})=-G_{\mathbf{i}+\mathbf{v}} {\cal W}(\mathbf{v})= - \bigg( \prod_{r=1}^d q_r^{i_r+v_r+1} \bigg) \,G_{-\mathbf{1}} \times {\cal W}(\mathbf{v}),\end{align*} $$

which holds for $\mathbf {v}$ in ${\mathbb Z}^d$ such that ${\cal W}(\mathbf {v})=0$ . Note that, in this case, since G is geometric in each direction, $G_{-\mathbf {1}}=\prod _{r=1}^d q_r^{-\overline {u_r}} \times G_{\bar {{\mathbf {u}}}-\mathbf {1}}$ ; therefore, $G^2_{-\mathbf {1}}=\prod _{r=1}^d q_r^{-\overline {u_r}}.$ This shows that $\prod _{r=1}^d q_r^{u_r}$ is a square.

For all r in $\{1,2,\ldots ,d \}$ , when $G_{-\mathbf {1}}$ is well defined, we set ${\cal A}_r=q_r$ and ${\cal C}=- ( \prod _{r=1}^d q_r^{i_r+1} ) G_{-\mathbf {1}}$ . Thus, we can write $ {\cal C}^2=\prod _{r=1}^d q_r^{2(i_r+1)}\times G_{-\mathbf {1}}^2=\prod _{r=1}^d {\cal A}_r^{u_r}$ (which is just $\xi ({\mathbf {u}})^2=\chi ({\mathbf {u}},{\mathbf {u}})$ ; see (2.5)). Hence, ${\cal W}({\mathbf {u}}+\mathbf {v})={\cal C} \prod _{r=1}^d {\cal A}_r^{v_r} \times {\cal W}(\mathbf {v})$ and a simple induction on k give the desired result (1.10). The formulas for ${\cal A}$ and ${\cal C}$ in (1.10) follow immediately from the existence of these quantities.

On the other hand, if we set ${\mathbf {u}}_1={({\mathbf {u}}-{\bar {\mathbf {u}}})}/{2}$ and ${\mathbf {u}}_2={({\mathbf {u}}+{\bar {\mathbf {u}}})}/{2}$ with possibly ${\mathbf {u}}_1=\mathbf {u_2}$ , we have $G_{-\mathbf {1}}={{\cal W}({\mathbf {u}}_1)}/{{\cal W}({\mathbf {u}}_2)}$ . Hence, $G_{-\mathbf {1}}$ is not defined if ${\mathbf {u}}=\pm \mathbf {1}$ or ${\mathbf {u}}={\mathbf {u}}_1+{\mathbf {u}}_2$ with ${\mathbf {u}}_1.\mathbf {P}=0_E$ and ${\mathbf {u}}_2.\mathbf {P}=0_E$ . Suppose that ${\mathbf {u}} \neq \pm \mathbf {1}$ . For s in $\{1,2,\ldots ,d\}$ , we have $G_{-\mathbf {e}_s-\mathbf {1} +{\bar {\mathbf {u}}}}=1/G_{-\mathbf {e}_s-\mathbf {1}}$ and thus $q_s^2 \prod _{r=1}^d q_r^{-{\bar {\mathbf {u}}_r}}=G_{-\mathbf {e}_s-\mathbf {1}}^2$ . We still have

$$ \begin{align*}{\cal W}({\mathbf{u}}+\mathbf{v})=-G_{\mathbf{i}+\mathbf{v}} {\cal W}(\mathbf{v})= - \bigg( \prod_{r=1}^d q_r^{i_r+v_r+1} \bigg)\, \frac{G_{\mathbf{e}_s-\mathbf{1}}}{q_s} \times {\cal W}(\mathbf{v}),\end{align*} $$

and so we set ${\cal A}_r=q_r$ and ${\cal C}=- ( \prod _{r=1}^d q_r^{i_r+1} {G_{\mathbf {e}_s-\mathbf {1}}}/{q_s})$ . Note that, for $s \neq s'$ , $G_{\mathbf {e}_s+\mathbf {e}_{s'}-\mathbf {1}}=G_{\mathbf {e}_s-\mathbf {1}}q_{s'}=G_{\mathbf {e}_{s'}-\mathbf {1}}q_s$ . Again, we obtain ${\cal C}^2=\prod _{r=1}^d {\cal A}_r^{u_r}$ .

For ${\mathbf {u}}=\mathbf {1}$ (the case ${\mathbf {u}}=-\mathbf {1}$ can be handled in the same manner), we write instead

$$ \begin{align*}{\cal W}({\mathbf{u}}+\mathbf{v})= - \bigg( \prod_{r=1}^d q_r^{i_r+v_r} \bigg)\, G_{-\mathbf{e}_s}q_s \times {\cal W}(\mathbf{v})= \bigg( \prod_{r=1}^d q_r^{v_r} \bigg)\, (-{\cal W}(\mathbf{1}-\mathbf{e}_s)q_s )\times {\cal W}(\mathbf{v})\end{align*} $$

and set ${\cal A}_r=q_r$ and ${\cal C}=-{\cal W}(\mathbf {1}-\mathbf {e}_s)q_s$ for s in $\{1,2,\ldots ,d\}$ . Note that, since $G_{-\mathbf {e}_s-\mathbf {e}_{s'}}=G_{-\mathbf {e}_{s'}-\mathbf {e}_{s}}$ for $s \neq s'$ , we have ${\cal W}(\mathbf {1}-\mathbf {e}_s)q_s= {\cal W}(\mathbf {1}-\mathbf {e}_{s'})q_{s'}$ . Moreover, ${\cal C}^2=q_1q_2{\cal W}(\mathbf {1}-\mathbf {e}_1){\cal W}(\mathbf {1}-\mathbf {e}_2)$ but

$$ \begin{align*} q_3 & =\frac{G_{-\mathbf{e}_1}}{G_{-\mathbf{e}_1-\mathbf{e}_3}}={\cal W}(\mathbf{1}-\mathbf{e}_1)\times \frac{{\cal W}(\mathbf{1}-\mathbf{e}_2-\mathbf{e}_4 -\cdots-\mathbf{e}_d)}{{\cal W}(\mathbf{e}_2+\mathbf{e}_4 +\cdots+\mathbf{e}_d)} \\ & = {\cal W}(\mathbf{1}-\mathbf{e}_1)\times G_{-\mathbf{e}_2-\mathbf{e}_4 -\cdots-\mathbf{e}_d}={\cal W}(\mathbf{1}-\mathbf{e}_1) \times (q_4\cdots q_d)^{-1} G_{-\mathbf{e}_2}, \end{align*} $$

and hence ${\cal C}^2=\prod _{r=1}^d q_r$ since $G_{-\mathbf {e}_2}={\cal W}(\mathbf {1}-\mathbf {e}_2)$ . This completes the proof of Theorem 1.4.

Moreover, this result includes [2, Theorem 1] for $u>3$ (see (2.6) for $u=2$ or $3$ ). If $u=2m$ then, ${\cal A}=q={\psi _{m+1}}/{\psi _{m-1}}=\omega $ and ${\cal C}=-q^{i+1}G_{-1}=-q^m$ , which gives $\psi _{ku+v}=(-1)^k\omega ^{k(v+km)}\psi _v$ . If $u =2m+1$ , then ${\cal A}=q=({\psi _{m+1}}/{\psi _m})^2=\omega ^2$ and ${\cal C}=-q^{i+1}G_{-1}=-q^{m+1}/\omega =-\omega ^{2m+1}$ , which gives $\psi _{ku+v}=(-1)^k\omega ^{k(2v+k(2m+1))}\psi _v$ .

Example 2.4. Over ${\mathbb Q}$ , the curve $y^2=x^3-4x+1$ with

$$ \begin{align*} P_1=(0,1),\quad P_2= (82264/505521, 213664697/359425431),\quad P_3=(4,7), \end{align*} $$

gives ${\mathbf {u}}=(3,1,2)$ and

$$ \begin{align*} {\cal C}= 255551481441/19041697792,\hspace{10pt} {\cal A}=(711/208, 359425431/297526528, 711/368).\end{align*} $$

We give some calculations to illustrate Theorem 1.4 in Table 1.

Table 1 Calculations illustrating Theorem 1.4 in characteristic zero.

According to the Lutz–Nagell theorem [3, Ch. 8], the only possible points of $E({\mathbb Q})_{tors}$ are $(0,1), (2,\pm 1)$ and $(-2, \pm 1)$ , which cannot arise according to Mazur’s theorem. As a result, none of the sequences $\psi _n(P_1); \psi _n(P_2); \psi _n(P_3)$ have a rank of zero-apparition.

Over ${\mathbb F}_{7919}$ , the curve $y^2=x^3+1562x+1805$ with the points $P_1=(4856,5835)$ , $P_2=(6128,7637)$ , $P_3=(3336,2121)$ and $P_4=(2415,7795)$ gives ${\mathbf {u}}=(18,17,12,17)$ and ${\cal C}=3648$ , ${\cal A}=(2664,4758,5312,531).$ Some calculations are given in Table 2.

Table 2 Calculations illustrating Theorem 1.4 in nonzero characteristic.

2.5. The latest known general result

We now link our results to [1, Theorem 1.13]. With the assumptions and the notation $\chi $ and $\xi $ of this theorem, one can write

$$ \begin{align*}{\cal W}({\mathbf{u}}+\mathbf{v})=\xi({\mathbf{u}}) \chi({\mathbf{u}},\mathbf{v}) {\cal W}(\mathbf{v}).\end{align*} $$

More precisely, with $\Lambda =\{ \mathbf {v} \in {\mathbb Z}^d \mid W(\mathbf {v})=0\}$ , the functions $\chi $ and $\xi $ are defined by

$$ \begin{align*} \delta : \Lambda \times ({\mathbb Z}^d \backslash \Lambda) & \ \rightarrow\ {\mathbb K}^* \\ ({\mathbf{u}},\mathbf{v}) & \ \mapsto\ \frac{{\cal W}({\mathbf{u}}+\mathbf{v})}{{\cal W}(\mathbf{v})} \end{align*} $$

and the relations

$$ \begin{align*} \chi : \Lambda \times {\mathbb Z}^d & \ \rightarrow\ {\mathbb K}^*, \\ ({\mathbf{u}},\mathbf{v}) & \ \mapsto\ \frac{\delta({\mathbf{u}},\mathbf{v}+\mathbf{v}')}{\delta({\mathbf{u}},\mathbf{v}')} \quad\text{where } \mathbf{v}' \in {\mathbb Z}^d \text{ but } \mathbf{v}',\mathbf{v}'+\mathbf{v} \not\in \Lambda, \\ \xi : \Lambda & \ \rightarrow\ {\mathbb K}^*, \\ {\mathbf{u}} &\ \mapsto\ \frac{\delta({\mathbf{u}},\mathbf{v})}{\chi({\mathbf{u}},\mathbf{v})} \quad\text{for any } \mathbf{v} \in {\mathbb Z}^d \backslash \Lambda. \end{align*} $$

We now relate the functions $\delta $ of (1.8) and $\chi , \xi $ of (1.9) to our notation. We have

$$ \begin{align*}\chi({\mathbf{u}},\mathbf{v})=\frac{{\cal W}({\mathbf{u}}+\mathbf{v}+\mathbf{v}')}{{\cal W}(\mathbf{v}+\mathbf{v}')}\frac{{\cal W}(\mathbf{v}')}{{\cal W}({\mathbf{u}}+\mathbf{v}')}= \prod_{r=1}^d {\cal A}_r^{v_r}.\end{align*} $$

So we deduce, for all k in $\{1,2,\ldots ,d\}$ , that $\chi ({\mathbf {u}},\mathbf {e}_k)={\cal A}_k$ , and, in the same way,

$$ \begin{align*}\xi({\mathbf{u}})={\cal C} \quad \mathrm{and} \quad \delta({\mathbf{u}},\mathbf{v})={\cal C} \prod_{r=1}^d {\cal A}_r^{v_r}= \xi({\mathbf{u}}) \chi({\mathbf{u}},\mathbf{v}).\end{align*} $$

Now, we recall the results of [1, Theorem 1.13, Lemma 4.2] to which we can give an immediate proof.

Theorem 2.5. The functions $\xi $ and $\chi $ have the following properties.

1. (1) $\chi $ is bilinear symmetric: that is, for all ${\mathbf {u}},{\mathbf {u}}^{(1)},{\mathbf {u}}^{(2)} \in \Lambda $ and $\mathbf {v},\mathbf {v}^{(1)}, \mathbf {v}^{(2)} \in {\mathbb Z}^d$ ,

   1. (a) $\chi ({\mathbf {u}},\mathbf {v}^{(1)}+\mathbf {v}^{(2)})=\chi ({\mathbf {u}},\mathbf {v}^{(1)})\chi ({\mathbf {u}},\mathbf {v}^{(2)})$ ,

   2. (b) $\chi ({\mathbf {u}}^{(1)}+{\mathbf {u}}^{(2)},\mathbf {v})=\chi ({\mathbf {u}}^{(1)},\mathbf {v})\chi ({\mathbf {u}}^{(2)},\mathbf {v})$ ,

   3. (c) $\chi ({\mathbf {u}}^{(1)},{\mathbf {u}}^{(2)})=\chi ({\mathbf {u}}^{(2)},{\mathbf {u}}^{(1)})$ ,

   4. (d) $\chi ({\mathbf {u}},-\mathbf {v})=\chi ({\mathbf {u}},\mathbf {v})^{-1}$ .

2. (2) $\xi ({\mathbf {u}}^{(1)}+{\mathbf {u}}^{(2)})=\xi ({\mathbf {u}}^{(1)})\xi ({\mathbf {u}}^{(2)})\chi ({\mathbf {u}}^{(1)},{\mathbf {u}}^{(2)})$ .

3. (3) $\xi (-{\mathbf {u}})=\xi ({\mathbf {u}})$ .

4. (4) $\xi ({\mathbf {u}})^2=\chi ({\mathbf {u}},{\mathbf {u}})$ .

5. (5) $\xi (n{\mathbf {u}})=\xi ({\mathbf {u}})^{n^2}$ , for all $n \in {\mathbb Z}$ .

Proof.

1. (1) (a) is obvious; (b) is obtained from (1.4) with $\mathbf {p}=\mathbf {e}_r$ , $\mathbf {q}=-{\mathbf {u}}^{(2)}$ , $\mathbf {r}=2\mathbf {e}_r$ and $\mathbf {s}={\mathbf {u}}^{(1)}+{\mathbf {u}}^{(2)}$ ; (c) is easily obtained from ${\cal W}({\mathbf {u}}^{(1)}+({\mathbf {u}}^{(2)}+\mathbf {v}))={\cal W}({\mathbf {u}}^{(2)}+({\mathbf {u}}^{(1)}+\mathbf {v}))$ ; and (d) is obvious.

2. (2) This is easily obtained from ${\cal W}(({\mathbf {u}}^{(1)}+{\mathbf {u}}^{(2)})+\mathbf {v})={\cal W}({\mathbf {u}}^{(1)}+({\mathbf {u}}^{(2)}+\mathbf {v}))$ .

3. (3) From (1.5) with $\mathbf {p}=2\mathbf {e}_r$ , $\mathbf {q}={\mathbf {u}}$ and $\mathbf {r}=\mathbf {e}_r$ , we deduce that $\chi (-{\mathbf {u}},\mathbf {v})=\chi ({\mathbf {u}},\mathbf {v})^{-1}$ so $\chi (-{\mathbf {u}},-\mathbf {v})=\chi ({\mathbf {u}},\mathbf {v})$ . The result comes from ${\cal W}(-{\mathbf {u}}-\mathbf {v})=-{\cal W}({\mathbf {u}}+\mathbf {v})$ .

4. (4) This follows from $1=\xi (0)=\xi ({\mathbf {u}}-{\mathbf {u}})=\xi ({\mathbf {u}})\xi (-{\mathbf {u}})\chi ({\mathbf {u}},-{\mathbf {u}})$ .

5. (5) This result can be deduced from the previous statements.

Example 2.6. Following [6, Section 5.1], we consider $Q=k.P$ on an elliptic curve E with P and Q of order m. The elliptic net associated to P and Q cancels at the points ${\mathbf {u}}=(-k,1), \mathbf {s}=(m,0)$ and $\mathbf {t}=(0,m)$ . With obvious notation,

$$ \begin{align*}\chi((-km,m),\mathbf{e}_r)=\chi(m(-k,1),\mathbf{e}_r)=\chi^m((-k,1),\mathbf{e}_r)=({\cal A}^{({\mathbf{u}})}_r)^m\end{align*} $$

and

$$ \begin{align*}\chi((-km,m),\mathbf{e}_r)=\chi^{-k}((m,0),\mathbf{e}_r)\chi((0,m),\mathbf{e}_r)=({\cal A}^{(\mathbf{s})}_r)^{-k}{\cal A}^{(\mathbf{t})}_r.\end{align*} $$

Thus, we easily obtain $({\cal A}^{({\mathbf {u}})}_r)^m=({\cal A}^{(\mathbf {s})}_r)^{-k}{\cal A}^{(\mathbf {t})}_r$ , which is [6, Equation (9)].

For the curve $y^2=x^3+x+1$ over ${\mathbb F}_{11}$ , with the points $P_1=(6,5)$ and $P_2=(3,3)$ of order seven, we have the values shown in Table 3.

Table 3 Calculations illustrating Theorem 2.5 for various $u \in \Lambda $ .

2.6. Points of order two or three

We return here to special cases related to the degeneracy conditions of ${\cal W}$ , namely, ${\cal W}(2e_i) \neq 0$ for $1\leq i \leq d$ and ${\cal W}(3e_1) \neq 0$ when $d=1$ . This, therefore, concerns cases where there are points of order two, or order three when $d=1$ , on the elliptic curve E. Note that $|{\mathbb Z}^d/\Lambda |=2$ occurs only in the case $d=1$ when $\mathbf {P}=P$ is of order two. We have $|{\mathbb Z}^d/\Lambda |=3$ if either $d=1$ and $\mathbf {P}=P$ is of order three, or $d=2$ and $\mathbf {P}=(P_1,P_2)$ are two points of order two and ${\mathbf {u}}=(2,2)$ .

For the case $d=1$ with $\mathbf {P}=P$ of order two on E, we have $u=2$ so $i=0$ and $j=2$ , and hence $G_\ell ={\psi _{2+\ell }}/{\psi _\ell }$ with $\ell $ odd. In (1.1) with $m=2\ell +1$ and $n=2$ , we obtain $G_{2\ell + 1}=-\psi _3 G_{2 \ell -1}$ . But we can easily show that, when $y=0$ , we have $\psi _3(x,y)=-( {(2ax+3b)}/{x})^2$ if $x\neq 0$ and $\psi _3(x,y)=-a^2$ if $x=0$ . Hence, in every case, we can write $-\psi _3=q^2$ with q in ${\mathbb K}$ . So, we deduce that $G_{2\ell +1}=q^{2\ell +2}G_{-1}=q^{2\ell +2}$ , and writing $2\ell +1=i+v=v$ for v odd in ${\mathbb Z}$ , since $G_{i+v}={\psi _{u+v}}/{\psi _{-v}}$ , we have $\psi _{u+v}=-q^{v+1}\psi _v$ . Finally, we set ${\cal C}=-q$ and ${\cal A}=q$ , to obtain ${\cal C}^2={\cal A}^u$ and $\psi _{ku+v}={\cal C}^{k^2}{\cal A}^{kv} \psi _v$ . We also find the result of [2, Theorem 1].

For the case $d=1$ with $\mathbf {P}=P$ of order three on E, we proceed in the same way. We have $u=3$ so $i=1$ and $j=2$ , and hence $G_\ell ={\psi _{2+\ell }}/{\psi _{1-\ell }}$ with $\ell \not \equiv 1 \bmod 3$ . In (1.1) with $m=\ell +1$ and $n=2$ , we obtain $G_{\ell + 1}=\psi _2^2 G_{\ell }$ for $\ell \equiv 2 \bmod 3$ . The rest follows in the same way as before with ${\cal C}=-\psi _2^3$ and ${\cal A}=\psi _2^2$ ( ${\cal C}^2={\cal A}^3={\cal A}^u$ ) or $w=\psi _2$ to obtain [2, Theorem 1] when $u=3$ .

For the case $d=2$ , with one or two points of order two, as already mentioned, if $G_{\ell }$ creates a problem, then the $G_{{\ell }'}$ are well defined for ${\ell }'=\ell \pm \mathbf {e}_r$ or $\ell + \mathbf {e}_s$ or $\ell +\mathbf {e}_s \pm \mathbf {e}_r$ with $r\neq s$ in $\{1,2,\ldots d\}$ , and we can then ‘bypass’ the index $\ell $ by setting ${G_\ell =({G_{\ell +\mathbf {e}_s-\mathbf {e}_r}}/{G_{\ell +\mathbf {e}_s}})G_{\ell -\mathbf {e}_r}=q_rG_{\ell -\mathbf {e}_r}}$ . Furthermore, $G_{\ell +\mathbf {e}_r}=q_s^{-1}G_{\ell +\mathbf {e}_r+\mathbf {e}_s}=q_s^{-1}q_r^2G_{\ell -\mathbf {e}_r+\mathbf {e}_s}=q_r^2G_{\ell -\mathbf {e}_r}$ , and hence $G_{\ell +\mathbf {e}_r}=q_rG_{\ell }$ .

For the case $d=3$ , we can have three points of order two but, in this case, ${{\mathbf {u}}=\mathbf {1}}$ , which we have already dealt with. For $d>3$ , we can always make sure that the geometric character of $G_\ell $ subsists with the same ratio through a problematic index with points of order two by ‘bypassing’ in another direction.