1. Introduction
This paper concerns a generalisation of a theorem of Ward [Reference Ward7] 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 [Reference Ward7] are only known for the cases
$d=1$
[Reference Stange, Lauter, Avanzi, Keliher and Sica6] and
$d=2$
[Reference Stange4, Reference Stange, Lauter, Avanzi, Keliher and Sica6]. 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 [Reference Ward7, Theorem 8.1], Stange [Reference Stange4, Theorem 10.2.2] and [Reference Stange, Lauter, Avanzi, Keliher and Sica6, Theorem 4], and the author [Reference Dewaghe2, Theorem 1]; for
$d=2$
, [Reference Stange4, Lemma 10.2.5] and [Reference Stange, Lauter, Avanzi, Keliher and Sica6, Theorem 5]; and for
$d>2$
, [Reference Stange4, Theorem 10.2.3] and Akbary et al. [Reference Akbary, Bleaney and Yazdani1, Theorems 1.12 and 1.13].
Let E be an elliptic curve over a field
${\mathbb K}$
(see [Reference Silverman3]). To simplify, we assume that the characteristic is different from
$2$
and
$3$
. Then
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu1.png?pub-status=live)
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 [Reference Washington8]) by
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu2.png?pub-status=live)
and for n a natural integer,
$\psi _{-n}=-\psi _n$
. Then, for all
$(m,n)$
in
${\mathbb Z}^2$
,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn1.png?pub-status=live)
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,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu3.png?pub-status=live)
for n in
${\mathbb Z}$
. Note also Stephen Nelson’s form (see [Reference Stange4, page 22]): for all
$(\alpha ,\beta ,\gamma ,\delta ) \in {\mathbb Z}^4$
,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn2.png?pub-status=live)
Division polynomials have partial periodicity, called symmetry.
Theorem 1.1 [Reference Dewaghe2]
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) 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) 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 [Reference Ward7]
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn3.png?pub-status=live)
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 [Reference Stange5], 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
${\mathbb Z}^d$
,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn4.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn5.png?pub-status=live)
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 [Reference Stange5], 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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu5.png?pub-status=live)
From [Reference Akbary, Bleaney and Yazdani1, 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,
$W_1=1$
[Reference Ward7, 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 [Reference Stange4, Ch. 3, page 22].
Proposition 1.3. For all
$(p,q,r,s) \in {\mathbb Z}^4$
,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn6.png?pub-status=live)
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
$(\alpha ,\beta ,\gamma ,\delta ) \in {\mathbb Z}^4,$
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn7.png?pub-status=live)
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
$d=1$
[Reference Stange4, Theorem 10.2.2], with
${\cal W}(u)=0\ (u \in {\mathbb Z})$
at a point P of E, we have, for all
$k \in {\mathbb Z}$
,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu6.png?pub-status=live)
For the case
$d=2$
[Reference Stange4, 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
$k \in {\mathbb Z}$
,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu7.png?pub-status=live)
There are some general results in the literature [Reference Stange4, Theorem 10.2.3] and [Reference Akbary, Bleaney and Yazdani1, 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 ([Reference Stange4], [Reference Akbary, Bleaney and Yazdani1, Theorem 1.12]), which deals with nondegenerate elliptic nets associated with an elliptic curve and a d-tuple of points on it,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn8.png?pub-status=live)
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 [Reference Stange4, Theorem 10.2.3, page 62] and a simplified version of its proof with ‘general’ elliptic nets W can be found in [Reference Akbary, Bleaney and Yazdani1, Theorem 1.13] with a factorised form of
$\delta $
into linear and quadratic forms: that is,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn9.png?pub-status=live)
To obtain their results, Ward and Stange use complex analysis, which requires the nondegeneracy hypothesis. The authors in [Reference Akbary, Bleaney and Yazdani1] 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 [Reference Akbary, Bleaney and Yazdani1] for more details).
The purpose of this article is to prove the following result that unifies [Reference Ward7, Theorem 9.2], [Reference Dewaghe2, Theorem 1], [Reference Stange4, Theorem 10.2.3] and [Reference Akbary, Bleaney and Yazdani1, 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
$\mathbf {v}=(v_1,v_2,\ldots ,v_d) \in {\mathbb Z}^d$
,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn10.png?pub-status=live)
with
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu8.png?pub-status=live)
We limit ourselves to elliptic nets of the form
${\cal W}$
. Indeed, Ward [Reference Ward7] showed that almost all elliptic divisibility sequences are of the form
${\cal W}=W_{E,P}=\psi _n(P)$
and Stange [Reference Stange, Lauter, Avanzi, Keliher and Sica6] 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 [Reference Akbary, Bleaney and Yazdani1], 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 [Reference Stange5, 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 [Reference Stange, Lauter, Avanzi, Keliher and Sica6, 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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu9.png?pub-status=live)
Then, the sequence
$G_{\pmb {\ell }}$
is geometric in each direction
$\mathbf {e}_k$
for
$k \in \{1,2,\ldots ,d\}$
, namely,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu10.png?pub-status=live)
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,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu11.png?pub-status=live)
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,
$\mathbf {u}.\mathbf {P}=u_1P_1+\cdots +u_dP_d=0_E$
[Reference Stange5, 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
$\{ 1,2,\ldots ,d \}$
,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn11.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu12.png?pub-status=live)
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),
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn12.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu13.png?pub-status=live)
From the previous section, with
$q_r={G_{\mathbf {e}_r}}/{G_{\mathbf {0}}}$
, we deduce that
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu14.png?pub-status=live)
Finally,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn13.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu15.png?pub-status=live)
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 [Reference Stange5, 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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn14.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn15.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn16.png?pub-status=live)
From (2.5) and (2.6), we deduce that
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu16.png?pub-status=live)
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,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu17.png?pub-status=live)
and for
$G_{-\mathbf {j}+k\mathbf {e}_r}$
,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu18.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqn17.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu19.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu20.png?pub-status=live)
Therefore, from (2.3), we obtain, in the cases where
$G_{-\mathbf {1}}$
is well defined,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu21.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu22.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu23.png?pub-status=live)
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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu24.png?pub-status=live)
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 [Reference Dewaghe2, 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
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu25.png?pub-status=live)
gives
${\mathbf {u}}=(3,1,2)$
and
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu26.png?pub-status=live)
We give some calculations to illustrate Theorem 1.4 in Table 1.
Table 1 Calculations illustrating Theorem 1.4 in characteristic zero.
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_tab1.png?pub-status=live)
According to the Lutz–Nagell theorem [Reference Silverman3, 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.
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_tab2.png?pub-status=live)
2.5. The latest known general result
We now link our results to [Reference Akbary, Bleaney and Yazdani1, Theorem 1.13]. With the assumptions and the notation
$\chi $
and
$\xi $
of this theorem, one can write
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu27.png?pub-status=live)
More precisely, with
$\Lambda =\{ \mathbf {v} \in {\mathbb Z}^d \mid W(\mathbf {v})=0\}$
, the functions
$\chi $
and
$\xi $
are defined by
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu28.png?pub-status=live)
and the relations
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu29.png?pub-status=live)
We now relate the functions
$\delta $
of (1.8) and
$\chi , \xi $
of (1.9) to our notation. We have
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu30.png?pub-status=live)
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,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu31.png?pub-status=live)
Now, we recall the results of [Reference Akbary, Bleaney and Yazdani1, 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)
$\chi $ is bilinear symmetric: that is, for all
${\mathbf {u}},{\mathbf {u}}^{(1)},{\mathbf {u}}^{(2)} \in \Lambda $ and
$\mathbf {v},\mathbf {v}^{(1)}, \mathbf {v}^{(2)} \in {\mathbb Z}^d$ ,
-
(a)
$\chi ({\mathbf {u}},\mathbf {v}^{(1)}+\mathbf {v}^{(2)})=\chi ({\mathbf {u}},\mathbf {v}^{(1)})\chi ({\mathbf {u}},\mathbf {v}^{(2)})$ ,
-
(b)
$\chi ({\mathbf {u}}^{(1)}+{\mathbf {u}}^{(2)},\mathbf {v})=\chi ({\mathbf {u}}^{(1)},\mathbf {v})\chi ({\mathbf {u}}^{(2)},\mathbf {v})$ ,
-
(c)
$\chi ({\mathbf {u}}^{(1)},{\mathbf {u}}^{(2)})=\chi ({\mathbf {u}}^{(2)},{\mathbf {u}}^{(1)})$ ,
-
(d)
$\chi ({\mathbf {u}},-\mathbf {v})=\chi ({\mathbf {u}},\mathbf {v})^{-1}$ .
-
-
(2)
$\xi ({\mathbf {u}}^{(1)}+{\mathbf {u}}^{(2)})=\xi ({\mathbf {u}}^{(1)})\xi ({\mathbf {u}}^{(2)})\chi ({\mathbf {u}}^{(1)},{\mathbf {u}}^{(2)})$ .
-
(3)
$\xi (-{\mathbf {u}})=\xi ({\mathbf {u}})$ .
-
(4)
$\xi ({\mathbf {u}})^2=\chi ({\mathbf {u}},{\mathbf {u}})$ .
-
(5)
$\xi (n{\mathbf {u}})=\xi ({\mathbf {u}})^{n^2}$ , for all
$n \in {\mathbb Z}$ .
Proof.
-
(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) 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) 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) This follows from
$1=\xi (0)=\xi ({\mathbf {u}}-{\mathbf {u}})=\xi ({\mathbf {u}})\xi (-{\mathbf {u}})\chi ({\mathbf {u}},-{\mathbf {u}})$ .
-
(5) This result can be deduced from the previous statements.
Example 2.6. Following [Reference Stange, Lauter, Avanzi, Keliher and Sica6, 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,
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu32.png?pub-status=live)
and
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_eqnu33.png?pub-status=live)
Thus, we easily obtain
$({\cal A}^{({\mathbf {u}})}_r)^m=({\cal A}^{(\mathbf {s})}_r)^{-k}{\cal A}^{(\mathbf {t})}_r$
, which is [Reference Stange, Lauter, Avanzi, Keliher and Sica6, 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 $
.
![](https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20250213055819578-0421:S0004972724000583:S0004972724000583_tab3.png?pub-status=live)
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 [Reference Dewaghe2, 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 [Reference Dewaghe2, 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.
Acknowledgement
The author would like to thank the anonymous referee for their comments.