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Relations between certain power sums of elliptic modular forms in characteristic two. (English) Zbl 0969.11025

Let \(E_a:y^2+xy=x^3+a\) be an ordinary (see Deuring) elliptic curve over an algebraically closed field \(K\) of characteristic \(p=2\) so that the discriminant is \(\Delta_a=a\neq 0\), \(a\in K\), and the modular invariant is \(j_a= {1\over a}\). Let \(G\subseteq E_a\) be a finite subgroup of \(E_a=E_a(K)\) of odd cardinality \(|G|= 2d+1\) and \({\mathcal O}=(\infty, \infty)=(0:1:0)\), \(\pm Q_1, \dots,\pm Q_d\) be a set of elements of \(G\). For \(r\in\mathbb{N}\), consider the finite sum of \(x\)-coordinates \({\mathcal E}_r(G) =\sum^d_{i=1} x(Q_i)^r\). Since for even \(r\), \({\mathcal E}_r(G)= {\mathcal E}_{r\over 2}(G)^2\), it suffices to consider \({\mathcal E}_r(G)\) for odd \(r\in\mathbb{N}\). This sum is the true analogue in characteristic 2 of the modular form of level \(N\) and weight \(2r\) \({\mathcal E}_r (G)= {1\over 2}\sum_{0\neq Q\in G}\wp (Q)^r\) in characteristic 0, where \(E(\mathbb{C}) =\mathbb{C}/(\mathbb{Z} +\tau\mathbb{Z})\) with \(\tau\) in the complex upper half plane, \(G \subseteq E(\mathbb{C})\) is the finite subgroup generated by \({1\over N}\), \(N\in \mathbb{N}\), and \(\wp(Q)= \wp(u_Q)\) is the Weierstraß \(\wp\)-function parametrizing the elliptic curve \(E\), so that in this case, \({\mathcal E}_r(G)= {1 \over 2}\sum^{N-1}_{k=1} \wp({k\over N})^r\). The quotient \(E_a/G \approx E_b\) is again an ordinary elliptic curve for some \(0\neq b=b(G)\in K\). If one puts (see section 2) \(x_G=\sum_{Q\in G}x\circ \tau_Q\) where \(x,y\) are the \(x\)- and \(y\)-coordinates on \(E_a\), \(\tau_Q\) designating the translation by \(Q\), and if one defines \(y_a\) in a similar way, then the map \[ \begin{aligned} E_a &\to E_b\\ (x,y) & \mapsto (x_G,y_G), \end{aligned} \] where \(b\in K^\times\), given by the relation \(y^2_G+x_Gy_G =x^3_G+b\), is a separable isogeny with kernel \(G\) (Lemma 2.1). This is shown in the paper by using ideas of Vélu.
The following is proved by the author as main results (see Section 1): \(b(G)=a+ {\mathcal E}_1(G)^2+ {\mathcal E}_1 (G)\) and, for every odd \(r\geq 3\), \(r\in\mathbb{N}\), that there is a polynomial \(P_r \in \mathbb{F}_2 [X_0,X_1, \dots,X_r]\) such that \(P_r(a,{\mathcal E}_1(G),\dots,{\mathcal E}_r (G))=0\) (Theorem 1). If, furthermore, \({\mathcal E}_r=\langle P_3,P_5, \dots, P_r \rangle\subseteq \mathbb{A}_{r+1}\) denotes the affine variety determined by the polynomials \(P_i\) \((i=3,5,\dots,r)\) in the affine space \(\mathbb{A}_{r+1}\) of dimension \(r+1\) over \(K\), and, for \(a,b\in K\), \({\mathcal E}_r(a,b)\) is the set of points in \({\mathcal E}_r\) satisfying the relations \(X_0=a\), \(X^2_1+X_1=a+b\), (whose cardinality is \(|{\mathcal E}_r(a,b) |=2^m\), where \(2^m<r+1\leq 2^{m+1})\) then, under the condition that \((x,y)\in G\Rightarrow (x^\sigma,y^\sigma)\in G\) for every automorphism \(\sigma\) of \(K\) over a subfield \(F\) of \(K\) containing \(a \neq 0\), the points of \({\mathcal E}_r(a,b)\), \(b=b(G)\), are rational over \(F\) (Theorem 2). Finally, there is an automorphism \(\gamma\) of \({\mathcal E}_r\) having a single orbit on \({\mathcal E}_r(a,b)\) for any \(a,b\in K\) (Theorem 3).
The results and corresponding proofs are based on the discovery that the power sums \({\mathcal E}_r (G)\) are closely related to the relations satisfied by the coefficients of homomorphisms of the associated formal group laws (see Section 3 and the above review). J. M. Couveignes [Quelques calculs en théorie des nombres (Bordeaux, 1995)] and N. D. Elkies [Elliptic and modular curves over finite fields and related computational issues, based on a talk given at a conference, Atkin (1996), AMS/IP Stud. Adv. Math. 7, 21-76 (1998; Zbl 0915.11036)] (see Lemma 2.2) have used formal groups to explicitly computing isogenies of elliptic curves over finite fields. (But the reviewer did not have the works [C] or [E] at hand.) In Section 4, the two theorems of Couveigne together with proofs are repeated (see the preceding review Zbl 0969.11024 and the above review). Couveigne’s theorems are useful for the proof of Theorem 3 in Section 6. The proof of Theorem 1 is given in Section 5. Theorem 2 is a consequence of Theorems 1 and 3.
The main value of the present paper is the detection of the close connection between modular forms in characteristic 2 and formal groups associated to elliptic curves over (finite) fields of characteristic 2.

MSC:

11G20 Curves over finite and local fields
11F11 Holomorphic modular forms of integral weight
14L05 Formal groups, \(p\)-divisible groups
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References:

[1] A. W. Bluher, Formal groups, elliptic curves, and some theorems of Couveignes, 1998; A. W. Bluher, Formal groups, elliptic curves, and some theorems of Couveignes, 1998 · Zbl 0969.11024
[2] J. M. Couveignes, Quelques calculs en theorie des nombres, Bordeaux, 1995; J. M. Couveignes, Quelques calculs en theorie des nombres, Bordeaux, 1995
[3] N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, based on a talk given at a conference, Atkin, 1996; N. D. Elkies, Elliptic and modular curves over finite fields and related computational issues, based on a talk given at a conference, Atkin, 1996
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