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Supersingular elliptic curves and congruences for Legendre polynomials. (English) Zbl 0655.14018
Elliptic curves and modular forms in algebraic topology, Proc. Conf., Princeton/NJ 1986, Lect. Notes Math. 1326, 69-93 (1988).
[For the entire collection see Zbl 0642.00007.]
This paper is motivated by the recently discovered theory of elliptic genera and elliptic cohomology. This has led to discussions on formal group laws associated to supersingular elliptic curves (e.g. Jacobi quartics), modular forms, and congruences for Legendre polynomials.
Let E be an elliptic curve given by a Jacobi quartic \((*)\quad Y^ 2=1- 2\delta X^ 2+\epsilon X^ 4=:R(X)\) with uniformizer X near the origin \((X,Y)=(0,1)\). E has the invariant differential \[ \frac{dX}{Y}=(\sum_{n\geq 0}P_ n(\delta,\epsilon)X^{2n})dX, \] where \(P_ n(\delta,\epsilon)\) are homogeneous Legendre polynomials. The formal group law, \(F_ E\), of E is defined over \({\mathbb{Z}}[1/2][\delta,\epsilon]\) and for any prime \(p>2\), a multiplication-by-p map on \(F_ E\) is given by a power series \([p](X)=pX+...+u_ 1X\) \(p+...+u_ 2X^{p^ 2}+..\). with \(u_ 1\equiv P_{(p-1)/2}(\delta,\epsilon)\) mod p, \(u_ 2\equiv (1/p)\{P_{(p^ 2- 1)/2}(\delta,\epsilon)-P_{(p-1)/2}(\delta,\epsilon)^{1+p}\}\) mod \((p,u_ 1).\) Put \[ v_ 1(X)=P_{(p-1)/2}(X),\quad v_ 2(X)=p^{- 1}\{P_{(p^ 2-1)/2}(X)-P_{(p-1)/2}(X)^{1+p}\}. \] The main results are the following congruences of Legendre polynomials:
Theorem 1. For any odd prime p, \(v_ 1(X)\) and \(v_ 2(X)\) satisfy the congruences:
(1) \(v_ 2(X)\equiv -p^{-1}=(-1)^{(p-1)/2}\) mod \((p,v_ 1(X)).\)
(2) \((X^ 2-1)^{(p^ 2-1)/4}\equiv 1\) mod \((p,v_ 1(X)).\)
(3) \((X^ 2-1)^{(p^ 2-1)/12}\equiv 1\) mod \((p,v_ 1(X))\) if \(p>3.\)
Corollary. For any odd prime p the following congruences hold in \({\mathbb{Z}}[\delta,\epsilon]\) mod \((p,P_{(p-1)/2}(\delta,\epsilon)):\)
\(u_ 2\equiv (-1)^{(p-1)/2}\epsilon^{(p^ 2-1)/4}\) mod \((p,u_ 1),\) and (\(\delta^ 2-\epsilon)^{(p^ 2-1)/4}\equiv \epsilon^{(p^ 2-1)/4}.\)
These results are proved transforming the assertions on supersingular elliptic curves of Weierstrass forms (theorems 2 and 3 below) to those on supersingular Jacobi quartics, by means of their formal group laws (theorem 4 below).
Now let E be an elliptic curve given by a Weierstrass equation \((**) y^ 2+a_ 1xy+a_ 3y=x^ 3+a_ 2x^ 2+a_ 4x+a_ 6\) over a field K of characteristic \(p>0.\) E is called supersingular if \([p](x)\) mod \((p,\deg(p+1))=u_ 1x\) p is zero in K. In this case, there is a relation between \(u_ 2\) and the discriminant, \(\Delta\), of E. Theorem 2. For a supersingular elliptic curve E of Weierstrass form (**) over a field of characteristic \(p>0\), the following relations hold:
(1) \(u^ 4_ 2=\Delta\) if \(p=2\).
(2) \(u^ 3_ 2=-\Delta^ 2\) if \(p=3\).
(3) \(u_ 2=-p^{-1}\Delta^{(p^ 2-1)/12}= (-1)^{(p- 1)/2}\Delta^{(p^ 2-1)/12}\) if \(p>3.\)
Theorem 3. For a supersingular elliptic curve E given by the classical Weierstrass form \((***)\quad y^ 2=4x^ 3-g_ 2x-g_ 3=4(x-e_ 1)(x- e_ 2)(x-e_ 3)\) over a field of characteristic \(p>3,\) the following relation holds: \(\Delta^{(p^ 2-1)/24}=(e_{\alpha}- e_{\beta})^{(p^ 2-1)/4}\) if \(\alpha\neq \beta\) and also in particular \(\Delta^{(p^ 2-1)/12}=(e_{\alpha}-e_{\beta})^{(p^ 2-1)/2}\) if \(\alpha\neq \beta.\)
Theorem 4. Given elements \(\delta\) and \(\epsilon\) of a field K of characteristic not 2 or 3 satisfying \(\Delta =\epsilon (\delta^ 2- \epsilon)^ 2\neq 0\), define \(g_ 2=(\delta^ 2+3\epsilon)/3\); \(g_ 3=\delta (\delta^ 2-9\epsilon)/27.\) Then \(X=f(z)=z-(1/3)\delta w(z)\) \((x=-2x/y\), \(w=-2/y=w(z))\) gives a strict isomorphism from the formal group law \(F_ w(z_ 1,z_ 2)\) of the elliptic curve (***) to the formal group law \(F_ E(X_ 1,X_ 2)\) of the Jacobi quartic (*). Moreover, the roots \(e_ 1, e_ 2, e_ 3\) of \(4x^ 3-q_ 2x-q_ 3=0\) may be taken to satisfy the relation \(\delta =3e_ 3\), \(\epsilon =(e_ 1-e_ 2)^ 2,\) \((\delta^ 2-\epsilon)/4=(e_ 1-e_ 3)(e_ 2- e_ 3)\).
Reviewer: N.Yui

MSC:
14L05 Formal groups, \(p\)-divisible groups
33C55 Spherical harmonics
14H45 Special algebraic curves and curves of low genus
14H52 Elliptic curves