On the Chow-forms of elliptic normal curves of degree 4

Abstract

In this note, we shall aim, using explicit theta constants,to describe the Chow-form of an elliptic normal curve of degree 4 embedded in $P^{3}$ by using theta functions, at any point of the upper half plane. In this case, the embedded elliptic curve is a complete intersection which is defined by two quadratic forms. In order to calculate its Chow-form, we use the elimination theory. Our main result is Th. 1' (§2). In our case, Chow-forms are divisors of the Grassmann variety of lines in $P^{3}$. From the theorem, we see that the Chow-forms of elliptic normal curves of degree 4 lie on a linear subspace $M$ (cf. 2.3) of dimension 23. §2 is the main part of this note.In §3, using the theory of elliptic modular forms of level 4, we consider the geometric meaning of Th. 1'. Then our theorem shows that the Chow point is given by modular forms of weight 2 (of weight 1 in the traditional sense) and of level 4. The compactification of the moduli space of level 4 is isomorphic to a plane conic. Thus, we see that the Chow points of the projective elliptic normal curves of degree 4 determined by the points of the upper half plane form a rational curve of degree 4 in $M$ which is essentially the image of the 2-uple embedding of the above plane conic (Th. 2, §3. cf. Comments after the proof of Th. 2). Furthermore, we see that the correspondence which assigns the Chow point to a point of the upper half plane can be extended to the inequivalent cusps of the principal congruence subgroup of level 4 (Cor. 1, §3). Finally, we give a remark about the structure of the corresponding component of the Chow variety parametrizing l-cycles of degree 4 in $P^{3}$ (Cor. 2, §3).