MATHEMATICS

I. R. SHAFAREVICH

EXPONENTS OF ELLIPTIC CURVES

(Presented by Academician I. M. Vinogradov on 26 XII 1956)

To an algebraic curve (\gamma), defined over a field (k), one may assign, besides the genus (g), several further integral invariants: the least positive degree (f) of a divisor on (\gamma); the least degree (d) of a curve birationally equivalent to (\gamma) over (k); the least degree (\nu) of a prime divisor on (\gamma). The relations between these invariants have been little studied in the case of arbitrary curves, but for elliptic curves ((g=1)) they are all expressed in terms of one of them; namely, as follows easily from the Riemann–Roch theorem, for an elliptic curve (\nu=f,\ d=f) for (f>2); (d=4) for (f=2) and (d=3) for (f=1). We shall call the invariant (f) the exponent of the curve (\gamma).

For a curve of genus (g), (f \mid 2g-2), and therefore for (g\ne 1) the exponent can assume only a finite number of values. For elliptic curves, as already noted (see, for example, ((^{1,2}))), it is unknown what values the exponent may assume. In this note it will be proved that over the field of rational numbers (R) there exist elliptic curves (\gamma) with arbitrarily large exponent, and moreover the Jacobian curve (\omega) of the curve (\gamma) can even be prescribed arbitrarily.

For the proof we use the group (H(\omega)) formed by the classes of curves birationally equivalent over (k) with a given Jacobian curve (\omega). This group was first defined by A. Weil ((^3)). In what follows we shall use the construction of the group (H(\omega)) given in my paper ((^4)). As shown in that paper, the exponent of the curve (\gamma) is a multiple of the order of (\gamma) as an element of the group (H(\omega)). It is also shown there that the subgroup (H(K,\omega)) of the group (H(\omega)), consisting of all curves in (H(\omega)) that have a prime divisor of first degree in a given normal extension (K) of the field (R), is isomorphic to the group (H^1(G,\mathfrak A_K)), where (G) is the Galois group (K/R); (\mathfrak A_K) is the group of points on (\omega) with coordinates in (K), and (H^1(G,\mathfrak A_K)) is the group of crossed homomorphisms of (G) into (\mathfrak A_K).

If (\mathfrak p) is a prime divisor of (K); (p) is the prime number divisible by it; (G_{\mathfrak p}) is the decomposition group of (\mathfrak p), and (K_{\mathfrak p}) and (R_p) are the corresponding local fields, then there is a natural embedding homomorphism:

[
\varphi_{\mathfrak p}:\quad H^1(G;\mathfrak A_K)\to H^1(G_{\mathfrak p},\mathfrak A_{K_{\mathfrak p}}).
\tag{1}
]

We shall first consider the group (H^1(G_{\mathfrak p},\mathfrak A_{K_{\mathfrak p}})).

Let the equation of (\omega) have the form

[
y^2=x^3+ax+b,\qquad \Delta=4a^3+27b^2\ne 0,
]

where (a) and (b) may be taken to be rational integers. Denote by (H_p) the subgroup of those elements of the group (H) whose orders are relatively prime to (p).

In studying the group (H^1(G_{\mathfrak p},\mathfrak A_{K_{\mathfrak p}})p) one may assume that the field (K) has no higher ramification, since otherwise one can

one could pass to some subgroup of it. Denote by (\mathfrak A'{K_p}) the subgroup (\mathfrak A) are integral. It is easy to prove that, for (r>1), the group (\mathfrak A'}), considered by Lutz ({}^{(5)}), consisting of the points ((x,y)) on (\omega) for which (xp^{2(r-1)}) and (yp^{3(r-1){K_p}/\mathfrak A)}^{\prime\,r+1}) is a (p)-group. From this it is easy to deduce that the group (H^1(G_p,\mathfrak A_{K_pp) is isomorphic to the group (H^1(G_p,\mathfrak A')_p).}/\mathfrak A_{K_p}^{\prime\,r

Suppose, in addition, that (p\nmid \Delta). Then, as shown in ({}^{(5)}), the group (\mathfrak A_{K_p}/\mathfrak A^1_{K_p}) is isomorphic to (\mathfrak A_{\mathfrak K_p}), where (\mathfrak K_p) is the residue class field of (K_p) modulo (\mathfrak p), and (\mathfrak A_{\mathfrak K_p}) is the group of points on the curve (\omega), considered modulo (p). Denote by (F_p) the inertia group of (\mathfrak p) in (K_p). Then (G_p/F_p) is the Galois group of the field (\mathfrak K_p), and (G_p) is a group of operators for (\mathfrak A_{\mathfrak K_p}), with (F_p) acting trivially. As was said,

[
H^1(G_p,\mathfrak A_{K_p})p \simeq H^1(G_p,\mathfrak A)_p.
]

Consider the restriction homomorphism

[
H^1(G_p,\mathfrak A_{\mathfrak K_p})p \to H^1(F_p,\mathfrak A})_p. \tag{2
]

The kernel of this mapping consists of homomorphisms that become (0) on (F_p) and, consequently, are homomorphisms of (G_p/F_p) into (\mathfrak A_{\mathfrak K_p}). Since for curves over finite fields, as is known ({}^{(6)}), (f=1), we have

[
H^1(G_p/F_p,\mathfrak A_{\mathfrak K_p})=0,
]

and therefore the homomorphism (2) is a monomorphism. It is easy to see that the image of this homomorphism coincides with the group (\operatorname{Hom}{G_p/F_p}(F_p,\mathfrak A). We arrive at the following result:})) of operator (G_p/F_p)-homomorphisms of (F_p) into (\mathfrak A_{\mathfrak K_p

Theorem 1. If (p\nmid \Delta) and the field (K_p) has no higher ramification, then

[
H^1(G_p,\mathfrak A_{K_p})p \simeq \operatorname{Hom}).}(F_p,\mathfrak A_{\mathfrak K_p
]

Now we shall construct a field (K) in which there is a crossed homomorphism (f\in H(G,\mathfrak A_K)) of a prescribed order (m). To do this, consider all algebraic points (P) on (\omega) for which (mP=0). They form a group (\mathfrak A_m), which is the sum of two cyclic groups of order (m). Denote by (T_m) the field obtained by adjoining to (R) the coordinates of (P\in\mathfrak A_m). This field is normal. Denote its Galois group by (\overline G). Obviously, for (\sigma\in\overline G), (P\in\mathfrak A_m) and (P^\sigma\in\mathfrak A_m), and (\sigma) defines an automorphism of the group (\mathfrak A_m). Consider the group (G), containing a normal divisor (A), isomorphic to (\mathfrak A_m), and which is the semidirect product of (\overline G) and (A) with automorphisms

[
\alpha^\sigma=\varphi^{-1}\bigl(\varphi(\alpha)^\sigma\bigr),\qquad \alpha\in A, \tag{3}
]

where (\varphi) is some fixed isomorphism of (A) onto (\mathfrak A_m).

Construct a field (K), containing (T_m), normal over (R), and having over it Galois group (G). The existence of such a field follows from the results of Scholz ({}^{(7)}) and Delone and Faddeev ({}^{(8)}). Constructing the field (K) by any of these methods, one may arrange that any prime divisor (\mathfrak p) of the field (T_m) has ramification exponent (m) in (K). We shall choose (\mathfrak p) relatively prime to the discriminant of (T_m/R), to (m), to (\Delta), and to (2).

Define a mapping (f) of the group (G) into (\mathfrak A_K) as follows:

[
f(\sigma\alpha)=\varphi(\alpha),\qquad \sigma\in\overline G,\quad \alpha\in A.
]

From (3) it is easy to derive that (f) is a crossed homomorphism, i.e. (f \in H'(G,\mathfrak A_K)). Obviously, (mf=0). We shall show that (f) has in (H'(G,\mathfrak A_K)) order exactly equal to (m). For this it suffices to prove that its image (\varphi_{\mathfrak p} f) under the homomorphism (\varphi_{\mathfrak p}), defined in (1), has order (m). By Theorem 1 this will be proved if we prove that (\varphi_{\mathfrak p} f), as an operator homomorphism (F_{\mathfrak p}) into (\mathfrak A_{\overline K_{\mathfrak p}}), has order (m). The latter assertion follows from the fact that, by the choice of (\mathfrak p), the group (F_{\mathfrak p}) is cyclic of order (m), and (f) maps its generator to an element (P) of (\mathfrak A_m) which has order (m); and since (\mathfrak p) is not critical in (T_m), the image of (P) in (\mathfrak A_{\overline K_{\mathfrak p}}) has the same order.

We have arrived at the final result:

Theorem 2. In the group of elliptic curves having over the field of rational numbers a prescribed Jacobian curve, there exist elements of any order.

Corollary. Among the elliptic curves having over the field of rational numbers a prescribed Jacobian curve, there exist curves whose exponent is divisible by any preassigned number.

All the proofs, with considerable simplifications, also carry over to the field of rational functions over the field of complex numbers. The fact itself was noted in this case without detailed proofs by Enriques(^9).

Let us note that for the field of (p)-adic numbers the analogous theorem is not true. Indeed, Theorem 1 shows that if the discriminant (\omega) is not divisible by (p), then the exponents of curves having (\omega) as their Jacobian curve can be divisible only by (p) and by the prime divisors of the number of points on (\omega) in the residue field modulo (p).

Received
20 XII 1956

CITED LITERATURE

(^1) H. Hasse, Zahlentheorie, Berlin, 1949.
(^2) A. Weil, Arch. d. Math., 5, No. 1—3, 197 (1954).
(^3) A. Weil, Am. J. Math., 77, No. 3, 493 (1955).
(^4) I. R. Shafarevich, DAN, 114, No. 2 (1957).
(^5) E. Lutz, J. f. reine u. angew. Math., 177, No. 4, 238 (1937).
(^6) F. K. Schmidt, Math. Zs., 33, No. 1, 1 (1931).
(^7) A. Scholz, Math. Zs., 30, 332 (1929).
(^8) B. N. Delaunay, D. K. Faddeev, Matem. sborn., 15 (57), 243 (1944).
(^9) F. Enriques, Math. Ann., 51, 134 (1899).