UDC 511

MATHEMATICS

V. A. DEM’YANENKO

RATIONAL POINTS OF A CLASS OF ALGEBRAIC CURVES

(Presented by Academician I. M. Vinogradov on 4 III 1966)

The results set forth in the present note concern the problem of finding all rational points of a class of curves of higher genus.

Let \(K\) be a field of algebraic numbers of degree \(n\) over the field of rational numbers. Consider the curve

\[ \Gamma:\ F(x,y)=0 \]

of genus \(>1\), admitting over the field \(K\) \(r\) linearly independent mappings \(\varphi_i\) \((i=1,2,\ldots,r)\) onto one and the same elliptic curve

\[ \Gamma_1:\ \Phi(u,v)=0. \]

The problem is to find all points of the curve \(\Gamma\) rational over the given field \(K\), under certain restrictions imposed on the curves \(\Gamma\) and \(\Gamma_1\).

It is known that the mappings of the curve \(\Gamma\) onto the curve \(\Gamma_1\) and the points of the curve \(\Gamma_1\) over \(K\) form groups \(G\) and \(U\) with a finite number of generators. To each point \(a\) of the curve \(\Gamma\) there correspond \(r\) points \(\varphi_i(a)\) \((i=1,2,\ldots,r)\) of the curve \(\Gamma_1\).

Two methods are indicated for solving the problem posed.

The first method consists in investigating the points \(\varphi_i(a)\) \((i=1,2,\ldots,r)\) for linear dependence; in doing so the Tate height of points and the properties of height expounded in the book \((^1)\) are used in an essential way. In this direction the following results have been obtained.

Let \(g(\varphi)\) be the degree of the mapping \(\varphi\); let \(e\) be a function taking only the values \(\pm 1\); and let \(H_a\) be the Tate height of the point \(a\). Then the following theorems hold.

Theorem 1. If

\[ g\left(\sum_{i=1}^{r} e_i\varphi_i\right)> g\left(\sum_{\substack{i=1\\ i\ne k}}^{r} e_i\varphi_i\right) \quad (k=1,2,\ldots,r) \]

for all values \(e_i\), then the points \(\varphi_i(a)\) \((i=1,2,\ldots,r)\), \(a\in\Gamma\), starting from some \(C\) when \(H_a>C\), are linearly independent.

Note that in Theorem 1 the mappings \(\varphi_i\) \((i=1,2,\ldots,r)\) are not assumed to be linearly independent; this already follows from the theorem itself.

Theorem 2. If the mappings \(\varphi_i\) \((i=1,2,\ldots,r)\) are linearly independent, then the points of the curve \(\Gamma_1\), \(\varphi_i(a)\) \((i=1,2,\ldots,r)\), \(a\in\Gamma\), starting from some \(C\) when \(H_a>C\), are also linearly independent.

Theorem 3 (main). If the curve \(\Gamma\) admits over the field \(K\) \(r\) linearly independent mappings onto one and the same elliptic curve \(\Gamma_1\) of rank \(<r\), then the curve \(\Gamma\) has a finite number of points rational over this field.

Note that if the mappings of the curve \(\Gamma\) onto the curve \(\Gamma_1\) are known, then, when the conditions of Theorem 3 are satisfied, the constant \(C\) can be effectively found, and consequently all points of the curve \(\Gamma\) rational over the given field can be found.

Particular cases of the proved theorem are the following examples, considered at the end of the paper \((^2)\):

Example 1. The curve \(x^4+y^4=A\) has a finite number of points rational over the field \(K\), \(A\in K\), if the curve \(u^3+v^2=A\) has rank \(\leq 1\) over \(K\).

Example 2. The curve \(x^4+y^4=A\) has a finite number of points rational over the field \(K\), \(A\in K\), if the curve \(u^4+v^2=A\) has rank \(\leq 1\) over \(K\).

With respect to the curves considered in Examples 1 and 2, we prove the following additional theorems.

Theorem 4. The curve \(x^6+y^6=A\) has a finite number of points rational over the field \(R(1)\), if the rank of one of the curves \(\Gamma_1: u^3+v^2=A\), \(\Gamma_2: Au^3-1=v^2\), \(\Gamma_3: u^3+1=Av^2\) over \(R(1)\) does not exceed 1, or if the ranks of any two of these curves do not exceed 2.

Theorem 5. If the rank of the curve \(u^4+v^2=A\) over the field \(R(1)\) does not exceed 2, and the rank of the curve \(u^4+1=Av^2-1\), then the curve \(x^4+y^4=A\) has a finite number of rational points.

In the proof of these theorems one uses the fact that the curves \(x^4+y^4=A\), \(x^6+y^6=A\) admit, respectively over the fields \(R(\sqrt[4]{-A})\), \(R(\sqrt[6]{-A})\), a greater number of mappings onto one and the same elliptic curve than over the field \(R(1)\).

In the second method, one studies the distribution of the points \(\varphi_i(a)\) \((i=1,2,\ldots,r)\) among the elements of the factor group \(U/2U\), without using the Tate height. In this method, curves of genus \(>1\) for which the order of the automorphism group is equal to 2 are considered in greater detail. The following theorems are proved.

Theorem 6. If the rank of one of the curves \(\Gamma_1,\Gamma_2,\Gamma_3\) over the field \(R(\varepsilon)\), \(\varepsilon^2+\varepsilon+1=0\), does not exceed 2, then the curve \(x^6+y^6=A\) has no points in this field, except for the cases: \(A=1\), \((x,y)=(\varepsilon_1,0),(0,\varepsilon_2)\); \(A=2\), \((x,y)=(\varepsilon_1,\varepsilon_2)\), \(\varepsilon_1^6=\varepsilon_2^6=1\).

Theorem 7. If the rank of the curve \(u^4+v^2=A\) over the field \(R(\sqrt{-1})\) does not exceed 2, then the curve \(x^4+y^4=A\) has no points in this field, except for the cases: \(A=1\), \((x,y)=(\varepsilon_1,0),(0,\varepsilon_2)\); \(A=2\), \((x,y)=(\varepsilon_1,\varepsilon_2)\); \(\varepsilon_1^4=\varepsilon_2^4=1\).

Theorem 8. If the rank of the curve \((A^2-4)u^4+2A(A+2)u^2+A^2-4=v^2\) over the field \(R(1)\) does not exceed 2, then the curve \(x^4+y^4+1=A(x^2y^2+x^2+y^2)\) has no rational points, except for the cases: \(A=(2a^4+1)/(a^4+2a^2)\), \((x,y)=(\pm a,\pm a)\); \(A=(a^4+2)/(2a^2+1)\), \((x,y)=(\pm a,\pm 1),(\pm 1,\pm a)\).

In the proof of Theorems 6, 7, 8 it is established: first, that none of the points \(\varphi_i(a)\) \((i=1,2,\ldots,r)\) of the corresponding elliptic curve can fall into the class \(2U\); second, that no two points among \(\varphi_i(a)\) can fall into one and the same class of the factor group \(U/2U\).

Under the condition that \(K=R(1)\), we put forward the following conjecture for the curves of Theorems 6, 7, 8: if the points \(\varphi_i(a),\varphi_i(b)\) \((i=1,2,\ldots,r)\), \(r\geq 2\), are distributed entirely in two classes of the factor group \(U/2U\), then \(a=b\). From the validity of this hypothesis the following results would follow:

1. The curves \(x^4+y^4=A\), \(x^4+y^4+1=A(x^2y^2+x^2+y^2)\) have in the field \(R(1)\), respectively, no more than \(2^{r_1-1}(2^{r_1}-1)\), \(2^{r_2-1}(2^{r_2}-1)\) points, where \(r_1,r_2\) are the ranks of the curves \(u^4+v^2=A\), \((A^2-4)u^4+2A(A+2)u^2+A^2-4=v^2\) over this field.

2. The curve \(x^6+y^6=A\) has in the field \(R(1)\) no more than \(2^{s-1}(2^s-1)\) points, where \(s=\min(r_1,r_2,r_3)\), and \(r_1,r_2,r_3\) are the ranks of the curves \(u^3+v^2=A\), \(Au^3-1=v^2\), \(u^3+1=Av^2\) over this field.

Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
24 II 1966

REFERENCES

1. S. Lang, Diophantine Geometry, 1962.
2. Yu. I. Manin, Izv. AN SSSR, ser. matem., No. 6, 1363 (1964).