Triangles and elliptic curves
The author obtains an infinite family of elliptic curves defined over an algebraic number field \(k\), such that every curve of it has positive Mordell-Weil rank with respect to \(k\): At first, take \(P = (a^ 2 + b^ 2 - c^ 2)/2\), \(Q = (a + b + c) (a + b - c) (a - b + c) (a - b - c)/16\), \(a,b,c \in k\), \(k\) a field of characteristic \(\neq 2\), and consider the elliptic curve \(E\): \(y^ 2 = x^ 3 + Px^ 2 + Qx\), then \(\pi = (x_ 0, y_ 0)\) with \(x_ 0 = (c/2)^ 2\), \(y_ 0 = c(a^ 2 - b^ 2)/8\) belongs to \(E\). Second, if \(k\) is an algebraic number field, \(a,b,c \in {\mathfrak o}\), the ring of integers of \(k\), and \[ a + b \equiv c \pmod 2,\quad c \not \equiv 0 \text{mod} {\mathfrak p} \quad \text{ for some } \quad {\mathfrak p} | 2, \tag{\(\circledast\)} \] by a generalized Nagell- Lutz theorem for number fields, the point \(\pi\) is no torsion point. When \(a^ 2 + b^ 2 = c^ 2\), we have \(P = 0\), and we are concerned with the congruent number problem \(y^ 2 = x^ 3 - A^ 2x\), \(A = (a \cdot b)/2\), if \(a,b,c \in \mathbb{Z}\). Because \(A\) is now the area of the right triangle with integer sides \(a,b,c, \circledast\) is fulfilled if \((a,b) = 1\). So we can speak of replacing right triangles in the congruent number problem by arbitrary triangles.
scientific article; zbMATH DE number 701461