Albertas
|
Registered User
|
|
|
May 6 |
comment |
Elliptic curve with no points in a number field It is not clear to me, however, if one can obtain from the above a curve of the form $y^2 = x^3 + Ax + B$ for some integers $A, B$ that had no points over $K$. |
|
May 6 |
comment |
Elliptic curve with no points in a number field It is not very difficult to show that for a prime number $q$ that splits completely in $K$ and any integer $a$ that is not a square modulo $q$, the curve defined by $y^2 = qx^2 + a$ does not have points over $K$. One then can obtain a genus one curve defined by $y^2 = qx^4 + a$ that has no points over $K$. The projective curve $y^2z^2 = qx^4 + az^4$ then would have precisely one point over $K$, namely, $(x : y : z) = (0 : 1 : 0)$. |
