This is how an Elliptic Curve is defined by my professor, which also appears in Silverman and Tate's book Rational Points on Elliptic Curves. The set of solutions to the equation (over a field) $y^2 = x^3 + ax^2 + bx +c$ is an Elliptic curve.
Are there any curves of this form which don't have ANY rational points? Because if so, then the set of all rational points will be empty and since the set of rational points forms a group, and an empty set is not a group, there is a contradiction here.
Moreover, the way in which group law is defined in this book, it assumes that the curve has a rational point because it defines that very point to be the identity element.
So am I missing something in the definition because at some places online I see that it is required for the curve to have a $k-$ rational point? And if the definition that I am using is not wrong then how is the contradiction above resolved?
Best Answer
You are right that an elliptic curve must always have a rational point, since at the very least you'll want to have a point that acts as the identity element.
Any elliptic curve can be put into Weiestrass form; with respect to this form, you always have the point at infinity, which is taken to be the identity element. Can you see how that is indeed a rational point?
In general, there are genus-1 curves without rational points, and a fortiori these cannot be put into Weierstrass form.