I want to learn some algebraic geometry, also in concrete examples. That's were my question comes from.
Let $X$ be a non-singular plane cubic projective curve over $\mathbb Q$. I wonder about the following:
(Q1) Are there methods to determine, for special families of such $X$, whether there is a rational point on $X$?
(Q2) Are there families of equations for which it is known that the corresponding $X$ has a rational point?
(Q3) Are there references about the existence of rational points on such $X$?
In the books I know, e.g., Silverman/Tate or Husemöller, the authors say something like: "We will ignore this difficult problem and assume that $X$ always has a rational point."
Thanks in advance!