Let $A$ be a abelian curve over a number field $K$. The Weil Chatelet group parametrizes the twists of $A$, modulo the twists with a $K$ rational point. We can assume that $A$ is a plane curve. My question is if this group can be seen as the rational points of the parameter space of curves projectivelly equivalent to $A$ in the projective plane. I ask this question because if the Weil chatelet gruop consist of the rational points in a abelian variety, maybe the points in the Tate Shafarevich group have been the torsion point of this variety, so it be finite. I belive that I'm room, but I want to know if this idea make sense.
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2$\begingroup$ You have to allow curves of arbitrarily large degree to represent the whole of the Weil-Chatelet group, plus the equivalence relation is complicated, so no it's not the set of points of a variety. Plus the group is torsion so it can't be the group of points of an abelian variety. $\endgroup$– Felipe VolochCommented Sep 7, 2015 at 2:21
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$\begingroup$ Thanks, another question, were i can found a accesible article about the Weil Chatelet Group, I want to understand the Kolinvagin work, but it is very complex. $\endgroup$– camiloCommented Sep 8, 2015 at 0:37
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$\begingroup$ If you haven't yet read Silverman "The arithmetic of elliptic curves", you should start there. Chapter 10 discusses the WC group. $\endgroup$– Felipe VolochCommented Sep 8, 2015 at 1:10
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