Let $K$ be a number field and let $E/K$ be an elliptic curve. Let $L/K$ be a finite extension. Consider the trace map $$ \operatorname{Tr}_{L/K}:E(L)\longrightarrow E(K),\qquad \operatorname{Tr}_{L/K}(P)=\sum_{\sigma\in G_{L/K}}P^\sigma. $$ Why does not the following way of finding $K$-rational points work? Choose any $y\in K$. Calculate $x$ which will lie in some finite extension $L$. Then calculate $\operatorname{Tr}_{L/K}((x,y))\in E(K)$. Is there a more sophisticated variant?
1
-
2$\begingroup$ Well, it works in the sense that if $L=K$ you have found a rational point. If $[L:K]=3$ then the three points $P^\sigma$ are colinear (same $y$ coordinate) and so sum to $\mathcal O$. $\endgroup$– Matthew BolanCommented Jan 18 at 18:36
Add a comment
|