Let $L/K$ be finite Galois ext. of number fields and $E/K$ an elliptic curve. Define trace $$Tr : E(L) \rightarrow E(K), \;\; P \mapsto \sum_{\sigma \in G_{L/K}} P^{\sigma}$$ When is this map surjective? (Assume rk $E(K) \geq 1$).
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3$\begingroup$ If $G$ is the Galois group then the failure to to be surjective is measured in $\hat H^0(G, E(L))$. So this will depend on the Galois module structure of $E(L)$. A lot of things can happen. If the rank doesn't grow, it can happen (only rarely) that the map is surjective, but usually it isn't. Instead if the rank grows it is much harder to know. Even the local situation for a cyclic extension is complicated. $\endgroup$– Chris WuthrichCommented May 13, 2021 at 7:59
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1$\begingroup$ $E(L)$ is a $G$-module of finite type and one may ask which $G$-modules arise in this way. Already for $G=\{1\}$ and $K=\mathbb{Q}$, this is highly non-trivial as it asks what are the possible Mordell-Weil groups of elliptic curves. $\endgroup$– François BrunaultCommented May 13, 2021 at 8:31
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1$\begingroup$ For a fixed $E/K$, if you know generators of $E(L)$ then you will be able to determine the $G$-module structure, and compute the image of the trace map by linear algebra. Also, you can consider the $L$-function $L(E/L,s)$ (base change of $E$ to $L$). It factors as a product of copies of $L(E \otimes \rho,s)$ where $\rho$ are the Artin representations of $G$. There is a generalisation of BSD conjecture for these $L$-functions (equivariant Tamagawa number conjecture for $L(E/L,s)$). $\endgroup$– François BrunaultCommented May 13, 2021 at 8:52
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1$\begingroup$ In theory the orders of vanishing of $L(E \otimes \rho,s)$ at $s=1$ should give you (only conjecturally) the $G$-module structure of $E(L) \otimes \mathbb{Q}$, and possibly the values (or derivatives) at $s=1$ will give $E(L)$ as $G$-module but I haven't checked. $\endgroup$– François BrunaultCommented May 13, 2021 at 8:56
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1$\begingroup$ @François Brunault I fear it is not that easy. You are right that the order of vanishing should give the rank, but that does not help with the finer question of what the Galois module is and the surjectivity of the trace will depend on that. If the extension is cyclic of prime degree $p$ and the rank grows from $1$ to $p$ and no torsion is present, there are two possible modules. The etnc will link the value of $L'(E,\chi,1)$ to the arithmetic, but only if all the other involved terms vanish it could distinguish the two cases. $\endgroup$– Chris WuthrichCommented May 13, 2021 at 9:34
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