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Let $L/K$ be an extension of number fields and $A/K$ an abelian variety (or an elliptic curve, or a modular abelian variety). Then what can we say about the structure of $A(L)/A(K)$ (or of $A(L)_{\text{tor}}/A(K)_{\text{tor}}$, or of $A(L)_{\text{non tor}}/A(K)_{\text{non tor}}$)? Are there any researches?

For example, if $L$ is of exponent $2$ and if $A$ is the jacobian variety of a hyperelliptic curve, then we know that the prime to $2$ part of $A(L)$ is isomorphic to the one of $\oplus_\chi A^\chi(K)$. (where the product takes all twist of $A/K$ over $L$.)

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    $\begingroup$ I doubt one can say much in this generality beyond simple representation theory. (Of course your statement for exponent $2$ generalises to split $A(L)\otimes \mathbb{Z}[1/\vert G\vert]$ when $L/K$ is Galois with group $G$). See $A(L)\otimes \mathbb{C}$ could be any representation of the Galois group $G$, probably, maybe, well this question mathoverflow.net/questions/280585/… hasn't been answered yet. $\endgroup$
    – Chris Wuthrich
    Commented Jul 16, 2021 at 23:03

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