Assume that BSD holds for number fields. Let $E/\mathbf{Q}$ be an elliptic curve. For simplicity, let's assume it has Mordell-Weil rank zero. Let $F\_1/\mathbf{Q}$ and $F\_2/\mathbf{Q}$ be finite, abelian, disjoint extensions, and let $F=F\_1.F\_2$ denote their compositum.



> 
> **Question:** Under what conditions can we conclude that
> $$ \mathrm{rank}\_{\mathbf{Z}}(E/F)=\mathrm{rank}\_{\mathbf{Z}}(E/F\_1) + \mathrm{rank}\_{\mathbf{Z}}(E/F\_2) ?$$
> 
> 
> In other words, are there any sufficient conditions to ensure that any Mordell-Weil rank growth in $F$ actually in arises either $F\_1$ or $F\_2$?
> 
> 
> 


On the one hand, this seems too good to be true. On the other hand, *if we assume BSD for number fields*, it seems that we should be able to make an argument using the factorization of $L$-functions as follows:


Let $\mathrm{Gal}(F\_1/\mathbf{Q})=G\_1$ and $\mathrm{Gal}(F\_2/\mathbf{Q})=G\_2$. Since $F\_1 \cap F\_2 = \mathbf{Q}$, we have $\mathrm{Gal}(F/\mathbf{Q})=G\_1 \times G\_2$. Using $^\hat{}$ to denote character groups, it seems we should have
\begin{align\*}
L(E, F, s) &= \prod\_{\chi\in \hat{G}} L(E, \mathbf{Q}, s, \chi)\\ &\simeq \prod\_{\chi\_1\in \hat{G\_1}}\prod\_{\chi\_2\in \hat{G\_2}} L(E, \mathbf{Q}, s, \chi\_1 \chi\_2)\\ &=L(E, F\_1, s)\times L(E, F\_2, s) \times\prod\_{\substack{\chi\_1\in \hat{G\_1} \\ \chi\_1\neq 1}}\prod\_{\substack{\chi\_2\in \hat{G\_2} \\ \chi\_2\neq 1}} L(E, \mathbf{Q}, s, \chi\_1 \chi\_2)
\end{align\*}
where the products are taken over the irreducible characters. Thus, we have the desired additivity if we can control the terms of the form $L(E, \mathbf{Q}, s, \chi\_1 \chi\_2)$.


Are there any hypotheses which push this across the finish line?




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Edit: Ariel Weiss has posted an answer which highlights the following situation:
$$E=X\_0(11), F\_1=\mathbf{Q}(\sqrt{-7}), F\_1=\mathbf{Q}(\sqrt{-8}), F=\mathbf{Q}(\sqrt{-7},\sqrt{-8})$$ Then we have


$$\mathrm{rank}\_{\mathbf{Z}}(E/\mathbf{Q})=0, \mathrm{rank}\_{\mathbf{Z}}(E/F\_1)=1, \mathrm{rank}\_{\mathbf{Z}}(E/F\_2)=1, \mathrm{rank}\_{\mathbf{Z}}(E/F)=2$$ This is precisely the sort of situation I'm looking to capture in general.



Additivity of Elliptic Curve Rank over Compositum of Fields
If you're willing to expand your question a bit, there are interesting relations to be found. More precisely, let $E/K$ be an elliptic curve defined over a number field, and let $L/K$ be a Galois extension such that $G(L/K)$ has a non-trivial idempotent relation (also sometimes called a Brauer relation). For example, this will be true if if $G(L/K)$ is a non-cyclic abelian group. Each idempotent relation yields a corresponding relation among the ranks of $E(F)$ as $F$ ranges over the fields lying between $K$ and $L$. This, and much more, may be found in the paper
*Kani, Ernst; Rosen, Michael*, [**Idempotent relations among arithmetic invariants attached to number fields and algebraic varieties**](http://dx.doi.org/10.1006/jnth.1994.1014), J. Number Theory 46, No. 2, 230-254 (1994). [ZBL0853.14011](https://zbmath.org/?q=an:0853.14011).


