Suppose $E/\mathbb{Q}$ is an elliptic curve with rank zero. According to the conjecture of Birch and Swinnerton-Dyer, the special value $L(1,E\_{/\mathbb{Q}})$ should be equal (up to some harmless factors) to the order of the Tate-Shafarevich group Sha$(E/\mathbb{Q})$. Now, suppose $\chi$ is a Dirichlet character, and $K/\mathbb{Q}$ is the cyclic extension cut out by $\chi$. My question is:


What is the precise (conjectural) relation between the *individual* special values $L(1,E\times\chi^i)$ and Sha$(E/K)$?


More precisely, we have $L(s,E\_{/K})=\prod\_{i=0}^{\mathrm{ord}(\chi)} L(s,E\times \chi^i)$. If $E/K$ has rank zero, is there a decomposition of Sha$(E/K)$ with respect to an action of $\mathrm{Gal}(K/\mathbb{Q})$ such that the individual pieces of this decomposition have orders given by the individual values $L(1,E \times \chi^i)$?



Decomposition of Tate-Shafarevich groups in field extensions
Fix a prime p which doesn't divide the degree of K over ${\mathbb Q}$, and let ${\mathcal O}$ denote the ring of integers of ${\mathbb Q}\_p(\chi)$ i.e. an extension of ${\mathbb Q}\_p$ containing the values of $\chi$. Then the group algebra ${\mathcal O}[G]$ decomposes into a direct sum of 1-dimensional pieces over ${\mathcal O}$, one for each power of $\chi$. 


Then $Sha(E/K)[p^\infty] \otimes {\mathcal O}$ being an ${\mathcal O}[G]$-module inherits such a decomposition. Concretely, the $\chi^i$-component of $Sha(E/K)[p^\infty] \otimes {\mathcal O}$ is the subset where $G$ acts by $\chi^i$. 


This $\chi^i$-component is then a reasonable candidate to compare to the $p$-adic valuation of the algebraic part of $L(E,\chi^i,1)$. 


Some further comments:


-Note that extending scalars to ${\mathcal O}$ increases the size of the modules so this has to be taken into account.


-The component corresponding to the trivial character is the invariants under $G$, and when $G$ has size prime-to-p this is simply $Sha(E/{\mathbb Q})[p^\infty] \otimes {\mathcal O}$ (which is good).


-To make a precise relationship between the $L$-value and Sha, you need to take into account the other terms in BSD. Namely:


\*The torsion-term should work out exactly as above (decomposing into $\chi$-components). 


\*The periods have to be considered (which was ignored above in my vague phrase "the algebraic part of"). 


\*The Tamagawa numbers give me pause -- possibly there is an analogous $\chi$-decomposition, but I don't see it now. 


\*Lastly, if K is ramified over ${\mathbb Q}$ then the discriminant of K appears in the BSD quotient (in the denominator which increases the size of Sha). To handle this, I imagine what should be done is that rather then considering the L-value alone, consider the L-value times the Gauss sum of the character. (By the conductor-discriminant formula this should give exactly the extra powers of p needed.)


