Is there some relationship between the $p$-adic regulators of isogenous curves over $\mathbb{Q}$? I've done some computations and their ratio seems to be related (equivalent in all calculations so far) to the ratio of their modular degree.


There doesn't seem to be too much literature on this. Or perhaps it's a just a corollary of some known theorem and thus not very interesting. If I'm looking at the wrong place, could somebody please advise on this?



$p$-adic Regulators
Whether you take the $p$-adic or the real regulator won't make a difference for this question. 


Let $\varphi :E\to E'$ be an isogeny defined over a number field $K$. Consider the map $\varphi\_K : E(K)\to E'(K)$. Then
$$ \frac{\operatorname{Reg}(E')}{(\# E'(K)\_{\mathrm{tors}} )^2 } \cdot \frac{\# \operatorname{coker} \varphi\_K}{\#\ker \varphi\_K} = \frac{\operatorname{Reg}(E)}{(\# E(K)\_{\mathrm{tors}} )^2 } \cdot \frac{\# \operatorname{coker} \hat\varphi\_K}{\#\ker \hat\varphi\_K}$$
where $\hat\varphi$ is the dual isogeny. This sort of formulae are used when proving that BSD is invariant under isogeny, see for instance Milne's Arithmetic Duality I.7. Papers by Dokchitsers contain variations of this formula.


