Let E be an elliptic curve over $\mathbb{Q}$. Is there an efficient algorithm which can solve an elliptic curve discrete logarithm in E?



Is there an efficient algorithm to solve ECDLP over global field?
As Joro says, you can use the height pairing. And it's worth pointing out that it is generally possible to compute canonical heights even when the coefficients of $E$ are so large that it's infeasible to factor the discriminant. 


* Computing canonical heights with little (or no) factorization, *Math. Comp.* **66** (1997), 787-805.


In particular, it should be possible even on curves whose coefficients are of cryptographic size. 


It's also possible to solve ECDLP over a global field using ideas based on the proof of the weak Mordell-Weil theorem. I briefly discuss this in 


* Lifting and elliptic curve discrete logarithms, Selected Areas of
Cryptography (SAC 2008), *Lecture Notes in Computer Science* **5381**,
Springer--Verlag, Berlin, 2009, 82-102.


