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


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*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 


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

A: Yes, it is related to the canonical height (assuming you can handle large points).
Here is a sage session:
sage: E=EllipticCurve(QQ,[1,1,1,1,0]);P=E(0,0);k=9;Q=k*P
sage: sqrt(Q.height()/P.height())
9.00000000000000

If $H(P)$ is the canonical height and $Q= k P$, $k=\sqrt{H(Q)/H(P)}$
