Let $C$ be a hyperelliptic curve, we know its jacobian (or $Pic^{0}C$) is in one to one correspondence to reduced divisors. In the reduced divisor representation, any class is expressed in the form $[u,v]$, where $u,v$ are rational functions on $C$. Also, we have the Cantor algorithm to compute the exact $[u,v]$ obtain from the divisor addition.
But I am wondering if I want to do some Diophantine estimate and using height theory, how can one make this convenient representation in calculating height? More precisely, it there a way to simply use $[u,v]$ to compute its height? The difficult part looks like in this representation theory only give the existence of $v$, I don't have a good idea how to come up with an appropriate $v$.
It seems to me that it should be possible since the representation is for classes and height functions don't change for linearly equivalent divisors.
Any comments are welcome!