I'm having trouble understanding why do we have a bijection between points on Jac(C) and divisors in Mumford representation, where C is a hyperelliptic curve.
So I found here https://en.wikipedia.org/wiki/Imaginary_hyperelliptic_curve#Reduced_divisors_and_their_Mumford_representation that if we have an imaginary hyperelliptic curve there is a 1-1 correspondence between reduced divisors and divisors in Mumford representation. But what happens if we have a real hyperelliptic curve? I found here https://en.wikipedia.org/wiki/Real_hyperelliptic_curve under the "Transformation from real hyperelliptic curve to imaginary hyperelliptic curve" section that if we have a $K-$rational point on a real hyperelliptic curve that we have a birational equivalence to an imaginary hyperelliptic curve. But what does that mean for us? That if we have a real hyperelliptic curve (with a $K-$rational point) we can birationally transform it to an imaginary hyperelliptic curve and then go through the same reasoning? It the existence of a $K-$rational point really enough or am I misunderstanding something?
What happens if a real hyperelliptic curve has no $K-$rational points? I am under the impression that we also have a bijection between points on Jac(C) and divisors in Mumford representation but I don't know why.
Does my reasoning make sense?