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Arakelov constructed a nice intersection theory on arithmetic surfaces. A key point is the notion of Green function for a Riemann surface, which will be involved in the ''part at infinity'' of the calculation of the intersection number between two Arakelov divisors.

Lately I've been reading the paper "P. Hriljac - Heights and Arakelov's intersection theory" and he defines the intersection theory on arithmetic surfaces by using just the notion of Neron function (on a Riemann surface). A Green function is in particular a Neron function and in the paper you may find the following comment:

The existence of Neron functions on curves over $\mathbb C$ (...) may be viewed as stemming from Arakelov, where he uses Green's functions in lieu of Neron functions. The Green's functions can be viewed as explicit realizations of Neron functions (...) Observe that there is no unique such family, which corresponds to choices of metrics as in [Arakelov main paper]. Among all Green's functions, there is one which can be selected to be "better" than all others (...)

Basically I don't understand what is the author saying with the above lines. In particular I don't understand why we choose Green functions as the "right" Neron functions involved in intersection theory.

This question may be relevant.

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A very short answer I learned from summer school - because they want to construct local-global correspondence. A Neron function correspond exactly to the local part and Riemann-Roch is a local-global theorem.

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