Potential theory on the Berkovich projective line by Baker M., Rumely R.

By Baker M., Rumely R.

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We define the “Berkovich disc of radius R about 0” by D(0, R) := M(K R−1 T ) . 1 that D(0, 1) is precisely the Berkovich unit disc studied in Chapter 1. It is also easy to see that for each R in the value group of K × , D(0, R) is homeomorphic to D(0, 1) as a topological space. 1. THE BERKOVICH AFFINE LINE A1Berk 21 on K r−1 T to the continuous multiplicative seminorm [ ]ιr,R (x) on K R−1 T defined by [f ]ιr,R (x) = [π(f )]x , where π : K R−1 T → K r−1 T is the natural K-algebra homomorphism. This map is easily checked to be continuous, with dense image, so ιr,R is injective.

3. Given x ∈ A1Berk , if x is of type I then Rx and [ ]x is a seminorm but not a norm. If x is of type II, III, or IV then Rx = K(T ) and [ ]x is a norm. Indeed, × ˜ ˜ (A) x is of type I iff Rx K(T ), [R× x ]x = |K |, and kx = K. × × ∼ ˜ ˜ (B) x is of type II iff Rx = K(T ), [Rx ]x = |K |, and kx = K(t) where ˜ t is transcendental over K. ˜ (C) x is of type III iff Rx = K(T ), [R× |K × |, and k˜x = K. x ]x × × ˜ ˜ (D) x is of type IV iff Rx = K(T ), [R ]x = |K |, and kx = K. x ˜ Proof. The possibilities for the triples (Rx , [R× x ]x , kx ) are mutually exclusive, so it suffices to prove all implications in the forward direction.

By construction, each disc D(a, r) ⊂ D(0, 1) of positive radius corresponds to a unique point of Λ. In other words, as a set Λ just consists of the points of type II or III in D(0, 1). To incorporate the points of type I and IV, we must enlarge Λ by adding “ends”. Consider a strictly decreasing sequence of nested discs x = {D(ai , ri )}, and put r = lim ri . The union of the lines of discs [ζai ,ri , ζGauss ] is a “half-open” line of discs, which we will write as (ζa,r , ζGauss ]. If i D(ai , ri ) is nonempty, then the intersection is either a disc D(a, r) of positive radius, in which case (ζa,r , ζGauss ] extends in a natural way to the closed path [ζa,r , ζGauss ], or a point a ∈ D(0, 1), in which case the half-open path (ζa,r , ζGauss ] extends to a closed path by adding on the corresponding type I point a as an endpoint.

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