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1. The image of the natural map k → A 1Berk is dense.

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Berkovich spaces, Problem List 4

Let (k, | · |) be an algebraically closed complete non-Archimedean non-trivial normed field and let A 1 Berk denote the Berkovich affine line over k.

1. The image of the natural map k → A 1 Berk is dense.

2. Let x ∈ A 1 Berk . Show the following.

(a) If x is of type (2), then the connected components of A 1 Berk \ {x}

are in a bijection with P 1 (k res ) (in particular, x is a branch point).

(b) If x is of type (3), then x is an ordinary point.

3. For x ∈ A 1 Berk , let

O x := {f ∈ k[T ] | |f | x 6 1}, m x := {f ∈ k[T ] | |f | x < 1}, k res x := O x /m x (we consider k res as a subfield of k res x ),

Γ := |k \ {0}|, Γ x := |k[T ] \ {0}| x . We define

E x := dim Q (Γ x /Γ) ⊗ Z Q, F x := trdeg k

res

k x res . Show the following.

(a) The group Γ x /Γ is cyclic.

(b) E x + F x 6 1 (a special case of Abhyankar’s inequality).

(c) x is of type (1) if and only if k res = k x res . (d) x is of type (2) if and only if F x = 1.

(e) x is of type (3) if and only if E x = 1.

(f) x is of type (4) if and only if E x = 0 = F x and x is not of type (1).

4. We define

D(0, 1) := x ∈ A 1 Berk | x 6 ζ 0,1 . Show that

O D(0,1) an ∼ = nX

a i T i ∈ k JT K | |a i | → 0 o .

5. Formulate and prove a generalization of the previous problem from A 1 Berk to A n Berk .

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