Notation

We set the following notation, which will be used throughout unless oth-

erwise specified. Symbols are listed roughly in the order they are introduced

in the book, except that related notations are grouped together.

Z the ring of integers.

N the set of natural numbers, {n ∈ Z : n ≥ 0}.

Q the field of rational numbers.

Q a fixed algebraic closure of Q.

R the field of real numbers.

C the field of complex numbers.

Qp the field of p-adic numbers.

Zp the ring of integers of Qp.

Cp the completion of a fixed algebraic closure of Qp for some

prime number p.

Fp the finite field with p elements.

Fp a fixed algebraic closure of Fp.

K a complete, algebraically closed non-Archimedean field.

K×

the set of nonzero elements in K.

| · | the non-Archimedean absolute value on K.

x, y the spherical distance on

P1(K)

associated to |·|, and also

the spherical kernel, its canonical upper semicontinuous

extension to PBerk

1

(see §4.3).

(x, y) the norm max(|x|, |y|) of a point (x, y) ∈

K2

(see §10.1).

qv a fixed real number greater than 1 associated to K, used

to normalize | · | and ordv(·).

logv(t) shorthand for logqv (t).

ordv(·) the normalized valuation − logv(| · |) associated to | · |.

|K×|

the value group of K, that is, {|α| : α ∈

K×}.

O the valuation ring of K.

m the maximal ideal of O.

˜

K the residue field O/m of K.

˜(T) g the reduction, in

˜

K (T), of a function g(T) ∈ O(T).

K[T] the ring of polynomials with coeﬃcients in K.

K(T) the field of rational functions with coeﬃcients in K.

K[[T]] the ring of formal power series with coeﬃcients in K.

K T the Tate algebra of formal power series converging on the

closed unit disc.

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