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.

the set of nonzero elements in K.
| · | the non-Archimedean absolute value on K.
x, y the spherical distance on
associated to |·|, and also
the spherical kernel, its canonical upper semicontinuous
extension to PBerk
(see §4.3).
(x, y) the norm max(|x|, |y|) of a point (x, y)
(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 | · |.
the value group of K, that is, {|α| : α
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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