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. 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 P1 Berk (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 logq v (t). ordv(·) the normalized valuation log v (| · |) 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. ˜ the residue field O/m of K. ˜(T) the reduction, in ˜ (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. xxix
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