TOPICS IN p-ADIC FUNCTION THEORY

The function ez shows that a complex entire function can indeed omit one value. Lately, it has become fashionable to prove p -adic versions of value distribution theorems, of which Picard’s Theorem is an example, though not a recent one. More recent examples can be found in the works listed in the references section. Recall that the p -adic absolute value | |p on the rational number field Q is defined as follows. If x ∈ Q is written pka/b, where p is a prime, k is an integer, and a and b are integers relatively prime to p, then |x|p = p−k. Completing Q with respect to this absolute value results in the field of p-adic numbers, denoted Qp. Taking the algebraic closure of Qp, extending | |p to it, and then completing once more results in a complete algebraically closed field, denoted Cp, and often referred to as the p -adic complex numbers. Recall that the absolute value | |p satisfies a very strong form of the triangle inequality, namely |x + y|p ≤ max{|x|p, |y|p}. This is referred to as a nonArchimedean triangle inequality, and this non-Archimedean triangle inequality is what accounts for most of the differences between function theory on Cp and on C. Recall that an infinite series ∑ an converges under a non-Archimedean norm if and only if lim n→∞ an = 0. By an entire function on Cp, one means a formal